® SAS Stat Studio 3.11 User’s Guide The correct bibliographic citation for this manual is as follows: SAS Institute Inc. 2009. SAS ® Stat Studio 3.11: User’s Guide. Cary, NC: SAS Institute Inc. SAS® Stat Studio 3.11: User’s Guide Copyright © 2009, SAS Institute Inc., Cary, NC, USA ISBN 978-1-59994-940-6 All rights reserved. Produced in the United States of America. For a hard-copy book: No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the publisher, SAS Institute Inc. For a Web download or e-book: Your use of this publication shall be governed by the terms established by the vendor at the time you acquire this publication. U.S. Government Restricted Rights Notice. Use, duplication, or disclosure of this software and related documentation by the U.S. government is subject to the Agreement with SAS Institute and the restrictions set forth in FAR 52.227-19 Commercial Computer Software-Restricted Rights (June 1987). SAS Institute Inc., SAS Campus Drive, Cary, North Carolina 27513. 1st electronic book, February 2009 1st printing, March 2009 SAS® Publishing provides a complete selection of books and electronic products to help customers use SAS software to its fullest potential. For more information about our e-books, e-learning products, CDs, and hard-copy books, visit the SAS Publishing Web site at support.sas.com/publishing or call 1-800-727-3228. SAS® and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. ® indicates USA registration. Other brand and product names are registered trademarks or trademarks of their respective companies. Contents Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones . . . . . . 11 Chapter 3. Creating and Editing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter 4. The Data Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter 5. Exploring Data in One Dimension . . . . . . . . . . . . . . . . . . . . . . . 53 Chapter 6. Exploring Data in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . 69 Chapter 7. Exploring Data in Three Dimensions . . . . . . . . . . . . . . . . . . . . . 93 Chapter 8. Interacting with Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Chapter 9. General Plot Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Chapter 10. Axis Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Chapter 11. Techniques for Exploring Data . . . . . . . . . . . . . . . . . . . . . . . . 151 Chapter 12. Plotting Subsets of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Chapter 13. Distribution Analysis: Descriptive Statistics . . . . . . . . . . . . . . . . . 187 Chapter 14. Distribution Analysis: Location and Scale Statistics . . . . . . . . . . . . . 195 Chapter 15. Distribution Analysis: Distributional Modeling . . . . . . . . . . . . . . . 203 Chapter 16. Distribution Analysis: Frequency Counts . . . . . . . . . . . . . . . . . . 217 Chapter 17. Distribution Analysis: Outlier Detection . . . . . . . . . . . . . . . . . . . 225 Chapter 18. Data Smoothing: Loess . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Chapter 19. Data Smoothing: Thin-Plate Spline . . . . . . . . . . . . . . . . . . . . . 247 Chapter 20. Data Smoothing: Polynomial Regression . . . . . . . . . . . . . . . . . . 257 Chapter 21. Model Fitting: Linear Regression . . . . . . . . . . . . . . . . . . . . . . 267 Chapter 22. Model Fitting: Robust Regression . . . . . . . . . . . . . . . . . . . . . . 285 Chapter 23. Model Fitting: Logistic Regression . . . . . . . . . . . . . . . . . . . . . . 297 Chapter 24. Model Fitting: Generalized Linear Models . . . . . . . . . . . . . . . . . . 317 Chapter 25. Multivariate Analysis: Correlation Analysis . . . . . . . . . . . . . . . . . 343 Chapter 26. Multivariate Analysis: Principal Component Analysis . . . . . . . . . . . . 353 Chapter 27. Multivariate Analysis: Factor Analysis . . . . . . . . . . . . . . . . . . . . 371 Chapter 28. Multivariate Analysis: Canonical Correlation Analysis . . . . . . . . . . . 389 Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis . . . . . . . . . . 399 Chapter 30. Multivariate Analysis: Discriminant Analysis . . . . . . . . . . . . . . . . 415 Chapter 31. Multivariate Analysis: Correspondence Analysis . . . . . . . . . . . . . . 425 Chapter 32. Variable Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Chapter 33. Running Custom Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Chapter 34. Configuring the Stat Studio Interface . . . . . . . . . . . . . . . . . . . . . 471 Appendix A. Sample Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 Appendix B. SAS/INSIGHT Features Not Available in Stat Studio . . . . . . . . . . . 499 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 Release Notes The following release notes pertain to SAS Stat Studio 3.11. • Stat Studio requires SAS 9.2. • This is an updated release of Stat Studio that enables access to remote SAS Workspace Servers. • If you need to open a data set containing Chinese, Japanese, or Korean characters, it is important that you configure the “Regional and Language Options” in the Windows Control Panel for the appropriate country. It is not necessary to change the Windows setting called “Language for non-Unicode programs,” which is also referred to as the system locale. vi Chapter 1 Introduction What Is Stat Studio? Stat Studio is a tool for data exploration and analysis. Figure 1.1 shows a typical Stat Studio analysis. You can use Stat Studio to do the following: • explore data through graphs linked across multiple windows • subset data • analyze univariate distributions • fit explanatory models • investigate multivariate relationships Figure 1.1. The Stat Studio Interface 2 Chapter 1. Introduction In addition, Stat Studio provides an integrated development environment that enables you to write, debug, and execute programs that combine the following: • the flexibility of the SAS/IML matrix language • the analytical power of SAS/STAT procedures • the data manipulation capabilities of Base SAS • dynamically linked graphics for exploratory data analysis The programming language in Stat Studio, which is called IMLPlus, is an enhanced version of the IML programming language. IMLPlus extends IML to provide new language features such as the ability to create and manipulate statistical graphics and to call SAS procedures. Stat Studio requires that you have a license for Base SAS, SAS/STAT, and SAS/IML. Stat Studio runs on a PC in the Microsoft Windows operating environment. Related Software and Documentation This book is one of three documents about Stat Studio. In this book you learn how to use the Stat Studio GUI to conduct exploratory data analysis and standard statistical analyses. A second book, Stat Studio for SAS/STAT Users, is intended for SAS/STAT programmers. In it, you learn how to use Stat Studio in conjunction with SAS/STAT in order to explore data and visualize statistical models. In particular, you learn to call procedures in other SAS products such as SAS/STAT or Base SAS by using the SUBMIT statement. The third source of documentation is the Stat Studio online Help. You can display the online Help by selecting Help I Help Topics from the main menu. The online Help includes documentation for all IMLPlus classes and associated methods. Stat Studio is closely related to the SAS/IML software. The language used to write programs in Stat Studio is called IMLPlus. This language consists of IML functions and subroutines, plus additional syntax to support the creation and manipulation of statistical graphics. The Stat Studio program windows color-code keywords in the IMLPlus language. Most IML programs run without modification in the IMLPlus environment. The Stat Studio online Help includes a list of differences between IML and IMLPlus. For your convenience in referencing related SAS software, the SAS/IML User’s Guide, the SAS/STAT User’s Guide, and the Base SAS Procedures Guide are available from the Stat Studio Help menu. Chapter 1. Introduction Exploratory Data Analysis Data analysis often falls into two phases: exploratory and confirmatory. The exploratory phase “isolates patterns and features of the data and reveals these forcefully to the analyst” (Hoaglin, Mosteller, and Tukey 1983). If a model is fit to the data, exploratory analysis finds patterns that represent deviations from the model. These patterns lead the analyst to revise the model, and the process is repeated. In contrast, confirmatory data analysis “quantifies the extent to which [deviations from a model] could be expected to occur by chance” (Gelman 2004). Confirmatory analysis uses the traditional statistical tools of inference, significance, and confidence. Exploratory data analysis is sometimes compared to detective work: it is the process of gathering evidence. Confirmatory data analysis is comparable to a court trial: it is the process of evaluating evidence. Exploratory analysis and confirmatory analysis “can—and should—proceed side by side” (Tukey 1977). How Many Observations Can You Analyze? Stat Studio provides the data analyst with interactive and dynamic statistical graphics. By definition, interactive graphics must respond quickly to the changes and manipulations of the analyst. This quick response restricts the size of data sets that can be handled while still maintaining interactivity. Wegman (1995) points out that the number of observations you can analyze depends on the algorithmic complexity of the statistical algorithms you are using. For example, if you have n observations, computing a mean and variance is O(n), sorting is O(n log n), and solving a least squares regression on p variables is O(np2 ). Furthermore, visualization of individual observations is limited by the number of pixels that can be represented on a display device. Wegman’s conclusion is that “visualization of data sets say of size 106 or more is clearly a wide open field.” More recently, Unwin, Theus, and Hofmann (2006) discuss the challenges of “visualizing a million,” including a chapter dedicated to interactive graphics. On a typical PC (for example, a 1.8 GHz CPU with 512 MB of RAM), Stat Studio can help you analyze dozens of variables and tens of thousands of observations. Visualization of data with graphics such as histograms and box plots remains feasible for hundreds of thousands of observations, although the interactive graphics become less responsive. Scatter plots of this many observations suffer from overplotting. Stat Studio uses the RAM on your PC to facilitate interaction and linking between plots and data tables. If you routinely analyze large data sets, increasing the RAM on your PC might increase Stat Studio’s interactivity. For example, if you routinely examine hundreds of thousands of observations in dozens of variables, 1 GB of RAM is preferable to 512 MB. 3 4 Chapter 1. Introduction Summary of Features Stat Studio provides tools for exploring data, analyzing distributions, fitting parametric and nonparametric regression models, and analyzing multivariate relationships. In addition, you can extend the set of available analyses by writing programs. To explore data, you can do the following: • identify observations in plots • select observations in linked data tables, bar charts, box plots, contour plots, histograms, line plots, mosaic plots, and two- and three-dimensional scatter plots • exclude observations from graphs and analyses • search, sort, subset, and extract data • transform variables • change the color and shape of observation markers based on the value of a variable To analyze distributions, you can do the following: • compute descriptive statistics • create quantile-quantile plots • create mosaic plots of cross-classified data • fit parametric and kernel density estimates for distributions • detect outliers in contaminated Gaussian data To fit parametric and nonparametric regression models, you can do the following: • smooth two-dimensional data by using polynomials, loess curves, and thinplate splines • add confidence bands for mean and predicted values • create residual and influence diagnostic plots • fit robust regression models, and detect outliers and high-leverage observations • fit logistic models • fit the general linear model with a wide variety of response and link functions • include classification effects in logistic and generalized linear models To analyze multivariate relationships, you can do the following: • calculate correlation matrices and scatter plot matrices with confidence ellipses for relationships among pairs of variables • reduce dimensionality with principal component analysis Chapter 1. Introduction • examine relationships between a nominal variable and a set of interval variables with discriminant analysis • examine relationships between two sets of interval variables with canonical correlation analysis • reduce dimensionality by computing common factors for a set of interval variables with factor analysis • reduce dimensionality and graphically examine relationships between categorical variables in a contingency table with correspondence analysis To extend the set of available analyses, you can do the following: • write, debug, and execute IMLPlus programs in an integrated development environment • add legends, curves, maps, or other custom features to statistical graphics • create new static graphics • animate graphics • execute SAS procedures or DATA steps from within your IMLPlus programs • develop interactive data analysis programs that use dialog boxes • call computational routines written in IML, C, FORTRAN, or Java Comparison with SAS/INSIGHT Stat Studio and SAS/INSIGHT have the same goal: to be a tool for data exploration and analysis. Both have dynamically linked statistical graphics. Both come with pre-written statistical analyses for analyzing distributions, regression models, and multivariate relationships. Figure 1.2 shows a typical SAS/INSIGHT analysis. Figure 1.3 shows the same analysis performed in Stat Studio. You can see that the analyses are qualitatively similar. 5 6 Chapter 1. Introduction Figure 1.2. A SAS/INSIGHT Analysis Figure 1.3. A Comparable Stat Studio Analysis Chapter 1. Introduction However, there are three major differences between the two products. The first is that Stat Studio runs on a PC in the Microsoft Windows operating environment. It is client software that can connect to SAS servers. The SAS server might be running on a different computer than Stat Studio. In contrast, SAS/INSIGHT runs on the same computer on which SAS is installed. A second major difference is that Stat Studio is programmable, and therefore extensible. SAS/INSIGHT contains standard statistical analyses that are commonly used in data analysis, but you cannot create new analyses. In contrast, you can write programs in Stat Studio that call any licensed SAS procedure, and you can include the results of that procedure in graphics, tables, and data sets. Because of this, Stat Studio is often referred to as the “programmable successor to SAS/INSIGHT.” A third major difference is that the Stat Studio statistical graphics are programmable. You can add legends, curves, and other features to the graphics in order to better analyze and visualize your data. Stat Studio contains many features that are not available in SAS/INSIGHT. General features that are unique to Stat Studio include the following: • Stat Studio can connect to multiple SAS servers simultaneously. • Stat Studio can run multiple programs simultaneously in different threads, each with its own WORK library. • Stat Studio sessions can be driven by a program and rerun. The following list presents features of Stat Studio data views (tables and plots) that are not included in SAS/INSIGHT: • Stat Studio provides modern dialog boxes with a native Windows look and feel. • Stat Studio provides a line plot in which the lines can be defined by specifying a single X and Y variable and one or more grouping variables. • Stat Studio supports a polygon plot that can be used to build interactive regions such as maps. • Stat Studio provides programmatic methods to draw legends, curves, or other decorations on any plot. • Stat Studio provides programmatic methods to attach a menu to any plot. After the menu is selected, a user-specified program is run. • Stat Studio supports arbitrary unions and intersections of observations selected in different views. Stat Studio also provides the following analyses and options that are not included in SAS/INSIGHT: • Stat Studio can be programmed to call any licensed SAS analytical procedure and any IML function or subroutine. 7 8 Chapter 1. Introduction • Stat Studio detects outliers in contaminated Gaussian data. • Stat Studio fits robust regression models and detects outliers and high-leverage observations. • Stat Studio supports the generalized linear model with a multinomial response. • Stat Studio creates graphical results for the analysis of logistic models with one continuous effect and a small number of levels for classification effects. • Stat Studio provides parametric and nonparametric methods of discriminant analysis. • Stat Studio provides common factor analysis for interval variables. • Stat Studio provides correspondence analysis for nominal variables. Features of SAS/INSIGHT that are not included in Stat Studio are presented in Appendix B, “SAS/INSIGHT Features Not Available in Stat Studio.” Typographical Conventions This documentation uses some special symbols and typefaces. • Field names, menu items, and other items associated with the graphical user interface are in bold; for example, a menu item is written as File I Open I Server Data Set. A field in a dialog box is written as the Anchor tick field. • Names of Windows files, folders, and paths are in bold; for example, C:\Temp\MyData.sas7bdat. • SAS librefs, data sets, and variable names are in Helvetica; for example, the age variable in the work.MyData data set. • Keywords in SAS or in the IMLPlus language are in all capitals; for example, the SUBMIT statement or the ORDER= option. This documentation is full of examples. Each step in an example appears in bold. =⇒ This symbol and typeface indicates a step in an example. Chapter 1. Introduction References Gelman, A. (2004), “Exploratory Data Analysis for Complex Models,” Journal of Computational and Graphical Statistics, 13(4), 755–779. Hoaglin, D. C., Mosteller, F., and Tukey, J. W., eds. (1983), Understanding Robust and Exploratory Data Analysis, Wiley series in probability and mathematical statistics, New York: John Wiley & Sons. Tukey, J. W. (1977), Exploratory Data Analysis, Reading, MA: Addison-Wesley. Unwin, A., Theus, M., and Hofmann, H. (2006), Graphics of Large Datasets, New York: Springer. Wegman, E. J. (1995), “Huge Data Sets and the Frontiers of Computational Feasibility,” Journal of Computational and Graphical Statistics, 4(4), 281–295. 9 10 Chapter 2 Getting Started: Exploratory Data Analysis of Tropical Cyclones This chapter describes how you can use Stat Studio for exploratory data analysis. The techniques presented in this section do not require any programming. This example shows how you can use Stat Studio to explore data about North Atlantic tropical cyclones. (A cyclone is a large system of winds that rotate about a center of low atmospheric pressure.) The data were recorded by the U.S. National Hurricane Center at six-hour intervals. The data set includes information about each storm’s location, sustained low-level winds, and atmospheric pressure, and also contains variables indicating the size of the storm. The cyclones from 1988 to 2003 are included. A full description of the Hurricanes data set is included in Appendix A, “Sample Data Sets.” The analysis presented here is based on Mulekar and Kimball (2004) and Kimball and Mulekar (2004). Opening the Data Set =⇒ Open the Hurricanes data set. This data set is distributed with Stat Studio. To use the GUI to open the data set, do the following: 1. Select File I Open I File from the main menu. The dialog box in Figure 2.1 appears. 2. Click Go to Installation directory near the bottom of the dialog box. 3. Double-click on the Data Sets folder. 4. Select the Hurricanes.sas7bdat file. 5. Click Open. 12 Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones Figure 2.1. Opening a Sample Data Set Creating a Bar Chart The category variable is a measure of wind intensity, corresponding to the Saffir-Simpson wind intensity scale in Table 2.1. Table 2.1. The Saffir-Simpson Intensity Scale Category TD TS Cat1 Cat2 Cat3 Cat4 Cat5 Description Tropical Depression Tropical Storm Category 1 Hurricane Category 2 Hurricane Category 3 Hurricane Category 4 Hurricane Category 5 Hurricane Wind Speed (knots) 22–33 34–63 64–82 83–95 96–113 114–134 135 or greater In this section you create a bar chart of the category variable and exclude observations that correspond to weak storms. =⇒ Select Graph I Bar Chart from the main menu. The bar chart dialog box in Figure 2.2 appears. =⇒ Select the variable category, and click Set X. Note: In most dialog boxes, double-clicking on a variable name adds the variable to the next appropriate field. Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones Figure 2.2. Bar Chart Dialog Box =⇒ Click OK. The bar chart in Figure 2.3 appears. Figure 2.3. A Bar Chart The bar chart shows the number of observations for storms in each Saffir-Simpson intensity category. In the next step, you exclude observations of less than tropical storm intensity (wind speeds less than 34 knots). =⇒ In the bar chart, click on the bar labeled with the symbol . 13 14 Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones This selects observations for which the category variable has a missing value. For these data, “missing” is equivalent to an intensity of less than tropical depression strength (wind speeds less than 22 knots). =⇒ Hold down the CTRL key and click on the bar labeled “TD.” When you hold down the CTRL key and click, you extend the set of selected observations. In this example, you select observations with tropical depression strength (wind speeds of 22–34 knots) without deselecting previously selected observations. This is shown in Figure 2.4. Figure 2.4. A Bar Chart with Selected Observations The row heading of the data table includes two special cells for each observation: one showing the position of the observation in the data set, and the other showing the status of the observation in analyses and plots. Initially, the status of each observation is indicated by the marker (by default, a filled square) and a χ2 symbol. The presence of a marker indicates that the observation is included in plots, and the χ2 symbol indicates that the observation is included in analyses (see Chapter 4, “The Data Table,” for more information about the data table symbols). =⇒ In the data table, right-click in the row heading of any selected observation, and select Exclude from Plots from the pop-up menu. The pop-up menu is shown in Figure 2.5. Notice that the bar chart redraws itself to reflect that all observations being displayed in the plots now have at least 34-knot winds. Notice also that the square symbol in the data table is removed from observations with relatively low wind speeds. Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones Figure 2.5. Data Table Pop-up Menu =⇒ In the data table, right-click in the row heading of any selected observation, and select Exclude from Analyses from the pop-up menu. Notice that the χ2 symbol is removed from observations with relatively low wind speeds. Future analysis (for example, correlation analysis and regression analysis) will not use the excluded observations. =⇒ Click in any data table cell to clear the selected observations. Creating a Histogram In this section you create a histogram of the latitude variable and examine relationships between the category and latitude variables. The figures in this section assume that you have excluded observations with low wind speeds as described in the “Creating a Bar Chart” section on page 12. =⇒ Select Graph I Histogram from the main menu. The histogram dialog box in Figure 2.6 appears. =⇒ Select the variable latitude, and click Set X. Figure 2.6. Histogram Dialog Box =⇒ Click OK. 15 16 Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones A histogram (Figure 2.7) appears, showing the distribution of the latitude variable for the storms that are included in the plots. Move the histogram so that it does not cover the bar chart or data table. Figure 2.7. Histogram of Latitudes of Storms Stat Studio plots and data tables are collectively known as data views. All data views are dynamically linked, meaning that observations that you select in one data view are displayed as selected in all other views of the same data. You have seen that you can select observations in a plot by clicking on observation markers. You can add to a set of selected observations by holding the CTRL key and clicking. You can also select observations by using a selection rectangle. To create a selection rectangle, click in a graph and hold down the left mouse button while you move the mouse pointer to a new location. =⇒ Drag out a selection rectangle in the bar chart to select all storms of category 3, 4, and 5. The bar chart looks like the one in Figure 2.8. Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones Figure 2.8. Selecting the Most Intense Storms Note that these selected observations are also shown in the histogram in Figure 2.9. The histogram shows the marginal distribution of latitude, given that a storm is greater than or equal to category 3 intensity. The marginal distribution shows that very strong hurricanes tend to occur between 11 and 37 degrees north latitude, with a median latitude of about 22 degrees. If these data are representative of all Atlantic hurricanes, you might conjecture that it would be relatively rare for a category 3 hurricane to strike north of the North Carolina–Virginia border (roughly 36.5◦ north latitude). Figure 2.9. Latitudes of Intense Storms 17 18 Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones Creating a Box Plot The data set contains several variables that measure the size of a tropical cyclone. One of these is the radius– eye variable, which contains the radius of a cyclone’s eye in nautical miles. (The eye of a cyclone is a calm, relatively cloudless central region.) The radius– eye variable has many missing values, because not all storms have well-defined eyes. In this section you create a box plot that shows how the radius of a cyclone’s eye varies with the Saffir-Simpson category. The figures in this section assume that you have excluded observations with low wind speeds as described in the “Creating a Bar Chart” section on page 12. =⇒ Select Graph I Box Plot from the main menu. The box plot dialog box appears as in Figure 2.10. Figure 2.10. Box Plot Dialog Box =⇒ Select the variable radius– eye, and click Set Y. =⇒ Select the variable category, and click Add X. =⇒ Click OK. A box plot appears. Move the box plot so that it does not cover the data table or other plots. The box plot summarizes the distribution of eye radii for each Saffir-Simpson category. The plot indicates that the median eye radius tends to increase with storm intensity for tropical storms, category 1, and category 2 hurricanes. Category 2–4 Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones storms have similar distributions, while the most intense hurricanes (Cat5) in this data set tend to have eyes that are small and compact. The box plot also indicates considerable spread in the radii of eyes. Recall that the radius– eye variable contains many missing values. The box plot displays only observations with nonmissing values, corresponding to storms with well-defined eyes. You might wonder what percentage of all storms of a given Saffir-Simpson intensity have well-defined eyes. You can determine this percentage by selecting all observations in the box plot and noting the proportion of observations that are selected in the bar chart. =⇒ Drag out a selection rectangle in the box plot around the category 1 storms. In the bar chart in Figure 2.11, note that approximately 25% of the bar for category 1 storms is displayed as selected, meaning that approximately one quarter of the category 1 storms in this data set have nonmissing measurements for radius– eye. Figure 2.11. Proportion of Category 1 Storms with Well-Defined Eyes =⇒ Drag the selection rectangle to select eye radii in other categories. The selected observations displayed in the bar chart reveal the proportion of storms in each Saffir-Simpson category that have nonmissing values for radius– eye. Note in particular that very few tropical storms have eyes, whereas almost all category 4 and 5 storms have well-defined eyes. =⇒ Click outside the plot area in any plot to deselect all observations. 19 20 Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones Creating a Scatter Plot In this section you examine the relationship between wind speed and atmospheric pressure for tropical cyclones. The National Hurricane Center routinely reports both of these quantities as indicators of a storm’s intensity. The figures in this section assume that you have excluded observations with low wind speeds as described in the “Creating a Bar Chart” section on page 12. =⇒ Select Graph I Scatter Plot from the main menu. The scatter plot dialog box appears as in Figure 2.12. Figure 2.12. Scatter Plot Dialog Box =⇒ Select the variable wind– kts, and click Set Y. =⇒ Select the variable min– pressure, and click Set X. =⇒ Click OK. A scatter plot appears as in Figure 2.13. Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones Figure 2.13. Wind Speed versus Minimum Pressure Modeling Variable Relationships In this section you model the relationship between wind speed and atmospheric pressure for tropical cyclones. The scatter plot in Figure 2.13 shows a strong negative correlation between wind speed and pressure. To compute the correlation between these variables, you can run Stat Studio’s correlation analysis. The results in this section assume that you have excluded observations with low wind speeds as described in the “Creating a Bar Chart” section on page 12. Note: You can select from the Analysis or Graph menu only when the active window is a data table or a graph. Click on a window’s title bar to make it the active window. =⇒ Select Analysis I Multivariate Analysis I Correlation Analysis from the main menu. The correlation dialog box appears as in Figure 2.14. =⇒ Click on the wind– kts variable. Hold down the CTRL key, click on the min– pressure, and click Add Y. Both variables are added to the list of Y variables. 21 22 Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones Figure 2.14. Correlations Analysis Dialog Box =⇒ Click the Plots tab. =⇒ Clear the Pairwise correlation plot check box. =⇒ Click OK. See Chapter 25, “Multivariate Analysis: Correlation Analysis,” for more information about the correlations analysis. An output window appears (Figure 2.15), showing the results from the CORR procedure. The output shows that the Pearson correlation between wind– kts and min– pressure is –0.92533. Figure 2.15. Output from the CORR Procedure Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones Suppose you want to compute a linear model that relates wind– kts to min– pressure. Several choices of parametric and nonparametric models are available from the Analysis I Model Fitting menu. If you are interested in a response due to a single explanatory variable, you can also choose from models available from the Analysis I Data Smoothing menu. Note: If the scatter plot of wind– kts versus min– pressure is the active window prior to your choosing an analysis from the Analysis I Data Smoothing menu, then the data smoother is added to the existing scatter plot. Otherwise, a new scatter plot is created by the analysis. =⇒ Activate the scatter plot of wind– kts versus min– pressure. Select Analysis I Data Smoothing I Polynomial Regression from the main menu. The polynomial regression dialog box appears as in Figure 2.16. Figure 2.16. Polynomial Smoother Dialog Box =⇒ Select the variable wind– kts, and click Set Y. =⇒ Select the variable min– pressure, and click Set X. =⇒ Click OK. A scatter plot appears (Figure 2.17), and output from the REG procedure is added at the bottom of the output window. 23 24 Chapter 2. Getting Started: Exploratory Data Analysis of Tropical Cyclones Figure 2.17. Least-Squares Regression The output from the REG procedure indicates an R-square value of 0.8562 for the line of least squares given approximately by wind– kts = 1222 − 1.177 × min– pressure. The scatter plot shows this line and a 95% confidence band for the predicted mean. The confidence band is very thin, indicating high confidence in the means of the predicted values. References Kimball, S. K. and Mulekar, M. S. (2004), “A 15-year Climatology of North Atlantic Tropical Cyclones. Part I: Size Parameters,” Journal of Climatology, 3555–3575. Mulekar, M. S. and Kimball, S. K. (2004), “The Statistics of Hurricanes,” STATS, 39, 3–8. Chapter 3 Creating and Editing Data The Stat Studio data table displays data in a tabular view. You can create small data sets by entering data into the table. You can edit cells to examine “what-if” scenarios. You can add new variables or observations, and cut and paste between cells of the data table and the Microsoft Windows clipboard. Entering Data This section describes how you can use the data table to enter small data sets. You learn how to do the following: • enter new variables • enter observations • copy, cut, and paste to and from the Windows clipboard Example: Creating a Small Data Set The data in this example are quarterly sales for two employees, June and Bob. =⇒ Create a new data set by choosing File I New I Data Set from the main menu. A dialog box prompts you for the name of the first variable. The first variable will contain the name of the sales staff. Fill in the dialog box (shown in Figure 3.1) as described in the following steps. =⇒ Type Employee in the Name field. The contents of this box must be a valid SAS variable name as specified in the section “Adding Variables” on page 28. =⇒ In the Type field, select Character. =⇒ Click OK. 26 Chapter 3. Creating and Editing Data Figure 3.1. Creating a Character Variable The second variable will indicate the quarter of the financial year for which sales are recorded. The only valid values for this numeric variable are the discrete integers 1–4. Thus you will create this next variable as a nominal variable. =⇒ Create a new variable by choosing Edit I Variables I New Variable from the main menu. Fill in the dialog box (shown in Figure 3.2) as described in the following steps. =⇒ Type Quarter in the Name field. =⇒ Select Nominal from the Measure Level menu. =⇒ Click OK. Figure 3.2. Creating a Nominal Numeric Variable The third variable will contain the revenue, in thousands of dollars, for each salesperson for each financial quarter. =⇒ Create a third variable by choosing Edit I Variables I New Variable from the main menu. Fill in the dialog box (shown in Figure 3.3) as described in the following steps. =⇒ Type Sales in the Name field. Chapter 3. Creating and Editing Data =⇒ In the Label field, type Sales (Thousands). =⇒ In the Format list, select DOLLAR. Type 4 in the W field. =⇒ Click OK. Figure 3.3. Creating a Numeric Variable with a Format Now you can enter observations for each variable. Note that the new data set was created with one observation containing a missing value for each variable. The first observation should be typed in the first row; subsequent observations are added as you enter them. Entering data in the data table row marked with an asterisk (S) creates a new observation. When you are entering (or editing) data, the ENTER key takes you down to the next observation. The TAB key moves the active cell to the right, whereas holding down the SHIFT key and pressing TAB moves the active cell to the left. You can also use the keyboard arrow keys to navigate the cells of the data table. =⇒ Enter the data shown in Table 3.1. Table 3.1. Sample Data Employee June Bob June Bob June Bob June Bob Quarter 1 1 2 2 3 3 4 4 Sales 34 29 24 18 28 25 45 32 Note: When you enter the data for the Sales variable, do not type the dollar sign. The actual data is {34, 29, . . . , 32}, but because the variable has a DOLLAR4. format, the data table displays a dollar sign in each cell. The data table looks like the table in Figure 3.4. 27 28 Chapter 3. Creating and Editing Data Figure 3.4. New Data Set At this point you can save your data. =⇒ Select File I Save as File from the main menu. Navigate to the Data Sets subdirectory of your personal files directory and save the file as sales.sas7bdat. Note: The default location of the personal files directory is given in the section “The Personal Files Directory” on page 485. When you want to open your data later, you can select File I Open I File from the main menu. The dialog box that appears has a button near the bottom that says Go to Personal Files directory. For this reason, it is convenient to save data in your personal files directory. Adding Variables When you add a new variable, the New Variable dialog box appears as shown in Figure 3.5. You can add a new variable by choosing Edit I Variables I New Variable from the main menu. Note: The Edit I Variables menu also appears when you right-click on a variable heading. Chapter 3. Creating and Editing Data Figure 3.5. The New Variable Dialog Box The following list describes each field of the New Variable dialog box. Name specifies the name of the new variable. This must be a valid SAS variable name. This means the name must satisfy the following conditions: • • • • must be at most 32 characters must begin with an English letter or underscore cannot contain blanks cannot contain special characters other than an underscore Label specifies the label for the variable. Type specifies the type of variable: numeric or character. Measure Level specifies the variable’s measure level. The measure level determines the way a variable is used in graphs and analyses. A character variable is always nominal. For numeric variables, you can choose from two measure levels: Interval The variable contains values that vary across a continuous range. For example, a variable measuring temperature would likely be an interval variable. Nominal The variable contains a discrete set of values. For example, a variable indicating gender would be a nominal variable. Format specifies the SAS format for the variable. For many formats you also need to specify values for the W (width) and D (decimal) fields associated with the format. For more information about formats see the SAS Language Reference: Dictionary. 29 30 Chapter 3. Creating and Editing Data Informat specifies the SAS informat for the variable. For many informats you also need to specify values for the W (width) and D (decimal) fields associated with the format. For more information about informats see the SAS Language Reference: Dictionary. Note: You can type the name of a format into the Format or Informat field, even if the name does not appear in the list. Adding and Editing Observations To add a new observation, type data into any cell in the last data table row. This row is marked with an asterisk (S). When you are entering (or editing) data, the ENTER key takes you down to the next observation. The TAB key moves the active cell to the right, whereas holding down the SHIFT key and pressing TAB moves the active cell to the left. You can also use the keyboard arrow keys to navigate the cells of the data table. It is possible to perform operations on a range of cells. If you select a range of cells, then you can do the following: • Delete the contents of the cells with the DELETE key. • Cut or copy the contents of the range of cells to the Windows clipboard, in tab-delimited format. This makes the contents of the cells available to all Windows applications (Excel, Word, etc.). • Paste from the Windows clipboard into the selected range of cells, provided that the data on the clipboard is in tab-delimited format. You can paste numeric data into cells in a character variable (the data are converted to text), but you cannot paste character data into cells in a numeric variable. Typing in a cell changes the data for that cell. Graphs that use that observation will update to reflect the new data. Caution: If you change data after an analysis has been run, you will need to rerun the analysis; the analysis does not automatically rerun to reflect the new data. Chapter 4 The Data Table The Stat Studio data table displays data in a tabular view. You can use the data table to change properties of a variable, such as a variable’s name, label, or format. You can also change properties of observations, including the shape and color of markers used to represent an observation in graphs. You can also control which observations are visible in graphs and which are used in statistical analyses. Context Menus The first two rows of the data table are column headings (also called variable headings). The first row displays the variable’s name or label. The second row indicates the variable’s measure level (nominal or interval), the default role the variable plays, and, if the variable is selected, in what order it was selected. Subsequent rows contain observations. The first two columns of the data table are row headings (also called observation headings). The first column displays the observation number (or some other label variable). The second column indicates whether the observation is included in plots and analyses. The effect of selecting a cell of the data table depends on the location of the cell. To select a variable, click on the column heading. To select an observation, click on the row heading. You can display a context menu as in Figure 4.1 by right-clicking when the mouse pointer is positioned over a column heading or row heading. A context menu means that you see different menus depending on where the mouse pointer is when you right-click. For the data table, the Variables menu differs from the Observations menu. 32 Chapter 4. The Data Table Figure 4.1. Data Table with the Variables Menu Variable Properties You can change the properties of a variable by using the Variables menu, as shown in Figure 4.2. You can access the Variables menu by clicking on the column heading and selecting Edit I Variables from the main menu. Alternatively, right-clicking on a variable heading (see Figure 4.1) selects that variable and displays the same menu. You can use the Variables menu to do the following: • change properties of existing variables • set the role of an existing variable • create a new variable • change the set of variables that are displayed in the data table • change the set of selected and unselected variables One variable property that might be unfamiliar is the role. You can assign three default roles: Label The values of the variable are used to label clicked-on markers in plots. Frequency The values of the variable are used as the frequency of occurrence for each observation. Weight The values of the variable are used as weights for each observation. If you assign a variable to a Frequency role, then that variable is automatically added to dialog boxes for analyses and graphs that support a frequency variable. The same is true for variables with a Weight role. Chapter 4. The Data Table There can be at most one variable for each role. A variable can play multiple roles. Figure 4.2. The Variables Menu The following list describes each item on the variable menu. Properties displays the Variable Properties dialog box, described in the section “Adding Variables” on page 28. The dialog box enables you to change most properties for the selected variable. However, you cannot change the type (character or numeric) of an existing variable. Interval/Nominal changes the measure level of the selected numeric variable. A character variable cannot be interval. Label makes the selected variable the label variable for plots. Frequency makes the selected variable the frequency variable for analyses and plots that support a frequency variable. Only numeric variables can have a Frequency role. Weight makes the selected variable the weight variable for analyses and plots that support a weight variable. Only numeric variables can have a Weight role. Ordering specifies how nominal variables are ordered. This affects the way that a variable is sorted and the order of categories in plots. If a variable has missing values, they are always ordered first. See the section “Ordering Categories of 33 34 Chapter 4. The Data Table a Nominal Variable” on page 155 for further details. The Ordering submenu is shown in Figure 4.3. You can order a variable in the following ways: Standard specifies that categories are arranged in ASCII order by their unformatted values. In ASCII order, numerals precede uppercase letters, which precede lowercase letters. by Frequency specifies that categories are arranged according to the descending frequency count of formatted values in each category. by Format specifies that categories are arranged in ASCII order by their formatted values. by Data specifies that categories are arranged according to the data order of formatted values. The data order is determined by traversing the values of a variable, starting from the first observation. The first (nonmissing) value you encounter is ordered first, the next unique (nonmissing) value of the variable is ordered second, and so on. Sorting the data table does not affect this ordering; it is based on the original order of observations. by Frequency (unformatted) specifies that categories are arranged according to the descending frequency count of unformatted values in each category. by Data (unformatted) specifies that categories are arranged according to the data order of unformatted values. Sorting the data table does not affect this ordering; it is based on the original order of observations. Custom specifies that this variable was ordered by calling the DataObject.SetVarValueOrder method. See the Stat Studio online Help for details about this method. Sort displays the Sort dialog box. The Sort dialog box is described in the section “Sorting Observations” on page 37. New Variable displays the New Variable dialog box (Figure 3.5) to create a new variable as described in the section “Adding Variables” on page 28. Delete deletes the selected variables. Display Name/Display Label toggles whether the column heading displays the name of variables or displays their labels. Hide hides the selected variables. The variables can be displayed at a later time by selecting Show All. Hidden variables cannot be selected. Show All displays all variables, including variables that were hidden. Chapter 4. The Data Table Invert Selection changes the set of selected variables. Unselected variables become selected, while selected variables become unselected. Generate – OBSTAT– Variable creates a new character variable called – OBSTAT– that encodes the current state of each observation. The values of the – OBSTAT– variable are described in the following paragraphs. Figure 4.3. The Ordering Menu The – OBSTAT– variable is a character variable of length 20. It was introduced in SAS/INSIGHT software as a way to capture the state of observations, including the color and shape of markers and whether an observation is selected. The first few characters encode the state of binary options such as whether an observation is selected. A character is ‘1’ if the corresponding property is true and ‘0’ if the related property is false. The properties are described in the following list: Character 1 stores whether the observation is selected. Character 2 stores whether the observation is included in plots. Character 3 stores whether the observation is included in analyses. Character 4 stores whether the observation has a label. Character 5 stores the marker shape for an observation. This is a value between 1 and 8 that corresponds to a shape, as given in the following table: 35 36 Chapter 4. The Data Table Value 1 2 3 4 5 6 7 8 Shape + ◦ ♦ × 4 5 ? Characters 6–20 store the RGB value of the fill color for an observation marker. The RGB color model represents colors as combinations of the colors red, green, and blue. Each component is a five-digit decimal number between 0 and 65535. Characters 6–10 store the red component. Characters 11–15 store the green component. Characters 16–20 store the blue component. If you read a data set for which there is no associated DMM file, and if that data set contains a variable named – OBSTAT– , then the state of each observation is determined by the corresponding value of the – OBSTAT– variable. If an – OBSTAT– variable already exists when you select Generate – OBSTAT– Variable from the variable menu, then the values of the variable are updated with the current state of the observations. The – OBSTAT– variable is often used to analyze observations with certain properties by using a SAS procedure. To use the – OBSTAT– variable outside Stat Studio, you can do the following: 1. Create an – OBSTAT– variable by selecting Generate – OBSTAT– Variable from the variable menu. 2. Save the augmented data set to a libref such as SASUSER. 3. Use the following DATA step to extract each observation property into its own variable: /* Create numerical variables from an _OBSTAT_ data MyData; set sasuser.MyData; IsSelected = inputn(substr(_obstat_, 1, 1), IsInPlots = inputn(substr(_obstat_, 2, 1), IsInAnalysis = inputn(substr(_obstat_, 3, 1), IsLabeled = inputn(substr(_obstat_, 4, 1), MarkerShape = inputn(substr(_obstat_, 5, 1), MarkerRed = inputn(substr(_obstat_, 6, 5), MarkerGreen = inputn(substr(_obstat_, 11, 5), MarkerBlue = inputn(substr(_obstat_, 16, 5), run; variable. */ 1.); 1.); 1.); 1.); 1.); 5.); 5.); 5.); Chapter 4. The Data Table 4. Use a WHERE clause to analyze only observations with a given set of properties. Sorting Observations This section describes how to sort a data table by one or more variables. You can select Edit I Variables I Sort from the main menu to open the Sort dialog box. Alternatively, you can right-click on a variable heading to select that variable and display the same menu, shown in Figure 4.2. The Sort dialog box is shown in Figure 4.4. The first time the Sort dialog box is created, any variables that are selected are automatically placed in the Sort by list. Subsequently, the Sort dialog box remembers the Sort by list from the last sort. Figure 4.4. The Sort Dialog Box The following list describes each item in the Sort dialog box. Variables lists the variables in the data set that are not yet in the Sort by list. Select variables in this list to transfer them to the Sort by list. ≫ transfers the selected variables from the Variables list to the Sort by list. ≪ removes selected variables from the Sort by list. Sort by lists the variables to sort by. Up moves a selected variable up one space in the Sort by list. Down moves a selected variable down one space in the Sort by list. 37 38 Chapter 4. The Data Table Ascending marks the selected variables in the Sort by list to be sorted in ascending order. Descending marks the selected variables in the Sort by list to be sorted in descending order. To carry out the sort operation, click OK. If a nominal variable has a nonstandard ordering, as described in the section “Variable Properties” on page 32, then the sort dialog box indicates that fact by marking the variable name with an asterisk. Observation Properties This section describes how to change properties of observations. You can do the following: • select observations • change the shapes and colors of markers associated with observations • change whether certain observations are included or excluded from plots and from analyses The row heading to the left of the data table gives the status of each observation. The heading indicates whether an observation is selected, which shape and color is used to represent the observation in plot, and whether the observation is included in analyses. You can change the properties of selected observations by using the Observations menu. You can access the Observations menu by selecting Edit I Observations from the main menu. Alternatively, right-clicking on the row heading of a selected observation displays the same Observations menu, shown in Figure 4.5. Figure 4.5. The Observations Menu Chapter 4. The Data Table The following list describes each item on the observation menu. Include in Plots includes the selected observations in graphs. Exclude from Plots excludes the selected observations from graphs. Include in Analyses includes the selected observations in statistical analyses. Exclude from Analyses excludes the selected observations from statistical analyses. Marker Properties displays the Marker Properties dialog box, shown in Figure 4.8. Label by Observation Number sets the label that is displayed in the left-most column of the data table to be the observation number. The observation number is also set as the default label that is displayed when you click on an observation marker in a graph. Label by Variable displays the Label by Variable dialog box, shown in Figure 4.9. Invert Selection changes the set of selected observations. Unselected observations become selected, while selected observations become unselected. Delete deletes the selected observations. Examine Selected Observations displays the Examine Selected Observations dialog box, shown in Figure 4.14. You can use this dialog box to view and compare the selected observations. Selecting Observations You can select observations in a data table by clicking on the row heading to the left of the data table. You can drag to select contiguous observations. You can click while holding down the CTRL key to select new observations without losing the ones already selected. Figure 4.6 shows selected observations. Note: To select observations, you must click or drag in the row headings on the left side of the data table. Highlighting a range of cells in the data table does not select the observations. The section “Adding and Editing Observations” on page 30 lists operations that you can perform on a range of cells. 39 40 Chapter 4. The Data Table Figure 4.6. Selected Observations Clicking in any of the four cells in the upper-left corner of the data table does the following: • Right-clicking in a cell brings up the Observations menu shown in Figure 4.5. Consequently, this is a safe place to right-click when you want to change properties of the selected observations, but no selected observations are currently visible. • Click in the upper-left or lower-right cell to deselect all observations and variables. • Click in the upper-right cell to deselect all observations and select all variables. • Click in the lower-left cell to deselect all variables and select all observations. If no observations are selected, the lower-left cell displays the total number of observations in the data table. If observations are selected, the lower-left cell displays (in brackets) the number of selected observations. If no variables are selected, the upper-right cell displays the total number of variables in the data table. If variables are selected, the upper-right cell displays (in brackets) the number of selected variables. Figure 4.7 illustrates two possibilities. The left portion of the figure indicates a data table that has 2,322 selected observations; none of the 36 variables are selected. The right portion of the figure indicates that 6 variables are selected, but none of the 6,188 observations are selected. Figure 4.7. Indicating Selected Observations (left) and Variables (right) Chapter 4. The Data Table Changing Marker Properties You can change the markers used to represent observations. You can use marker shapes and colors to represent observations that share common properties. Marker shapes often used to discriminate observations with different values of a categorical variable (for example, male versus female). Marker colors can also be used for this purpose, or can represent a continuous variable. Chapter 9, “General Plot Properties,” describes coloring markers by a continuous variable. Select Edit I Observations I Marker Properties from the main menu to open the Marker Properties dialog box, shown in Figure 4.8. Figure 4.8. The Marker Properties Dialog Box The following list describes each item in the dialog box. Shape sets the marker shape for the observations. Outline sets the marker outline color for the observations. Fill sets the marker fill color for the observations. Sample shows what the marker with the specified shape and colors looks like. Apply to specifies the set of observations whose markers will change. By default, changes are applied to only the selected observations. Changing Observation Labels You can change the label displayed in the left-most column of the data table. Observation numbers are shown by default. You can select Edit I Observations I Label by Variable from the main menu to open the Label by Variable dialog box shown in Figure 4.9. You can use this dialog box to select the variable whose values are displayed in the left-most column of the data table. The variable is also set as the default label that is displayed when you click on an observation marker in a graph. 41 42 Chapter 4. The Data Table Figure 4.9. The Label by Variable Dialog Box The Hide Label Variable check box hides the label variable, because its values are displayed in the left-most column of the data table. This is especially useful if the label variable is one of the first variables in the data table. Including and Excluding Observations You can choose which observations appear in plots and which are used in analyses. To include or exclude observations, first select the observations. From the Edit I Observations menu, you can then select Include in Plots, Exclude from Plots, Include in Analyses, or Exclude from Analyses. The row heading of the data table shows the status of an observation in analyses and plots. A marker symbol indicates that the observation is included in plots; observations excluded from plots do not have a marker symbol shown in the data table. Similarly, the χ2 symbol is present if and only if the observation is included in analyses. If an observation is excluded from analyses but included in plots, then the marker symbol changes to the × symbol. For example, Figure 4.10 shows what the data table would look like if you excluded some observations. In this example, the second observation was included in plots but excluded from analyses. The third observation was excluded from plots but included in analyses. The fourth observation was excluded from both plots and analyses. Figure 4.10. Excluded Observations Chapter 4. The Data Table Examining Data This section describes how to do the following: • find observations that satisfy certain conditions • examine selected observations • copy selected observations into a separate data set In analyzing data, you might want to find observations that satisfy certain conditions. For example, you might want to select all sales to a particular company. Or you might want to select all patients with high blood pressure. After you have found the observations, you can examine the observations or copy them to a new data set. Finding Observations You can select observations in the data table by using the Find dialog box. (For a way to graphically and interactively select observations that satisfy multiple constraints, see Chapter 11, “Techniques for Exploring Data.” ) You can open the Find dialog box (shown in Figure 4.11) by selecting Edit I Find from the main menu. Figure 4.11. The Find Dialog Box The following list describes each item in the Find dialog box. Variable chooses the variable whose values are examined. The list includes each variable in the data set. Operation selects the logical operation used to compare each observation with the contents of the Value field. 43 44 Chapter 4. The Data Table Value specifies the value used to select observations. Apply variable’s informat to value applies the variable’s informat to the contents of the Value field. If the variable does not have an informat, then this item is inactive. Apply format to each value during search applies the variable’s format to the variable and then compares the formatted data to the contents of the Value field. If the variable does not have a format, then this item is inactive. Match case specifies that each observation is compared to the contents of the Value field in a case-sensitive manner. If the variable is numeric, then this item is inactive. Use tolerance of specifies that a tolerance, , is used in comparing each observation to the contents of the Value field. Table 4.1 specifies how is used. If the chosen variable is a character variable, then this item is inactive. Clear existing selection specifies that all observations are searched, but only the observations that match the search criterion are selected. Search within existing selection specifies that only the observations that are selected are searched. You can use this option to perform logical AND operations. Add to existing selection specifies that all observations are searched, but observations that were selected prior to the search remain selected. You can use this option to perform logical OR operations. For numeric variables, let v be the value of the Value field and let be the value of the Use tolerance of field. (If you are not using a tolerance, then = 0.) Table 4.1 specifies whether an observation with value x for the chosen variable matches the query. Table 4.1. Find Operations for Numeric Variables Operation Equals Less than Greater than Not equals Less than or equals Greater than or equals Is missing Values Found x ∈ [v − , v + ] x<v+ x>v− x∈ / [v − , v + ] x≤v+ x≥v− x is missing Missing Selected? No Yes No Yes Yes No Yes Chapter 4. The Data Table To remember whether missing values match the query, recall that SAS missing values are represented as large negative numbers. Table 4.1 is consistent with the WHERE clause in the SAS DATA step. For character variables, comparisons are performed according to the ASCII order of characters. In particular, all uppercase letters [A–Z] precede lowercase characters [a–z]. Let v be the value of the Value field and let v ≺ x indicate that v precedes x in ASCII order. Table 4.2 specifies whether an observation with value x for the chosen variable matches the query. Table 4.2. Find Operations for Character Variables Operation Equals Less than Greater than Not equals Less than or equals Greater than or equals Is missing Contains Does not contains Begins with Values Found x=v x≺v v≺x x 6= v xv vx x is missing x contains v x does not contain v x begins with v Missing Selected? No Yes No Yes Yes No Yes No Yes No To help remember whether character missing values match the query, think of the character missing value as being a zero-length string that contain no characters. Table 4.2 is consistent with the WHERE clause in the SAS DATA step. As a first example, Figure 4.11 shows how to find observations in the Hurricanes data set whose latitude variable is contained in the interval [28, 32]. This is a quick way to find observations with latitudes between 28 and 32 in a single search. A second example is shown in Figure 4.12. This search finds observations for which the date variable strictly precedes 07AUG1988. Note that the date variable has a DATE9. informat, so you can use that informat to make it more convenient to input the contents of the Value field. (Without the informat, you would need to search for the value 10445, the SAS date value corresponding to 06AUG1988.) Note that the date variable is a numeric variable, even though the formatted values appear as text. 45 46 Chapter 4. The Data Table Figure 4.12. Searching for Dates A related example is shown in Figure 4.13. This search finds all observations for which the date variable contains the text “AUG”. Note that to perform this search you must check Apply format to each value during search. This forces the Find dialog box to apply the DATE9. format to the date variable, which means comparing strings (character data) instead of numbers (numeric data). You can then select Contains from the Operation list. Each formatted string is searched for the value “AUG”. Figure 4.13. Matching Text in a Formatted Variable Chapter 4. The Data Table Examining Selected Observations You can examine a set of selected observations. To do this, select Edit I Observations I Examine Selected Observations from the main menu. Figure 4.14 shows the dialog box that appears. By clicking on observation numbers in the left-hand list (or by using the UP and DOWN arrow keys), you can examine each selected observation in turn. Figure 4.14. Examining Selected Observations Copying Selected Data You can subset your data by copying selected observations or variables to a separate data set. (You can select variables without losing selected observations by holding down the CTRL key while you click.) You can then analyze or save this new data set. If no variables are selected, all variables are copied. If no observations are selected, all observations are copied. After you have selected observations and/or variables, select File I New I Data Set from Selected Data from the main menu. A new data table (Figure 4.15) appears, containing only the selected subset of the original data. 47 48 Chapter 4. The Data Table Figure 4.15. Copying Selected Data Saving Data If you save data after changing variable or observation properties, then the changes are saved as well. Most variable properties (for example, formats) are saved with the SAS data set, whereas observation properties (for example, marker shapes) are saved in a separate metadata file. The metadata file is stored on the client PC and has the same name as the data set, but with a dmm extension. For example, if you save a data set named MyData to your PC, then a file named MyData.dmm is also created in the same Windows folder as the MyData.sas7bdat file. If you have changed the data and try to exit Stat Studio, you are prompted to save the data set if you have done any of the following actions: • edited cells in the data table • changed a variable’s properties (name, label, format, informat) • changed a variable’s measure level (nominal, interval) • sorted a data set • added or deleted a variable • included or excluded observations • changed an observation’s marker properties (shape, color) • added or deleted an observation Chapter 4. The Data Table Properties of Data Tables When a data table is the active window, you can do the following: • create additional copies of the data table • change the default properties of data tables in the current workspace You can select Windows I New Window from the main menu to create a copy of the current data table. (The new table might appear on top of the existing data table, so drag it to a new location, if necessary.) This second data table can be scrolled independently from the first. This is useful, for example, if you are interested in examining several variables or observations whose positions in the data table vary widely. You can examine different subsets of the data simultaneously by using two or more tabular views of the same data. By default, if you sort one data table, then other data tables are also sorted in the same order. This is because a sort typically changes the order of the underlying data. (As mentioned in the section “Saving Data” on page 48, when you exit Stat Studio you are prompted to save the data if you have sorted it.) However, there might be instances when it is useful to view the same data, but sorted in a different order. To accomplish this, you can locally sort a data table. To locally sort a data table, select Edit I Properties from the main menu, which displays the dialog box in Figure 4.16. Figure 4.16. Data Table Ordering Properties 49 50 Chapter 4. The Data Table The Ordering tab has the following items: Changes in observation order affect gives you two choices. If you select Actual data, then sorting the data table results in a global sort that reorders the observation. If you select This view only, then sorting the data table results in a local sort that does not reorder the observations but only changes the view of the data in the current data table. Default sort order gives you two choices. Your selection of Ascending or Descending determines the default order in which variables are sorted. The Selections tab has a single item, as shown in Figure 4.17. If you select Scroll selected observations into view, then the data table automatically scrolls to a selected observation each time an observation is selected. To manually scroll a selected item into view, use the F3 key. Figure 4.17. Data Table Selection Properties Chapter 4. The Data Table 51 Keyboard Shortcuts in Data Tables Some keys in a data table are associated with certain actions, as shown in Table 4.3. Table 4.3. Keys and Actions in Data Tables Key ESC ESC F1 F3 SHIFT+F3 F10 TAB SHIFT+TAB ENTER ALT+RIGHT ARROW ALT+LEFT ARROW ALT+DOWN ARROW ALT+UP ARROW SHIFT+ALT+RIGHT ARROW SHIFT+ALT+LEFT ARROW SHIFT+ALT+DOWN ARROW SHIFT+ALT+UP ARROW SHIFT+RIGHT ARROW SHIFT+LEFT ARROW SHIFT+DOWN ARROW SHIFT+UP ARROW HOME END CTRL+SPACEBAR CTRL+HOME CTRL+END CTRL+INSERT DELETE Action When editing data, abort the current edit and deselect cells. Deselect any selected observations and variables. Display the online Help system. Move the active cell to the row of the next selected observation. Move the active cell to the row of the previous selected observation. If observations are selected, display the Observations menu. If variables are selected, display the Variables menu. If observations and variables are selected, display the Observations menu followed by the Variables menu. Move the active cell to the right. Move the active cell to the left. Move the active cell down one row. Toggle selection of a variable without changing the active cell. Toggle selection of an observation without changing the active cell. Toggle selection of a variable and move the active cell to the next or previous variable. Toggle selection of an observation and move the active cell to the next or previous observation. Extend the selection of a range of cell columns. Extend the selection of a range of cell rows. Edit the active cell and place the cursor at the beginning of the cell. Edit the active cell and place the cursor at the end of the cell. Clear selected observations and variables. Set the active cell to the first row and first column. Set the active cell to the last row and last column. Display the New Variable dialog box. If observations or variables are selected, delete the selected variables or observations. If cells are selected, delete the contents of the selected cells. 52 Chapter 4. The Data Table In addition, the data table supports the arrow keys for navigating cells, and supports the standard Microsoft control sequences shown in Table 4.4. Table 4.4. Standard Control Sequences in Data Tables Key CTRL+A CTRL+C CTRL+F CTRL+P CTRL+V CTRL+X CTRL+Y CTRL+Z Action Select all observations. Copy contents of selected cells to Windows clipboard. Display the Find dialog box. Print the data table. Paste contents of Windows clipboard to cells. Cut contents of selected cells and paste to Windows clipboard. Redo last undo. Undo last operation. Chapter 5 Exploring Data in One Dimension You can explore the distributions of nominal variables by using bar charts. You can explore the univariate distributions of interval variables by using histograms and box plots. Bar Charts This section describes how to use a bar chart to visualize the distribution of a nominal variable. A bar chart shows the relative frequency of unique values of a variable. The height of each bar is proportional to the number of observations with each given value. Example In this section you create a bar chart of the category variable of the Hurricanes data set. The category variable gives the Saffir-Simpson wind intensity category for each observation. The category variable is encoded according to the value of wind– kts, as shown in Table 5.1. Table 5.1. The Saffir-Simpson Intensity Scale Category TD TS Cat1 Cat2 Cat3 Cat4 Cat5 Description Tropical Depression Tropical Storm Category 1 Hurricane Category 2 Hurricane Category 3 Hurricane Category 4 Hurricane Category 5 Hurricane Wind Speed (knots) 22–33 34–63 64–82 83 –95 96 –113 114 –134 135 or greater The category variable also has missing values, representing weak intensities (wind speed less than 22 knots). =⇒ Open the Hurricanes data set. =⇒ Select Graph I Bar Chart from the main menu, as shown in Figure 5.1. 54 Chapter 5. Exploring Data in One Dimension Figure 5.1. Selecting a Bar Chart A dialog box appears as in Figure 5.2. =⇒ Select the category variable, and click Set X. =⇒ Click OK. Note: The bar chart also supports an optional frequency variable. Figure 5.2. The Bar Chart Dialog Box A bar chart appears (Figure 5.3), showing the unique values of the category variable. The chart shows that most of the observations in the data set are for tropical storms and tropical depressions. There are relatively few category 5 hurricanes. Chapter 5. Exploring Data in One Dimension Figure 5.3. A Bar Chart The category variable has missing values. The set of missing values are grouped together and represented by a bar labeled with the symbol. You can click on a bar to select the observations contained in that bar. You can click while holding down the CTRL key to select observations in multiple bars. You can drag out a selection rectangle to select observations in contiguous bars. You can create bar charts of any nominal variable, numeric or character. Bar Chart Properties This section describes the Bars tab associated with a bar chart. To access the bar chart properties, right-click near the center of a plot, and select Plot Area Properties from the pop-up menu. The Bars tab controls attributes of the bar chart. The Bars tab is shown in Figure 5.4. Fill sets the fill color for each bar. Fill: Use blend sets the fill color for each bar according to a color gradient. Outline sets the outline color for each bar. Outline: Use blend sets the outline color for each bar according to a color gradient. 55 56 Chapter 5. Exploring Data in One Dimension Fill bars specifies whether each bar is filled with a color. Otherwise, only the outline of the bar is shown. Show labels specifies whether each bar is labeled with the height of the bar. Y axis represents specifies whether the vertical scale represents frequency counts or percentage. “Other” threshold (%) sets a cutoff value for determining which observations are placed into an “Others” category. Figure 5.4. Plot Area Properties for a Bar Chart For a discussion of the remaining tabs, see Chapter 9, “General Plot Properties.” Bar Charts of Selected Variables If one or more nominal variables are selected in a data table when you select Graph I Bar Chart, then the bar chart dialog box (Figure 5.2) does not appear. Instead bar charts are created of the selected nominal variables. You can also select nominal and interval variables and select Graph I Bar Chart. A bar chart appears for each nominal variable; a histogram appears for each interval variable. If you create a matrix of plots from selected variables, you can close the matrix by pressing the F11 key while any plot is active and selecting from the pop-up menu. Alternatively, you can use the Workspace Explorer (see the section “Workspace Explorer” on page 165) to quickly close plots. Chapter 5. Exploring Data in One Dimension If a variable in the data table has a Frequency role, it is automatically used as the frequency variable for the plots; the frequency variable should not be one of the selected variables. Variables with a Weight role are ignored when you are creating bar charts. Histograms This section describes how to use a histogram to visualize the distribution of a continuous (interval) variable. A histogram is an estimate of the density of data. The range of the variable is divided into a certain number of subintervals, or bins. The height of the bar in each bin is proportional to the number of data points that have values in that bin. A histogram is determined not only by the bin width, but also by the choice of an anchor (or origin). Example In this section you create a histogram of the latitude variable of the Hurricanes data set. The latitude variable gives the latitude of the center of each tropical cyclone observation. =⇒ Open the Hurricanes data set. =⇒ Select Graph I Histogram from the main menu, as shown in Figure 5.5. Figure 5.5. Selecting a Histogram A dialog box appears as in Figure 5.6. =⇒ Select the latitude variable, and click Set X. =⇒ Click OK. Note: The histogram also supports an optional frequency variable. 57 58 Chapter 5. Exploring Data in One Dimension Figure 5.6. The Histogram Dialog Box A histogram appears (Figure 5.7), showing the distribution of latitudes for the tropical cyclones in this data set. The histogram shows that most Atlantic tropical cyclones occur between 10 and 40 degrees north latitude. The data distribution looks bimodal: one mode near 15 degrees and the other near 30 degrees of latitude. Figure 5.7. A Histogram If a variable has missing values, those values are not included in the histogram. You can click on a histogram bar to select the observations contained in that bin. You can click while holding down the CTRL key to select observations in multiple bins. You can drag out a selection rectangle to select observations in contiguous bins. Chapter 5. Exploring Data in One Dimension Histogram Properties This section describes the Bars tab associated with a histogram. To access the histogram properties, right-click near the center of a plot, and select Plot Area Properties from the pop-up menu. The Bars tab controls attributes of the histogram. The Bars tab is shown in Figure 5.8. Fill sets the fill color for each bar. Fill: Use blend sets the fill color for each bar according to a color gradient. Outline sets the outline color for each bar. Outline: Use blend sets the outline color for each bar according to a color gradient. Fill bars specifies whether each bar is filled with a color. Otherwise, only the outline of the bar is shown. Show labels specifies whether each bar is labeled with the height of the bar. Y axis represents specifies whether the vertical scale represents frequency counts, percentage, or density. Figure 5.8. Plot Area Properties for a Histogram For a discussion of the remaining tabs, see Chapter 9, “General Plot Properties.” 59 60 Chapter 5. Exploring Data in One Dimension Histograms of Selected Variables If one or more interval variables are selected in a data table when you select Graph I Histogram, then the histogram dialog box (Figure 5.6) does not appear. Instead histograms are created of the selected interval variables. You can also select nominal and interval variables and select Graph I Histogram. A bar chart appears for each nominal variable; a histogram appears for each interval variable. If a variable has a Frequency role, it is automatically used as the frequency variable for the plots; the frequency variable does not need to be selected. If you create a matrix of plots from selected variables, you can close the matrix by pressing the F11 key while any plot is active and selecting from the pop-up menu. Alternatively, you can use the Workspace Explorer (see the section “Workspace Explorer” on page 165) to quickly close plots. Histogram Binning: Setting Tick Positions By default, Stat Studio produces histograms with an anchor location and bin width chosen according to an algorithm by Terrell and Scott (1985). This section describes how you can choose a different anchor location or bin width for a histogram. The example in this section is a continuation of the example in “Example”, in which you created a histogram of the latitude variable in the Hurricanes data set. For a histogram, the major tick unit is also the width of the histogram bins. For example, the tick marks for the histogram in Figure 5.7 are anchored at 6.25 and have a tick unit of 2.5. The following steps show you how to change the location of the histogram ticks so that the bins show the frequency of observations in the intervals 5–10, 10–15, 15–20, and so on. =⇒ Right-click on the horizontal axis of the histogram, and select Axis Properties from the pop-up menu, as shown in Figure 5.9. Figure 5.9. The Axis Pop-up Menu The Axis Properties dialog box appears as in Figure 5.10. Note that this is a quick way to determine the anchor location, tick unit, and tick range for an axis. =⇒ Change the Major tick unit value to 5. Chapter 5. Exploring Data in One Dimension =⇒ Change the Anchor tick value to 10. Figure 5.10. Dialog Box for Specifying Histogram Bins =⇒ Click OK. The histogram updates to reflect the new histogram bin locations. The revised histogram is shown in Figure 5.11. The Tick Range field shown in Figure 5.10 is automatically widened, if necessary, so that all data are contained in bins. Figure 5.11. Histogram with Customized Bins 61 62 Chapter 5. Exploring Data in One Dimension Interactive Histogram Binning Sometimes it is useful to explore how the shape of a histogram varies with different combinations of anchor locations and bin widths. Interactively changing the histogram can help you determine if apparent modes in the data are real or are an artifact of a specific binning. To interactively change the anchor location and bin width, right-click in the middle of the histogram and select Bin Tool from the pop-up menu, as shown in Figure 5.12. Figure 5.12. The Histogram Pop-up Menu Note that the mouse pointer changes its shape, as shown in Figure 5.13. If you drag the pointer around in the plot area, then the histogram rebins. Dragging the pointer horizontally changes the anchor position. Dragging the pointer vertically changes the bin width. When the pointer is near the top of the plot area, the bin widths are relatively small; when the pointer is near the bottom, the bin widths are larger. Figure 5.13. Interactively Rebinning a Histogram Chapter 5. Exploring Data in One Dimension Box Plots A box plot summarizes the distribution of data sampled from a continuous numeric variable. The central line in a box plot indicates the median of the data, while the edges of the box indicate the first and third quartiles (that is, the 25th and 75th percentiles). Extending from the box are whiskers that represent data that are a certain distance from the median. Beyond the whiskers are outliers: observations that are relatively far from the median. These features are shown in Figure 5.14. Figure 5.14. Schematic Description of a Box Plot This section describes how to use a box plot to visualize the distribution of a continuous (interval) variable. You can also use box plots to see how the distribution changes across levels of one or more nominal variables. Example In this section you create a box plot of the latitude variable of the Hurricanes data set, grouped by levels of the category variable. The latitude variable gives the latitude of the center of each tropical cyclone observation. The category variable gives the Saffir-Simpson wind intensity category for each observation. The category variable also has missing values, representing weak intensities (wind speed less than 22 knots). =⇒ Open the Hurricanes data set. =⇒ Select Graph I Box Plot from the main menu, as shown in Figure 5.15. 63 64 Chapter 5. Exploring Data in One Dimension Figure 5.15. Selecting a Box Plot A dialog box appears as in Figure 5.16. =⇒ Select the latitude variable, and click Set Y. =⇒ Select the category variable, and click Add X. =⇒ Click OK. Note: X variables are optional. If you do not select an X variable, you get a box plot of the Y variable. Only nominal variables can be selected as an X variable. Note: The box plot also supports an optional frequency variable. Figure 5.16. The Box Plot Dialog Box A box plot appears (Figure 5.17), showing the distribution of the latitude variable for each unique value of the category variable. The plot shows that the most intense cyclones occur in a relatively narrow band of southern latitudes. Intense hurricanes have median latitudes that are farther south than weaker hurricanes. There is also less variance in the latitudes of the intense hurricanes. Tropical storms and tropical depressions do not obey these general trends, and have the largest spread in latitude. Chapter 5. Exploring Data in One Dimension Figure 5.17. A Box Plot The category variable has missing values. The set of missing values are grouped together and represented by a bar labeled with the symbol. You can click on any box, whisker, or outlier to select the observations contained in that box. You can click while holding down the CTRL key to select observations in multiple boxes. You can drag out a selection rectangles to select observations in adjacent boxes. Box Plot Properties This section describes the Boxes tab associated with a box plot. To access the box plot properties, right-click near the center of a plot, and select Plot Area Properties from the pop-up menu. The Boxes tab controls attributes of the box plot. The Boxes tab is shown in Figure 5.18. Box: Whisker length sets the length of the whiskers. length of w means that whiskers are drawn from the quartiles to the farthest observation not more than w times the interquartile distance (Q3–Q1). Box: with serifs specifies whether each whisker is capped with a horizontal line segment. Box: with notches specifies whether each box is drawn with notches. The medians of two box plots are significantly different at approximately the 0.05 level if the corresponding notches do not overlap. 65 66 Chapter 5. Exploring Data in One Dimension Mean: with one standard deviation specifies whether each box is draw with mean markers extending one standard deviation from the mean. The central line of the mean marker indicates the mean. The upper and lower extents of the mean marker indicate the mean plus or minus one standard deviation. Mean: with two standard deviations specifies whether each box is draw with mean markers extending two standard deviation from the mean. Mean: Shape specifies whether the mean markers should be drawn as a diamond or an ellipse. Color: Fill sets the fill color for each box. Color: Outline sets the outline color for each box. Color: Mean sets the color for mean markers. Fill boxes specifies whether each box is filled with a color. Otherwise, only the outline of the box is shown. Figure 5.18. Plot Area Properties for a Box Plot For a discussion of the Observations tab, see Chapter 6, “Exploring Data in Two Dimensions.” For a discussion of the remaining tabs, see Chapter 9, “General Plot Properties.” Chapter 5. Exploring Data in One Dimension Box Plots of Selected Variables If one or more interval variables are selected in a data table when you select Graph I Box Plot, then the box plot dialog box (Figure 5.16) does not appear. Instead box plots are created for each selected interval variable. You can also select nominal and interval variables and select Graph I Box Plot. A box plot appears for each interval variable; nominal variables are assigned to the X axis. If a variable has a Frequency role, it is automatically used as the frequency variable for the plots; the frequency variable does not need to be selected. If you create a matrix of plots from selected variables, you can close the matrix by pressing the F11 key while any plot is active and selecting from the pop-up menu. Alternatively, you can use the Workspace Explorer (see the section “Workspace Explorer” on page 165) to quickly close plots. References Terrell, G. R. and Scott, D. W. (1985), “Oversmoothed Nonparametric Density Estimates,” Journal of the American Statistical Association, 80, 209–214. 67 68 Chapter 6 Exploring Data in Two Dimensions You can explore the relationship between two (or more) nominal variables by using a mosaic chart. You can explore the relationship between two variables by using a scatter plot. Usually the variables in a scatter plot are interval variables. If you have a time variable, you can observe the behavior of one or more variables over time with a line plot. You can also use line plots to visualize a response variable (and, optionally, fitted curves and confidence bands) versus values of an explanatory variable. You can create and explore maps with a polygon plot. Mosaic Plots This section describes how to use a mosaic plot to visualize the cells of a contingency table. A mosaic plot displays the frequency of data with respect to multiple nominal variables. A mosaic plot is a set of adjacent bar plots formed first by dividing the horizontal axis according to the proportion of observations in each category of the first variable and then by dividing the vertical axis according to the proportion of observations in the second variable. For more than two nominal variables, this process can be continued by further horizontal or vertical subdivision. The area of each block is proportional to the number of observations it represents. Example In this section you create a mosaic plot of the nation and industry variables of the Business data set. The nation variable gives the nation of each business listed in the data set, while the industry variable assigns each business to a category that describes the business. =⇒ Open the Business data set. =⇒ Select Graph I Mosaic Plot from the main menu, as shown in Figure 6.1. 70 Chapter 6. Exploring Data in Two Dimensions Figure 6.1. Selecting a Mosaic Plot A dialog box appears as in Figure 6.2. =⇒ Select the nation variable, and click Set Y. =⇒ Select the industry variable, and click Add X. =⇒ Click OK. Note: The mosaic also supports an optional frequency variable. Figure 6.2. The Mosaic Plot Dialog Box A mosaic plot appears (Figure 6.3), showing the relative proportions of businesses in this data set as grouped by nation and industry. The mosaic plot shows that the U.S. food companies make up the largest subset, because that cell has the largest area. Other large cells include Japanese automobile companies, Japanese electronics Chapter 6. Exploring Data in Two Dimensions companies, and U.S. oil companies. The plot also shows that there are no German food companies in the data set. Figure 6.3. A Mosaic Plot You can click on a cell to select the observations contained in that cell. Note that clicking on a cell also shows you the number of observations in that cell. You can click while holding down the CTRL key to select observations in multiple cells. You can drag out a selection rectangle to select observations in contiguous cells. You can create mosaic plots of any nominal variables, numeric or character. However, the variables should have a small to moderate number of levels. Note that the cells in this mosaic plot represent the count (number of observations) of businesses in each nation and industry. However, you might be more interested in comparing the revenue generated by these businesses. You can make this comparison by re-creating the mosaic plot and adding sales as a frequency variable. =⇒ Select Graph I Mosaic Plot from the main menu. A dialog box appears. =⇒ Select the nation variable, and click Set Y. =⇒ Select the industry variable, and click Add X. =⇒ Select the sales variable, and click Set Freq. =⇒ Click OK. A mosaic plot appears (Figure 6.4), showing the relative proportions of sales for each nation and industry. The mosaic plot shows that the U.S. oil companies generate the most revenue, followed by the U.S. and Japanese automobile companies. Companies from the U.S. and Japan account for over two thirds of the sales. 71 72 Chapter 6. Exploring Data in Two Dimensions Figure 6.4. A Mosaic Plot with a Frequency Variable Similarly, if you were interested in comparing the number of employees in these businesses, you could use employs as a frequency variable. However, note that you could not compare profits in this way, because some profits are negative and the mosaic plot ignores any observation whose frequency is negative. You should also make sure that the frequency variable contains integers; noninteger values are truncated. Mosaic Plot Properties This section describes the Mosaic tab associated with a mosaic plot. To access the mosaic plot properties, right-click near the center of a plot, and select Plot Area Properties from the pop-up menu. The Mosaic tab controls attributes of the mosaic plot. The Mosaic tab is shown in Figure 6.5. “Other” threshold (%) sets a cutoff value for determining which observations are placed into an “Others” category. Layout sets the method by which cells are formed from the X and Y variables. 2 way In this layout scheme, the X variables determine groups, and the mosaic plot displays a stacked bar chart of the Y variable for each group. N way This layout scheme is available only if there are exactly two X variables. In this layout scheme, the plot subdivides in the horizontal direction by the first X variable, then subdivides in the vertical direction by the Y variable, and finally subdivides in the horizontal direction by the second X variable. Chapter 6. Exploring Data in Two Dimensions Show labels for all tiles specifies whether each cell is labeled with the proportion it represents. Show labels as specifies whether a cell represents frequency or percentage. Figure 6.5. Plot Area Properties for a Mosaic Plot For a discussion of the remaining tabs, see Chapter 9, “General Plot Properties.” Mosaic Plots of Selected Variables If one or more nominal variables are selected in a data table when you select Graph I Mosaic Plot, then the mosaic plot dialog box (Figure 6.2) does not appear. Instead mosaic plots are created for each pair of the selected nominal variables. If you create a matrix of plots from selected variables, you can close the matrix by pressing the F11 key while any plot is active and selecting from the pop-up menu. Alternatively, you can use the Workspace Explorer (see the section “Workspace Explorer” on page 165) to quickly close plots. If a variable in the data table has a Frequency role, it is automatically used as the frequency variable for the plots; the frequency variable should not be one of the selected variables. Variables with a Weight role are ignored when you are creating mosaic plots. 73 74 Chapter 6. Exploring Data in Two Dimensions Scatter Plots This section describes how to use a scatter plot to visualize the relationship between two variables. Usually each variable is continuous (interval), but that is not a requirement. Example In this section you create a scatter plot of the wind– kts and min– pressure variables of the Hurricanes data set. The wind– kts variable is the wind speed in knots; the min– pressure variable is the minimum central pressure for each observation. The min– pressure variable has a few missing values; those observations are not included in the scatter plot. =⇒ Open the Hurricanes data set. =⇒ Select Graph I Scatter Plot from the main menu, as shown in Figure 6.6. Figure 6.6. Selecting a Scatter Plot A dialog box appears as in Figure 6.7. =⇒ Select the variable wind– kts, and click Set Y. =⇒ Select the variable min– pressure, and click Set X. =⇒ Click OK. Chapter 6. Exploring Data in Two Dimensions Figure 6.7. The Scatter Plot Dialog Box A scatter plot appears (Figure 6.8) showing the bivariate data. The plot shows a strong negative correlation (ρ = −0.93) between wind speed and pressure. The plot also shows that most, although not all, wind speeds are rounded to the nearest 5 knots. Figure 6.8. A Scatter Plot You can click on any observation marker to select the observation. You can click while holding down the CTRL key to select multiple observations. You can drag out a selection rectangle to select a group of observations. 75 76 Chapter 6. Exploring Data in Two Dimensions Scatter Plot Properties This section describes the Observations tab associated with a scatter plot. To access the scatter plot properties, right-click near the center of a plot, and select Plot Area Properties from the pop-up menu. The Observations tab controls attributes of the scatter plot. The Observations tab is shown in Figure 6.9. Marker Attributes: Shape sets the shape of the marker for each observation. Marker Attributes: Outline specifies the color of the marker boundary. If the Blend list is set to None, the Outline list enables you to specify the outline color of observation markers. If the Blend list is not set to None, the Outline list enables you to specify the color blend to be used to color the outlines of observation markers. Marker Attributes: Blend (Outline) sets the variable whose values should be used to perform color blending for the outline colors of observation markers. If this value is set to None, color blending is not performed. Marker Attributes: Fill specifies the color of the marker interior. If the Blend list is set to None, the Fill list enables you to specify the fill color of observation markers. If the Blend list is not set to None, the Fill list enables you to specify the color blend to be used to color the interiors of observation markers. Marker Attributes: Blend (Fill) sets the variable whose values should be used to perform color blending for the fill colors of observation markers. If this value is set to None, color blending is not performed. Marker Attributes: Apply to specifies whether marker shape and color changes should be applied to all observations, or just the ones currently selected. Marker Attributes: Size specifies the size of observation markers. All observation markers in a plot are drawn at the same size. Selecting Auto causes the size of markers to change according to the size of the plot. Show only selected observations specifies whether observation markers are shown only for selected observations. Label all observations specifies whether labels are displayed next to each observation marker. Label observations by specifies the variable to use to label observations. Chapter 6. Exploring Data in Two Dimensions Figure 6.9. Plot Area Properties for a Scatter Plot For a discussion of the remaining tabs, see Chapter 9, “General Plot Properties.” Scatter Plots of Selected Variables If one or more variables are selected in a data table when you select Graph I Scatter Plot, then the scatter plot dialog box (Figure 6.7) does not appear. Instead, a scatter plot matrix is created showing each pair of the selected variables (Figure 6.10). If you create a matrix of plots from selected variables, you can close the matrix by pressing the F11 key while any plot is active and selecting from the pop-up menu. Alternatively, you can use the Workspace Explorer (see the section “Workspace Explorer” on page 165) to quickly close plots. Variables with a Frequency or Weight role are ignored when you are creating scatter plots. 77 78 Chapter 6. Exploring Data in Two Dimensions Figure 6.10. A Matrix of Scatter Plots Line Plots This section describes how to use a line plot to observe the behavior of one or more variables over time. You can also use line plots to visualize a response variable (and, optionally, fitted curves and confidence bands) versus values of an explanatory variable. You can create line plots when your data are in one of two configurations. The first configuration (Table 6.1) is when you have an X variable and one or more Y variables. Each Y variable has the same number of observations as the X variable. (Some of the Y values might be missing.) In this configuration there are as many lines in the plot as there are Y variables. Table 6.1. A Data Configuration for a Line Plot X 1 2 3 4 5 Y1 1 3 2 4 5 Y2 4 3 3 2 1 In the second configuration (Table 6.2), there is a single X and a single Y variable, but there are one or more group variables that specify which line each observation Chapter 6. Exploring Data in Two Dimensions belongs to. In this configuration there are as many lines in the plot as there are unique values of the group variables. Table 6.2. An Alternative Data Configuration for a Line Plot X 1 1 2 2 3 3 4 4 5 5 Y 1 4 3 3 2 3 4 2 5 1 Group A B A B A B A B A B The X variable does not need to be sorted in either configuration. Any data arranged in the first configuration can be rewritten in the second. For example, Table 6.2 represents the same data as Table 6.1. The second configuration is more useful if you have different values of the X variable for each group. Example: Multiple Y Variables In this section you create a line plot of the co and wind variables versus the datetime variable of the Air data set. The co variable is a measurement of carbon monoxide. The wind variable is a measurement of wind speed. The datetime variable is the hour and date of each measurement. =⇒ Open the Air data set. =⇒ Select Graph I Line Plot from the main menu, as shown in Figure 6.11. Figure 6.11. Selecting a Line Plot A dialog box appears as in Figure 6.12. =⇒ Select the co variable. Hold down the CTRL key and select the wind variable. Click Add Y. =⇒ Select the variable datetime, and click Set X. 79 80 Chapter 6. Exploring Data in Two Dimensions =⇒ Click OK. Figure 6.12. The Line Plot Dialog Box A line plot appears (Figure 6.13), showing the carbon monoxide and wind measurements for each hour of a seven-day period. By default, the two lines are displayed in different colors. You can change the color and line style of the lines, as shown in the remainder of this example. Figure 6.13. A Line Plot Chapter 6. Exploring Data in Two Dimensions =⇒ Right-click near the center of the plot, and select Plot Area Properties from the pop-up menu. A dialog box appears as in Figure 6.14. Figure 6.14. Plot Area Properties for a Line Plot =⇒ Select the wind line in the left-hand list. =⇒ Change the line style to dashed and the color to blue. =⇒ Click the large left arrow in the center of the dialog box to apply the changes to the wind line. =⇒ Click OK. Figure 6.15. A Line Plot with Line Colors and Styles 81 82 Chapter 6. Exploring Data in Two Dimensions The line plot now looks like the plot in Figure 6.15. The carbon monoxide line shows periodic behavior for the first half of the week, followed by extremely low values for the second half of the week. The wind values are low for the first half of the week, but much stronger for the second half. These data might indicate that sufficiently strong winds can blow away carbon monoxide. You can click on any observation marker to select the observation. You can click while holding down the CTRL key to select multiple observations. You can drag out a selection rectangle to select a group of observations. You can also select the lines themselves by clicking on a line segment that is away from any observation. If you open the dialog box shown in Figure 6.14, selected lines in the line plot are also selected in the left-hand list in the dialog box. Note: If you plot multiple Y variables, then an observation in the data table is represented by multiple markers in the line plot. Clicking on any marker in the plot selects the entire corresponding observation. Example: A Group Variable In this example you use the same data set and the co variable, but this time you plot the variable over a 24-hour period for each day of the week. =⇒ In the data table, right-click on the day variable, and select nominal from the pop-up menu, as shown in Figure 6.16. Figure 6.16. Changing the Role of a Variable Nominal variables can be used as group variables in the construction of a line plot. =⇒ Press the ESC key to deselect the day variable. =⇒ Select Graph I Line Plot from the main menu. A dialog box appears (Figure 6.17). Chapter 6. Exploring Data in Two Dimensions =⇒ Select the variable co, and click Add Y. =⇒ Select the variable hour, and click Set X. =⇒ Select the variable day, and click Add Group. =⇒ Click OK. Figure 6.17. Specifying a Group Variable The line plot that appears (Figure 6.18) has seven lines, one for each day of the week. For several days early in the week, the daily carbon monoxide peaked during the morning and evening commuting times: roughly 8 a.m. and 6–7 p.m. 83 84 Chapter 6. Exploring Data in Two Dimensions Figure 6.18. A Line Plot with a Group Variable To better visualize each day’s carbon dioxide, you can use a bar chart to select each day individually. =⇒ Select Graph I Bar Chart from the main menu. The Bar Chart dialog box appears. =⇒ Select the day variable, and click Set X. =⇒ Click OK. The resulting plots are shown in Figure 6.19. You can now select each day of the week and examine the observations for that day. Chapter 6. Exploring Data in Two Dimensions Figure 6.19. Exploring Data for Each Day Line Plot Properties This section describes the Line Styles tab associated with a line plot. To access the line plot properties, right-click near the center of a plot, and select Plot Area Properties from the pop-up menu. The Line Styles tab controls attributes of the lines displayed on a line plot. The Line Styles tab is shown in Figure 6.14. Lines displays each line in the plot. You can select one or more items in the list to change their properties. ⇐= (large left arrow) applies the current set of properties to the lines selected in the Lines list. You must click the large left arrow to transfer the line attributes to the selected items in the Lines list. Attributes: Style sets the line style. Attributes: Width sets the line width. 85 86 Chapter 6. Exploring Data in Two Dimensions Attributes: Color sets the line color. Attributes: Marker sets the markers for the line. The default marker is the marker shown in the data table for each observation. Line markers are independent from observation markers. Attributes: Connect observations specifies whether the line connects adjacent observations with a line segment. Attributes: Show observations specifies whether observations are shown along the line. Attributes: Sort observations specifies whether observations along the line are sorted according to the value of the X variable. For a discussion of the Observations tab, see the section “Scatter Plot Properties” on page 76. For a discussion of the remaining tabs, see Chapter 9, “General Plot Properties.” Line Plots of Selected Variables If one or more variables are selected in a data table when you select Graph I Line Plot, then the line plot dialog box (Figure 6.12) does not appear. Instead, a line plot is created. The rules for constructing the line plot are as follows: 1. If one variable is selected, create a line plot of Y versus Y . 2. If exactly two variables are selected, the first is used as the Y variable and the second as the X variable. 3. If there are k > 2 variables selected, then count the number of selected nominal variables. (a) If no nominal variables are selected, create a line plot of (Y1 , Y2 , . . . , Yk−1 ) versus Yk . (b) If there are nominal variables selected, then count the number of selected interval variables. i. If there are no interval variables selected, then plot the first selected variables as Y, plot the second selected variables as X, and use the remaining selected variables as group variables. ii. If there is exactly one interval variable, then plot it as Y if it was chosen first, and otherwise plot it as X. The first nominal variable is assigned to the X or Y role, and the remaining selected variables are used as group variables. iii. If there are exactly two interval variables, then plot the first selected interval variable as Y, plot the second as X, and use the remaining selected variables as group variables. Chapter 6. Exploring Data in Two Dimensions iv. If there are more than two interval variables, then ignore the nominal variables and plot the interval variables as in rule 1. Variables with a Frequency or Weight role are ignored when you are creating line plots. Polygon Plots This section describes how to use a polygon plot to visualize map data. A polygon plot displays polygons that are linked to levels of one or more categorical variables. The polygon plot can display arbitrary polylines and polygons. To create a polygon plot, you need to specify at least three variables. The coordinates of vertices of each polygon (or vertices of a piecewise-linear polyline) are specified with X and Y variables. The polygon is drawn in the order in which the coordinates are specified. A third nominal variable specifies an identifier to which each coordinate belongs. In some instances, a polygon is composed of subpolygons. For example, a continent is composed of countries, a country is composed of individual provinces or states, and some of those states are composed of disconnected landmasses (such as islands). The polygon plot supports this hierarchical structure by allowing multiple nominal variables that identify the continent, state, and island to which each coordinate pair belongs. Example In this section you create a polygon plot of the lat and lon variables of the States48 data set. The lat variable gives the latitude of state boundaries for the lower 48 contiguous United States. The lon variable gives the corresponding longitude. =⇒ Open the States48 data set. =⇒ Select Graph I Polygon Plot from the main menu, as shown in Figure 6.20. Figure 6.20. Selecting a Polygon Plot A dialog box appears as in Figure 6.21. =⇒ Select the lon variable, and click Set X. 87 88 Chapter 6. Exploring Data in Two Dimensions =⇒ Select the lat variable, and click Set Y. =⇒ Select the state variable. Hold down the CTRL key and select the segment variable. Click Add ID. =⇒ Click OK. Note: The order of the ID variables is important. The second variable should be nested in the first variable. Figure 6.21. The Polygon Plot Dialog Box A polygon plot appears (similar to Figure 6.24) showing the contiguous 48 United States. The color of a region (in this example, a state) is determined by the first observation encountered for that region. The observation’s fill color determines the color of the interior of the polygon; the outline color determines the color of the region’s outline. For these data, the observations are all black. To make the polygon plot look more like a map, you can color observations by the value of the state variable. =⇒ Right-click near the center of the plot, and select Plot Area Properties from the pop-up menu. A dialog box appears, as shown in Figure 6.22. Chapter 6. Exploring Data in Two Dimensions Figure 6.22. Plot Area Properties for a Polygon Plot =⇒ Click the Observations tab, as shown in Figure 6.23. Figure 6.23. The Observations Tab =⇒ Select state from the Fill: Blend menu. =⇒ Select a gradient colormap from the Fill menu. =⇒ Click OK. The polygon plot (Figure 6.24) is now colored according to your choice of colormap. 89 90 Chapter 6. Exploring Data in Two Dimensions Figure 6.24. A Polygon Plot The polygon plot supports the selection of polygonal regions. For example, you can click on a state to select the observations that define the boundary of that state. You can click while holding down the CTRL key to select observations defining multiple states. You can also drag out a selection rectangle to select observations defining contiguous states. Note that if a state is composed of two or more components, you can click on each component independently. For example, you can select just the upper peninsula of Michigan, or select only Long Island, New York. You can also color each region independently. Polygon Plot Properties This section describes the Polygons tab associated with a polygon plot. To access the polygon plot properties, right-click near the center of a plot, and select Plot Area Properties from the pop-up menu. The Polygons tab controls attributes of the polygon plot. The Polygons tab is shown in Figure 6.22. Close polygons specifies whether a line segment is drawn from the last observation in each region to the first observation in that region. Show outline specifies whether the outline of a region is displayed. Show fill specifies whether the interior of a region is displayed. Chapter 6. Exploring Data in Two Dimensions For a discussion of the Observations tab, see the section “Scatter Plot Properties” on page 76. For a discussion of the remaining tabs, see Chapter 9, “General Plot Properties.” Polygon Plots of Selected Variables If variables are selected in a data table when you select Graph I Polygon Plot, then the polygon plot dialog box (Figure 6.21) does not appear. The first selected interval variable is used for the X variable; the second is used for the Y variable. Any interval variables after the second variable are ignored. Any nominal variables are assigned the ID role in the order in which they were selected. Variables with a Frequency or Weight role are ignored when you are creating polygon plots. 91 92 Chapter 7 Exploring Data in Three Dimensions You can explore the relationships between three variables by using a rotating scatter plot. Often the three variables are interval variables. If one of the variables can be modeled as a function of the other two variables, then you can add a response surface to the rotating plot. Similarly, you can visualize contours of the response variable by using a contour plot. Rotating Plots This section describes how to use a rotating plot to visualize the relationships between three variables. Often each variable is continuous (interval), but that is not a requirement. Example: A Rotating Scatter Plot In this section you create a rotating plot to explore the relationships between the wind– kts, latitude, and longitude variables of the Hurricanes data set. The wind– kts variable gives the wind speed in knots for each observation. None of the variables in this example have missing values. If an observation has a missing value for any of the three variables in the rotating plot, that observation is not plotted. =⇒ Open the Hurricanes data set. =⇒ Select Graph I Rotating Plot from the main menu, as shown in Figure 7.1. Figure 7.1. Selecting a Rotating Plot A dialog box appears as in Figure 7.2. =⇒ Select the wind– kts variable, and click Set Z. 94 Chapter 7. Exploring Data in Three Dimensions =⇒ Select the latitude variable, and click Set Y. =⇒ Select the longitude variable, and click Set X. =⇒ Click OK. Figure 7.2. The Rotating Plot Dialog Box A rotating plot appears (Figure 7.3), showing a cloud of points. You can rotate the plot by clicking the icons on the left side of the plot. The top two buttons rotate the plot about a horizontal axis. The next two buttons rotate the plot about a vertical axis. The last two buttons rotate the plot clockwise and counterclockwise. The slider below the buttons controls the speed of rotation. Alternatively, you can rotate the plot by moving the mouse pointer into a corner of the plot until the pointer changes (to ). You can interactively rotate the plot by holding down the left mouse button while you move the mouse. Chapter 7. Exploring Data in Three Dimensions Figure 7.3. A Rotating Plot You can click on an observation in a rotating plot to select the observation. You can click while holding down the CTRL key to select multiple observations. You can also drag out a selection rectangle to select multiple observations. You can create rotating plots of any variables, numeric or character. Because there are so many observations in the rotating plot, some observations obscure others—a phenomenon known as overplotting. It also can be difficult to discern the coordinates of observations as they are positioned in three-dimensional space. That is, which observations are “closer” to the viewer? A visualization technique that sometimes helps distinguish observations with similar projected coordinates is to color the observations. For these data, you can color the observations according to the wind– kts variable. =⇒ Right-click near the center of the plot, and select Plot Area Properties from the pop-up menu. The dialog box in Figure 7.4 appears. 95 96 Chapter 7. Exploring Data in Three Dimensions Figure 7.4. Plot Area Properties for a Rotating Plot =⇒ Click the Observations tab, as shown in Figure 7.5. Figure 7.5. Observations Tab for a Rotating Plot =⇒ Select wind– kts from the Outline: Blend list. =⇒ Select a gradient colormap from the Outline list. =⇒ Select the same options for the Fill: Blend and Fill lists. =⇒ Select name from the Label observations by list. The last step specifies that the name of the cyclone should appear when you click on an observation. By default the observation number is used as a label. You can update the plot to apply the options you have selected so far. Chapter 7. Exploring Data in Three Dimensions =⇒ Click Apply. You can optionally use two additional features to aid in visualizing these data. =⇒ Click the Reference Lines tab, as shown in Figure 7.6. =⇒ Select Show Z reference lines. =⇒ Click Apply. When you click Apply, the plot updates to show reference lines at each tick on the axis for the Z variable (in this case, wind– kts). The reference lines are displayed in Figure 7.8. Figure 7.6. Reference Lines Tab for a Rotating Plot =⇒ Click the Plot Area tab, as shown in Figure 7.7. =⇒ Select Show plot frame box. =⇒ Click OK. The rotating plot updates (Figure 7.8) to reflect the options you selected. You can rotate the plot to observe how wind speeds in these tropical cyclones vary according to latitude and longitude. You can click on interesting observations and see the name of the storms they represent. 97 98 Chapter 7. Exploring Data in Three Dimensions Figure 7.7. Plot Area Tab for a Rotating Plot You can see that the storms with the strongest winds tend to occur west of 45 degrees west latitude, and roughly between 12 and 32 degrees north latitude. You can also see that many storm tracks appear to begin in southern latitudes heading west or northwest, then later turn north and northeast as they approach higher latitudes. The wind speed along a track tends to increase over warm water and decrease over land or cooler water. Figure 7.8. A Rotating Plot with Selected Observations Chapter 7. Exploring Data in Three Dimensions Example: A Rotating Surface Plot In the previous example you created a rotating scatter plot. A rotating scatter plot does not presume any relationship between the Z variable and the X and Y variables. In this section you create a rotating plot in which you assume that the Z variable is functionally related to the X and Y variables. That is, the Z variable can be modeled as a response variable of X and Y. A typical use of the rotating surface plot is to visualize the response surface for a regression model of two continuous variables. If you model a response variable by using an analysis chosen from the Analysis I Model Fitting menu, you can add the predicted values of the model to the data table. Then you can plot the predicted values as a function of the two regressor variables. In this example you examine three variables in the Climate data set. You explore the functional relationship between the elevationFeet variable and the latitude and longitude variables. The elevationFeet variable gives the elevation in feet above mean sea level for each of 40 cities in the continental United States. =⇒ Open the Climate data set. =⇒ Select Graph I Rotating Plot from the main menu. A dialog box appears as in Figure 7.9. =⇒ Select the elevationFeet variable, and click Set Z. =⇒ Select the latitude variable, and click Set Y. =⇒ Select the longitude variable, and click Set X. =⇒ Click OK. 99 100 Chapter 7. Exploring Data in Three Dimensions Figure 7.9. The Rotating Plot Dialog Box A rotating plot appears (Figure 7.10), showing a cloud of points. You can rotate the plot as explained in the previous example. Figure 7.10. A Rotating Plot You can visualize elevation as a function over longitude and latitude by adding a surface to these data. =⇒ Right-click near the center of the plot, and select Plot Area Properties from the pop-up menu. Chapter 7. Exploring Data in Three Dimensions =⇒ Select Smooth color mesh from the group of radio buttons labeled Surface drawing modes. =⇒ Click OK. The rotating plot (Figure 7.11) updates to show a rough approximation to an elevation map of the continental United States. There are only 40 data points in the plot, so the surface map is understandably coarse. Having more data points distributed uniformly across the country would result in a surface that is a better approximation of actual elevations. Nevertheless, the surface helps you to identify cities near the Rocky Mountains with high elevations (Cheyenne, WY, and Albuquerque, NM), one city in the Appalachian Mountains (Asheville, NC), and the coastal cities. Figure 7.11. A Rotating Plot Caution: You can add a surface to any rotating scatter plot, but you should first determine whether it is appropriate to do so. Surface plots might not be appropriate for data with replicated measurements. Surface plots of highly correlated data can be degenerate. Rotating Plot Properties This section describes the property tabs associated with a rotating plot. To access the rotating plot properties, right-click near the center of a plot, and select Plot Area Properties from the pop-up menu. Tabs not discussed in this chapter are discussed in Chapter 9, “General Plot Properties.” 101 102 Chapter 7. Exploring Data in Three Dimensions The Surface Tab The Surface tab (Figure 7.4) controls attributes of the rotating plot. You can use the Surface tab to control the placement of axes, the type of surface that is drawn, and whether contours of the data are shown in the (X,Y) plane. Axis Locations specifies the location of axes. At midpoint specifies that the origin of each axis is placed at the midpoint of the range of the variable for that axis. At minima specifies that the origin of each axis is placed at the minimum value of the variable for that axis. 3 sections specifies that each axis is placed on an edge of the bounding cube surrounding the data so that the axis interferes as little as possible with viewing the data. Off specifies that no axes are displayed. Surface Drawing Modes specifies the attributes of the surface added to the rotating plot. Transparent mesh specifies that the surface is drawn as a wire mesh, but hidden-line removal is not performed. Opaque mesh specifies that the surface is drawn as a wire mesh and hidden-line removal is performed. Block color mesh specifies that the surface is drawn as a patch of rectangles in which each rectangle is a single color. Smooth color mesh specifies that the surface is drawn as a patch of triangles in which each triangle is a single color. Stacked contours specifies that the surface is not drawn, but that contour levels are drawn. Stacked contours mesh specifies that the surface is drawn as for Opaque mesh and also that contour levels are added as for Stacked contours. Off specifies that no surface is displayed. Contour Projections specifies whether contours of the data are shown in the (X,Y) plane. Show contour lines specifies that contours for the surface are shown projected onto the (X,Y) plane. Show contour areas specifies that region between projected contours are filled with color. Show contour values specifies whether projected contour lines are labeled by the value of the Z axis variable. The Contours Tab The Contours tab controls attributes of the projected contours for the surface. The Contours tab is described in the section “Contour Plot Properties” on page 113. Chapter 7. Exploring Data in Three Dimensions The Grid Tab The Grid tab (Figure 7.12) controls the size and color of the grid used to construct a surface and to compute contours for the surface. Show grid specifies whether to display a grid on the displayed surface. Colors specifies the color of the grid when seen from the front (positive Z direction) or back (negative Z direction). Resolution specifies the resolution of the computational grid used to fit a surface to the data. The algorithm that computes the surface uses a grid superimposed on the (X,Y) plane. This grid consists of evenly spaced subdivisions along the X and Y axes. Generally, having more subdivisions results in a smoother surface, whereas having fewer subdivisions results in a rougher surface. Figure 7.12. The Grid Tab The Observations Tab The Observations tab (Figure 7.5) controls the attributes of markers in the rotating plot. The Observations tab for the rotating plot contains all the controls documented in the section “Scatter Plot Properties” on page 76. In addition, the Observations tab for the rotating plot includes the following check boxes: Show rays specifies whether lines are drawn from the center of the bounding cube to each observation marker. 103 104 Chapter 7. Exploring Data in Three Dimensions Show depth perception specifies whether observation markers are drawn in varying sizes to indicate their distance from the viewer. The Reference Lines Tab The Reference Lines tab (Figure 7.6) controls the attributes of reference lines in the rotating plot. The Reference Lines tab for the rotating plot contains all the controls documented in the section “The Reference Lines Tab” on page 140. In addition, the Reference Lines tab for the rotating plot includes the following check box: Show Z reference lines specifies whether to show reference lines for the Z axis. The Plot Area Tab The Plot Area tab (Figure 7.7) controls the attributes of plot area in the rotating plot. Background specifies the color of the background of the plot area. Show plot frame box specifies whether to display a framing box surrounding the plot area. Rotating Plots of Selected Variables If one or more variables are selected in a data table when you select Graph I Rotating Plot, then the rotating plot dialog box (Figure 7.2) does not appear. Instead rotating plots are created of the selected variables. All threefold combinations of the selected variables are plotted. That is, if you select n n ≥ 3 variables, then you see a matrix of 3 rotating plots. If you create a matrix of plots from selected variables, you can close the matrix by pressing the F11 key while any plot is active and selecting from the pop-up menu. Alternatively, you can use the Workspace Explorer (see the section “Workspace Explorer” on page 165) to quickly close plots. Variables with a Frequency or Weight role are ignored when you are creating rotating plots. Figure 7.13 shows a matrix of rotating plots for four selected variables: wind– kts, min– pressure, radius– eye, and latitude. Chapter 7. Exploring Data in Three Dimensions Figure 7.13. A Matrix of Rotating Plots Contour Plots In this section you create a contour plot. A contour plot assumes that the Z variable is functionally related to the X and Y variables. That is, the Z variable can be modeled as a response variable of X and Y. A typical use of a contour plot is to visualize the response for a regression model of two continuous variables. If you model a response variable by using an analysis chosen from the Analysis I Model Fitting menu, you can add the predicted values of the model to the data table. Then you can create a contour plot of the predicted values as a function of the two regressor variables. Contour plots are most useful when the X and Y variables are nearly uncorrelated. The contour plot fits a piecewise-linear surface to the data, modeling Z as a response function of X and Y. The contours are level curves of the response function. By default, the minimum and maximum values of the Z variable are used to compute the contour levels. The three variables in a contour plot must be interval variables. 105 106 Chapter 7. Exploring Data in Three Dimensions Example In this example you examine three variables in the Climate data set. You explore the functional relationship between the elevationFeet variable and the latitude, and longitude variables. The elevationFeet variable gives the elevation in feet above mean sea level for each of 40 cities in the continental United States. None of the variables in this example have missing values. If an observation has a missing value for any of the three variables in the contour plot, that observation is not plotted. =⇒ Open the Climate data set. =⇒ Select Graph I Contour Plot from the main menu, as shown in Figure 7.14. Figure 7.14. Selecting a Contour Plot A dialog box appears as in Figure 7.15. =⇒ Select the elevationFeet variable, and click Set Z. =⇒ Select the latitude variable, and click Set Y. =⇒ Select the longitude variable, and click Set X. =⇒ Click OK. Chapter 7. Exploring Data in Three Dimensions Figure 7.15. A Contour Plot Dialog Box A contour plot appears (Figure 7.16), showing a scatter plot of the longitude and latitude variables. Contours of the elevationFeet variable are shown overlaid on the scatter plot. Figure 7.16. A Contour Plot You can double-click on an observation to display the variable values associated with that observation. (See the section “The Observation Inspector” on page 123 for further details.) In this way, you can identify cities and find out their exact elevations. 107 108 Chapter 7. Exploring Data in Three Dimensions It is somewhat difficult to guess where the state boundaries are in Figure 7.16, so Figure 7.17 overlays the outline of the continental United States onto the contour plot. The figure was created by using the DrawPolygonsByGroups module, which is documented in the Stat Studio online Help chapter titled “IMLPlus Module Reference.” Figure 7.17. A Contour Plot Caution: You can create a contour plot of any three continuous variables, but you should first determine whether it is appropriate to do so. Contour plots might not be appropriate for data with replicated measurements or for data with highly correlated X and Y variables. If you display the contour plot property dialog box, you can examine the values associated with each contour. (To display plot properties, right-click near the center of a plot, and select Plot Area Properties from the pop-up menu.) Figure 7.18 shows that there are 10 evenly spaced contours in the range of the elevationFeet variable. The minimum and maximum values of elevationFeet are 3 and 6126. Chapter 7. Exploring Data in Three Dimensions Figure 7.18. Default Contours Changing the Contours and Colors The default contours are usually adequate for obtaining a qualitative feel for the response surface. However, sometimes you might want to manually specify the levels of the contours. You might need to conform to some standard (for example, 50-meter contour intervals) or include a critical level (for example, a control limit). Suppose you decide that you want the contour levels of elevationFeet to be “round numbers,” such as multiples of 100. You can change the set of contours by doing the following: • Remove the old contours. • Add new contours. • Color the new contours. To remove the old contours, do the following: =⇒ Select the first contour (labeled “3”). Scroll the Contour Levels list to the last contour. Hold down the SHIFT key while clicking on the last contour (labeled “5819.85”) to select all contours in the list. =⇒ Click Remove selected levels, as shown in Figure 7.19. 109 110 Chapter 7. Exploring Data in Three Dimensions Figure 7.19. Removing Contours To add a new set of uniformly spaced contours, do the following: =⇒ Click the Levels subtab. =⇒ Type 10 in the Number field. =⇒ Type 0 in the Minimum field. The value for this field is typically a “round number” near the minimum value of the Z variable. =⇒ Type 6000 in the Maximum field. The value for this field is typically a “round number” near the maximum value of the Z variable. =⇒ Click the large left arrow (⇐=) to create the contours, as shown in Figure 7.20. Chapter 7. Exploring Data in Three Dimensions Figure 7.20. Adding Evenly Spaced Contours The Contour Levels list is filled with the values 300, 900, . . . , 5700. These values do not include the minimum and maximum specified values (0 and 6000), because contours at the extreme values are often degenerate. By default, the region between the new contours is gray. You can change the colors of contours by doing the following: =⇒ Click the Contour Styles subtab. =⇒ Select a gradient colormap from the Blend list. =⇒ Click the large left arrow (⇐=) to color the selected contours according to the gradient colormap, as shown in Figure 7.21. Figure 7.21. Coloring Contours 111 112 Chapter 7. Exploring Data in Three Dimensions =⇒ Click Apply to update the contour plot. You can also add individual contours for specific levels. For example, some investigators might want to see the “sea level” contour, Z = 0. Adding an individual contour is similar to adding a set of contours: =⇒ Click the Levels subtab. =⇒ Type 1 in the Number field. =⇒ Type 0 in the Minimum field. =⇒ Type 0 in the Maximum field. If the minimum and maximum values are the same, then a single contour is created at the common value. =⇒ Click the large left arrow (⇐=) to create the contour, as shown in Figure 7.22. Figure 7.22. Adding a Single Contour =⇒ Click OK to apply the changes. The contour plot looks like the plot in Figure 7.23. Note that the contour plot has not qualitatively changed from Figure 7.16. The new contour values are within a few hundred feet of their previous values, so the new contour curves are close to the previous contours. The primary change is that the new contours correspond to “round numbers” of elevationFeet. The colors are also slightly different. Chapter 7. Exploring Data in Three Dimensions Figure 7.23. A Plot with Custom Contours Caution: In this example you added a single contour at Z = 0. While Stat Studio permits you to add contours at any level of the Z variable, you should usually choose evenly spaced levels. A standard usage of contour maps is to locate regions in which the contours are densely packed. These regions correspond to places where the gradient of Z is large; that is, the function is changing rapidly in these regions. If you add contours that are not evenly spaced in the range of Z, then you risk creating contours that are close together even though the gradient of Z is not large. Contour Plot Properties This section describes the property tabs associated with a contour plot. To access the rotating plot properties, right-click near the center of a plot, and select Plot Area Properties from the pop-up menu. For a discussion of the Observations tab, see Chapter 6, “Exploring Data in Two Dimensions.” For a discussion of the remaining tabs, see Chapter 9, “General Plot Properties.” The Contours Tab The Contours tab controls attributes of the contours. You can use this tab to remove contours, create contours, and change the color or line styles of contours. The Contours tab is shown in Figure 7.18. The Contours tab has two subtabs: the Contour Styles subtab and the Levels subtab. Contour Levels displays each contour in the plot. The contours are labeled by values of the Z variable. You can select one or more items in the list to change their properties or to remove them from the list. 113 114 Chapter 7. Exploring Data in Three Dimensions Remove selected levels removes contours that are selected in the Contour Levels list. ⇐= (large left arrow) applies the current set of properties to the contours selected in the Contour Levels list. You must click on the large left arrow to transfer the contour attributes to the selected items in the Contour Levels list, or to create new contours. Contour Styles specifies the contour colors and line styles. These attributes are not applied until you click the large left arrow (⇐=). Color specifies a single color for the selected contour levels. Blend specifies a gradient colormap for a range of contour levels. Style specifies a line style for the selected contours. Width specifies a line width for the selected contours. Levels specifies the number and range of contour levels. These contours are created when you click the large left arrow (⇐=). Number specifies the number of contours to create. Minimum specifies a value zL used in the creation of new contours. Maximum specifies a value zR used in the creation of new contours. You can create a set of contours by using the Levels subtab, as shown in Figure 7.20. Let n be the value in the Number field, and let zL and zR be the values in the Minimum and Maximum fields. These values specify that the interval between contours is δ = (zR − zL )/n. When you click the large left arrow (⇐=), contours are created for the levels zi = zL + δ/2 + δi, for i = 0, . . . , (n − 1). This implies that the first level is zL + δ/2 and the last level is zR − δ/2. Note that no contours appear for the zL and zR levels, because levels for extreme values are often degenerate. (For example, if z = x2 + y 2 on the domain [−1, 1] × [−1, 1], then the minimum value of z is 0, and the contour for that level is a single point.) If, instead, you know that you want the first contour to be at the level z0 and you want the contour interval to be δ, then it is straightforward to compute values of n, zL , and zR that satisfy those conditions. You can choose zL = z0 − δ/2 and zR = zL + nδ, where n is an integer. If you want the contours to encompass all of your data, then you can compute n = d(zmax − zL )/δe, where zmax is the largest data value for the Z variable, and dxe is the least integer greater than x. You should also choose z0 so that |z0 − zmin | ≤ δ/2. For example, if the range of your data is [3, 97], and you want a contour interval of δ = 10 with the first contour at z0 = 5, then you can choose zL = 5 − 10/2 = 0, n = d(97 − 0)/10e = 10, and zR = 0 + 10 ∗ 10 = 100. Chapter 7. Exploring Data in Three Dimensions The Contour Options Tab The Contour Options tab controls attributes of the contour plot. You can also use this tab to control the size of the grid used to construct contours. The Contour Options tab is shown in Figure 7.24. Grid Sizes specifies the resolution of the computational grid used to construct contours from the data. The algorithm that computes the surface uses a grid superimposed on the (X,Y) plane. This grid consists of evenly spaced subdivisions along the X and Y axes. Generally, having more subdivisions results in smoother contours, whereas having fewer subdivisions results in a rougher contours. Show contour lines specifies whether contours are shown. Show contour values specifies whether contour lines are labeled by the value of the Z axis variable. Show contour areas specifies whether the region between contours is filled with color. Figure 7.24. The Contour Options Tab 115 116 Chapter 7. Exploring Data in Three Dimensions Contour Plots of Selected Variables If one or more interval variables are selected in a data table when you select Graph I Contour Plot, then the contour plot dialog box (Figure 7.15) does not appear. Instead, the first selected variable is used as the Z variable. Contour plots are created for Z as a function of each pair of remaining interval variables. If you create a matrix of plots from selected variables, you can close the matrix by pressing the F11 key while any plot is active and selecting from the pop-up menu. Alternatively, you can use the Workspace Explorer (see the section “Workspace Explorer” on page 165) to quickly close plots. Variables with a Frequency or Weight role are ignored when you are creating contour plots. Figure 7.25 shows a matrix of contour plots for four selected variables. The TotalAvePrecipIn variable is plotted as a function of longitude, latitude, and elevationFeet. Figure 7.25. A Matrix of Contour Plots Chapter 8 Interacting with Plots In this chapter you learn how you can interact with plots. These interactions include selecting observations, panning, and zooming. You learn how to display text on a plot and how to adjust the margins in a plot. You also learn about the observation inspecector, a window that displays values of all variables for an observation. Interaction Tools The simplest way to interact with plots is by using the mouse to click or drag in the plot. Each plot supports tools that control the way that clicking or dragging affects the plot. You can see the interaction tools for a plot by right-clicking in a plot. For example, Figure 8.1 shows the tools available for a histogram. Selecting a tool item from the pop-up menu changes the shape of the mouse pointer and determines how the plot interprets a mouse click. Figure 8.1. Some Available Tools The default tool for all plots is the select tool. The various tools and their effects on the plots are summarized in the following sections. The Select Tool When you choose the select tool, the mouse pointer looks like a diagonally pointing arrow. Clicking on a plot marker selects the corresponding observation. Clicking on a bar in a histogram or bar chart selects all observations represented by that bar. Clicking on a box plot quartile or whisker selects all observations in that quartile or whisker. By holding down the SHIFT or CTRL key, you can select multiple graphical elements. 118 Chapter 8. Interacting with Plots Dragging a rectangle selects all observations within that rectangle. The rectangle is also known as a brush. After a brush is created, you can move it by placing the mouse pointer inside the rectangle and dragging it to a new location. As the brush passes over observations, those observations are automatically selected, as shown in Figure 8.2. If you hold down the CTRL key while moving the brush, observations outside the brush are not deselected. Figure 8.2. Selecting Observations with a Brush It is also possible to throw the brush. Release the mouse button while dragging the brush, and the brush begins moving freely in the direction in which you last dragged it. The brush bounces off the sides of the graph area. Throwing the brush can be computationally intensive when you are working with large data sets. Note: If you click on an observation, it is labeled in the plot. Details are given in the section “Labeling Observations” on page 138. Observations selected by using a brush are not labeled. The Pan Tool When you choose the pan tool, the mouse pointer looks like four arrows meeting at right angles. By dragging the pointer, you can translate the contents of the plot. The rotating plot does not support the pan tool. The Zoom Tool When you choose the zoom tool, the mouse pointer looks like a magnifying glass. Clicking in a plot fixes the relative position of the pointer and expands the scale of the plot by a factor of 1.5. Clicking while holding down the SHIFT key shrinks the scale of the plot by a factor of 1.5. Chapter 8. Interacting with Plots If you drag out a rectangle with the zoom tool, the region inside the rectangle expands to fill the plot area. If you drag out a rectangle with the zoom tool while holding down the SHIFT key, the plot area is shrunk down into the rectangle. The rotating plot does not support the zoom tool. The Spin Tool When you choose the spin tool, the mouse pointer looks like a circular arrow (). Only the rotating plot supports the spin tool. Clicking in the plot causes the plot to rotate toward the pointer by an amount proportional to the distance between the pointer and the center of the plot. Dragging the pointer rotates the plot. If you release the mouse button while the pointer is in motion, the plot freely rotates. Click anywhere in the plot to stop the rotation. The Bin Tool When you choose the bin tool, the mouse pointer looks like a double-headed arrow between a pair of lines. Only the histogram supports the bin tool. Clicking or dragging the bin tool shifts the location of the histogram bins. Clicking near the horizontal axis reduces the number of bins and makes the bars wider. Clicking near the top of the plot increases the number of bins and makes the bars narrower. Dragging the mouse pointer horizontally does not change the number of bins but changes the position at which the bins start. When the pointer is at the left edge of the histogram, the bins start at an integral multiple of the bin width. When you move the pointer toward the right, the bins are offset by an amount proportional to the distance between the pointer and the left edge of the histogram. The Level Tool When you choose the level tool, the mouse pointer looks like a pencil. Only the contour plot supports the level tool. Clicking and dragging the level tool near a contour changes the value of the Z variable associated with the contour. You can insert a new contour by clicking the level tool away from existing contours. Resetting the Plot View In many cases, you can reset a plot to its original view of the data. Right-click in the plot and select Reset View from the pop-up menu to reset changes to a plot that were made by the pan tool, zoom tool, or spin tool. Changes made by the bin tool or level tool are not affected by Reset View. 119 120 Chapter 8. Interacting with Plots Inserting Annotations In this section you learn how to display text on a plot. For example, you might want to draw attention to an outlier or display statistics associated with the plot. To add text to a plot, right-click in the plot and select Insert Annotation from the pop-up menu, as shown in Figure 8.3. Example In this example, you insert text that displays certain statistics related to a scatter plot of two variables. =⇒ Open the Hurricanes data set, and create a scatter plot of wind– kts versus min– pressure. The scatter plot (Figure 8.4) shows a strong negative correlation between wind speed and pressure. A correlation analysis reveals the following: • There are 6185 observations for which both variables are nonmissing. • The correlation between these two variables is approximately –0.93. You can display these statistics on the plot. =⇒ Right-click in the plot, and select Insert Annotation from the pop-up menu. Figure 8.3. Creating an Annotation The mouse pointer changes its shape. It looks like a pencil with the letter “A” next to it. =⇒ Drag out a rectangle with the mouse pointer, as shown in Figure 8.4. Chapter 8. Interacting with Plots Figure 8.4. Inserting an Annotation =⇒ Type text into the text box, as shown in Figure 8.5. Click outside the text box to finish editing the text. You can resize or move the text rectangle after it is created, if necessary. You can also right-click on the text box to change properties of the text or the text box. For example, in Figure 8.5 the text box is displayed with a border around it. The annotation properties are discussed in the section “Annotation Properties” on page 122. Figure 8.5. An Inset Containing Statistics 121 122 Chapter 8. Interacting with Plots If you decide to delete the annotation, click on the text box to select it. Then right-click outside the text box, and select Delete Annotation from the pop-up menu, as shown in Figure 8.6. Caution: If you right-click inside the text box, you get a different menu, as discussed in the following section. Figure 8.6. Deleting an Annotation Annotation Properties You can change properties of an annotation. Click on the annotation text box to select it. Right-click inside the text box, and select Properties from the pop-up menu. A dialog box appears. The dialog box has two tabs. You can use the Font tab to set attributes of the font used to display an annotation. The Font tab is described in “Common Plot Properties.” You can use the Text Editor tab (Figure 8.7) to set attributes of the text box containing the text. The Text Editor tab has the following fields: Text Alignment specifies the alignment of the text within the text box. Horizontal specifies the horizontal position of the text box within the graph area or plot area. Vertical specifies the vertical position of the text box within the graph area or plot area. Background specifies the color of the text box background. Show border specifies whether to display a frame around the text box. Chapter 8. Interacting with Plots Figure 8.7. Text Editor Tab Adjusting Graph Area Margins You can interactively resize the plot area when the select tool is active. Hover the mouse pointer at the edge of the plot area until the pointer changes to a double-headed arrow. Then click and drag the plot area to resize it. When you resize the plot area, you are actually changing the graph area margins, as described in the section “Common Graph Area Properties” on page 143. You cannot adjust the graph area margins if the plot has a fixed aspect ratio. The Observation Inspector You can interactively query plots to display the values of variables for the observations beneath the mouse pointer. The discussion in this section applies to plots that show individual markers for each observation. The observation inspector window (Figure 8.8) displays the values of all the variables for a particular observation. You can display the observation inspector window in one of three ways: • Hold down the F2 key. The observation inspector window appears for any observations beneath the mouse pointer. • Press the F2 key while holding down the SHIFT key. You are now in observation inspector mode. If you hover the mouse pointer over an observation, the observation inspector window appears. To exit observation inspector mode, press the ESC key while the observation inspector is active, or press SHIFT+F2 a second time. • Double-click on an observation. 123 124 Chapter 8. Interacting with Plots Figure 8.8. The Observation Inspector Window The top portion of the observation inspector window displays the variables used by the plot. For example, the observation inspector window for a scatter plot displays the X- and Y-axis variables first. If observations are labeled by some variable, that label variable also appears. The observation inspector window next displays a horizontal line, followed by the observation number (in the current sort order), followed by all variables in the order in which they appear in the data set. If there are many variables, it is possible that not all of the variables fit into the observation inspector window. You can scroll the observation inspector window by using the HOME, END, PAGE UP, PAGE DOWN, UP ARROW, and DOWN ARROW keys. If there are multiple observation markers near the mouse pointer (as in Figure 8.8), the observation inspector creates a list of all the nearby observations and displays the text “Value (1 of N)” in its column heading. You can display the next observation in the list by pressing the RIGHT ARROW key. You can go back to a previous observation by pressing the LEFT ARROW key. Pressing RIGHT ARROW or LEFT ARROW while holding down the SHIFT key jumps forward or backward to the observation that is approximately N/5 entries away in the list. Copying Plots to the Windows Clipboard You can copy a plot to the Windows clipboard by selecting Edit I Copy from the main menu when the plot is active. Alternatively, you can press CTRL+C. You can paste plots into the Stat Studio output document or into other applications such as Microsoft Word and PowerPoint. Chapter 8. Interacting with Plots Keyboard Shortcuts in Plots All plots support the standard Microsoft Windows control sequences listed in Table 8.1. Table 8.1. Standard Control Sequences in Plots Key CTRL+A CTRL+C CTRL+P Action Select all observations that are included in the plots. Copy the plot to Windows clipboard. Print the plot. All plots support the keyboard shortcuts listed in Table 8.2. Table 8.2. Keys and Actions in All Plots Key 0, 1–9 A B C E G I L X, Y, Z Action Set the color of selected observations to the color specified in Table 8.3. If no observations are selected, set the marker color of all observations. Select all observations that are included in the plots. Apply a color blend according to values of the X variable. (Plots with multiple X variables ignore this key.) Bar charts and histograms also color each bin according to the color blend. Select the complement of the selected observations. Exclude selected observations from plots and analyses. Toggle a reference line grid. Include selected observations in plots and analyses. Toggle labels on bars or observations. Display axis property dialog box for the corresponding axis. Ten predefined colors are associated with number keys. Table 8.3 lists the color associated with the digits 0–9. Table 8.3. Keys and Colors in Plots Key 0 1 2 3 4 5 6 7 8 9 Color Black Red Green Blue Gold Magenta Olive Steel Brown Violet 125 126 Chapter 8. Interacting with Plots Area plots (histogram, bar chart, and mosaic plot) support the keyboard shortcuts listed in Table 8.4. Table 8.4. Keys and Actions in Area Plots Key H P CTRL+digit Action Toggle filling the bars. Cycle through displaying frequency, percentages, and (for histograms) density on the Y axis. Set the percentage threshold for the “Others” category. For example, CTRL+4 sets the threshold to 4%, whereas CTRL+0 sets the threshold to 0% and therefore turns off the “Others” category. (The histogram ignores this key.) Point plots (any plot that displays individual observations) support the keyboard shortcuts listed in Table 8.5. Table 8.5. Keys and Actions in Point Plots Key F2 SHIFT+F2 H [ or ] (square bracket) CTRL+UP ARROW, CTRL+DOWN ARROW ALT+UP ARROW ALT+DOWN ARROW Action Display the observation inspector for any observations beneath the mouse pointer. Toggle observation inspector mode, as described in the section “The Observation Inspector” on page 123. Toggle the “show only selected observations” option. Toggle fixed aspect ratio. (The box plot ignores this key.) Increase or decrease the size of markers. Increase or decrease the size difference between selected and unselected markers. Box plots support the keyboard shortcuts listed in Table 8.6. Table 8.6. Keys and Actions in Box Plots Key M N HYPHEN Action Toggle displaying the mean and standard deviation. Toggle displaying the notches that measure the significance of the difference between two medians. Toggle displaying serifs. Chapter 8. Interacting with Plots Line plots support the keyboard shortcuts listed in Table 8.7. Table 8.7. Keys and Actions in Line Plots Key 0–9 ESC CTRL+PAGE UP, CTRL+PAGE DOWN CTRL+UP ARROW, CTRL+DOWN ARROW CTRL+LEFT ARROW, CTRL+RIGHT ARROW Action Set the color of the selected lines. The colors are listed in Table 8.3. Deselect all lines. Select the previous or next line. (Select the first line if no line is selected.) Increase or decrease the width of selected lines. Cycle through line styles for the selected lines. Rotating plots support the keyboard shortcuts listed in Table 8.8. Table 8.8. Keys and Actions in Rotating Plots Key UP ARROW, DOWN ARROW LEFT ARROW, RIGHT ARROW PAGE UP, PAGE DOWN CTRL+B CTRL+D CTRL+G CTRL+R Action Rotate up or down. Rotate left or right. Rotate about an axis perpendicular to the computer monitor. Toggle displaying the frame box. Toggle depth perception. Toggle displaying the surface graph. Toggle displaying rays from the origin. Polygon plots support the keyboard shortcut listed in Table 8.9. Table 8.9. Keys and Actions in Polygon Plots Key CTRL+ALT+F Action Toggle filling the polygons. 127 128 Chapter 9 General Plot Properties In this chapter you learn about basic properties of plots. Knowing how to change the default plot properties enables you to better visualize and explore your data. In this chapter you learn how to do the following: • display different menus by clicking in different regions of a plot • change the shape of markers • change the color of markers • display only selected observations • label observations • change common plot properties such as reference lines, fonts, and plot margins • change common graph properties such as margins, titles, and footnotes Context Areas Right-clicking inside a plot window brings up a context menu. This means that the contents of the pop-up menu depend on the location of the mouse pointer when you right-click. Figure 9.1 shows six nonoverlapping regions in a plot: the plot area, the graph area, two axis areas, and two axis label areas. Each region has its own context menu. The figure applies to all plots except for the rotating plot, which lacks the “axis area” regions. The rotating plot behaves differently because the position of the axes changes as the plot rotates. 130 Chapter 9. General Plot Properties Figure 9.1. Context Areas for a Two-Dimensional Plot The dialog box for the plot area has controls that affect the appearance of the plot. Which tabs the dialog box displays depends on the plot type (histogram, scatter plot, box plot, etc.). Properties common to all plots are discussed in this chapter. Plot-specific properties are discussed in Chapter 5, “Exploring Data in One Dimension,” Chapter 6, “Exploring Data in Two Dimensions,” Chapter 7, “Exploring Data in Three Dimensions,” and Chapter 8, “Interacting with Plots.” By using the dialog box for the graph area, you can change general properties that affect the way the plot appears. This dialog box is discussed in the section “Common Graph Area Properties” on page 143. Finally, the dialog box for an axis has controls that affect the scale, font, and placement of tick marks for that axis. The dialog box for an axis label has controls that affect the font and text used to label that axis. These dialog boxes are discussed in Chapter 10, “Axis Properties.” Changing Marker Shapes Not every plot shows individual observations. Some plots, such as histograms, bar charts, and mosaic plots, aggregate observations into a group and represent that group with a bar or box. The discussion in this section applies to plots that show individual markers. When a graph is printed on a gray-scale printer, it is often easier to discern observations that have different marker shapes than it is to discern markers that have different colors. Even on a computer screen, marker shape is sometimes preferred for classifying markers according to a small number of discrete values. For example, marker shape is an ideal way to encode gender. Chapter 9. General Plot Properties You can change the marker shape for all observations, or just for observations that are selected. You can select observations by using graphical techniques or by using the Find dialog box in a data table, as discussed in the section “Finding Observations” on page 43. Example In this example, you use a bar chart of a categorical variable to select observations, and you change the marker shape of the selected observations. =⇒ Open the GPA data set, and create a scatter plot of satm versus satv. The scatter plot appears in Figure 9.2. Figure 9.2. A Scatter Plot Each observation in this data set represents a student. You can use marker shape to indicate each student’s gender. =⇒ Create a bar chart of the sex variable. If necessary, move the bar chart so that it does not overlap the scatter plot. =⇒ Select all the male students in the bar chart, as shown in Figure 9.3. 131 132 Chapter 9. General Plot Properties Figure 9.3. A Bar Chart with Male Students Selected While the bar chart is convenient for selecting all the male students, you need to return to the scatter plot in order to change the marker shapes of the selected observations. =⇒ Right-click near the center of the scatter plot, and select Plot Area Properties from the pop-up menu. A dialog box appears, as shown in Figure 9.4. You can use the Observations tab to change marker shapes, colors, and sizes. The section “Scatter Plot Properties” on page 76 gives a complete description of the options available on the Observations tab. Figure 9.4. The Observations Tab Chapter 9. General Plot Properties =⇒ Select a triangle (4) from the Shape list. Note that Apply to defaults to Selected whenever there are selected observations. This means that the Shape, Outline, and Fill options are applied only to the selected observations. (You can, of course, override this default and apply changes to all observations.) =⇒ Select 6 from the Size list. Note that the Size list is not in the same group box as Apply to. All markers in a plot have a common scale; size differences are used to distinguish between selected and unselected observations. When a plot is active, you can increase the size difference between selected and unselected markers by pressing the UP ARROW key while holding down the ALT key. =⇒ Click OK. The scatter plot updates, as shown in Figure 9.5. The SAT scores of male students are represented by triangles; scores of female students are represented by squares. Figure 9.5. Using Marker Shape to Indicate Gender Changing Marker Colors You can use the color of markers to indicate observations of interest (for example, outliers) or to color observations according to the value of some variable. The discussion in this section applies to plots that show individual markers. The simplest use of color is to assign a color to one or more selected observations. For example, you can repeat the example of the section “Changing Marker Shapes” on page 130, but use color to indicate the male students. You can color markers according to values of a nominal or interval variable. In the next example you color markers according to an interval variable. This technique is 133 134 Chapter 9. General Plot Properties sometimes useful for visualizing trivariate data by using a scatter plot to visualize two variables and using color to visualize the third. Example =⇒ Open the GPA data set, and create a scatter plot of satm versus satv. The scatter plot appears in Figure 9.2. You can use color to visualize the grade point average (GPA) for each student. =⇒ Right-click near the center of the plot, and select Plot Area Properties from the pop-up menu. A dialog box appears, as shown in Figure 9.6. You can use the Observations tab to change marker shapes, colors, and sizes. The section “Scatter Plot Properties” on page 76 gives a complete description of the options available on the Observations tab. Figure 9.6. The Observation Tab =⇒ Select gpa from the Outline: Blend and Fill: Blend lists. Select a gradient color map (the same one) from the Outline and Fill lists. Make sure that Apply to is set to All. =⇒ Select 6 from the Size list. Note that the Size list is not in the same group box as Apply to. All markers in a plot have a common scale; size differences are used to distinguish between selected and unselected observations. =⇒ Click OK. The scatter plot updates, as shown in Figure 9.7. These data do not seem to indicate a strong relationship between a student’s college grade point average and SAT scores. Chapter 9. General Plot Properties Figure 9.7. Using Color to Indicate Grade Point Average Displaying Only Selected Observations The discussion in this section applies to plots that show individual markers. The default Stat Studio behavior is to show all observations in a plot. Selected observations are displayed at a larger size than unselected observations. You can choose instead to display only selected observations. This is useful when there are so many points in a plot that the selected observations are not distinguishable (a phenomenon known as overplotting). You can also examine subsets of the data by displaying only selected observations. This technique is called slicing. You can slice dynamically to explore multivariate relationships. Example In this example, you visualize the distribution of points in a scatter plot, as subset by values of a categorical variable. =⇒ Open the Hurricanes data set, and create a scatter plot of latitude versus longitude. The scatter plot appears in Figure 9.8. The plot shows the position of Atlantic cyclones during a 16-year period. There is considerable overplotting in this scatter plot, particularly along a path between the Cape Verde Islands (lower-right corner of the plot) and the Caribbean Sea (near the coordinates (−75, 20)). 135 136 Chapter 9. General Plot Properties Figure 9.8. A Scatter Plot The overplotting prevents the clear examination of rare events such as category 4 and category 5 hurricanes. You can modify the scatter plot so that it displays only selected observations. This makes it easier to examine these storms. =⇒ Right-click near the center of the plot, and select Plot Area Properties from the pop-up menu. A dialog box appears, as shown in Figure 9.9. Figure 9.9. The Observations Tab =⇒ Select Show only selected observations. =⇒ Click OK. Chapter 9. General Plot Properties The scatter plot updates. All of the observations disappear because none are selected. You can use another plot or the data table’s Find dialog box (see the section “Finding Observations” on page 43) to select data of interest. =⇒ Create a bar chart of the category variable. =⇒ Select all category 4 and 5 hurricanes in the bar chart, as shown in Figure 9.10. Figure 9.10. A Bar Chart with Category 4 and 5 Hurricanes Selected The selected observations appear in the scatter plot, as shown in Figure 9.11. Most of the selected storms appear in the Gulf of Mexico, the Caribbean Sea, and the Atlantic Ocean east of the Greater Antilles. Figure 9.11. Displaying Only Selected Observations 137 138 Chapter 9. General Plot Properties Labeling Observations The discussion in this section applies to plots that show individual markers. If you click on an observation in a plot, a label appears near the selected observation. By default, the label is the observation number (position in the data table). You can choose instead to label observations by the value of any variable, called the label variable. You can set a default label variable that will be used for all plots, or you can set a label variable for a particular plot that overrides the default label variable. Example In this example, you label observations in a scatter plot according to values of a third variable. =⇒ Open the Hurricanes data set and create a scatter plot of wind– kts versus min– pressure. The scatter plot appears, as shown in Figure 9.12. =⇒ Click on an observation. The selected observation is labeled by its position in the data table. Figure 9.12. A Scatter Plot =⇒ Right-click near the center of the plot, and select Plot Area Properties from the pop-up menu. A dialog box appears, as shown in Figure 9.13. Chapter 9. General Plot Properties Figure 9.13. The Observations Tab =⇒ Select name from the Label observations by list. =⇒ Click OK. The label for the selected observation updates, as shown in Figure 9.14. If you click on subsequent observations, each label displays a storm name. Figure 9.14. Labeling Only Selected Observations Note: Only the scatter plot is affected by selecting Label observations by on the Observations tab of the Plot Area Properties dialog box. If you create a second plot, that new plot defaults to using observation numbers to label observations. You can also set a default label variable that is used for all plots. In the data table, right-click on a variable heading. Select Label from the pop-up menu, as shown in 139 140 Chapter 9. General Plot Properties Figure 9.15. The values of the selected variable are displayed when you click on observations in a plot (unless that plot overrides the default). Figure 9.15. The Variables Menu Common Plot Properties This section presents plot properties that are common to multiple plots. These properties are found in the Plot Area Properties dialog box. You can access the properties by right-clicking near the center of the plot and selecting Plot Area Properties from the pop-up menu. The Reference Lines Tab You can use the Reference Lines tab (Figure 9.16) to set attributes of reference lines displayed in the background of a plot. Figure 9.16. The Reference Lines Tab Style specifies the style of the line used for reference lines. Chapter 9. General Plot Properties Width specifies the width of the line used for reference lines. Color specifies the color of the line used for reference lines. Show X reference lines specifies whether to show reference lines for the X axis. These are vertical lines originating at each tick mark on the X axis. Show Y reference lines specifies whether to show reference lines for the Y axis. These are horizontal lines originating at each tick mark on the Y axis. The Font Tab You can use the Font tab (Figure 9.17) to set attributes of the font used to display observation labels in plots. The section “Labeling Observations” on page 138 discusses observation labels. Figure 9.17. The Font Tab Font specifies the font used for text in the plot area. Font style specifies the style of the font used for text in the plot area. Size specifies the point size of the text in the plot area. Color specifies the color of the text in the plot area. 141 142 Chapter 9. General Plot Properties Sample shows what text with the specified properties looks like. Strikeout specifies whether a line is drawn through text in the plot area. Underline specifies whether a line is drawn below text in the plot area. The Plot Area Tab You can use the Plot Area tab (Figure 9.18) to set attributes of the plot area. Figure 9.18. The Plot Area Tab Left specifies the distance between the left edge of the plot area and the minimum value of the visible axis range for the X axis. The distance is specified as a fraction of the plot area’s width. The value must be in the range 0 to 0.8. Right specifies the distance between the right edge of the plot area and the maximum value of the visible axis range for the X axis. The distance is specified as a fraction of the plot area’s width. The value must be in the range 0 to 0.8. Top specifies the distance between the top edge of the plot area and the maximum value of the visible axis range for the Y axis. The distance is specified as a fraction of the plot area’s height. The value must be in the range 0 to 0.8. Bottom specifies the distance between the bottom edge of the plot area and the minimum value of the visible axis range for the Y axis. The distance is Chapter 9. General Plot Properties specified as a fraction of the plot area’s height. The value must be in the range 0 to 0.8. Background specifies the background color of the plot area. Show plot frame specifies whether the plot area’s frame is displayed. Note: Because the plot area has margins, the edges of the plot area do not correspond to the minimum and maximum values of the axis. Let xL and xR be the minimum and maximum values of the horizontal axis. Let mL and mR be the left and right margin fractions. Define s = (xR − xL )/(1 − mL − mR ). Then the left edge of the plot area is located at xL − smL , and the right edge of the plot area is located at xR + smR . For example, if xL = 0, xR = 1, mL = 1/20, and mR = 1/10, then s = 20/17. The left edge of the plot area is located at −1/17 ≈ −0.0588, while the right edge is located at 19/17 ≈ 1.118. Common Graph Area Properties This section presents graph area properties that are common to multiple plots. These properties are found in the Graph Area Properties dialog box. You can access the properties by right-clicking near a corner of the graph area and selecting Graph Area Properties from the pop-up menu. The Graph Area Tab You can use the Graph Area tab (Figure 9.19) to set attributes of the graph area. Figure 9.19. The Graph Area Tab 143 144 Chapter 9. General Plot Properties Left specifies the distance between the left edge of the graph and the left edge of the plot area. The distance is specified as a fraction of the graph’s width. The value must be in the range 0 to 1. Right specifies the distance between the right edge of the graph and the right edge of the plot area. The distance is specified as a fraction of the graph’s width. The value must be in the range 0 to 1. Top specifies the distance between the top edge of the graph and the top edge of the plot area. The distance is specified as a fraction of the graph’s height. The value must be in the range 0 to 1. Bottom specifies the distance between the bottom edge of the graph and the bottom edge of the plot area. The distance is specified as a fraction of the graph’s height. The value must be in the range 0 to 1. Fixed aspect ratio specifies a fixed ratio between units on the Y axis and units on the X axis. When you select this check box, you can specify the ratio. If a plot has a fixed aspect ratio, then the Graph Margins are not active. Background specifies the background color of the graph area. Show title specifies whether the graph’s title is displayed. Show footnote specifies whether the graph’s footnote is displayed. If you select Show title, the graph initially displays a default title. Click on the title to edit it. You can also change the title’s font or position by right-clicking on the title and selecting Properties from the pop-up menu. The section “Annotation Properties” on page 122 describes the dialog box that appears. A default footnote appears when you select Show footnote. To edit the footnote, follow the preceding instructions. If you do not want to display a plot’s title or footnote, open the Graph Area Properties dialog box, and clear the appropriate check boxes on the Graph Area tab. Chapter 10 Axis Properties In this chapter you learn about basic properties of axes. You learn how to change view ranges, tick marks, and labels on axes. Adjusting Axes and Ticks In this section you learn how to change the axis range and tick marks for plots. The section “Histogram Binning: Setting Tick Positions” on page 60 discusses adjusting tick marks for a histogram. For a histogram, the major tick unit is also the width of each histogram bin. Therefore, changing the major tick unit is equivalent to rebinning. Example You can change the default tick marks for the axis of an interval variable by following these steps. =⇒ Open the Hurricanes data set, and create a scatter plot of wind– kts versus latitude. The scatter plot appears as in Figure 10.1. Note that the latitude axis has only a few tick marks. You might decide to add a few additional tick marks. Figure 10.1. A Scatter Plot 146 Chapter 10. Axis Properties =⇒ Right-click on the horizontal axis of the plot, and select Axis Properties from the pop-up menu, as shown in Figure 10.2. Figure 10.2. The Axis Pop-up Menu The Axis Properties dialog box appears, as shown in Figure 10.3. Note that this is a quick way to determine the anchor location, tick unit, and tick range for an axis. Figure 10.3. Axis Properties Dialog Box =⇒ Change the Anchor tick value to 0. =⇒ Change the Major tick unit value to 10. =⇒ Change the Number of minor ticks value to 1. =⇒ Click OK. The latitude axis updates, as shown in Figure 10.4. Chapter 10. Axis Properties Figure 10.4. A Scatter Plot with Custom Tick Marks Axis Properties The Axis Properties dialog box controls the appearance of an axis. For an interval variable, major tick marks are placed on an axis within the interval [L, R] at locations x0 ± iδ for integer i. The value x0 is called the anchor tick. The positive quantity δ is called the major tick unit. The interval [L, R] is called the tick range. The Axis Properties dialog box has the following tabs: Scales, Font, and Line. The Scales tab (Figure 10.3) appears only for interval variables. You can use the Scales tab to set tick marks. The Font tab is used to change the font and size of labels on an axis. The Line tab is used to set the line styles for an axis. The Scales tab has the following fields: Major tick unit sets the distance between tick marks. Anchor tick sets the value of one tick mark from which the positions of other ticks are computed. Number of minor ticks sets the number of unlabeled tick marks to appear between consecutive major ticks. Visible Axis Range: Minimum sets the minimum value of the axis range. Visible Axis Range: Maximum sets the maximum value of the axis range. 147 148 Chapter 10. Axis Properties Tick Range: Minimum sets the minimum value of a tick mark. Ticks with values less than this value are not displayed. Tick Range: Maximum sets the maximum value of a tick mark. Ticks with values greater than this value are not displayed. Note: The minimum and maximum values for Visible Axis Range do not necessarily correspond to the edges of the plot area. The plot area also has plot area margins. The computation to find the edges of the plot area is described in the section “The Plot Area Tab” on page 142. Changing an Axis Label An axis label is text near an axis that identifies the axis variable. You can change the axis label. By default, plots display the name of a variable as the label. However, you might prefer that the plot display a variable’s label instead of its name. Or you might prefer to customize the axis label in some other way. To change the axis label properties, right-click while the mouse pointer is on the axis label. You can then select Axis Label Properties from the pop-up menu. The Axis Label Properties dialog box appears, as shown in Figure 10.5. Figure 10.5. Axis Label Properties Dialog Box You can display a variable’s label instead of the variable’s name by selecting Variable label. If the variable does not have a label defined, or if you prefer to display a different label, you can select Custom and type your own label. This is shown in Figure 10.6. Chapter 10. Axis Properties Figure 10.6. Specifying a Custom Label One instance in which you might want to define your own label is for a line plot that has multiple Y variables. If the Y variables all measure different aspects of a single quantity, you can replace the multiple Y labels with a single custom label. For example, Figure 10.7 shows a line plot of the co, o3, and so2 variables versus datetime for the Air data set. Each of the Y variables is a kind of pollutant, so the three Y labels are replaced with a single custom label. Figure 10.7. A Custom Label for the Y Axis 149 150 Chapter 10. Axis Properties Suppressing the Display of Axes By default, axes are shown for all plots. However, you can suppress the display of all axes. Right-click in a plot and select Show X Axis or Show Y Axis from the pop-up menu to toggle the display of an axis and the variable label. Chapter 11 Techniques for Exploring Data This chapter describes some useful techniques for analyzing data in Stat Studio. The following techniques are presented in this chapter: • copying selected observations or variables to a new data table • excluding observations from plots or analyses • ordering categories of a nominal variable • graphically selecting observations that satisfy complex criteria • managing graphs and workspaces with the Workspace Explorer • copying plots to the Window clipboard and pasting them to another application, such as Microsoft Word or PowerPoint Subsetting Data This section describes how to copy selected observations or variables to a new data table. The new data table is not dynamically linked to the original data. The original data are not changed. You can copy selected data by selecting File I New I Data Set from Selected Data from the main menu, as shown in Figure 11.1. Figure 11.1. Creating a New Data Table from Selected Data When you select File I New I Data Set from Selected Data, Stat Studio performs one of the following actions: 152 Chapter 11. Techniques for Exploring Data • If no variables or observations are selected, the Choose Variables dialog box (Figure 11.2) opens and prompts you to select one or more variables. When you click OK, the selected variables are copied to a new data table. The variables are copied in the order in which they appear in the original data table. Figure 11.2. The Choose Variables Dialog Box • If no variables are selected, but there are selected observations, the selected observations (for all variables) are copied to a new data table. You can use this technique to copy data that satisfy certain conditions. • If variables are selected, but there are no selected observations, the selected variables are copied to a new data table. The variables are copied in the order in which they were selected. You can use this technique to reorder variables. • If both variables and observations are selected, the selected observations for the selected variables are copied to a new data table. The variables are copied in the order in which they were selected. For example, in Figure 11.3 the variables were selected in the order longitude, latitude, and category. (Note that the column headings display numbers that indicate the order in which you selected the variables.) If you copy the data to a new data table, the new data table will contain 12 observations for the longitude, latitude, and category variables, in that order. Chapter 11. Techniques for Exploring Data Figure 11.3. Selected Variables and Observations Excluding Observations This section describes how to exclude selected observations from plots and from statistical analyses. The data table must be the active window in order for you to exclude observations. Select Edit I Observations I Exclude from Plots from the main menu to exclude selected observations from plots. Select Edit I Observations I Exclude from Analyses to exclude selected observations from analyses. Alternatively, you can right-click on the row heading of any selected observation in the data table and select Exclude from Plots or Exclude from Analyses from the pop-up menu, as shown in Figure 11.4. Figure 11.4. Data Table Pop-up Menu The row heading of the data table shows the status of an observation in analyses and plots. A marker symbol indicates that the observation is included in plots; observations excluded from plots do not have a marker symbol shown in the data table. Similarly, the χ2 symbol is present if and only if the observation is included in analyses. For example, the first, fifth, and sixth observations in Figure 11.5 are included in plots and analyses. 153 154 Chapter 11. Techniques for Exploring Data Figure 11.5. Excluded Observations If you exclude observations from plots, all plots linked to the current data table automatically redraw themselves. (For example, excluding an extreme value might result in a new range for an axis.) The row headings for the excluded observations no longer show the observation marker. For example, the third and fourth observations in Figure 11.5 are excluded from plots. If you exclude observations from analyses, the row headings for the excluded observations no longer show the χ2 symbol. For example, the second and fourth observations in Figure 11.5 are excluded from analyses. Caution: If you change the observations included in analyses, previously run analyses and statistics are not automatically rerun. If an observation is excluded from analyses but included in plots, then the marker symbol changes to the × symbol. This combination is useful if you want to fit a regression model to data but also want to exclude outliers or high-leverage observations prior to modeling. The regression model does not use the excluded observations, but the observations show up (as ×) on diagnostic plots for the regression. An example of including some observations in plots but not in analyses is shown in Figure 11.6. The figure shows data from the Mining data set—the results of an experiment to determine whether drilling time was faster for wet drilling or dry drilling. The plot shows the time required to drill the last five feet of a hole plotted against the depth of the hole. A loess fit is plotted only for the wet drilling trials (open circles). This is accomplished by excluding the observations for dry drilling (markers with the × shape) before running the loess analysis. Chapter 11. Techniques for Exploring Data Figure 11.6. Loess Fit of a Subset of Data Although Stat Studio analyses do not support BY-group processing, you can restrict an analysis to a single BY group by excluding all other BY groups. For data with many BY groups, this is tedious to do using the Stat Studio GUI, but you can write an IMLPlus program to automate the processing of BY groups. You easily restore all observations into plots and analyses: 1. Activate the data table. Press CTRL+A. This selects all observations in the table. 2. Select Edit I Observations I Include in Plots from the main menu. 3. Select Edit I Observations I Include in Analyses from the main menu. Ordering Categories of a Nominal Variable This section describes how to specify the order of categories for a nominal variable. You cannot change the order of values for interval variables. By default, numeric nominal variables are ordered numerically, whereas character nominal variables are arranged in ASCII order. In ASCII order, numerals precede uppercase letters, which precede lowercase letters. Even if a variable has a SAS format, Stat Studio determines the default order of categories by using the ASCII order of the unformatted values. When the data table is active, you can use the Edit I Variables I Ordering menu to change the order of categories for a nominal variable. You can order nominal variables in three ways: according to the ASCII order of values, the frequency count of values, or the data order of values. For each ordering, you can specify whether to base the order on formatted or unformatted values. Therefore, there are six possible ways to order a nominal variable. Four of these orderings are the same as provided 155 156 Chapter 11. Techniques for Exploring Data by the ORDER= option of the FREQ procedure. An ordering determines the order of categories in a plot (for example, a bar chart) and also the order of sorted observations when sorting a variable in a data table. As an example, consider the data presented in Table 11.1. Table 11.1. Sample Data Observation 1 2 3 4 5 6 Y C B C a a a The Y variable has three categories: a, B, and C. The ASCII order of this data is {B, C, a}, because uppercase letters precede lowercase letters. The data order is {C, B, a}, because as you traverse the data from top to bottom, C is the first value you encounter, followed by B, followed by a. The order by frequency count is {a, C, B}, because there are three observations with the value a, two with the value C, and one with the value B. If you specify an ordering based on formatted values when the variable does not have a SAS format, then Stat Studio applies either a BEST12. format (for numeric variables) or a $w. format (for character variables). When a variable has missing values, the missing values are always ordered first. Example In this section you create a bar chart of the category variable in the Hurricanes data set. =⇒ Open the Hurricanes data set. Note that the column heading for the category variable displays Nom to indicate that the variable is nominal. =⇒ Create a bar chart of the category variable. The bar chart is shown in Figure 11.7. Note that the first category consists of missing values, and the other categories appear in standard ASCII order. Chapter 11. Techniques for Exploring Data Figure 11.7. Standard Ordering of the Category Data When exploring data, it is useful to be able to reorder data categories. The next step arranges the bar chart categories according to frequency counts. =⇒ Right-click in the data table on the column heading for the category variable. Select Ordering I by Frequency, as shown in Figure 11.8. Figure 11.8. Ordering by Frequency Count The bar chart automatically updates, as shown in Figure 11.9. Note that the first bar still represents missing values, but that the remaining bars are ordered by their frequency counts. This presentation of the plot makes it easier to compare the relative frequencies of categories. 157 158 Chapter 11. Techniques for Exploring Data Note that the column heading for the category variable now displays Ord to indicate that this variable has a nonstandard ordering. Figure 11.9. The Category Data Ordered by Frequency Count The next step arranges the bar chart categories according to the data order of the seven nonmissing categories. =⇒ Right-click in the data table on the column heading for the category variable. Select Ordering I by Data, as shown in Figure 11.10. Figure 11.10. Ordering by Data Set Position The bar chart automatically updates, as shown in Figure 11.11. As always, the first bar represents missing values. The TD category is ordered next, because TD is the first nonmissing value for the category variable. The next category is TS, because as you traverse the data starting from the top, the next unique value you encounter is Chapter 11. Techniques for Exploring Data TS (the eighth observation). The remaining categories are Cat1 (the 72nd observation), Cat2 (the 148th observation), Cat3 (the 149th observation), Cat4 (the 155th observation), and Cat5 (the 157th observation). Figure 11.11. The Category Data Ordered by Data Set Position Arranging values by their data order is sometimes useful when the values are inherently ordered. For example, suppose you have a variable Y with the values Low, Medium, and High. The ASCII order for these categories is {High, Low, Medium}. A plot displaying the categories in this order might be confusing. One way to deal with this problem is to do the following: 1. Create a numerical indicator variable with the values {1, 2, 3} corresponding to observations with the values {Low, Medium, High} for Y. The section “Custom Transformations” on page 456 describes how to create an indicator variable. 2. Sort the data by the indicator variable. 3. Save the sorted data. 4. Close your workspace. 5. Open the sorted data. 6. Right-click on the column heading for the variable, and select Ordering I by Data. Plots of the Y variable will display the categories in the order {Low, Medium, High}. Although you can use the previous steps to order any single variable, you might not be able to order multiple variables simultaneously using this method. In that case, you should consult the online Help and read about the DataObject.SetVarValueOrder method. 159 160 Chapter 11. Techniques for Exploring Data Local Selection Mode This section describes how to use graphical methods to visualize observations that satisfy multiple conditions simultaneously. Overview of Global and Local Selection Modes Stat Studio supports two techniques for selecting observations. Global selection mode is the traditional selection technique used in SAS/INSIGHT and other products. This is the default selection mode in Stat Studio. In global selection mode, all data views (that is, plots or data tables) share a common selection state for observations. When you select an observation in one view, that observation is treated as selected in all other views. Global selection mode enables you to graphically subset data by interacting with a single data view. For example, if you have three plots called A, B, and C, selecting observations in plot A causes plots B and C to display those same observations as selected. In contrast, local selection mode enables you to subset data by interacting with multiple data views. In local selection mode, you specify each data view to be either a selector or an observer. You configure an observer to display either the union or the intersection of the selected observations in all selector views. For example, if you have three plots called A, B, and C, you can configure plot C to be the “observer of the intersection” of the other plots. This means that an observation is selected in plot C only if it is selected in both plot A and plot B. You can manually select observations in selector views. You cannot manually select observations in an observer view. An observer view displays an observation as selected based on the observation’s selection state in the selector views. An “observer of the union” displays an observation as selected if the observation is selected in any of the selector views. An “observer of the intersection” displays an observation as selected if the observation is selected in all of the selector views. Example In this section, you create several plots of variables in the Hurricanes data set. You use local selection mode to display the wind speed and pressure of tropical cyclones that satisfy certain spatial and temporal criteria. =⇒ Open the Hurricanes data set. =⇒ Create a histogram of the latitude variable. The histogram will become one of the selector views. The next plot to create is a bar chart of the month variable. By default, the month variable is an interval (continuous) variable. In order to create a bar chart, you first need to change the measure level from interval to nominal. =⇒ Scroll the data table horizontally until you see the month variable. Chapter 11. Techniques for Exploring Data =⇒ Right-click on the heading of the month column, and select Nominal from the pop-up menu. =⇒ Create a bar chart of the month variable. Move the bar chart so that it does not overlap other data views. The bar chart will become a second selector view. =⇒ Create a scatter plot of wind– kts versus min– pressure. Move the plot so that it does not overlap other data views. The scatter plot will become an observer view. The workspace now looks like Figure 11.12. Figure 11.12. Global Selection Mode =⇒ Close the data table. =⇒ Right-click on the plot area in the scatter plot. Select Selection Mode from the pop-up menu. The dialog box shown in Figure 11.13 appears. 161 162 Chapter 11. Techniques for Exploring Data Figure 11.13. Selection Mode Dialog Box =⇒ Click on Local Selection Mode, Observer View, and Intersection. Click OK. The workspace now looks like Figure 11.14. The scatter plot is an observer view. All of the other data views were set to be selector views when you entered local selection mode. Note that selector views are indicated by an arrow icon in the upper-left corner of the view. Observer views are indicated by an icon that looks like an eye looking at the mathematical symbol for intersection (or union). Chapter 11. Techniques for Exploring Data Figure 11.14. Local Selection Mode Caution: If you forgot to close the data table, then it, too, is a selector view. A common error is to leave the data table open. If the data table is left open, then no observations are selected in the observer scatter plot unless they are selected in all other selector views, including data tables. =⇒ In the bar chart, click on the bar labeled “10” to select observations that correspond to the tenth month (October). Note that the histogram does not display any observations because it is a selector view. The scatter plot does not display any observations because it is an observer view: it displays observations as selected only if they are selected in all selector views. =⇒ Create a selection rectangle in histogram. Move it around the plot. The workspace now looks like Figure 11.15. 163 164 Chapter 11. Techniques for Exploring Data Figure 11.15. Displaying the Intersection of Multiple Selector Views The observations displayed as selected in the scatter plot are those that are selected in both the bar chart and the histogram. The selected observations in the scatter plot in Figure 11.15 are those tropical storms that occurred in October (month = 10) of any year and whose position was between 33.75 and 46.25 degrees north latitude. Details This section describes the Selection Mode dialog box, shown in Figure 11.13. To open the Selection Mode dialog box, right-click on a plot or data table, and select Selection Mode from the pop-up menu. Alternatively, click on a data view’s title bar to activate it, and select Edit I Selection Mode from the main menu. The Selection Mode dialog box has the following fields: Global selection mode sets the selection mode to be global selection mode. Local selection mode sets the selection mode to be local selection mode. The active window will become either a selector view or an observer view. All other data views linked to the active window will become selector views. Selector View sets the active window to be a selector view. Chapter 11. Techniques for Exploring Data Observer View sets the active window to be an observer view. Union sets the active window to be an observer of the union of selector views. An observation is displayed as selected if it is selected in any selector view. Intersection sets the active window to be an observer of the intersection of selector views. An observation is displayed as selected if it is selected in all selector views. The following list presents a few additional details about using local selection mode: • There is a limit of 31 selector views that can be linked to an observer view. There is no limit to the number of observer views. • It is often useful to have multiple selector views but only one observer view. In this case it is quickest to activate the plot that is to become the observer view, and then to select Edit I Selection Mode from the main menu. Configure that plot as a local observer view, and click OK. All of the other data views are automatically changed to selector views. This technique was used in the example. • If the observer view is a plot that displays individual observation markers (for example, a scatter plot), it is often useful to configure the plot to show only the selected observations. See the section “Displaying Only Selected Observations” on page 135 for details. This technique is sometimes called graphical filtering, because selected observations do not “reach” the observer view until they have passed through all of the “filters” (criteria) imposed by the selector views. Workspace Explorer In Stat Studio, it is easy to generate a large number of plots. Keeping track of the plots associated with an analysis can be a challenge. Manually closing or minimizing a large number of plots can be tedious. Finding a particular plot from among a large number of plots can be cumbersome. The Workspace Explorer helps solve all of these potential problems. Example In this section, you create many plots of variables in the Hurricanes data set. You use the Workspace Explorer to manage the display of plots. =⇒ Open the Hurricanes data set. =⇒ Scroll the data table horizontally until the min– pressure variable appears. Hold down the CTRL key while you select the min– pressure, wind– kts, longitude, and latitude variables, in that order. 165 166 Chapter 11. Techniques for Exploring Data Figure 11.16 shows the selected variables. Note that the column headings display numbers that indicate the order in which you selected the variables. Figure 11.16. Selecting Variables =⇒ Select Graph I Scatter Plot from the main menu. A matrix of scatter plots appears, as shown in Figure 11.17. Chapter 11. Techniques for Exploring Data Figure 11.17. A Matrix of Scatter Plots The scatter plot of wind– kts versus min– pressure show a strong negative correlation (ρ = −0.93) between wind speed and pressure. In the following steps, you model the linear relationship between these two variables and create plots of the fit residuals. =⇒ Select Analysis I Data Smoothing I Polynomial Regression from the main menu. The dialog box shown in Figure 11.18 appears. 167 168 Chapter 11. Techniques for Exploring Data Figure 11.18. The Polynomial Regression Dialog Box =⇒ Select the variable wind– kts, and click Set Y. Select the variable min– pressure, and click Set X. =⇒ Click the Plots tab, as shown Figure 11.19. Figure 11.19. The Plots Tab =⇒ Select all plots. Clear the check boxes Confidence limits for means and Prediction limits for individuals. Click OK. The analysis creates the five requested plots and an output window, as shown in Figure 11.20. Some of the plots produced by the analysis might be hidden beneath other plots. Chapter 11. Techniques for Exploring Data Figure 11.20. Output and Plots from Polynomial Regression Your workspace now has a data table, a matrix of six scatter plots, five plots associated with an analysis, and an output window, for a total of 13 windows. The Workspace Explore enables you to manage these windows. =⇒ Press ALT+X to open the Workspace Explorer. The Workspace Explorer is shown in Figure 11.21. Figure 11.21. The Workspace Explorer You can use the Workspace Explorer to do the following: • bring a window or group of windows to the front of other windows 169 170 Chapter 11. Techniques for Exploring Data • hide a window or group of windows • close a window or group of windows For example, if you want to see all of the windows associated with the scatter plot matrix, you can do the following. =⇒ Click on the node labeled Scatter Plot Matrix, and click View. This step is shown in Figure 11.22. The matrix of scatter plots becomes visible. Figure 11.22. Viewing a Group of Windows You also can view a particular plot. For example, the following steps activate the plot containing the least squares line. =⇒ In the Workspace Explorer, expand the node labeled Smoothing Polynomial Analysis of wind– kts, if it is not already expanded. =⇒ Click on the item labeled Observed vs. min– pressure. This step is shown in Figure 11.23. Note that the icon to the left of the plot name indicates that the plot is a scatter plot. The icons in the Workspace Explorer match the icons on the Graph main menu. Figure 11.23. Activating a Window Note that the Activate button is now active, whereas it was previously inactive. This is because the selected item is an individual window instead of a group of windows. Activate behaves similarly to View, but also closes the Workspace Explorer and makes the selected window the active window. Chapter 11. Techniques for Exploring Data =⇒ Click Activate. When you are finished viewing a group of plots, the Workspace Explorer makes it easy to close them. You can close workspaces in the same way. =⇒ Press ALT+X to open the Workspace Explorer. Click on the node labeled Analysis windows. Click Close Window(s). This step is shown in Figure 11.24. Stat Studio closes all of the plots created in this example. Figure 11.24. Closing a Group of Windows In summary, the Workspace Explorer enables you to view (or hide) windows. The following list describes each button in the Workspace Explorer. Activate makes the selected window visible and active. Selecting this button also closes the Workspace Explorer. View makes the selected window or group of windows visible. Hide Window(s) hides the selected window or group of windows. Close Window(s) closes the selected window or group of windows. You can also press the DELETE key to close the selected window or group of windows. Close closes the Workspace Explorer. 171 172 Chapter 11. Techniques for Exploring Data Copying Plots to the Windows Clipboard It is easy to copy a plot to the Windows clipboard, and to paste the plot from the clipboard to the Stat Studio output document window or to another application, such as Microsoft Windows or PowerPoint. To copy a plot to the clipboard, activate the plot and select Edit I Copy or press CTRL+C. You can paste to most applications by selecting Edit I Paste or pressing CTRL+V. Stat Studio places the plot on the clipboard in three graphics formats: Windows Enhanced Metafile Format (EMF) stores the image as a series of 32-bit Windows drawing commands. This is the best format for exporting plots from Stat Studio, because the file size is small and the image is faithful to the original. However, not all Windows applications support the EMF format. Specifically, Stat Studio’s output document window does not support the EMF format. Microsoft Word and PowerPoint do support the EMF format. Windows Metafile Format (WMF) stores the image as a series of 16-bit Windows drawing commands. This format is supported by virtually all Windows applications. However, the WMF format does not support wide patterned lines—lines that are not solid and have a width greater than one pixel. The WMF format represents a wide patterned line as a solid line of the same width. Windows Device Independent Bitmap Format (BMP) stores the image as a bitmap. This format is supported by virtually all Windows applications. Plots stored in the BMP format require much more memory than those stored in either the EMF or WMF format. Note: When you paste a plot from the clipboard to a Stat Studio output document window, Stat Studio pastes the plot by using the BMP format. If the plot you are pasting does not make use of wide patterned lines, you can save memory by selecting Edit I Paste Special to paste the plot by using the WMF format. Chapter 12 Plotting Subsets of Data When your data contains categorical variables, you might be interested in comparing subsets of data defined by values of those variables. For example, if your data contains a gender variable, you might want to compare the characteristics of males with those of females. In Stat Studio you can create plots of subsets of data defined by values of one or more categorical variables. The variables whose values define the subsets are called BY variables in SAS, and the subsets are known as BY groups. The BY groups are, by definition, mutually disjoint. Consequently, these plots are not dynamically linked to each other. In Stat Studio, these plots are also not linked to the original data. When you select any graph from the main Graph menu, a dialog box appears that has multiple tabs. You can use the Variables tab to define variable used by the plot. If you click OK, the plot is created on the full data and is linked to other plots and views of that data. Alternatively, you can click the BY Variables tab (Figure 12.1) and define one or more BY variables. (The BY variables are usually nominal variables.) When you click OK, the data are subsetted into BY groups, and a plot is created for each BY group. You can specify options for the BY-group plots from the BY Options tab. Figure 12.1. A Plot Dialog Box 174 Chapter 12. Plotting Subsets of Data A Simple Example Suppose that you are interested in visualizing the location of tropical cyclones for each month (irrespective of the year). That is, you want to examine a scatter plot showing the location of all April cyclones, another showing the locations of May cyclones, etc. There are at least two methods to accomplish this. One approach is to create a bar chart of months, select a bar (that is, a particular month) in the bar chart, and look at the selected observations in a scatter plot of wind– kts versus latitude. This technique is illustrated in Figure 12.2. Figure 12.2. Selecting Cyclones in Certain Months This works well for many data sets. However, the selected observations might not be visible when the scatter plot suffers from overplotting (like Figure 12.2), or when the number of selected observations is small relative to the total number of observations. A variation of this technique is to show only the selected observations. See the “Displaying Only Selected Observations” section on page 135 for a complete example illustrating this approach. Overplotting can also make it difficult to compare features of the data across months. For example, in Figure 12.2, do early-summer cyclones originate in the same regions as autumn cyclones? Does the general shape of cyclone trajectories vary by month? A second visualization approach, known as BY-group processing, attempts to circumvent these problems by abandoning the concept of viewing all of the data in one plot. The idea behind BY group processing is simple: instead of using a single scatter plot linked to a bar chart, you subset the data into mutually exclusive BY groups and make a scatter plot for each subset. This enables you to see each month’s data in isolation, rather than superimposed on a single plot. In this section you create scatter plots of the latitude and longitude variables of the Hurricanes data set. The scatter plots are made for subsets of the hurricane data corresponding to the nine values of the month variable. (The data set does not contain any cyclones for January, February, or March.) =⇒ Open the Hurricanes data set. =⇒ Select Graph I Scatter Plot from the main menu. Chapter 12. Plotting Subsets of Data A dialog box appears as in Figure 12.3. Figure 12.3. Selecting Scatter Plot Variables =⇒ Select the latitude variable and click Set Y. Select the longitude variable and click Set X. =⇒ Click the BY Variables tab. The BY Variables tab is shown in Figure 12.4. Figure 12.4. Selecting BY Variables =⇒ Scroll down in the list of variables and select the month variable. Click Add BY. 175 176 Chapter 12. Plotting Subsets of Data =⇒ Click the BY Options tab. The BY Options tab is shown in Figure 12.5. Figure 12.5. Subsetting Data and Plotting BY Groups =⇒ Select 3x3 for the Layout option. Click OK. Nine scatter plots appear, one for each month 4–12, as shown in Figure 12.13. Figure 12.6. Scatter Plots of Location by Month Note that the X and Y axes are all set to a common range. This makes it easier to Chapter 12. Plotting Subsets of Data compare data characteristics across BY groups. If you want each plot to scale its axes independently, you can deselect Set uniform axis range for interval variables in the BY Options tab. A few features of the data are apparent. • Many tropical cyclones occur in September (month=9). • There is no apparent relationship between month and the shape of cyclone trajectories. It is not clear from this display whether the origin of cyclones varies with the month. Perhaps storms in May (month=6) originate farther west than September storms (month=9), but more investigation is needed. The next example continues this investigation. Example 2: Setting Marker Attributes This example illustrates the fact that observation properties (such as the color and shape of markers) are copied to each BY group during the subsetting of the data. One way to visualize the location in which tropical cyclones originate is to mark the origin of each storm with a special symbol. Figure 12.7 shows the first few observations of the Hurricanes data set. Observations 1–13 correspond to a time series for Tropical Storm Alberto. Observations 14–25 correspond to Beryl. Observations 26–63 correspond to Chris, and so on. The values of the latitude and longitude variables for observations 1, 14, 26, 64, . . ., are the origins of the cyclones. It would be useful to mark these observations so that they are noticeable in the BY-group plots. Figure 12.7. Hurricane Data 177 178 Chapter 12. Plotting Subsets of Data This example has three parts. The first part creates an indicator variable that enumerates the observations for each cyclone. In particular, an observation for which the indicator variable is ‘1’ represents the origin of the storm. The second part of the example assigns a special marker property to the origins. The third part creates plots of BY group, as in the previous example. =⇒ If you have not already done so, open the Hurricanes data set. Creating an Indicator Variable There is an easy way to create a variable that enumerates the observations for each cyclone by using the DATA step. That is the approach taken in this section. The following steps use the Variable Transformation Wizard to create the indicator variable. See “Custom Transformations” for details on the Variable Transformation Wizard. =⇒ Select Analysis I Variable Transformation from the main menu. The Variable Transformation Wizard appears, as shown in Figure 12.8. Figure 12.8. Selecting a Custom Transformation =⇒ Select Custom from the Family list and click Next. The second page of the wizard provides a window for you to enter DATA step code. =⇒ Type in the following DATA step code, prior to the RUN statement, as shown in Figure 12.9. by name notsorted; if first.name then Count=0; Count+1; /* implicitly RETAINS the Count value */ Chapter 12. Plotting Subsets of Data Figure 12.9. Entering DATA Step Code =⇒ Click Finish. A new variable, Count, is added to the data table. The variable enumerates the observations for each cyclone. In particular, Count=1 indicates the first observation for each cyclone. Figure 12.10 shows the new variable. (Some variables in the table are hidden.) Figure 12.10. Hurricane Data With a New Variable 179 180 Chapter 12. Plotting Subsets of Data Changing Marker Properties The following steps select observations where Count=1 and change the shape and color of those observations. =⇒ Select Edit I Find from the main menu. =⇒ Fill out the dialog box to find observations where Count equals 1, as shown in Figure 12.11. Click OK. Figure 12.11. The Find Dialog Box =⇒ Select Edit I Observations I Marker Properties from the main menu. The Marker Properties dialog box appears, as shown in Figure 12.12. Figure 12.12. The Marker Properties Dialog Box =⇒ Change Shape to a star (∗). Change the Outline and Fill to blue. Click OK. The observations with Count=1 are now selected and represented by blue star-shaped markers. Creating BY Group Plots The last part of this example is the same as for the previous example. =⇒ Select Graph I Scatter Plot from the main menu. =⇒ Select the latitude variable and click Set Y. Select the longitude variable and click Set X. Chapter 12. Plotting Subsets of Data =⇒ Click the BY Variables tab. =⇒ Scroll down in the list of variables and select the month variable. Click Add BY. The BY Options tab should be populated with your choices from the previous example. =⇒ Click OK. Nine scatter plots appear, one for each month 4–12, as shown in Figure 12.13. Figure 12.13. Scatter Plots of Location by Month Note that marker properties such as color, shape, and selected status are copied to each of the BY groups. In particular, the selected blue stars enable you to see the origin of each cyclone. A few new features of the data are apparent. • The origin of cyclones varies with the month. • Cyclones early in the season (May–June) and late in the season (October–November) often originate in the Gulf of Mexico (81–98 degrees west longitude and 18–30 degrees north latitude) or Caribbean Sea • In August and September quite a few Cape Verde-type cyclones are apparent. Cape Verde-type cyclones originate between the Cape Verde islands (23 181 182 Chapter 12. Plotting Subsets of Data degrees west longitude and 15 degrees north latitude) and the Lesser Antilles (60 degrees west longitude). • A large number of cyclones originate in the mid-Atlantic (25–35 degrees north latitude) in September, although mid-Atlantic origins are also seen in other months. The next section describes how you can use the Workspace Explorer to view, hide, close, and compare BY-group plots. Techniques for Managing BY Group Plots You can use BY-group plots more effectively if you understand a few details about the way BY-group plots are implemented in Stat Studio. When you create BY-group plots, the following steps occur: 1. A new variable, – ObsNum– , is added to the current data table. 2. The observations corresponding to each BY groups are identified. 3. The observations in each BY group are copied to a new DataObject. (See Stat Studio for SAS/STAT Users for details on the DataObject class.) The variables that are copied depend on the Individual plots can reference all variables option on the BY Options tab, shown in Figure 12.5. 4. The plots are created. If all observations in a BY group are excluded from plots, the BY group is not copied and no plot is created. The BY-group plots are not dynamically linked to the original data. Consequently, selections made to the original data are not reflected in the BY groups. However, you can use an action menu to select observations in the original data that correspond to selected observations in a BY-group plot. See the online Help for a description of action menus. Figure 12.14 illustrates the action menu. Press the F11 key to display the action menu in a BY-group plot. When you select the action menu item, Stat Studio looks at the values of the – ObsNum– variable for the selected observations. Stat Studio then selects observations in the original data that contain the same values of – ObsNum– , as shown in the right-hand portion of Figure 12.14. Using the action menu to select observations is a cumulative process: if an observation in the original data was selected prior to this action, it remains selected after the action. You can clear selections in the data table the usual way: press the ESC key or click in the upper-left cell of the data table. Chapter 12. Plotting Subsets of Data Figure 12.14. Selecting Observations from a BY Group Plot The Layout field shown in Figure 12.5 determines how many BY-group plots are displayed on the screen. If you create more BY-group plots than can fit on the screen, then the remaining plots are created as hidden windows. You can use the Workspace Explorer to manage BY-group plots. The Workspace Explorer is described in “Workspace Explorer”. For example, if you recreate the previous example, but select 2x2 for the Layout field, then only the first four plots are displayed. You can select Windows I Workspace Explorer from the main menu to display the Workspace Explorer, as shown in Figure 12.15. You can select “Panel 2” and click View to see the next four plots. You can also hide an entire panel by clicking Hide Window(s). Finally, you can compare plots belonging to different panels by selecting each individual plot and clicking View. Figure 12.15. Managing BY Group Plots with Workspace Explorer Note: The number of plots that you can display on the screen at one time is limited by Windows resources. The number of plots you can create depends on 183 184 Chapter 12. Plotting Subsets of Data characteristics of your PC, but a typical PC can create a few hundred. Stat Studio prevents you from creating more than 128 BY-group plots on the screen. If you need to create more plots than this limit, use the options on the BY Options tab to write the plots to the output document or to send the plots to files. BY Options Properties This section describes the BY Options tab associated with plots. The BY Options tab controls how data are subsetted and how the plots are displayed. The BY Options tab is shown in Figure 12.5. Individual plot windows specifies whether to display plots on the screen. Layout specifies how plots are arranged on the screen. Output document specifies whether to copy plots to the output document. Graphic type specifies the image type for plots copied to the output document. Files specifies whether to write plots to files on the client (or a networked drive). Directory specifies the directory for writing plots to files. Filename root specifies the prefix used for writing plots to files. The plots are named Root001, Root002, etc. The suffix of each file corresponds to an enumeration of the BY groups. Existing files with the same name are overwritten. File type specifies the image type for plots written to files. Data order determines BY groups This option corresponds to the NOTSORTED option in the BY statement in SAS procedures. If this option is selected, then no sorting is done prior to forming the BY groups. If this option is not selected, then the BY variables are internally sorted and the BY groups consist of observations corresponding to the unique values of the BY variables. Individual plots can reference all variables (slow) If this option is selected, then all variables are copied when forming BY groups. If this option is not selected, then the BY groups contain only the variables specified on the Variables and BY Variables tabs. This option is available only when Individual plot windows is selected. Chapter 12. Plotting Subsets of Data Set uniform axis range for interval variables If this option is selected, then the axes of interval variables are set to a common range. If this option is not selected, each axis is scaled individually according to the data in each BY group. This option is ignored for a rotating plot and for nominal axes. This option does not affect the frequency axis for histograms or bar charts. Figure 12.16. BY Group Options 185 186 Chapter 13 Distribution Analysis: Descriptive Statistics You can use the Descriptive Statistics analysis to compute descriptive statistics for a numeric variable. You can compute basic statistics such as the mean, median, variance, and interquartile range for the selected variable. You can also compute quantiles and extreme values. Finally, you can produce a histogram and box plot that are dynamically linked to the data. You can run a Descriptive Statistics analysis by selecting Analysis I Distribution Analysis I Descriptive Statistics from the main menu. When you request descriptive statistics, Stat Studio calls the UNIVARIATE procedure in Base SAS. See the UNIVARIATE procedure documentation in the Base SAS Procedures Guide for additional details. Example In this example, you generate descriptive statistics for the pressure– outer– isobar variable of the Hurricanes data set. The Hurricanes data set contains 6188 observations of tropical cyclones in the Atlantic basin. The pressure– outer– isobar variable gives the sea-level atmospheric pressure for the outermost closed isobar of a cyclone. This is a measure of the atmospheric pressure at the outermost edge of the storm. =⇒ Open the Hurricanes data set. =⇒ Select Analysis I Distribution Analysis I Descriptive Statistics from the main menu, as shown in Figure 13.1. Figure 13.1. Selecting the Descriptive Statistics Analysis A dialog box appears as in Figure 13.2. You can select a variable for the univariate analysis by using the Variables tab. =⇒ Select the variable pressure– outer– isobar, and click Set Y. 188 Chapter 13. Distribution Analysis: Descriptive Statistics Figure 13.2. Selecting a Variable =⇒ Click the Tables tab. The Tables tab (Figure 13.3) becomes active. =⇒ Select Extreme Values. =⇒ Select Missing Values. =⇒ Click OK. Chapter 13. Distribution Analysis: Descriptive Statistics Figure 13.3. Selecting Tables The analysis calls the UNIVARIATE procedure, which uses the options specified in the dialog box. The procedure displays tables in the output document, as shown in Figure 13.4. In addition to displaying basic statistics such as the mean, median, and standard deviation, the tables also display a few extreme values that seem incongruous. The Extreme Values table shows that there is one low value (998) and one high value (1032) that require investigation. The Missing Values table reveals that almost 25% of the values for this variable are missing. Two plots are created. One plot shows a histogram of the selected variable; the other shows a box plot. One plot might be hidden beneath the other. 189 190 Chapter 13. Distribution Analysis: Descriptive Statistics Figure 13.4. Output from a Descriptive Statistics Analysis For the pressure– outer– isobar variable, the box plot and the Extreme Values table reveal many outliers. It is often useful to investigate outliers to determine whether they are spurious or miscoded data, or to better understand the extreme limits of the data. =⇒ In the box plot, click on the outlier with the highest value of pressure– outer– isobar. This selects the observation in all views of the data, including the data table. You can use the F3 key to scroll through the data table to the next selected observations. =⇒ Activate the data table by clicking on the title bar. Use the F3 key to scroll the selected observation into view. The selected observation corresponds to Hurricane Isadore, September 28, 1996. Scrolling through the data table reveals that the observations before and after the selected observation had a value of 1012 for pressure– outer– isobar. This might indicate that the outlier value of 1032 is a misrecorded value. You can examine other outliers similarly. =⇒ In the box plot, click on the outlier with the lowest value of pressure– outer– isobar. =⇒ Activate the data table by clicking on its title bar. Use the F3 key to scroll the selected observation into view. Chapter 13. Distribution Analysis: Descriptive Statistics This selected observation corresponds to a pressure of 988 hPa for the outermost closed isobar of Hurricane Hugo, September 23, 1989. The data table shows that the observations before the selected observation had considerably larger values of pressure– outer– isobar. Furthermore, the value of min– pressure for the selected observation is 990 hPa, which is larger than the value being investigated. This violates the fact that for a low pressure system, the minimum central pressure should be less than the pressure of the outermost closed isobar. Therefore, the 988 hPa value is most likely misrecorded. You can exclude misrecorded observations by using the Exclude from Plots and Exclude from Analysis features of the data table (see Chapter 4, “The Data Table”). Excluding an observation affects all variables. You can also exclude a single misrecorded value by doing the following: replace the erroneous value with a missing value by typing “.” (or “ ” for a character variable) into the data table cell. Save the data if you want to make the change permanent. Specifying the Descriptive Statistics Analysis This section describes the dialog box options associated with the Descriptive Statistics analysis. The Descriptive Statistics analysis calls the UNIVARIATE procedure in Base SAS. The Variables Tab You can use the Variables tab to specify the variable for the analysis. Only a single variable can be analyzed at a time. The Variables tab is shown in Figure 13.2. The Plots Tab You can use the Plots tab (Figure 13.5) to create a histogram and a box plot of the chosen variable. The histogram can include a kernel density estimate. You can determine the bandwidth for the kernel density method by selecting an option from the Selection method list. The options are as follows: MISE specifies that the kernel bandwidth is chosen to minimize an approximate mean integrated square error. Sheather-Jones specifies that the kernel bandwidth is chosen by a plug-in formula of Sheather and Jones. Manual sets the kernel bandwidth to the value of the Bandwidth field. 191 192 Chapter 13. Distribution Analysis: Descriptive Statistics Figure 13.5. Selecting Plots Note: Stat Studio adds a kernel density estimate to an existing histogram when both of the following conditions are satisfied: • The histogram is the active window when you select the analysis. • The histogram variable and the analysis variable are the same. The Tables Tab You can use the Tables tab to display tables that summarize the results of the univariate analysis. The Tables tab is shown in Figure 13.3. You can choose from the following tables: Sample moments displays sample moments and related statistics, including the mean, variance, skewness, and kurtosis. Basic measures displays statistics related to the central location and the spread of the data. Quantiles displays quantile information. Confidence limits for normal data adds confidence limits to the Quantiles table, based on the assumption that the data are normally distributed. Chapter 13. Distribution Analysis: Descriptive Statistics Distribution-free confidence limits adds confidence limits to the Quantiles table, based on order statistics. Extreme observations displays the observations with the highest and lowest values for the selected variable. Extreme values displays the extreme values (highest and lowest) for the selected variable. Missing values displays the frequency and percentage of missing values for the selected variable. Caution: The observation numbers in the Extreme Observations table reflect the observations that are included in the analysis. If you exclude observations from the analysis, the observation numbers reported in the Extreme Observations table might not correspond to the same observations in the data table. The Roles Tab You can use the Roles tab to specify a frequency variable for the analysis. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, each observation represents n observations, where n is the value of the frequency variable. Analysis of Selected Variables If an interval variable is selected in a data table when you run the analysis, then that variable is automatically entered in the Y Variable field of the Variables tab. If any variable in the data table has a Frequency role, it is automatically entered in the Frequency Variable field of the Roles tab. 193 194 Chapter 14 Distribution Analysis: Location and Scale Statistics Univariate data are often summarized by computing statistics that estimate location and scale. The mean, median, mode, trimmed mean, and Winsorized mean are all statistics that describe the location (or central tendency) of data. Statistics that describe the scale (or variability) include the standard deviation, interquartile range, Gini’s mean difference, and median absolute deviation from the median (MAD). You can use the Location and Scale Statistics analysis to compute location and scale estimates for a single numeric variable. You can also test the hypothesis that the population mean equals a particular value. You can run a Location and Scale Statistics analysis by selecting Analysis I Distribution Analysis I Location and Scale Statistics from the main menu. When you request location and scale estimates, Stat Studio calls the UNIVARIATE procedure in Base SAS. See the UNIVARIATE procedure documentation in the Base SAS Procedures Guide for additional details. Example In this example, you compute statistics that estimate the location and scale for the pressure– outer– isobar variable of the Hurricanes data set. The Hurricanes data set contains 6188 observations of tropical cyclones in the Atlantic basin. The pressure– outer– isobar variable gives the sea-level atmospheric pressure for the outermost closed isobar of a cyclone. This is a measure of the atmospheric pressure at the outermost edge of the storm. The pressure– outer– isobar variable contains 4669 nonmissing values. =⇒ Open the Hurricanes data set. =⇒ Create a histogram of the pressure– outer– isobar variable. A histogram appears, as shown in Figure 14.1. 196 Chapter 14. Distribution Analysis: Location and Scale Statistics Figure 14.1. A Histogram The histogram indicates that there are outliers in these data. Consequently, you might decide to compute robust estimates of location and scale for this variable, in addition to traditional estimates. =⇒ Select Analysis I Distribution Analysis I Location and Scale Statistics from the main menu, as shown in Figure 14.2. Figure 14.2. Selecting the Location and Scale Statistics Analysis A dialog box appears as in Figure 14.3. You can select a variable for the univariate analysis by using the Variables tab. =⇒ Select the variable pressure– outer– isobar, and click Set Y. Chapter 14. Distribution Analysis: Location and Scale Statistics Figure 14.3. Selecting a Variable =⇒ Click on the Tables tab. The Tables tab (Figure 14.4) becomes active. =⇒ Select Modes. The following steps compute robust estimates for the location and scale of these data: =⇒ Select Robust location (trimmed/Winsorized mean). =⇒ Select Robust scale. =⇒ Click OK. 197 198 Chapter 14. Distribution Analysis: Location and Scale Statistics Figure 14.4. Selecting Tables The analysis calls the UNIVARIATE procedure, which uses the options specified in the dialog box. The procedure displays tables in the output document, as shown in Figure 14.5. Chapter 14. Distribution Analysis: Location and Scale Statistics Figure 14.5. Output from a Location and Scale Statistics Analysis For the pressure– outer– isobar variable, the location statistics are in the range of 1011–1012 hPa. Most of the scale statistics are in the range of 2–3 hPa. The mean is a nonrobust statistic, whereas the median, trimmed mean, and Winsorized mean are robust. Note that there is not much difference between the nonrobust and robust statistics of location for these data. The pressure– outer– isobar variable has outliers with extreme high and extreme low values. Therefore, the outliers did not appreciably change the mean. In general, the mean is affected by outliers. The standard deviation is a nonrobust statistic, whereas robust statistics are listed in the Robust Measures of Scale table (not shown in Figure 14.5). The table has two columns. The first column lists the value of each robust statistic, whereas the second column scales the statistics to estimate the normal standard deviation under the assumption that the data are from a normal sample. The “Details” section of the UNIVARIATE procedure documentation presents details of the statistics in this table. 199 200 Chapter 14. Distribution Analysis: Location and Scale Statistics The values of the interquartile range and the MAD statistics should be interpreted with caution for these data because the values of the pressure– outer– isobar variable are discrete integers. More important, meteorologists traditionally display on weather maps only the isobars corresponding to even values. For these data, more than 81% of the nonmissing data are even integers. Specifying the Location and Scale Statistics Analysis This section describes the dialog box tabs associated with the Location and Scale analysis. The Location and Scale Statistics analysis calls the UNIVARIATE procedure in Base SAS. The Variables Tab You can use the Variables tab to specify the variable for the analysis. Only a single variable can be analyzed at a time. The Variables tab is shown in Figure 14.3. The Tables Tab You can use the Tables tab to display tables that summarize the location and scale estimates. The Tables tab is shown in Figure 14.4. The following list describes the tables that can be displayed by the analysis. Basic measures displays statistics related to the central location and the spread of the data. Modes displays the most frequently occurring value or values. Basic confidence intervals displays confidence limits for the mean, standard deviation, and variance, under the assumption that the data are normally distributed. Robust location (trimmed/Winsorized mean) displays information and statistics for a two-sided trimmed mean and a two-sided Winsorized mean. You can use the Cutoff field to enter the percentage or number of observations to trim or Winsorize. Robust scale displays various robust scale statistics. Tests for location displays various tests for the hypothesis that the mean or median is equal to a given value. You can use the Location field to specify the value. The value is also used in the tables for the trimmed and Winsorized means. Chapter 14. Distribution Analysis: Location and Scale Statistics The Roles Tab You can use the Roles tab to specify a frequency variable for the analysis. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. Analysis of Selected Variables If an interval variable is selected in a data table when you run the analysis, then that variable is automatically entered in the Y Variable field of the Variables tab. If any variable in the data table has a Frequency role, it is automatically entered in the Frequency Variable field of the Roles tab. 201 202 Chapter 15 Distribution Analysis: Distributional Modeling You can use the Distributional Modeling analysis to fit parametric distributions to univariate data. You can estimate parameters for the fitted distributions, compute goodness-of-fit statistics, and display quantiles of the fitted distributions. You can use this analysis to create a histogram overlaid with up to five density curves. You can create a quantile-quantile (Q-Q) plot to help you determine how well a given distribution fits the data. You can also create a plot of the empirical cumulative distribution function. You can run a Distributional Modeling analysis by selecting Analysis I Distribution Analysis I Distributional Modeling from the main menu. When you request distributional modeling, Stat Studio calls the UNIVARIATE procedure in Base SAS. See the UNIVARIATE procedure documentation in the Base SAS Procedures Guide for additional details. Example In this example, you fit a normal distribution to the pressure– outer– isobar variable of the Hurricanes data set. The Hurricanes data set contains 6188 observations of tropical cyclones in the Atlantic basin. The pressure– outer– isobar variable gives the sea-level atmospheric pressure for the outermost closed isobar of a cyclone. This is a measure of the atmospheric pressure at the outermost edge of the storm. The plots and statistics in the Distributional Modeling analysis can help you answer questions such as the following: • Can these data be modeled by a parametric distribution? For example, are the data normally distributed? • If not, which characteristics of the data depart from the fitted distribution? For example, is the data distribution long-tailed? Is it skewed? • What proportion of the data is within a given range of values? Answers to these questions for the pressure– outer– isobar variable appear at the end of this example. =⇒ Open the Hurricanes data set. =⇒ Create a histogram of the pressure– outer– isobar variable. 204 Chapter 15. Distribution Analysis: Distributional Modeling A histogram appears, as shown in Figure 15.1. Figure 15.1. A Histogram From the shape of the histogram, you might wonder if the data distribution can be modeled by a normal distribution. If not, how do these data deviate from normality? The following steps add a normal curve to the histogram, and create other plots and statistics. =⇒ Select Analysis I Distribution Analysis I Distributional Modeling from the main menu, as shown in Figure 15.2. Figure 15.2. Selecting the Distributional Modeling Analysis A dialog box appears as in Figure 15.3. You can select a variable for the univariate analysis by using the Variables tab. =⇒ Select the variable pressure– outer– isobar, and click Set Y. Chapter 15. Distribution Analysis: Distributional Modeling Figure 15.3. Selecting a Variable =⇒ Click the Estimators tab. The Estimators tab is shown in Figure 15.4. 205 206 Chapter 15. Distribution Analysis: Distributional Modeling Figure 15.4. Selecting a Distribution Family The Estimators tab enables you to select distributions to fit to the data. For each distribution, you can enter known parameters, or indicate that the parameters should be estimated by maximum likelihood. The section “Specifying Multiple Density Curves” on page 209 describes how to create a histogram overlaid with more than one density curve. For this example, you select a single distribution to fit to the data. The normal distribution appears in the Estimators list by default. Also by default, the Automatic radio button is selected. This specifies that the location and scale parameters for the normal distribution be determined by using maximum likelihood estimation. Accept these defaults and proceed to the next tab. =⇒ Click the Plots tab. =⇒ Select all plots, as shown in Figure 15.5. Chapter 15. Distribution Analysis: Distributional Modeling Figure 15.5. Selecting Plots =⇒ Click OK. The analysis calls the UNIVARIATE procedure, which uses the options specified in the dialog box. The procedure displays tables in the output document, as shown in Figure 15.6. Figure 15.6. Output from a Distributional Modeling Analysis Several plots are created. These plots can help answer the questions posed earlier. Are the Data Normal? The histogram (the upper-right plot in Figure 15.6) is overlaid with a normal density curve. The curve does not fit the data in several locations. The curve predicts more observations in the [1006, 1008] bin than actually occur, and underestimates the count in the [1012, 1014] bin. 207 208 Chapter 15. Distribution Analysis: Distributional Modeling How Do the Data Deviate from Normality? A normal Q-Q plot appears as the upper-left plot in Figure 15.6. A Q-Q plot graphically indicates whether there is agreement between quantiles of the data and quantiles of a theoretical distribution. The Q-Q plot for the normal distribution shows several points to the left that are below the diagonal line. These points indicate that the data distribution has a longer left tail than would be expected from normally distributed data. The point to the right that is above the line might indicate an outlier in the data. Table 15.1 describes how to interpret common features of a Q-Q plot. The goodness-of-fit table in the output document shows that the p-values for the goodness-of-fit tests are very small. The null hypothesis for the goodness-of-fit tests is that the data are from a specified theoretical distribution. The smaller the p-value, the stronger the evidence against the null hypothesis. The small p-values in this example indicate that the normal distribution is not an adequate model to describe these data. Note: The pressure– outer– isobar variable contains 4669 nonmissing values. For a sample of this size, the goodness-of-fit tests can detect small departures from normality, so it is not surprising that these tests reject the null hypothesis. What Proportion of the Data Satisfies Certain Conditions? A CDF plot appears as the lower-right plot in Figure 15.6. The CDF plot shows a graph of the empirical cumulative distribution function. You can use the CDF plot to examine relationships between data values and data proportions. For example, Figure 15.7 graphically answers the question, What observations are contained in the upper quintile (20%) of the data? The selected observations answer the question: data values greater than or equal to 1013 hPa. Similarly, you can ask a converse question: What percentage of the data has values less than or equal to 1000 hPa? The answer (0.4%) can also be obtained by interacting with the CDF plot. Figure 15.7. A CDF Plot Chapter 15. Distribution Analysis: Distributional Modeling The CDF plot also shows how data are distributed. For example, the long vertical jumps in the CDF that occur at even values (1008, 1010, and 1012 hPa) indicate that there are many observations with these values. In contrast, the short vertical jumps at odd values (for example, 1009, 1011, and 1013 hPa) indicate that there are not many observations with these values. This fact is not apparent from the histogram, because the default bin width is 2 hPa. Specifying Multiple Density Curves You can overlay two (or more) density curves on a single histogram. The curves can be different distributions from the same family or distributions from different families. In this section, you fit a lognormal distribution and a Weibull distribution to data in the radius– eye variable. The radius– eye variable gives the radius of a cyclone’s eye (if an eye exists), in nautical miles. (The eye of a cyclone is a calm, relatively cloudless central region.) Note: There are often scientific or engineering considerations that lead to the choice of either a lognormal or a Weibull model. This example does not have a scientific basis; it merely illustrates how you can add multiple curves to a histogram. =⇒ Select Analysis I Distribution Analysis I Distributional Modeling from the main menu. =⇒ Select the variable radius– eye, and click Set Y. =⇒ Click the Estimators tab. The normal distribution appears in the Estimators list. The next step changes this item to a lognormal distribution. =⇒ Select the first item (“Normal”) in the Estimators list. Select Lognormal from the Family list, as shown in Figure 15.8. 209 210 Chapter 15. Distribution Analysis: Distributional Modeling Figure 15.8. Selecting a Lognormal Distribution The lognormal distribution has three parameters. By default, the thresholdparameter is set to zero, and the scale and shape parameters are estimated by maximum likelihood. The next step adds a Weibull distribution to the Estimators list. =⇒ Select the second item (a dashed line) in the Estimators list. =⇒ Select Weibull from the Family list. Chapter 15. Distribution Analysis: Distributional Modeling Figure 15.9. Selecting Multiple Distributions The Weibull distribution also has three parameters. Again, the threshold parameter defaults to zero, whereas the other parameters are estimated by maximum likelihood. Accept these defaults, as shown in Figure 15.9. =⇒ Click OK. Two density curves are added to the histogram, as shown in Figure 15.10. If these were competing scientific models, you could analyze and compare the merits of the models. 211 212 Chapter 15. Distribution Analysis: Distributional Modeling Figure 15.10. Multiple Density Curves Specifying the Distributional Modeling Analysis This section describes the dialog box options associated with the Distributional Modeling analysis. The Distributional Modeling analysis calls the UNIVARIATE procedure in Base SAS. The Variables Tab You can use the Variables tab to specify the variable for the analysis. Only a single variable can be analyzed at a time. The Variables tab is shown in Figure 15.3. The Estimators Tab You can use the Estimators tab (Figure 15.4) to specify parametric distributions to fit to the data. The options for the Estimators tab correspond to options for the HISTOGRAM statement in the UNIVARIATE procedure. See the documentation in the Base SAS Procedures Guide for details. For each distribution, you can enter values for one or more parameters, and estimate the remaining parameters with maximum likelihood estimation (MLE). The analysis typically creates a histogram overlaid with density curves, one for each specified distribution. To add a new distribution to the Estimators list, click on a blank item and select a distribution from the Family list. To delete a distribution from the Estimators list, click on an existing distribution and select None from the Family list. Chapter 15. Distribution Analysis: Distributional Modeling To change a distribution in the Estimators list, click on the distribution and select a new distribution from the Family list. Threshold parameters are set to zero unless you clear the Use threshold of 0 check box, in which case the threshold parameter is estimated by MLE. Other parameters in a distribution are estimated from the data by using MLE, unless you select Manual parameter selection. The Estimator tab has the following items: Estimators displays a list of distributions that will be fitted to the data. Clicking on an item in this list enables you to change the distribution or to specify parameters for the distribution. You can specify up to five distributions. Family specifies the distribution for the selected item in the Estimators list. Parameter Selection specifies how to determine parameters of the selected distribution in the Estimators list. If Automatic is selected, then parameters are estimated by using MLE. If Manual is selected, then you can enter one or more known parameters. Unspecified parameters are estimated by using MLE. Use threshold of 0 specifies whether the threshold parameter is set to zero for the current distribution. If you clear this check box, then the threshold parameter is estimated by using MLE. Caution: Maximum likelihood estimation of two parameters does not always converge. Three-parameter estimation often does not converge. Three-parameter estimation is attempted if you clear the Use threshold of 0 check box while Automatic is selected. The Plots Tab You can use the Plots tab to create the following plots: Histogram with density estimators creates a histogram overlaid with density curves for the parametric distributions specified on the Estimators tab. Quantile-quantile plots creates one Q-Q plot for each parametric distribution specified on the Estimators tab. Empirical cumulative distribution function (CDF) creates a plot of the empirical cumulative distribution function. 213 214 Chapter 15. Distribution Analysis: Distributional Modeling Note: Stat Studio adds a density curve to an existing histogram when both of the following conditions are satisfied: • The histogram is the active window when you select the analysis. • The histogram variable and the analysis variable are the same. Q-Q Plots A Q-Q plot graphically indicates whether there is agreement between quantiles of the data and quantiles of a theoretical distribution. If the quantiles of the theoretical and data distributions agree, the plotted points fall along a straight line. For most distributions, the slope of the line is the value of the scale parameter, and the intercept of the line is the value of the threshold or location parameter. (For the lognormal distribution, the slope is eζ , where ζ is the value of the scale parameter.) The parameter estimates for the distribution that best fits the data appear in an inset in the Q-Q plot. Table 15.1 presents reasons why the points in a Q-Q plot might not be linear. Table 15.1. Interpretation of Q-Q Plots Description of Point Pattern All but a few points fall on a line Left end of pattern is below the line; right end of pattern is above the line Left end of pattern is above the line; right end of pattern is below the line Curved pattern with slope increasing from left to right Curved pattern with slope decreasing from left to right Most points are not near line ax + b with scale parameter a and location parameter b Possible Interpretation Outliers in the data Long tails at both ends of the data distribution Short tails at both ends of the data distribution Data distribution is skewed to the right Data distribution is skewed to the left Data do not fit the theoretical distribution Caution: When the variable being graphed has repeated values, the Q-Q plot produced by Stat Studio is different from the Q-Q plot produced by the UNIVARIATE procedure. The UNIVARIATE procedure arbitrarily ranks the repeated values and assigns a quantile for the theoretical distribution based on the ranks. Two observations with the same value are assigned different quantiles. If a variable has many repeated values, the Q-Q plot produced by the UNIVARIATE procedure looks like a staircase. However, Stat Studio (and SAS/INSIGHT) averages the ranks of repeated values. Two observations with the same value are assigned the same quantiles for the theoretical distribution. CDF Plots A CDF plot shows the empirical cumulative distribution function. You can use the CDF plot to examine relationships between data values and data proportions. For example, you can determine whether a given percentage of your data is below some Chapter 15. Distribution Analysis: Distributional Modeling upper control limit. You can also determine what percentage of the data has values within a given range of values. The inset for the CDF plot displays two statistics. The first is the number of nonmissing observations for the plotted variable. The second is labeled D∗ . If D is the 95% quantile for Kolmogorov’s D distribution (D ≈ 1.36) and N is the number of nonmissing observations, then (D’Agostino and Stephens 1986) D∗ = D/ √ √ N + 0.12 + 0.11/ N The 95% confidence limits in the CDF plot are obtained by adding and subtracting D∗ from the empirical CDF. They form a confidence band around the estimate for the cumulative distribution function. The Tables Tab You can use the Tables tab to display the following tables that summarize the results of the univariate analysis: Parameter estimates displays parameter estimates for the specified theoretical distribution. Goodness-of-fit tests displays goodness-of-fit statistics that test whether the data come from the specified theoretical distribution. Quantiles of fitted distribution displays quantile information for the data and theoretical distributions. The Roles Tab You can use the Roles tab to specify a frequency variable for the analysis. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. Analysis of Selected Variables If an interval variable is selected in a data table when you run the analysis, then that variable is automatically entered in the Y Variable field of the Variables tab. If any variable in the data table has a Frequency role, it is automatically entered in the Frequency Variable field of the Roles tab. 215 216 Chapter 15. Distribution Analysis: Distributional Modeling References D’Agostino, R. and Stephens, M. (1986), Goodness-of-Fit Techniques, New York: Marcel Dekker, Inc. Chapter 16 Distribution Analysis: Frequency Counts You can use the Frequency Counts analysis to produce one-way frequency tables and compute chi-square statistics to test for equal proportions. You can use the analysis to tabulate the number of observations in each category of a variable. For nominal variables, you can also create a bar chart of the variable. You can run a Frequency Counts analysis by selecting Analysis I Distribution Analysis I Frequency Counts from the main menu. When you request one-way frequency table and associated statistics, Stat Studio calls the FREQ procedure in Base SAS. Example In this example, you create a one-way frequency table for the category variable of the Hurricanes data set. The Hurricanes data set contains 6188 observations of tropical cyclones in the Atlantic basin. The category variable gives the Saffir-Simpson category of the tropical cyclone for each observation. A missing value of the category variable means that the storm had an intensity of less than tropical depression strength (wind speeds less than 22 knots) at the time of observation. =⇒ Open the Hurricanes data set. =⇒ Select Analysis I Distribution Analysis I Frequency Counts from the main menu, as shown in Figure 16.1. Figure 16.1. Selecting the Frequency Counts Analysis A dialog box appears as in Figure 16.2. You can select a variable for the analysis by using the Variables tab. =⇒ Select the variable category, and click Set Y. 218 Chapter 16. Distribution Analysis: Frequency Counts Figure 16.2. Specifying a Variable For nominal variables, you can produce a bar chart of the categories of the chosen variable. =⇒ Click the Plots tab. The Plots tab (Figure 16.3) becomes active. =⇒ Select Bar chart. =⇒ Click OK. Figure 16.3. Selecting Plots Figure 16.4 shows the results of this analysis. The analysis calls the FREQ procedure, which uses the options specified in the dialog box. The procedure displays a frequency table in the output document. The table shows the frequency and percent of each Saffir-Simpson category for these data. Hurricanes of category 3 or higher account for only 7% of the nonmissing data, whereas almost half of the observations are classified as tropical storms. Chapter 16. Distribution Analysis: Frequency Counts Figure 16.4. Output from a Frequency Counts Analysis The bar chart shows a graphical view of the category variable. You can create a graphical version of the output table by labeling the bars in the bar chart with their frequencies or percentages. To add labels to the bar chart, do the following: =⇒ Right-click near the center of the plot area. Select Plot Area Properties from the pop-up menu. A dialog box appears, as shown in Figure 16.5. The Bars tab controls attributes of the bar chart. =⇒ Click Show labels. =⇒ Click Y axis represents: Percentage. =⇒ Click OK. Note: You can also label the bar chart by using keyboard shortcuts. Activate the bar chart. Press the “l” key (lowercase “L”) to toggle labels. Press the “p” key to alternate between displaying frequency and percentage. 219 220 Chapter 16. Distribution Analysis: Frequency Counts Figure 16.5. Bar Chart Properties The percentages displayed on the bar chart do not match the percentages in the one-way frequency table. That is because the bar chart includes the 366 missing observations in the total number of observations, whereas the analysis does not include those observations by default. (The counts for each bar do match the counts in the table; only the percentages differ.) If you want to exclude missing values from the bar chart, then you can do the following: 1. Select the missing observations by clicking on the first bar in the bar chart. 2. Select the data table to make it the active window. 3. Select Edit I Observations I Exclude from Plots. The bar chart now omits the missing values as shown in Figure 16.6. Chapter 16. Distribution Analysis: Frequency Counts Figure 16.6. The Bar Chart without Missing Values Alternatively, if you want to include missing values as a valid category, then you can specify that the one-way table should include a category of missing values. When you specify options for the Frequency Counts analysis, do the following: 1. Click the Tables tab, as shown in Figure 16.7. 2. In the Missing values list, select the option Include in tables and statistics. This option specifies that missing values should be regarded as a valid category. If you run (or rerun) the analysis with this option, the one-way table includes missing values as a valid category. The frequency table produced with this option agrees with the default bar chart. 221 222 Chapter 16. Distribution Analysis: Frequency Counts Figure 16.7. The Tables Tab Specifying the Frequency Counts Analysis This section describes the dialog box tabs associated with the Frequency Counts analysis. The Frequency Counts analysis calls the FREQ procedure in Base SAS to compute counts and percentages of each unique value of a variable. The Variables Tab You can use the Variables tab to specify the variable for the TABLES statement of the FREQ procedure. Only a single variable can be analyzed at a time. The Variables tab is shown in Figure 16.2. The Plots Tab You can use the Plots tab (Figure 16.3) to create a bar chart if the chosen variable is nominal. If the chosen variable is not nominal, the analysis prints a warning message to the log. (Note that you can convert an interval variable to nominal. In the data table, right-click on the variable’s column heading and select Nominal from the pop-up menu.) The Tables Tab You can use the Tables tab, shown in Figure 16.7, to specify the options used to produce the one-way frequency table. Each of these options corresponds to an option in the FREQ procedure, as indicated in the following list. Include percents specifies that a column of percents be included in the one-way frequency table. Chapter 16. Distribution Analysis: Frequency Counts Include cumulatives specifies that a column of cumulative percents be included in the one-way frequency table. Order by specifies the order in which the values of the variable appear in the frequency table. This corresponds to the ORDER= option in the PROC FREQ statement. Chi-square test for equal proportions requests a chi-square goodness-of-fit test for equal proportions. This corresponds to the CHISQ option in the TABLES statement. Missing values specifies the treatment of missing values. The following options are supported: Exclude from tables and statistics specifies that missing values be excluded from the analysis. Include in tables; Exclude from statistics specifies that missing value frequencies be displayed, even though the frequencies are not used in the calculation of statistics. This corresponds to the MISSPRINT option in the TABLES statement. Include in tables and statistics specifies that missing values be treated the same as nonmissing values: they are included in calculations of percentages and other statistics. This corresponds to the MISSING option in the TABLES statement. The Roles Tab You can use the Roles tab to specify a weight variable for the analysis. The weight variable in the FREQ procedure is a numeric variable whose value represents the frequency of the observation. If you use a weight variable, the FREQ procedure assumes that each observation represents n observations, where n is the value of the weight variable. For further information, see the documentation for the FREQ procedure in the SAS/STAT User’s Guide. Analysis of Selected Variables If a variable is selected in a data table when you run the analysis, then that variable is automatically entered in the Y Variable field of the Variables tab. If any variable in the data table has a Weight role, it is automatically entered in the Weight Variable field of the Roles tab. 223 224 Chapter 17 Distribution Analysis: Outlier Detection The Outlier Detection analysis computes outliers in contaminated normally distributed data. This analysis defines outliers as values that are sufficiently far from an estimate of the central tendency of the data. More formally, suppose the data are normally distributed with location parameter µ and scale parameter σ. Let µ̂ be an estimate of the location parameter. Let σ̂ be an estimate of the scale parameter. Then a value x is considered an outlier if |x − µ̂| > cσ̂ where c is a constant that you can specify. The constant c is called the scale multiplier. The basic idea is that if the data are normally distributed, then about 99% of the data are within three standard deviations of the mean. Therefore, if you can accurately estimate the mean (location parameter) and standard deviation (scale parameter), you can identify values in the tails of the distribution. However, if the data contain outliers, then you need to use robust estimators of the location and scale parameters. Robust estimates are described in the “Details” section of the documentation for the UNIVARIATE procedure, in the Base SAS Procedures Guide. You can use the analysis to specify traditional or robust estimates of location and scale parameters for a numerical variable. You can create a histogram with a normal curve overlaid. You can create an indicator variable that has the value 1 for observations that are sufficiently far from the location estimate. You can run an Outlier Detection analysis by selecting Analysis I Distribution Analysis I Outlier Detection from the main menu. When you request outlier detection, Stat Studio calls the UNIVARIATE procedure in Base SAS to compute location and scale estimates. SAS/IML is then used to compute the outliers. Example In this example, you detect outliers for the pressure– outer– isobar variable of the Hurricanes data set. The Hurricanes data set contains 6188 observations of tropical cyclones in the Atlantic basin. The pressure– outer– isobar variable gives the sea-level atmospheric pressure for the outermost closed isobar of a cyclone. This is a measure of the atmospheric pressure at the outermost edge of the storm. There are 4662 nonmissing values of pressure– outer– isobar. 226 Chapter 17. Distribution Analysis: Outlier Detection =⇒ Open the Hurricanes data set. =⇒ Select Analysis I Distribution Analysis I Outlier Detection from the main menu, as shown in Figure 17.1. Figure 17.1. Selecting the Outlier Detection Analysis A dialog box appears as in Figure 17.2. You can select a variable for the analysis by using the Variables tab. =⇒ Select the variable pressure– outer– isobar, and click Set Y. Figure 17.2. Specifying a Variable You can specify how the location and scale parameters are estimated by using the Method tab. =⇒ Click the Method tab. The Method tab (Figure 17.3) becomes active. The default is to estimate the Chapter 17. Distribution Analysis: Outlier Detection location with the median of the data, and to estimate the scale with the median absolute deviation from the median (MAD). Each estimate is described in the documentation for the UNIVARIATE procedure in the SAS/STAT User’s Guide. The default scale multiplier is 3. You can accept the default method parameters for this example. Figure 17.3. Specifying the Method =⇒ Click the Plots tab. The Plots tab (Figure 17.4) becomes active. =⇒ Select Normal quantile-quantile plot. =⇒ Click OK. Figure 17.4. Selecting Plots Figure 17.5 shows the results of this analysis. The analysis calls the UNIVARIATE procedure, which uses the options specified in the dialog box. The procedure displays tables in the output document. The tables show several estimates of the location and scale parameters. For this example, the median is 1012 hPa with a scale estimate of 2.965. SAS/IML is then used to read in the specified estimates and to compute values of pressure– outer– isobar that are more than 3 × 2.965 = 8.895 units away from 1012. Two plots are created. One shows a histogram of the selected variable. The histogram is overlaid with a normal curve with µ = 1012 and σ = 2.965. A vertical line at 1012 indicates the location estimate, and shading indicates regions more than 8.965 units from 1012. The other plot is a normal Q-Q plot of the data. 227 228 Chapter 17. Distribution Analysis: Outlier Detection Figure 17.5. Output from an Outlier Detection Analysis By default, the analysis adds an indicator variable to the data table. The indicator variable is named Outlier– Y, where Y is the name of the chosen variable. You can select all observations that are marked as outliers by doing the following: =⇒ Select the data table window to make it active. =⇒ Select Edit I Find from the main menu. The Find dialog box appears as in Figure 17.6. =⇒ Select Outlier– pressure– outer– isobar from the Variable list. =⇒ Select Equals from the Operation list. =⇒ Type 1 in the Value field. =⇒ Click OK. There are 36 observations marked as outliers. If the data table is active, you can use the F3 key to advance to the next selected observation. (Alternatively, you can use Edit I Observations I Examine Selected Observations to examine each selected observation in turn.) The normal Q-Q plot (Figure 17.5) shows that the quantiles of the unselected observations fall along a straight line, indicating that those observations appear to be normally distributed. The selected observations (the outliers) deviate from the line. Chapter 17. Distribution Analysis: Outlier Detection Figure 17.6. Finding Outliers Specifying the Outlier Detection Analysis This section describes the dialog box tabs associated with the Outlier Detection analysis. The Outlier Detection analysis calls the UNIVARIATE procedure in Base SAS to compute estimates of the location and scale. SAS/IML is then used to determine which values are sufficiently far from the location estimate. The Variables Tab You can use the Variables tab to specify the variable for the analysis. Only a single variable can be analyzed at a time. The Variables tab is shown in Figure 17.2. The Method Tab You can use the Method tab to specify the following options for estimating the location and scale parameters for the data, and for specifying the scale multiple. The Method tab is shown in Figure 17.3. Location estimate lists statistics used to estimate the location parameter for the data. Each statistic is described in the “Details” section of the UNIVARIATE procedure documentation in the SAS/STAT User’s Guide. The statistics are as follows: Mean estimates the location parameter by using the mean of the data. (Caution: The mean is not a robust statistic; it is influenced by outliers.) Median estimates the location parameter by using the median of the data. Trimmed mean estimates the location parameter by using the trimmed mean of the data. Winsorized mean estimates the location parameter by using the Winsorized mean of the data. 229 230 Chapter 17. Distribution Analysis: Outlier Detection Trimmed/Winsorized cutoff specifies the number of observations or proportion of observations used to estimate a trimmed or Winsorized mean. Scale estimate lists the statistics for estimating the scale parameter for the (uncontaminated) data. The statistics are as follows: Standard deviation estimates the scale parameter by using the standard deviation of the data. (Caution: The standard deviation is not a robust statistic; it is influenced by outliers.) MAD estimates the scale parameter by using 1.4826 times the median absolute deviation from the median of the data. Sn estimates the scale parameter by using a constant times the robust statistic Sn of the data. Qn estimates the scale parameter by using a constant times the robust statistic Qn of the data. Interquartile range estimates the scale parameter by using the interquartile range of the data divided by 1.34898. √ Gini’s mean difference estimates the scale parameter by using π/2 times Gini’s mean difference. Scale multiplier specifies the constant used to multiply the scale estimate. The resulting product, d, determines outliers: all values whose distance to the location estimate is greater than d are labeled as outliers. The Plots Tab You can use the Plots tab (Figure 17.4) to create a histogram and a normal Q-Q plot of the chosen variable. If you select Overlay normal curve and outlier cutoff, then the histogram includes an overlaid normal curve (Figure 17.5). The parameters for the normal curve are the location and scale estimates of the data. A vertical reference line in the histogram indicates the location estimate, and shading indicates regions more than cσ̂ units from the location estimate, where c is the scale multiplier and σ̂ is the scale estimate. The Output Variables Tab You can use the Output Variables tab to add an indicator variable to the data table. The indicator variable is named Outlier– Y, where Y is the name of the chosen variable. The indicator variable is 1 for observations that are classified as outliers. Chapter 17. Distribution Analysis: Outlier Detection The Roles Tab You can use the Roles tab to specify a frequency variable for the analysis. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. Analysis of Selected Variables If an interval variable is selected in a data table when you run the analysis, then that variable is automatically entered in the Y Variable field of the Variables tab. If any variable in the data table has a Frequency role, it is automatically entered in the Frequency Variable field of the Roles tab. 231 232 Chapter 18 Data Smoothing: Loess The Loess analysis is intended for scatter plot smoothing. Given bivariate data (xi , yi ), i = 1..n, the Loess analysis fits a regression function f whose value at a point x is obtained by evaluating a local regression function at that point. This function is constructed based on data within a neighborhood of x. Although the fit in each local neighborhood is parametric, the construction of the function f depends on many neighborhoods. Consequently, the resulting function is nonparametric. You can run a Loess analysis by selecting Analysis I Data Smoothing I Loess from the main menu. The computation of the loess regression function, confidence limits, and related statistics is implemented by calling the LOESS procedure in SAS/STAT. See the LOESS procedure documentation in the SAS/STAT User’s Guide for additional details. Note: Fitting a loess curve to data sets with more than several thousand observations might require you to wait a while for the computation to finish, especially if you are computing confidence limits or performing an exhaustive search to find the optimal value of the smoothing parameter. Because of this, the Loess analysis presents a warning message (shown in Figure 18.1) when your data contain more than 5000 observations. A similar warning appears if you are performing an exhaustive search and there are more than 1000 observations. Figure 18.1. A Warning Message Example In this example, you fit a loess curve to data in the miningx data set. The miningx data set contains 80 observations corresponding to a single test hole in the mining data set. The driltime variable is the time to drill the last five feet of the current depth, in minutes; the current depth is recorded in the depth variable. =⇒ Open the miningx data set. =⇒ Select Analysis I Data Smoothing I Loess from the main menu, as shown in Figure 18.2. 234 Chapter 18. Data Smoothing: Loess Figure 18.2. Selecting the Loess Analysis The Loess dialog box appears. You can select variables for the analysis by using the Variables tab, shown in Figure 18.3. =⇒ Select the variable driltime, and click Set Y. =⇒ Select the variable depth, and click Set X. Figure 18.3. Selecting Variables =⇒ Click the Plots tab. The Plots tab (Figure 18.4) becomes active. You can use this tab to request additional plots. =⇒ Select Raw residuals vs. Explanatory. Chapter 18. Data Smoothing: Loess For this example it is useful to request a plot of the smoothing criterion versus the smoothing parameter. The loess smoothing parameter determines the percentage of observations used to fit a weighted regression in each local neighborhood. Small values of the smoothing parameter often correspond to undersmoothed curves with many undulations; large values correspond to oversmoothed curves with few undulations. The parameter value that minimizes the smoothing criterion represents a compromise between model fit and model complexity. =⇒ Select Smoothing criterion vs. Smoothing parameter. =⇒ Click OK. Figure 18.4. Selecting Plots The Loess analysis calls the LOESS procedure with the options specified in the dialog box. The procedure displays two tables in the output document, as shown in Figure 18.5. The first table shows that the minimum value of the bias-corrected Akaike information criterion (AICC) was achieved for a smoothing parameter of 0.13125. The second table summarizes the options used by the LOESS procedure and also summarizes the loess fit. Three plots are created. Some plots might be hidden beneath others. If so, move the plots so that the workspace looks like Figure 18.5. 235 236 Chapter 18. Data Smoothing: Loess Figure 18.5. Output from a Loess Analysis One plot (upper left in Figure 18.5) shows the AICC for each value of the smoothing parameter evaluated by the LOESS procedure. Note that the selected smoothing parameter is the one that minimizes the AICC. A second plot (upper right in Figure 18.5) shows a scatter plot of driltime versus depth, with a loess smoother overlaid. The undulations in the smoother might correspond to depths at which variations in rock hardness affect the drilling time. In particular, it is known that the decrease in drilling time at 250 feet is due to encountering a layer of soft copper-nickel ore (Penner and Watts 1991). The third plot shows the residuals versus depth. The spread of the residuals suggests that the variance of the drilling time is a function of the depth of the hole being drilled. The next example creates a second curve that smooths out some of the undulations. This is accomplished by restricting the smoothing parameter to relatively large values. Specifically, the next example specifies that at least 50% of the points in the data set should be used for each local weighted regression. Chapter 18. Data Smoothing: Loess Example: Comparing Smoothers The “Details” section of the LOESS procedure documentation describes how the LOESS procedure computes predicted values. The predicted value at a point x is determined by a weighted average of observations near x. The number of observations used to form the predicted value depends on the smoothing parameter. Recall that the response variable in the previous example is the length of time required to drill the last five feet of a hole that is depth feet deep. For these data, the optimal smoothing parameter was approximately 0.131. This value results in a smoother that varies with the hardness of the underlying rock strata. However, you might want to average out the variations in rock hardness to get a better indication of how the drilling time varies with depth. While 0.131 is a global minimum of the AICC function, there might be a local minimum at a larger value of the smoothing parameter. Using a larger value results in a smoother that is less sensitive to local variation in rock hardness. This example computes another possible loess fit and compares it to the smoother with the parameter 0.131. The example assumes you have completed the previous example and your workspace looks like Figure 18.5. Recall that Stat Studio adds a smoother to an existing scatter plot when both of the following conditions are satisfied: • The scatter plot is the active window when you select the analysis. • The scatter plot variables match the analysis variables. =⇒ Click on the scatter plot of driltime versus depth to activate that window. =⇒ Select Analysis I Data Smoothing I Loess from the main menu. The loess dialog box appears. The dialog box remembers the variables you used in the last analysis. =⇒ Make sure that driltime is selected as the Y variable and depth is selected as the X variable. By examining the AICC plot from the previous example (upper left in Figure 18.5), you might guess that the AICC is an increasing function of the smoothing parameter on the interval [0.131, 0.5]. Thus, if there is a local minimum for AICC at a larger value of the smoothing parameter, it must occur on the interval [0.5, 1]. In the following steps you search for a local minimum of AICC restricted to this interval. =⇒ Click the Method tab. The Method tab is activated, as shown in Figure 18.6. =⇒ Click Exhaustive search for minimum. =⇒ Click Restrict search range and type 0.5 for the Lower bound. 237 238 Chapter 18. Data Smoothing: Loess Figure 18.6. The Method Tab Note: The Exhaustive search for minimum option is computationally expensive. It corresponds to the GLOBAL modifier of the SELECT= option in the LOESS MODEL statement. For the current example, which has 80 observations, the option results in evaluating loess models with at least 40 (0.5 × 80) points in the local neighborhoods. Thus, this option causes the LOESS procedure to evaluate many separate models: one with 40 points in the local neighborhoods, one with 41 points, and so on, up to 80 points. For a data set with 10,000 observations, the same options would result in evaluating up to 5,000 models. =⇒ Click the Plots tab. The Plots tab is activated, as shown in Figure 18.7. =⇒ Clear Raw residuals vs. Explanatory. Chapter 18. Data Smoothing: Loess Figure 18.7. Selecting Plots =⇒ Click OK. As shown in Figure 18.8, the scatter plot of driltime versus depth updates to display the new loess smoother. The AICC plot now shows that the chosen smoothing parameter is approximately 0.631, which corresponds to using 50 (≈ 0.631 × 80) points in the local neighborhoods. 239 240 Chapter 18. Data Smoothing: Loess Figure 18.8. Rerunning a Loess Analysis Note: This second Loess analysis creates a predicted value variable named LoessP– driltime. This variable overwrites the variable of the same name that was created by the first Loess analysis. If you want to compare the predicted values for these two models, you need to rename the first variable prior to running the second analysis. Removing a Smoother If you are trying to determine the relationship between drilling time and depth while averaging out variations in the rock strata, you might prefer the second smoother to the first. If so, you might want to remove the first smoother. When Stat Studio adds a smoother, it also adds an action menu to remove that smoother. You can access this menu by pressing the F11 key while the plot is active. =⇒ Activate the scatter plot of driltime versus depth and press the F11 key. An action menu appears, as shown in Figure 18.9. Chapter 18. Data Smoothing: Loess Figure 18.9. Removing a Smoother =⇒ Select the first menu item: Remove Loess Fit (Smooth=0.131). The first smoother vanishes. The plot now looks like the one in Figure 18.10. Figure 18.10. A Single Smoother The new loess smoother indicates that the drilling time varies roughly linearly at depths between 0 and 200 feet, and linearly (with a different slope) at depths greater than 300 feet. Between 200 and 300 feet, the response varies nonlinearly. Penner and Watts (1991) suggest that air forced through the drill shaft is able to expel debris from the hole at depths less than 200 feet, but at greater depths more and more debris falls back into the hole, thus reducing the drill’s efficiency. 241 242 Chapter 18. Data Smoothing: Loess You can use the techniques presented in this example to compare the loess model to other smoothers. For example, you might decide to compare the loess curve to a quadratic polynomial. If the predictions are nearly the same, you might favor the simpler model. Specifying the Loess Analysis This section describes the Loess analysis dialog box options. The Loess analysis calls the LOESS procedure in SAS/STAT. See the LOESS procedure documentation in the SAS/STAT User’s Guide for details. The Variables Tab You can use the Variables tab to specify the response and explanatory variables for the LOESS MODEL statement. The Y variable specifies the dependent (response) variable, and the X variable specifies the independent (explanatory) variable. The Stat Studio Loess analysis supports only a single dependent variable and a single smoothing variable. The Method Tab You can use the Method tab to specify options for the loess algorithm. The following options are available: Selection method specifies how to choose the loess smoothing parameter. This option corresponds to the SELECT= option in the MODEL statement. AICC selects the smoothing parameter that minimizes the corrected Akaike information criterion. GCV selects the smoothing parameter that minimizes the generalized cross validation criterion. Approx. model DF selects the smoothing parameter for which the trace of the prediction matrix is closest to the Target model DF. This corresponds to the SELECT=DF1 option in the MODEL statement. Manual enables you to specify the value in the Smoothing parameter field. Exhaustive search for minimum specifies that a global minimum be found within the range of smoothing parameter values examined. This corresponds to the GLOBAL modifier to the SELECT= option in the MODEL statement. This option is computationally expensive. Chapter 18. Data Smoothing: Loess Restrict search range specifies that only smoothing parameters greater than or equal to Lower bound and less than or equal to Upper bound be examined. Robust reweighting iterations specifies the number of iterative reweighting steps. Stat Studio counts the initial fit as the 0th reweighting iteration. This differs from the LOESS procedure, which counts the initial fit as the first iteration. Thus if you type n in this field, the option corresponds to ITERATIONS=n + 1 in the MODEL statement. Interpolating polynomial specifies whether the interpolating polynomial is linear or cubic. This corresponds to the INTERP= option in the MODEL statement. Calculation of lookup DF specifies the method used to calculate the “lookup” degrees of freedom used in performing statistical inference. This corresponds to the DFMETHOD= option in the MODEL statement. Local regression polynomial specifies the degree of the local polynomial to use for each local regression. The choice is linear or quadratic. This corresponds to the DEGREE= option in the MODEL statement. The Plots Tab You can use the Plots tab (Figure 18.4) to create plots that graphically display results of the Loess analysis. The raw residuals are computed as Y − Ŷ , where Ŷ indicates the variable containing the predicted values of the response. Creating a plot often adds one or more variables to the data table. The following plots are available: Observed vs. Explanatory with smoother creates a scatter plot of the X and Y variables, overlaid with a smoother. Confidence limits for means specifies whether to add 95% upper and lower confidence limits to the Observed vs. Explanatory plot. Observed vs. Predicted creates a scatter plot of the Y variable versus the predicted values. Raw residuals vs. Predicted creates a scatter plot of the residuals versus the predicted values. Raw residuals vs. Explanatory creates a scatter plot of the residuals versus the X variable. Residual normal QQ creates a normal Q-Q plot of the residuals. 243 244 Chapter 18. Data Smoothing: Loess Smoothing criterion vs. Smoothing parameter creates a scatter plot of the smoothing criterion (for example, AICC) versus the smoothing parameter value for all smoothing parameter values examined in the selection process. The value that minimizes the criterion is indicated by a star-shaped marker. Note: Stat Studio adds a smoother to an existing scatter plot when both of the following conditions are satisfied: • The scatter plot is the active window when you select the analysis. • The scatter plot variables match the analysis variables. The Tables Tab You can use the Tables tab to display tables that summarize the results of the analysis. The Tables tab is shown in Figure 18.11. The following tables are available: Fit summary summarizes the fit and the fit parameters. Smoothing criterion displays the selected smoothing parameter and the corresponding criterion value. Figure 18.11. The Tables Tab The Output Variables Tab You can use the Output Variables tab (Figure 18.12) to add analysis variables to the data table. If you request a plot that uses one of the output variables, then that variable is automatically created even if you did not explicitly select the variable on the Output Variables tab. The following list describes each output variable and indicates how it is named. Y represents the name of the response variable. Predicted values adds predicted values. The variable is named LoessP– Y . Chapter 18. Data Smoothing: Loess Confidence limits for means adds 95% confidence limits for the expected value (mean). The variables are named LoessLclm– Y and LoessUclm– Y . Raw residuals adds residuals, calculated as observed minus predicted values. The variable is named LoessR– Y . Figure 18.12. The Output Variables Tab Analysis of Selected Variables If one or more interval variables are selected in a data table when you run the analysis, then the following occur: • The first selected interval variable is automatically entered in the Y Variable field of the Variables tab. • The second selected interval variable is automatically entered in the X Variable field. No role variables are used for this analysis. 245 246 Chapter 18. Data Smoothing: Loess References Penner, R. and Watts, D. G. (1991), “Mining Information,” The American Statistician, 45(1), 4–9. Chapter 19 Data Smoothing: Thin-Plate Spline The Thin-Plate Spline analysis is intended for scatter plot smoothing. The Thin-Plate Spline analysis uses a penalized least squares method to fit a nonparametric regression model. You can use the generalized cross validation (GCV) function to select the amount of smoothing. You can run the Thin-Plate Spline analysis by selecting Analysis I Data Smoothing I Thin-Plate Spline from the main menu. The computation of the fitted spline function, confidence limits, and related statistics is implemented by calling the TPSPLINE procedure in SAS/STAT. See the TPSPLINE procedure documentation in the SAS/STAT User’s Guide for additional details. Example In this example, you fit a thin-plate spline curve to data in the miningx data set. These data were discussed in Chapter 18, “Data Smoothing: Loess.” The miningx data set contains 80 observations corresponding to a single test hole in the mining data set. The driltime variable is the time to drill the last five feet of the current depth, in minutes; the hole depth is recorded in the depth variable. =⇒ Open the miningx data set. =⇒ Select Analysis I Data Smoothing I Thin-Plate Spline from the main menu, as shown in Figure 19.1. Figure 19.1. Selecting the Thin-Plate Spline Analysis The Thin-Plate Spline dialog box appears. You can select variables for the analysis by using the Variables tab, shown in Figure 19.2. =⇒ Select the variable driltime, and click Set Y. =⇒ Select the variable depth, and click Set X. 248 Chapter 19. Data Smoothing: Thin-Plate Spline Figure 19.2. Selecting Variables =⇒ Click the Plots tab. The Plots tab (Figure 19.3) becomes active. By default, the analysis creates a scatter plot of Y versus X with the smoother overlaid. The smoothing penalty parameter is chosen to minimize the generalized cross validation (GCV) criterion. You can visualize how the smoothing parameter affects the GCV criterion by selecting the following option: =⇒ Select GCV vs. log(n*lambda). =⇒ Click OK. Chapter 19. Data Smoothing: Thin-Plate Spline Figure 19.3. Selecting Plots The Thin-Plate Spline analysis calls the TPSPLINE procedure with the options specified in the dialog box. The procedure displays three tables in the output document, as shown in Figure 19.4. The first table shows information about the number of observations. The second table summarizes model options used by the TPSPLINE procedure. The third table summarizes the fit, including the smoothing value (2.7433) chosen to optimize the selection criterion. Two plots are created, as shown in Figure 19.4. 249 250 Chapter 19. Data Smoothing: Thin-Plate Spline Figure 19.4. Output from a Loess Analysis The upper-left plot in Figure 19.4 shows the GCV criterion for a range of smoothing parameter values. Note that the selected smoothing parameter (2.7433) is the one that minimizes the GCV. A second plot overlays a scatter plot of driltime versus depth with a thin-plate smoother. As discussed in Chapter 18, “Data Smoothing: Loess,” the undulations in the smoother correspond to geological variations in the rock strata. Chapter 18 also discusses how to display multiple smoothers in a single scatter plot, and how to remove smoothers from a scatter plot. Specifying the Thin-Plate Spline Analysis This section describes the Thin-Plate Spline analysis dialog box options. The Thin-Plate Spline analysis calls the TPSPLINE procedure in SAS/STAT. See the TPSPLINE procedure documentation in the SAS/STAT User’s Guide for details. Chapter 19. Data Smoothing: Thin-Plate Spline The Variables Tab You can use the Variables tab to specify the response and explanatory variables for the TPSPLINE MODEL statement. The Y variable specifies the dependent (response) variable, and the X variable specifies the independent (smoothing) variable. The Thin-Plate Spline analysis supports a single dependent variable and a single smoothing variable. Semiparametric fits are not supported: if your data have a polynomial trend, you should subtract the trend and use thin-plate splines to model the residuals. The Method Tab You can use the Method tab (Figure 19.5) to specify options for the thin-plate spline algorithm. The following options are available: Selection method specifies how to choose the smoothing penalty parameter. This option corresponds to the SELECT= option in the MODEL statement. GCV selects the smoothing parameter that minimizes the generalized cross validation criterion. Approx. model DF selects the smoothing parameter for which the trace of the prediction matrix is closest to the Target model DF. This corresponds to SELECT=DF option. Manual enables you to specify the value in the log(n*lambda) field. This corresponds to the LOGNLAMBDA= option. Maximum number of unique design points specifies a limit on the number of unique design points, Nx , in the model. This option corresponds to the DISTANCE= option in the MODEL statement in the following way: the value in this field is used to compute a value for the DISTANCE= option so that there are at most Nx design points. This option is useful for large data sets, since the TPSPLINE procedure is computationally expensive. 251 252 Chapter 19. Data Smoothing: Thin-Plate Spline Figure 19.5. The Method Tab The Plots Tab You can use the Plots tab (Figure 19.3) to create plots that graphically display results of the Thin-Plate Spline analysis. The raw residuals are computed as Y − Ŷ , where Ŷ indicates the variable containing the predicted values of the response. Creating a plot often adds one or more variables to the data table. The following plots are available: Observed vs. Explanatory with smoother creates a scatter plot of the X and Y variables, overlaid with a smoother. Confidence limits for means specifies whether to add 95% upper and lower confidence curves to the Observed vs. Explanatory plot. The meaning of the curves is described in the section “Computational Formulas” in the TPSPLINE documentation. Observed vs. Predicted creates a scatter plot of the Y variable versus the predicted values. Raw residuals vs. Predicted creates a scatter plot of the residuals versus the predicted values. Chapter 19. Data Smoothing: Thin-Plate Spline Raw residuals vs. Explanatory creates a scatter plot of the residuals versus the X variable. Residual normal QQ creates a normal Q-Q plot of the residuals. GCV vs. log(n*lambda) creates a scatter plot of the GCV criterion versus the smoothing parameter value for a range of smoothing parameter values. Minimum log(n*lambda) specifies the minimum value of the smoothing parameter to consider. Maximum log(n*lambda) specifies the maximum value of the smoothing parameter to consider. Number of subdivisions specifies the number of smoothing parameters to consider. The value in this field is combined with the values in the previous two fields to form a list of values for the LOGNLAMBDA= option. Note: Stat Studio adds a smoother to an existing scatter plot when both of the following conditions are satisfied: • The scatter plot is the active window when you select the analysis. • The scatter plot variables match the analysis variables. Chapter 18, “Data Smoothing: Loess,” discusses how to display multiple smoothers in a single scatter plot, and how to remove smoothers from a scatter plot. The Tables Tab You can use the Tables tab to display tables that summarize the results of the analysis. The Tables tab is shown in Figure 19.6. The following tables are available: Data summary summarizes information about the number of observations. Fit summary summarizes the model parameters. Fit statistics summarizes the fit, including the smoothing value that optimizes the selection criterion. 253 254 Chapter 19. Data Smoothing: Thin-Plate Spline Figure 19.6. The Tables Tab The Output Variables Tab You can use the Output Variables tab (Figure 19.7) to add analysis variables to the data table. If you request a plot that uses one of the output variables, then that variable is automatically created even if you did not explicitly select the variable on the Output Variables tab. The following list describes each output variable and indicates how it is named. Y represents the name of the response variable. Predicted values adds predicted values. The variable is named TPSplP– Y . Confidence limits for means adds 95% confidence limits for the expected value (mean). The variables are named TPSplLclm– Y and TPSplUclm– Y . Raw residuals adds residuals, calculated as observed minus predicted values. The variable is named TPSplR– Y . Chapter 19. Data Smoothing: Thin-Plate Spline Figure 19.7. The Output Variables Tab Analysis of Selected Variables If one or more interval variables are selected in a data table when you run the analysis, then the following occurs: • The first selected interval variable is automatically entered in the Y Variable field of the Variables tab. • The second selected interval variable is automatically entered in the X Variable field. No role variables are used for this analysis. 255 256 Chapter 20 Data Smoothing: Polynomial Regression The Polynomial Regression analysis fits a low-order polynomial regression function to bivariate data by using ordinary least squares. This is a global parametric fit, whereas the other Stat Studio smoothers are modern local nonparametric smoothers. You can run a Polynomial Regression analysis by selecting Analysis I Data Smoothing I Polynomial Regression from the main menu. The computation of the regression function, confidence limits, and related statistics is implemented by calling the REG procedure in SAS/STAT. See the documentation for the REG procedure in the SAS/STAT User’s Guide for additional details. Note that general multivariate regression is available by selecting Analysis I Model Fitting I Linear Regression from the main menu. Example In this example, you create a polynomial regression analysis of wind– kts as a function of min– pressure in the Hurricanes data set. The wind– kts variable is the wind speed in knots; the min– pressure variable is the minimum central pressure for each observation. A scatter plot of these variables indicates that the relationship between these variables is approximately linear, as shown in Figure 20.1, so this example fits a line to the data. 258 Chapter 20. Data Smoothing: Polynomial Regression Figure 20.1. Linearly Related Variables =⇒ Open the Hurricanes data set. =⇒ Select Analysis I Data Smoothing I Polynomial Regression from the main menu, as shown in Figure 20.2. Figure 20.2. Selecting Variables A dialog box appears as in Figure 20.3. =⇒ Select the variable wind– kts, and click Set Y. =⇒ Select the variable min– pressure, and click Set X. Chapter 20. Data Smoothing: Polynomial Regression Figure 20.3. The Variables Tab =⇒ Click the Plots tab. The Plots tab becomes active. This tab controls which graphs are produced by the analysis, and the options for each graph (for example, whether to display confidence limits). By default, the analysis creates a scatter plot of the observed X and Y variables, with a smoother added. You can decide whether or not to plot confidence limits. =⇒ Clear Confidence limits for means. =⇒ Select Residual normal QQ. =⇒ Click OK. 259 260 Chapter 20. Data Smoothing: Polynomial Regression Figure 20.4. The Plots Tab Several plots appear, along with output from the REG procedure (Figure 20.5). A scatter plot shows the bivariate data and the requested linear smoother. The analysis also creates a normal Q-Q plot of the residuals. The Q-Q plot indicates that quite a few observations have wind speeds that are substantially lower than would be expected by assuming a linear model with normally distributed errors. In Figure 20.5 these observations are selected, and the corresponding markers in the scatter plot are highlighted. Chapter 20. Data Smoothing: Polynomial Regression Figure 20.5. Results from the Polynomial Regression Analysis Output from the REG procedure appears in the output document. The output informs you that min– pressure has three missing values; those observations are not included in the analysis. The parameter estimates table indicates that when the central atmospheric pressure of a cyclone decreases by 1 HPa, you can expect the wind speed to increase by about 1.3 knots. Specifying the Polynomial Regression Analysis This section describes the dialog box tabs associated with the Polynomial Regression analysis. The Polynomial Regression analysis calls the REG procedure in SAS/STAT. See the REG procedure documentation in the SAS/STAT User’s Guide for details. The Variables Tab You can use the Variables tab to specify the variables for the Polynomial Regression analysis. The Variables tab is shown in Figure 20.3. The Y variable is the response variable. The dialog box supports a single X (explanatory) variable. To analyze a response that depends on multiple explanatory variables, you can use the Linear Regression (Chapter 21, “Model Fitting: Linear Regression”) or the Generalized Linear Models (Chapter 24, “Model Fitting: Generalized Linear Models”) analysis. 261 262 Chapter 20. Data Smoothing: Polynomial Regression The Method Tab You can use the Method tab (Figure 20.6) to specify the degree of the polynomial used to model the data. The following options are available: Linear specifies a first-degree (linear) polynomial. Quadratic specifies a second-degree polynomial. Cubic specifies a third-degree polynomial. Figure 20.6. The Method Tab The Plots Tab You can use the Plots tab (Figure 20.4) to create plots that graphically display results of the analysis. The raw residuals are computed as Y − Ŷ , where Ŷ indicates the variable containing the predicted values of the response. Creating a plot often adds one or more variables to the data table. The following plots are available: Observed vs. Explanatory with smoother creates a scatter plot of the X and Y variables, overlaid with the polynomial smoother. Confidence limits for means specifies whether to add 95% upper and lower confidence limits to the Observed vs. Explanatory plot. Prediction limits for individuals specifies whether to add 95% upper and lower individual prediction limits to the Observed vs. Explanatory plot. Observed vs. Predicted creates a scatter plot of the Y variable versus the predicted values. Chapter 20. Data Smoothing: Polynomial Regression Raw residuals vs. Predicted creates a scatter plot of the residuals versus the predicted values. Raw residuals vs. Explanatory creates a scatter plot of the residuals versus the X variable. Residual normal QQ creates a normal Q-Q plot of the residuals. Note: Stat Studio adds a smoother to an existing scatter plot when both of the following conditions are satisfied: • The scatter plot is the active window when you select the analysis. • The scatter plot variables match the analysis variables. Chapter 18, “Data Smoothing: Loess,” discusses how to display multiple smoothers in a single scatter plot, and how to remove smoothers from a scatter plot. The Tables Tab You can use the Tables tab to display tables that summarize the results of the analysis. The Tables tab is shown in Figure 20.7. The following tables are available: Analysis of variance displays an ANOVA table. Summary of fit displays a table of model fit statistics. Estimated covariance displays the covariance of the parameter estimates. Estimated correlation displays the correlation of the parameter estimates. Parameter estimates displays estimates for the model parameters. Confidence limits for parameters adds 95% confidence limits for the parameter estimates. Standardized parameter estimates adds standardized parameter estimates. 263 264 Chapter 20. Data Smoothing: Polynomial Regression Figure 20.7. The Tables Tab The Output Variables Tab You can use the Output Variables tab (Figure 20.8) to add analysis variables to the data table. If you request a plot that uses one of the output variables, then that variable is automatically created even if you did not explicitly select the variable on the Output Variables tab. The following list describes each output variable and indicates how it is named. Y represents the name of the response variable. Predicted values adds predicted values. The variable is named PolyP– Y . Confidence limits for means adds 95% confidence limits for the expected value (mean). The variables are named PolyLclm– Y and PolyUclm– Y . Prediction limits for individuals adds 95% confidence limits for an individual prediction. The variables are named PolyLcli– Y and PolyUcli– Y . Raw residuals adds residuals, calculated as observed minus predicted values. The variable is named PolyR– Y . Chapter 20. Data Smoothing: Polynomial Regression Figure 20.8. The Output Variables Tab Analysis of Selected Variables If one or more interval variables are selected in a data table when you run the analysis, then the following occurs: • The first selected interval variable is automatically entered in the Y Variable field of the Variables tab. • The second selected interval variable is automatically entered in the X Variable field. No role variables are used for this analysis. 265 266 Chapter 21 Model Fitting: Linear Regression The Linear Regression analysis fits a linear regression model by using ordinary least squares. You can write the multiple linear regression equation for a model with p explanatory variables as Y = b0 + b1 X 1 + b2 X 2 + . . . + bp X p where Y is the response variable, the Xi ’s are explanatory variables, and the bi ’s are regression coefficients. You can run a Linear Regression analysis by selecting Analysis I Model Fitting I Linear Regression from the main menu. The computation of the regression function, confidence limits, and related statistics is implemented by calling the REG procedure in SAS/STAT. See the documentation for the REG procedure in the SAS/STAT User’s Guide for additional details. Example In this example you fit a linear regression model to predict the 1987 salaries of Major League Baseball players as a function of several explanatory variables in the Baseball data set. The response variable is salary. The example examines three explanatory variables: two measures of hitting performance and one measure of longevity. The explanatory variables are described in the following list: • no– hits, the number of hits in 1986 • no– home, the number of home runs in 1986 • yr– major, the number of years that the player has been in the major leagues The example has four major steps: 1. Apply a logarithmic transformation to the response variable. 2. Set name to be the variables whose values are used to label observations. 3. Run the Linear Regression analysis. 4. Discuss the various plots that the analysis can produce. =⇒ Open the Baseball data set. 268 Chapter 21. Model Fitting: Linear Regression Transforming the Response The salary variable ranges from 67.5 to 2460 (measured in thousands of dollars). Since the variation of salaries is much greater for the higher salaries, it is appropriate to apply a logarithmic transformation to the salaries before fitting the model. You can use the Variable Transformation Wizard to transform the salary variable, as described in Chapter 32, “Variable Transformations.” =⇒ Select Analysis I Variable Transformation from the main menu. The Variable Transformation Wizard in Figure 21.1 appears. =⇒ Select the log10(Y+a) transformation from the Transformations list. Figure 21.1. Selecting a Log10 Transformation =⇒ Click Next. The wizard displays the page shown in Figure 21.2. Figure 21.2. Selecting a Variable and Parameters =⇒ Scroll to the end of the variable list. Select the salary variable, and click Set Y. Chapter 21. Model Fitting: Linear Regression The parameter a is an offset that is useful if your variable contains nonpositive values. For these data, you can accept the default value of 0. =⇒ Click Finish. Because there are missing values for the salary variable, a warning message appears (Figure 21.3) informing you that the transformed values for these observations are set to missing values. Figure 21.3. A Warning Message =⇒ Click OK to dismiss the warning message. Stat Studio adds the new variable, Log10– salary, as the last variable in the data set. Selecting a Variable Used to Label Observations For these data, each observation represents a player. It will be convenient to use the name of each player to identify observations in residual plots and diagnostic plots. The following step sets the value of the name variable to be the label you see when you click on an observation. =⇒ Right-click on the variable heading for name to display the Variables menu. Select Label, as shown in Figure 21.4. 269 270 Chapter 21. Model Fitting: Linear Regression Figure 21.4. Selecting a Variable Used to Label Observations Specifying the Mode The following steps model Log10– salary as a function of three explanatory variables. =⇒ Select Analysis I Model Fitting I Linear Regression from the main menu, as shown in Figure 21.5. Figure 21.5. Selecting a Linear Regression A dialog box appears as in Figure 21.6. =⇒ Scroll to the end of the variable list. Select Log10– salary, and click Set Y. =⇒ Select no– hits. While holding down the CTRL key, select no– home, and yr– major. Click Add X. Chapter 21. Model Fitting: Linear Regression Figure 21.6. The Variables Tab =⇒ Click the Plots tab. The Plots tab becomes active, as shown in Figure 21.7. This tab controls which graphs are produced by the analysis. =⇒ Select Cook’s D vs. Observation number. Figure 21.7. The Plots Tab =⇒ Click the Tables tab. The Tables tab becomes active, as shown in Figure 21.8. =⇒ Click Confidence limits for parameters. 271 272 Chapter 21. Model Fitting: Linear Regression Figure 21.8. The Tables Tab =⇒ Click OK. Several plots appear, along with output from the REG procedure. Some plots might be hidden beneath others. Move the windows so that they are arranged as in Figure 21.9. The plot of residuals versus predicted values does not show any obvious trends in the residuals, although possibly the residuals are slightly higher for predicted values near the middle of the predicted range. The plot of the observed versus predicted values shows a reasonable fit, with a few exceptions. In the output window you can see that R square is 0.5646, meaning that the model accounts for 56% of the variation in the data. The no– home term is not significant (t = 1.38, p = 0.1677) and thus can be removed from the model. This is also seen by noting that the 95% confidence limits for the coefficient of no– home include zero. The plot of Cook’s D shows how deleting any one observation would change the parameter estimates. (Cook’s D and other influence statistics are described in the “Influence Diagnostics” section of the documentation for the REG procedure.) A few influential observations have been selected in the plot of Cook’s D; these observations are seen highlighted in the other plots. Three players (Steve Sax, Graig Nettles, and Steve Balboni) with high Cook’s D values also have large negative residuals, indicating that they were paid less than the model predicts. Two other players (Darryl Strawberry and Pete Rose) are also highlighted. These players are discussed in the next section. Chapter 21. Model Fitting: Linear Regression Figure 21.9. Results from the Linear Regression Analysis Interpreting Linear Regression Plots You can use the Linear Regression analysis to create a variety of residual and diagnostic plots, as indicated by Figure 21.7. This section briefly presents the types of plots that are available. To provide common reference points, the same five observations are selected in each set of plots. Partial Leverage Plots Partial leverage plots are an attempt to isolate the effects of a single variable on the residuals (Rawlings, Pantula, and Dickey 1998, p. 359). A partial regression leverage plot is the plot of the residuals for the dependent variable against the residuals for a selected regressor, where the residuals for the dependent variable are calculated with the selected regressor omitted, and the residuals for the selected regressor are calculated from a model where the selected regressor is regressed on the remaining regressors. A line fit to the points has a slope equal to the parameter estimate in the full model. Confidence limits for each regressor are related to the confidence limits for parameter estimates (Sall 1990). Partial leverage plots for the previous example are shown in Figure 21.10. The lower-left plot shows residuals of no– home. The confidence bands in this plot contain the horizontal reference line, which indicates that the coefficient of no– home is not significantly different from zero. 273 274 Chapter 21. Model Fitting: Linear Regression Figure 21.10. Partial Leverage Plots Plots of Residuals versus Explanatory Variables Figure 21.11 shows the residuals plotted against the three explanatory variables in the model. Note that the plot of residuals versus yr– major shows a distinct pattern. The plot indicates that players who have recently joined the major leagues earn less money, on average, than their veteran counterparts with 5–10 years of experience. The mean salary for players with 10–20 years of experience is comparable to the salary that new players make. This pattern of residuals suggests that the example does not correctly model the effect of the yr– major variable. Perhaps it is more appropriate to model log10– salary as a nonlinear function of yr– major. Also, the low salaries of Steve Sax, Graig Nettles, and Steve Balboni might be unduly influencing the fit. Chapter 21. Model Fitting: Linear Regression Figure 21.11. Residual versus Explanatory Plots More Residual Plots Figure 21.12 shows several residual plots. The Q-Q plot (upper left in Figure 21.12) shows that the residuals are approximately normally distributed. Three players with large negative residuals (Steve Sax, Graig Nettles, and Steve Balboni) are highlighted below the diagonal line in the plot. These players seem to be outliers for this model. The residuals versus predicted plot is located in the upper-right corner of Figure 21.12. As noted in the example, the residuals show a slight “bend” when plotted against the predicted value. Figure 21.13 makes the trend easier to see by adding a loess smoother to the residual plot. (See Chapter 18, “Data Smoothing: Loess,” for more information about adding loess curves.) As discussed in the previous section, this trend might indicate the need for a nonlinear term involving yr– major. Alternatively, excluding or downweighting outliers might lead to a better fit. 275 276 Chapter 21. Model Fitting: Linear Regression Figure 21.12. Residual Plots Figure 21.13. A Loess Smoother of the Residuals The lower-right plot in Figure 21.12 is a graph of externally studentized residuals versus predicted values. The externally studentized residual (known as RSTUDENT in the documentation of the REG procedure) is a studentized residual in which the error variance for the ith observation is estimated without including the ith Chapter 21. Model Fitting: Linear Regression observation. You should examine an observation when the absolute value of the studentized residual exceeds 2. The lower-left plot in Figure 21.12 is a graph of (externally) studentized residuals versus the leverage statistic. The leverage statistic for the ith observation is also the ith element on the diagonal of the hat matrix. The leverage statistic indicates how far an observation is from the centroid of the data in the space of the explanatory variables. Observations far from the centroid are potentially influential in fitting the regression model. Observations whose leverage values exceed 2p/n are called high leverage points (Belsley, Kuh, and Welsch 1980). Here p is the number of parameters in the model (including the intercept) and n is the number of observations used in computing the least squares estimates. For the example, n = 263 observations are used. There are three regressors in addition to the intercept, so p = 4. The cutoff value is therefore 0.0304. The plot of studentized residuals versus leverage has a vertical line that indicates high leverage points and two horizontal lines that indicate potential outliers. In Figure 21.12, Pete Rose is an observation with high leverage (due to his 24 years in the major leagues), but not an outlier. Graig Nettles and Steve Sax are outliers and leverage points. Steve Balboni is an outlier because of a low salary relative to the model, whereas Darryl Strawberry’s salary is high relative to the prediction of the model. You should be careful in interpreting results when there are high leverage points. It is possible that Pete Rose fits the model precisely because he is a high leverage point. Chapter 22, “Model Fitting: Robust Regression,” describes a robust technique for identifying high leverage points and outliers. Influence Diagnostic Plots Previous sections discussed plots that included Cook’s D statistic and the leverage statistic. Both of these statistics are influence diagnostics. (See Rawlings, Pantula, and Dickey 1998, p. 361, for a summary of influence statistics.) Figure 21.14 show other plots that are designed to identify observations that have a large influence on the parameter estimates in the model. For each plot, the horizontal axis is the observation number. The upper-left plot displays the leverage statistic along with the cutoff 2p/n. The upper-right plot displays the PRESS residuals. The PRESS residual for observation i is the residual that would result if you fit the model without using the ith observation. A large press residual indicates an influential observation. Pete Rose does not have a large PRESS residual. The lower-left plot displays the covariance ratio. The covariance ratio measures the change in the determinant of the covariance matrix of the estimates by deleting the ith observation. Influential observations have |c − 1| ≥ 3p/n, where c is the covariance ratio (Belsley, Kuh, and Welsch 1980). Horizontal lines on the plot mark the critical values. Pete Rose has the largest value of the covariance ratio. 277 278 Chapter 21. Model Fitting: Linear Regression The lower-right plot displays p the DFFIT statistic, which is similar to Cook’s D. The observations outside of ± p/n are influential (Belsley, Kuh, and Welsch 1980). Pete Rose is not influential by this measure. Figure 21.14. Influence Diagnostics Plots Specifying the Linear Regression Analysis This section explains the dialog box tabs associated with the Linear Regression analysis. The Linear Regression analysis calls the REG procedure in SAS/STAT. See the REG documentation in the SAS/STAT User’s Guide for details. The Variables Tab You can use the Variables tab to specify the variables for the Linear Regression analysis. The Variables tab is shown in Figure 21.6. The Y variable is the response variable. The dialog box supports multiple X (explanatory) variables. All X and Y variables must be interval variables. The Linear Regression analysis does not support nominal classification variables, nor does it support specifying interaction effects such as X1*X2 or higher-order polynomial effects such as X12 . You can create models with these features by using the Generalized Linear Models (Chapter 24, “Model Fitting: Generalized Linear Models”) analysis. Chapter 21. Model Fitting: Linear Regression The Plots Tab You can use the Plots tab (Figure 21.15) to create plots that graphically display results of the analysis. There are plots that help you to visualize the fit, the residuals, and various influence diagnostics. Creating a plot often adds one or more variables to the data table. The following plots are available: Observed vs. Predicted creates a scatter plot of the Y variables versus the predicted values, overlaid with the diagonal line that represents a perfect fit. Partial leverage creates a partial leverage plot for each regressor and for the intercept. A line in the plot has a slope equal to the parameter estimate in the full model. Confidence limits for each regressor are related to the confidence limits for parameter estimates Raw residuals vs. Predicted creates a scatter plot of the residuals versus the predicted values. Raw residuals vs. Explanatory creates scatter plots of the residuals versus the X variables. Externally studentized residuals vs. Predicted creates a scatter plot of the studentized residuals versus the predicted value. Externally studentized residuals vs. Leverage (H) creates a scatter plot of the studentized residuals versus the leverage statistic. Residual normal QQ creates a normal Q-Q plot of the residuals. Cook’s D vs. Observation number creates a scatter plot of Cook’s D statistic for each observation. Leverage (H) vs. Observation number creates a scatter plot of the leverage statistic for each observation. PRESS residuals vs. Observation number creates a scatter plot of the PRESS residual for each observation. COVRATIO vs. Observation number creates a scatter plot of the covariance ratio for each observation. DFFITS vs. Observation number creates a scatter plot of the DFFIT statistic for each observation. 279 280 Chapter 21. Model Fitting: Linear Regression Figure 21.15. The Plots Tab The Tables Tab The Tables tab is shown in Figure 21.8. You can use the Tables tab to display the following tables that summarize the results of the analysis: Analysis of variance displays an ANOVA table. Summary of fit displays a table of model fit statistics. Estimated covariance displays the covariance of the parameter estimates. Estimated correlation displays the correlation of the parameter estimates. X’X matrix displays the X’X crossproducts matrix for the model. The crossproducts matrix is bordered by the X’Y and Y’Y matrices. Collinearity diagnostics displays a detailed analysis of collinearity among the regressors. Parameter estimates displays estimates for the model parameters. Confidence limits for parameters adds 95% confidence limits for the parameter estimates. Chapter 21. Model Fitting: Linear Regression Standardized parameter estimates adds standardized parameter estimates. Figure 21.16. The Tables Tab The Output Variables Tab You can use the Output Variables tab (Figure 21.17) to add analysis variables to the data table. If you request a plot that uses one of the output variables, then that variable is automatically created even if you did not explicitly select the variable on the Output Variables tab. The following list describes each output variable and indicates how the output variable is named. Y represents the name of the response variable. Predicted values adds predicted values. The variable is named RegP– Y . Confidence limits for means adds 95% confidence limits for the expected value (mean). The variables are named RegLclm– Y and RegUclm– Y . Prediction limits for individuals adds 95% confidence limits for an individual prediction. The variables are named RegLcli– Y and RegUcli– Y . Raw residuals adds residuals, calculated as observed minus predicted values. The variable is named RegR– Y . 281 282 Chapter 21. Model Fitting: Linear Regression Internally studentized residuals adds internally studentized residuals, which are the residuals divided by their standard errors. (These correspond to the STUDENT= option in the OUTPUT statement.) The variable is named RegIntR– Y . Externally studentized residuals adds externally studentized residuals, which are studentized residuals with the current observation deleted. (These correspond to the RSTUDENT= option in the OUTPUT statement.) The variable is named RegExtR– Y . Cook’s D adds Cook’s D influence statistic. The variable is named RegCooksD– Y . Leverage (H) adds the leverage statistic. The variable is named RegH– Y . PRESS residuals adds the PRESS residuals. This is the ith residual divided by 1 − h, where h is the leverage, and where the model has been refit without the ith observation. The variable is named RegPRESS– Y . COVRATIO (influence on covariance of coefficients) adds the covariance ratio. This is the ith residual divided by 1 − h, where h is the leverage, and where the model has been refit without the ith observation. The variable is named RegCovRatio– Y . DFFITS (influence on predicted values) adds the standard influence of observation on the predicted value. The variable is named RegDFFITS– Y . DFBETAS (influence on coefficients) adds p variables, where p is the number of parameters in the model. The variables are scaled measures of the change in each parameter estimate and are calculated by deleting the ith observation. Large values of DFBETAS indicate observations that are influential in estimating a given parameter. √ Belsley, Kuh, and Welsch (1980) recommend 2/ n as a size-adjusted cutoff. The variables are named DFB– Xj , where Xj is the name of the jth regressor (including the intercept). Chapter 21. Model Fitting: Linear Regression Figure 21.17. The Output Variables Tab The Roles Tab You can use the Roles tab to specify a frequency variable or weight variable for the analysis. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. A weight variable is a numeric variable with values that are relative weights for a weighted least squares fit. Analysis of Selected Variables If one or more interval variables are selected in a data table when you run the analysis, then the following occurs: • The first selected interval variable is automatically entered in the Y Variable field of the Variables tab. • The remaining selected interval variables are automatically entered in the X Variable field of the Variables tab. Any variable in the data table with a Frequency or Weight role is automatically entered in the appropriate field of the Roles tab. 283 284 Chapter 21. Model Fitting: Linear Regression References Belsley, D. A., Kuh, E., and Welsch, R. E. (1980), Regression Diagnostics, New York: John Wiley & Sons. Rawlings, J. O., Pantula, S. G., and Dickey, D. A. (1998), Applied Regression Analysis: A Research Tool, Springer Texts in Statistics, Second Edition, New York: Springer-Verlag. Sall, J. (1990), “Leverage Plots for General Linear Hypotheses,” The American Statistician, 44(4), 308–315. Chapter 22 Model Fitting: Robust Regression The Robust Regression analysis fits a linear regression model that is robust in the presence of outliers and high leverage points. You can use robust regression to identify observations that are outliers and high leverage points. Once these observations are identified, they can be reweighted or excluded from nonrobust analyses. You can run a Robust Regression analysis by selecting Analysis I Model Fitting I Robust Regression from the main menu. The computation of the robust regression function and the identification of outliers and leverage points are implemented by calling the ROBUSTREG procedure in SAS/STAT. See the documentation for the ROBUSTREG procedure in the SAS/STAT User’s Guide for additional details. Example The example in Chapter 21, “Model Fitting: Linear Regression,” models 1987 salaries of Major League Baseball players as a function of several explanatory variables in the Baseball data set by using ordinary least squares regression. In that example, two conclusions are reached: • no– home, the number of home runs is not a significant variable in the model. • Several players are high leverage points. Pete Rose has the highest leverage because of his 25 years in the major leagues. Graig Nettles and Steve Sax are leverage points and also outliers. However, the model fitted by using ordinary least squares is influenced by high leverage points and outliers. Robust regression is a preferable method of detecting influential observations. This example uses the Robust Regression analysis to identify leverage points and outliers in the Baseball data. This example models the logarithm of salary by using no– hits and yr– major as explanatory variables. =⇒ Open the Baseball data set. The following two steps are the same as for the example in the section “Example” on page 267 in Chapter 21, “Model Fitting: Linear Regression”: =⇒ Use the Variable Transformation Wizard to create a new variable, Log10– salary, containing the logarithmic transformation of the salary variable. =⇒ Choose name to be the label variable for these data. 286 Chapter 22. Model Fitting: Robust Regression The following steps model Log10– salary as a function of two explanatory variables. =⇒ Select Analysis I Model Fitting I Robust Regression from the main menu, as shown in Figure 22.1. Figure 22.1. Selecting a Robust Regression A dialog box appears as in Figure 22.2. =⇒ Scroll to the end of the variable list. Select the Log10– salary, and click Set Y. =⇒ Select no– hits. While holding down the CTRL key, select yr– major. Click Add X. Figure 22.2. The Variables Tab =⇒ Click the Method tab. The Method tab becomes active, as shown in Figure 22.3. There are four robust estimation methods. The default method, known as M estimation, is not robust in the presence of high leverage points. The LTS and MM methods are better suited for handling high leverage points. =⇒ Select MM for the method. Chapter 22. Model Fitting: Robust Regression Note: If you use M estimation on data that contain leverage points, the ROBUSTREG procedure prints the following message to the error log: WARNING: The data set contains one or more high leverage points, for which M estimation is not robust. It is recommended that you use METHOD=LTS or METHOD=MM for this data set. Figure 22.3. The Method Tab =⇒ Click the Plots tab. The Plots tab becomes active, as shown in Figure 22.4. This tab controls which graphs are produced by the analysis. One plot is selected by default. For this example, select the following additional plots: =⇒ Select Observed vs. Predicted. =⇒ Select Robust residuals vs. Predicted. 287 288 Chapter 22. Model Fitting: Robust Regression Figure 22.4. The Plots Tab =⇒ Click the Output Variables tab. The Output Variables tab becomes active, as shown in Figure 22.5. This tab controls which analysis variables are added to the data table. =⇒ Select Final Weights (M and MM methods only). Note that the Outlier indicator and Leverage indicator options are selected by default. These options create indicator variables in the data table that you can use to identify outliers and leverage points. Figure 22.5. The Output Variables Tab Chapter 22. Model Fitting: Robust Regression =⇒ Click OK to run the analysis. Several plots appear, along with output from the ROBUSTREG procedure. Some plots might be hidden beneath others. Move the windows so that they are arranged as in Figure 22.6. In the figure, five players are selected to facilitate comparison with Figure 21.9 and Figure 21.12. The plots involving predicted values are similar to those in Figure 21.9. The plot of residuals versus predicted values does not show any obvious trends. The plot of observed versus predicted values shows a reasonable fit, with a few exceptions. The plot of (internally) studentized robust residuals versus robust distance (known as an RD plot) identifies which observations are outliers and which are high leverage points. Observations outside the horizontal lines at ±3 are outliers; observations to the right of the vertical line at 2.7162 are leverage points. The values of the outlier and leverage cutoffs are displayed in the “Diagnostics Summary” table in the output window. You can control these values from the Method tab. The robust regression model identifies Steve Sax as an outlier and identifies 19 other players (including Pete Rose and Graig Nettles) as leverage points. As displayed in the “Diagnostics Summary” table, these 19 players represent 7.2% of the 263 observations used in the analysis. (For comparison, the analysis in Chapter 21, “Model Fitting: Linear Regression,” suggests 11 outliers and 16 leverage points.) Figure 22.6. Results from the Robust Regression Analysis 289 290 Chapter 22. Model Fitting: Robust Regression Using the Results of Robust Regression Frequently, robust regression is used to identify outliers and leverage points. You can easily select outliers and leverage points by using the mouse to select observations in the RD plot, or by using the Find dialog box. (You can display the Find dialog box by choosing Edit I Find from the main menu.) The analysis added two indicator variables to the data table. The variable RobLev– Log10– salary has the value 1 for observations that are high leverage points. The variable RobOut– Log10– salary has the value 1 for the single observations that is an outlier. Figure 22.7 shows how you can select all of the leverage points. After the observations are selected, you can examine their values, exclude them, change the shapes of their markers, or otherwise give them special treatment. Figure 22.7. Finding Leverage Points Similarly, you can select outliers. The following steps indicate a typical analysis of data contaminated with outliers: 1. Examine the outliers. 2. If it makes sense to exclude the observation from future analyses, select Edit I Observations I Exclude from Analyses from the main menu. 3. Use ordinary least squares regression to model the data without the presence of outliers. Note: You can select Final least squares estimates after excluding outliers on the Tables tab. The parameter estimates in this table are the ordinary least squares estimates after excluding outliers. A second approach involves using the “Final Weights” variable that you requested on the Output Variables tab. The MM method uses an iteratively reweighted least squares algorithm to compute the final estimate, and the RobWt– Log10– salary variable contains the final weights. Figure 22.8 shows the relationship between the weights and the studentized residuals. The graph shows that observations with large residuals (in absolute value) Chapter 22. Model Fitting: Robust Regression receive little or no weight during the reweighted least squares algorithm. In particular, Steve Sax receives no weight, and so his salary was not used in computing the final estimate. For this example, Tukey’s bisquare function was used for the χ function in the MM method; if you use the Yohai function instead, Figure 22.8 looks different. Figure 22.8. Weights versus Studentized Residuals You can use the final weights to duplicate the parameter estimates by using ordinary least squares regression. For example, if you run the REG procedure on the Baseball data and use RobWt– Log10– salary as a WEIGHT variable, you get approximately the same parameter estimates table as displayed by the ROBUSTREG procedure: log10 (salary) = 1.7146 + 0.0047 no– hits + 0.0472 yr– major Specifying the Robust Regression Analysis This section explains the dialog box tabs associated with the Robust Regression analysis. The Robust Regression analysis calls the ROBUSTREG procedure in SAS/STAT. See the ROBUSTREG documentation in the SAS/STAT User’s Guide for details. The Variables Tab You can use the Variables tab to specify the variables for the Robust Regression analysis. The Variables tab is shown in Figure 22.2. The Y variable is the response variable. The dialog box supports multiple X (explanatory) variables. All X and Y variables must be interval variables: the analysis does not support choosing a nominal classification variable. 291 292 Chapter 22. Model Fitting: Robust Regression The Method Tab You can use the Method tab to specify options for one of four robust regression algorithms. The Method tab is shown in Figure 22.3. Each of the following options corresponds to an option in the ROBUSTREG procedure. Method specifies the algorithm used for the robust regression. The choices are M, LTS, S, and MM. This corresponds to the METHOD= option in the PROC ROBUSTREG statement. Outlier multiplier specifies the multiplier of the robust estimate of scale to use for outlier detection. This corresponds to the CUTOFF= option in the MODEL statement. Leverage alpha specifies a cutoff value for leverage-point detection. This corresponds to the CUTOFFALPHA= suboption of the LEVERAGE option in the MODEL statement. The various methods each have options associated with them. When you select a method, the relevant options become active. Options with Method=M With METHOD=M, you can specify the following additional suboptions: Estimation of scale specifies a method for estimating the scale parameter. This corresponds to the SCALE= option. Weight function specifies the weight function used for the M estimate. This corresponds to the WF= option. Options with Method=LTS With METHOD=LTS, you can specify the following additional suboptions: Intercept adjustment specifies the intercept adjustment method in the LTS algorithm. Choosing “Default” corresponds to omitting the IADJUST= option. The other choices correspond to IADJUST=ALL or IADJUST=NONE. Chapter 22. Model Fitting: Robust Regression Options with Method=S With METHOD=S, you can specify the following additional suboptions: Chi function specifies the choice of the χ function for the S estimator. This corresponds to the CHIF= option. Refine S estimate specifies whether to refine for the S estimate. This corresponds to the NOREFINE option. Options with Method=MM With METHOD=MM, you can specify the following additional suboptions: Initial estimator specifies the initial estimator for the MM estimator. This corresponds to the INITEST= option. Chi function specifies the choice of the χ function for the MM estimator. This corresponds to the CHIF= option. Compute bias test specifies whether to display the bias test for the final MM estimate. This corresponds to the BIASTEST option. The Plots Tab You can use the Plots tab (Figure 22.4) to create plots that graphically display results of the analysis. There are plots that help you to visualize the fit, the residuals, and various influence diagnostics. Creating a plot often adds one or more variables to the data table. The following plots are available: Observed vs. Predicted creates a scatter plot of the Y variables versus the predicted values, overlaid with the diagonal line that represents a perfect fit. Robust residuals vs. Predicted creates a scatter plot of the residuals versus the predicted values. Robust residuals vs. Explanatory creates scatter plots of the residuals versus the X variables. Residual normal QQ creates a normal Q-Q plot of the residuals. 293 294 Chapter 22. Model Fitting: Robust Regression Robust residuals vs. Robust MCD distance creates a scatter plot of the internally studentized residuals versus the robust distance. The robust distance is a measure of the distance between an observation and a robust estimate of location. The distance function uses robust estimates of scale and location computed by the minimum covariance determinant (MCD) method. Robust MCD distance vs. Mahalanobis distance creates a scatter plot of the robust distance versus the Mahalanobis distance. The Tables Tab The Tables tab is shown in Figure 22.9. You can use the Tables tab to display the following tables that summarize the results of the analysis: Summary statistics displays summary statistics for model variables. The statistics include robust estimates of the location and scale for each variable. Parameter estimates displays estimates for the model parameters. Diagnostics summary displays a summary of the outlier and leverage diagnostics. Goodness of fit displays goodness-of-fit statistics. Method profile (LTS, S, and MM methods only) displays a summary of the options used by the method. Final least squares estimates after excluding outliers displays least squares estimates computed after deleting the detected outliers. This corresponds to the FWLS option in the PROC ROBUSTREG statement. The parameter estimates reported in this table are the same as the estimates you get if you exclude the outliers reported by ROBUSTREG and then run the REG procedure on the remaining observations. Chapter 22. Model Fitting: Robust Regression Figure 22.9. The Tables Tab The Output Variables Tab You can use the Output Variables tab (Figure 22.5) to add analysis variables to the data table. If you request a plot that uses one of the output variables, then that variable is automatically created even if you did not explicitly select the variable on the Output Variables tab. The following list describes each output variable and indicates how the output variable is named. Y represents the name of the response variable. Predicted values adds predicted values. The variable is named RobP– Y . Final weights (M and MM methods only) adds the final weights used in the iteratively reweighted least squares algorithm. The variable is named RobWt– Y . Robust residuals adds residuals, calculated as observed minus predicted values. The variable is named RobR– Y . Internally studentized robust residuals adds internally studentized residuals, which are the residuals divided by their standard errors. The variable is named RobIntR– Y . Robust MCD distance adds a robust measure of distance between an observation and a robust estimate of location. The variable is named RobRD– Y . 295 296 Chapter 22. Model Fitting: Robust Regression Mahalanobis distance adds the Mahalanobis distance between an observation and the multivariate mean of the data. The variable is named RobMD– Y . Outlier indicator adds an indicator variable for outliers. The variable is named RobOut– Y . Leverage indicator adds an indicator variable for leverage points. The variable is named RobLev– Y . The Roles Tab You can use the Roles tab to specify a weight variable for the analysis. A weight variable is a numeric variable with values that are relative weights for the regression. Analysis of Selected Variables If one or more interval variables are selected in a data table when you run the analysis, then the following occurs: • The first selected interval variable is automatically entered in the Y Variable field of the Variables tab. • The remaining selected interval variables are automatically entered in the X Variable field of the Variables tab. Any variable in the data table with a Weight role is automatically entered in the appropriate field of the Roles tab. Chapter 23 Model Fitting: Logistic Regression The Logistic Regression analysis fits a logistic regression model by using the method of maximum likelihood estimation. If Xi are explanatory variables and p is the response probability to be modeled, the logistic model has the form log(p/(1 − p)) = b0 + b1 X1 + b2 X2 + . . . + bm Xm where the bi are regression coefficients. The explanatory variables in the Logistic Regression analysis can be interval variables or nominal variables (also known as classification variables). You can also specify more complex model terms such as interactions and nested terms. Any term specified in the model is referred to as an effect, whether it is the main effect of a variable, or a classification variable, or an interaction, or a nested term. You can run a Logistic Regression analysis by selecting Analysis I Model Fitting I Logistic Regression from the main menu. The computation of the estimated regression coefficients, confidence limits, and related statistics is implemented by calling the LOGISTIC procedure in SAS/STAT. See the documentation for the LOGISTIC procedure in the SAS/STAT User’s Guide for additional details. Example Neuralgia is pain that follows the path of specific nerves. Neuralgia is most common in elderly persons, but it can occur at any age. In this example, you use a logistic model to compare the effects of two test treatments and a placebo on a dichotomous response: whether or not the patient reported pain after the treatment. In particular, the example examines three explanatory variables: • Treatment, the administered treatment. This variable has three values: A and B represent the two test treatments, while P represents the placebo treatment. • Sex, the patient gender • Age, the patient’s age, in years, when treatment began Some questions that you might ask regarding these data include the following: • Is either treatment better than the placebo at reducing neuralgia? • How does age or gender affect the results? 298 Chapter 23. Model Fitting: Logistic Regression =⇒ Open the Neuralgia data set. =⇒ Select Analysis I Model Fitting I Logistic Regression from the main menu, as shown in Figure 23.1. Figure 23.1. Selecting a Logistic Regression A dialog box appears as in Figure 23.2. You can model the probability that a patient reports no pain after treatment in order to determine whether the treatments are effective. =⇒ Select Pain, and click Add Y. The Treatment and Sex variables are both classification variables, whereas Age is a quantitative (that is, interval) variable. =⇒ Select Treatment. While holding down the CTRL key, select Sex. Click Add Class. =⇒ Select Age, and click Add Quant. Note: Alternatively, you can double-click on a variable to automatically add it as an explanatory variable. Nominal variables are automatically added as classification variables; interval variables are automatically added as quantitative variables. Chapter 23. Model Fitting: Logistic Regression Figure 23.2. The Variables Tab =⇒ Click the Method tab. The Method tab becomes active, as shown in Figure 23.3. You can use this tab to set options for the analysis. The first option on this tab indicates that the analysis will predict the probability of the smallest ordered response. The responses for this example are “Yes” and “No.” Since “No” precedes “Yes” in alphabetical ordering, the smaller ordered response is “No.” This example predicts the probability that a patient will report no pain. This example includes data for a placebo treatment. It is easier to interpret the parameters of the model if you choose a reference parameterization for the coding of the classification variable. (For further details on parameterizations, see the section “CLASS Variable Parameterization” in the “Details” section of the documentation for the LOGISTIC procedure.) =⇒ Select Reference for the Classification variables parameterization option. 299 300 Chapter 23. Model Fitting: Logistic Regression Figure 23.3. The Method Tab =⇒ Click the Plots tab. The Plots tab becomes active, as shown in Figure 23.4. This tab controls which graphs are produced by the analysis. By default, the analysis creates three plots. The following step reduces the number of plots that the analysis creates by omitting a residual plot: =⇒ Clear Change in Pearson chi-square residuals vs. Predicted. Figure 23.4. The Plots Tab =⇒ Click OK. Chapter 23. Model Fitting: Logistic Regression Two plots appear, along with output from the LOGISTIC procedure. One plot might be hidden beneath the other. Move the plots so that they are arranged as in Figure 23.5. The tables created by the LOGISTIC procedure appear in the output window. The “Model Fit Statistics” table indicates that the model with the specified explanatory variables is preferable to an intercept-only model. The “Type 3 Analysis of Effects” table indicates that all explanatory variables in this model are significant. The “Analysis of Maximum Likelihood Estimates” table displays estimates for the parameters in the logistic model. The p-values for Treatment A and B (0.0017 and 0.0010, respectively) indicate that these treatments are significantly better at treating neuralgia than the placebo. The negative estimate for the age effect indicates that older patients in the study responded less favorably to treatment than younger patients. The “Odds Ratio Estimate” table enables you to quantify how changes in an explanatory variable affect the likelihood of the response outcome, assuming the other variables are fixed. Figure 23.5. Results from the Logistic Regression Analysis 301 302 Chapter 23. Model Fitting: Logistic Regression For an interval explanatory variable, the odds ratio approximates how much a unit change in the explanatory variable affects the likelihood of the outcome. For example, the estimate for the odds ratio for Age is 0.767. This indicates that the outcome of eliminating neuralgia occurs only 77% as often among patients of age x + 1, as compared with those of age x. In other words, neuralgia in older patients is less likely to go away than neuralgia in younger patients. For a categorical explanatory variable, the odds ratio compares the odds for the outcome between one level of the explanatory variable and the reference level. The estimate of the odds ratio for treatment A is 24.022. This means that eliminating neuralgia occurs 24 times as often among patients receiving treatment A as among those receiving the placebo. Similarly, eliminating neuralgia occurs more than 41 times as often in patients receiving treatment B, compared to the placebo patients. In the same way, eliminating pain occurs six times more often in females than in males. For a detailed description of how to interpret the odds ratio, including a discussion of various parameterization schemes, see the “Odds Ratio Estimation” section of the documentation for the LOGISTIC procedure. The results of the analysis are summarized by the line plot of predicted probability versus Age. Each line corresponds to a joint level of Treatment and Sex. The line colors indicate levels of Treatment; marker shapes indicate gender. The line plot graphically illustrates a few conclusions from the “Analysis of Maximum Likelihood Estimates” table: • Given a gender and an age, treatment A and treatment B are better at treating neuralgia than the placebo. • Given a treatment and an age, females tend to report less pain than males. • The efficacy of the treatments decreases with the age of the patient. This analysis did not include an interaction term between treatment and gender, so no conclusions are possible regarding whether the treatments affect pain differently in men and women. Also, this analysis did not compare treatment A with treatment B. The other graph in Figure 23.5 plots the confidence interval (CI) displacement diagnostic versus the observation numbers. The CI displacement measures the influence of individual observations on the regression estimates. Observations with large CI displacement values are influential to the prediction. Often these observations are outliers for the model. For example, the observation with the largest CI displacement value is selected in Figure 23.5. (You can double-click on an observation to display the observation inspector, described in Chapter 8, “Interacting with Plots.” ) This patient is a 74-year-old male who was given a placebo. He reported no pain after the treatment, in spite of the fact that the model predicts only a 2% probability that this would happen. The patient with the next largest CI displacement value (not selected in the figure) was a 69-year-old female receiving treatment A. She reported that her pain persisted, although the model predicted a 93% probability that she would not report pain. Chapter 23. Model Fitting: Logistic Regression Specifying the Logistic Regression Analysis This section describes the dialog box tabs associated with the Logistic Regression analysis. The Logistic Regression analysis calls the LOGISTIC procedure in SAS/STAT. See the LOGISTIC documentation in the SAS/STAT User’s Guide for details. The Variables Tab You can use the Variables tab to specify the variables for the Logistic Regression analysis. The Variables tab is shown in Figure 23.2. The analysis handles two types of models. For single-trial syntax, you specify a single binary variable as the response variable. This variable can be character or numeric. For events/trials syntax, you specify two numeric variables that contain count data for a binomial experiment. The value of the first variable is the number of positive responses (or events). The value of the second variable is the number of trials. The dialog box supports multiple explanatory variables. You can include nominal variables in the model by adding them to the Classification variables list. You can include interval variables in the model by adding them to the Quantitative variables list. When you add an explanatory variable, that main effect is added to the Effects tab. You can add interaction effects and nested effects by using the Effects tab. The Effects Tab You can use the Effects tab to add several different types of effects to your model. specifying All effects appear in the Effects in Model list. You can specify the following types of effects: • main effects • crossed effects • nested effects You can also use the tab to quickly create certain standard effects: factorial effects, polynomial effects, and multivariate polynomial effects. The notation for an effect consists of variable names, asterisks, and at most one pair of parentheses. The asterisks denote interactions; the parentheses denote nested effects. There are two rules to follow when specifying effects: 1. A nominal variable can appear in an effect at most once. 2. An interval variable cannot appear inside parentheses. The following text describes how to specify effects on the Effects tab. In the descriptions, assume that A, B, and C are classification variables and that X and Y are interval variables. 303 304 Chapter 23. Model Fitting: Logistic Regression Specifying Main Effects The notation for a main effect is just the name of the variable itself. To specify a main effect, do the following: 1. Select Main from the Standard Effects list. 2. Select one or more variables from the Explanatory Variables list. 3. Click Add. The effects are added to the Effects in Model list, as shown in Figure 23.6. Each main effect appears on a line by itself in the Effects in Model list. Because main effects are automatically added to this list when you select a variable on the Variables tab, you usually do not need to add main effects. Figure 23.6. Specifying Main Effects Specifying Crossed Effects The notation for a crossed effect is two or more variable names joined with asterisks. A crossed effect can involve one or more interval variables (such as X*X and X*Y) or two or more nominal variables (such as A*B, B*C, and A*B*C). You cannot cross a nominal variable with itself, but you can for effects that involve both interval variables and nominal variables, such as X*A. To specify a crossed effect in which each variable appears once (such as X*Y), do the following: 1. Select Cross from the Standard Effects list. 2. Select two or more variables from the Explanatory Variables list. 3. Click Add. Chapter 23. Model Fitting: Logistic Regression For example, the preceding steps were used to create the X*Y effect shown in Figure 23.7. To cross variables with effects already in the model, do the following: 1. Select Cross from the Standard Effects list. 2. Select one or more variables from the Explanatory Variables list. 3. Select one or more effects from the Effects in Model list. 4. Click Cross, located to the right of the Effects in Model list. For example, Figure 23.7 shows one way to create the effect X*X*Y. You can select the X variable from the Explanatory Variables list and the X*Y effect from the Effects in Model list. The X*X*Y effect is created when you click Cross. Figure 23.7. Specifying Crossed Effects Specifying Nested Effects The notation for a nested effect contains two parts. The first part is a main effect or crossed effect. The second part consists of a classification variable or an interaction between classification variables. The second part is enclosed in parentheses. The main effect or crossed effect is said to be “nested within” the effects in parentheses. For example, A(B*C) means “effect A is nested within the levels of the factors B and C.” The Standard Effects value is ignored when you specify nested effects. To create a nested effect, the effect outside the parentheses must already be specified in the Effects in Model list. To create a nested effect, do the following: 1. Select one or more nominal variables from the Explanatory Variables list. These variables will appear inside the parentheses. 305 306 Chapter 23. Model Fitting: Logistic Regression 2. Select one or more effects from the Effects in Model list. These variables will appear outside the parentheses. Make sure that the nominal variables selected in the Explanatory Variables list do not appear in any of the effects selected in the Effects in Model list. 3. Click Nest, located to the right of the Effects in Model list. 4. The effects in the Effects in Model list are replaced with the nested effects. For example, Figure 23.8 shows one way to create the effect A(B*C). Select the B and C variables from the Explanatory Variables list, and select the A main effect from the Effects in Model list. The A(B*C) effect is created when you click Nest. It replaces the A effect that is currently in the list. Figure 23.8. Specifying Nested Effects Specifying Factorial Effects Factorial effects are k-way interactions between a set of variables. To create factorial effects, do the following: 1. Select Factorial from the Standard Effects list. 2. Enter the Degree of the model. 3. Select two or more variables from the Explanatory Variables list. 4. Click Add. 5. The factorial effects are added to the Effects in Model list. Any effects already in the model (for example, main effects) are highlighted, although their position in the Effects in Model list does not change. For example, Figure 23.9 shows how to create a full three-way factorial model with the variables A, B, and C. The following effects are added to the Effects in Model list: A, B, C, A*B, A*C, B*C, and A*B*C. Chapter 23. Model Fitting: Logistic Regression Figure 23.9. Specifying Factorial Interaction Effects Specifying Polynomial Effects Interactions of an interval variable with itself are called polynomial effects. Each term is a monomial in one variable. To create polynomial effects, do the following: 1. Select Polynomial from the Standard Effects list. 2. Enter the Degree of the model. (The maximum degree is 10.) 3. Select one or more variables from the Explanatory Variables list. 4. Click Add. 5. The polynomial effects are added to the Effects in Model list. Any effects already in the model (for example, main effects) are highlighted, although their position in the Effects in Model list does not change. For example, Figure 23.10 shows how to create all terms in a degree-three polynomial in the variable X. The following effects are added to the Effects in Model list: X, X*X, and X*X*X. 307 308 Chapter 23. Model Fitting: Logistic Regression Figure 23.10. Specifying Polynomial Effects Specifying Multivariate Polynomial Effects Multivariate polynomial effects are polynomial and interaction effects among a group of variables. If you select m variables and request effects from a degree-d multivariate polynomial, then each term is a multivariate monomial, with degree at most min(k, d). To create multivariate polynomial interaction effects, do the following: 1. Select Multivariate Polynomial from the Standard Effects list. 2. Enter the Degree of the model. (The maximum degree is 4.) 3. Select one or more variables from the Explanatory Variables list. 4. Click Add. 5. The polynomial effects are added to the Effects in Model list. Any effects already in the model (for example, main effects) are highlighted, although their position in the Effects in Model list does not change. For example, Figure 23.11 shows how to create all main effects and valid two-way interactions among the three variables X, Y, and A. The following effects are added to the Effects in Model list: X, Y, A, X*X, Y*Y, X*Y, X*A, and Y*A. The term A*A is not created because A is a classification variable. Chapter 23. Model Fitting: Logistic Regression Figure 23.11. Specifying Polynomial Interaction Effects Reordering Effects You can reorder and remove effects in the Effects in Model list. The order that effects appear in the list is the order in which the effects appear in the MODEL statement of SAS/STAT procedures. Move Up moves selected effects up one position in the Effects in Model list. Move Down moves selected effects down one position in the Effects in Model list. Remove removes the selected effects from the Effects in Model list. Reset deletes all effects and then adds main effects to the Effects in Model list. The Method Tab You can use the Method tab (Figure 23.3) to set the following options for the analysis: Predict probability of specifies whether to model the probability of the first or last level of the response variable. For example, if the response variable has levels 0 and 1, then you would select Largest ordered response to model the probability of 1. This corresponds to the DESCENDING option in the PROC LOGISTIC statement. 309 310 Chapter 23. Model Fitting: Logistic Regression Classification variables parameterization specifies the parameterization method for the classification variables. This corresponds to the PARAM= option in the CLASS statement. The dialog box supports the GLM, effect, and reference coding schemes. Estimate scale parameter as specifies the method for estimating the dispersion parameter. This corresponds to the SCALE= option in the MODEL statement. Aggregate specifies the subpopulations on which certain test statistics are calculated. This corresponds to the AGGREGATE option in the MODEL statement. The Plots Tab You can use the Plots tab (Figure 23.4) to create plots that graphically display results of the analysis. There are plots that help you to visualize the fit, the residuals, and various influence diagnostics. Creating a plot often adds one or more variables to the data table. The following plots are available: Predicted probability vs. One continuous covariate creates a line plot of the predicted probability versus the continuous explanatory variable. This plot is created only if the following conditions are satisfied: • There is exactly one continuous explanatory variable. • There are three or fewer classification variables. • There are 12 or fewer joint levels of the classification variables. ROC curve creates a line plot that shows the trade-off between sensitivity and specificity. Models that fit the data well correspond to an ROC curve that has an area close to unity. A completely random predictor would produce an ROC curve that is close to the diagonal and has an area close to 0.5. Pearson chi-square residuals vs. Predicted creates a scatter plot of the Pearson chi-square residuals versus the predicted probabilities. Deviance residuals vs. Predicted creates a scatter plot of the deviance residuals versus the predicted probabilities. Change in Pearson chi-square vs. Predicted creates a scatter plot of the DIFCHISQ statistic versus the predicted probabilities. Chapter 23. Model Fitting: Logistic Regression Change in deviance vs. Predicted creates a scatter plot of the DIFDEV statistic versus the predicted probabilities. Confidence interval displacement (C) vs. Predicted creates a scatter plot of the confidence interval displacement diagnostic (C) versus the predicted probabilities. Confidence interval displacement (C) vs. Observation number creates a scatter plot of the confidence interval displacement diagnostic (C) for each observation. Leverage (H) vs. Observation number creates a scatter plot of the leverage statistic for each observation. The Tables Tab The Tables tab is shown in Figure 23.12. You can use the Tables tab to display the following tables that summarize the results of the analysis: Simple descriptive statistics displays a table of summary statistics for the explanatory variables. Model fit statistics displays a table of model fit statistics. Generalized R-square displays generalized R-square statistics. Parameter estimates displays estimates for the model parameters. Confidence intervals for parameters displays estimates of 95% confidence intervals for the model parameters. Odds ratios estimates displays the odds ratio estimates. Confidence intervals for odds ratios displays estimates of 95% confidence intervals for the odds ratios. Hosmer-Lemeshow goodness-of-fit test displays partition information and statistics for the Hosmer-Lemeshow goodness-of-fit test. 311 312 Chapter 23. Model Fitting: Logistic Regression Figure 23.12. The Tables Tab The Output Variables Tab You can use the Output Variables tab (Figure 23.13) to add analysis variables to the data table. If you request a plot that uses one of the output variables, then that variable is automatically created even if you did not explicitly select the variable on the Output Variables tab. The following list describes each output variable and indicates how the output variable is named. Y represents the name of the response variable. If you use events/trials syntax, then Y represents the name of the events variable. Proportions for events/trials adds a variable named Proportion– ET , where E is the name if the events variable and T is the name of the trials variable. The value of the variable is the ratio E/T . This variable is added only when you use events/trials syntax. Predicted probabilities adds predicted probabilities. The variable is named LogiP– Y . Confidence limits for predicted probabilities adds 95% confidence limits for the predicted probabilities. The variables are named LogiLclm– Y and LogiUclm– Y . Linear predictor (log odds) adds the linear predictor values. The variable is named LogiXBeta– Y . Pearson chi-square residuals adds the Pearson chi-square residuals. The variable is named LogiChiSqR– Y . Chapter 23. Model Fitting: Logistic Regression Deviance residuals adds the deviance residuals. The variable is named LogiDevR– Y . Confidence interval displacement (C) adds the confidence interval displacement diagnostic, C. The variable is named LogiC– Y . Scaled confidence interval displacement (CBAR) adds the confidence interval displacement diagnostic, C̄. The variable is named LogiCBar– Y . Leverage (H) adds the leverage statistic. The variable is named LogiH– Y . DIFCHISQ (influence on chi-square goodness-of-fit) adds the change in the chi-square goodness-of-fit statistic attributed to deleting the individual observation. The variable is named LogiDifChiSq– Y . DIFDEV (influence on deviance) adds the change in the deviance attributed to deleting the individual observation. The variable is named LogiDifDev– Y . DFBETAS (influence on coefficients) adds m variables, where m is the number of parameters in the model. The variables are scaled measures of the change in each parameter estimate and are calculated by deleting the ith observation. Large probabilities of DFBETAS indicate observations that are influential in estimating a given parameter. The variables are named DFBETA– X, where X is the name of an interval regressor (including the intercept). For classification variables, the variables are named DFBETA– CL, where C is the name of the variable and L represents a level. Figure 23.13. The Output Variables Tab 313 314 Chapter 23. Model Fitting: Logistic Regression The Roles Tab You can use the Roles tab (Figure 23.14) to specify a frequency variable or weight variable for the analysis. You can also specify an offset variable. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. A weight variable is a numeric variable with values that weigh each observation in the regression. An offset variable is a special explanatory variable. The regression coefficient for this variable will be fixed at 1. This corresponds to the OFFSET= option in the MODEL statement. Figure 23.14. The Roles Tab Analysis of Selected Variables If one or more interval variables are selected in a data table when you run the analysis, then the following occurs: • The first selected nominal variable is automatically entered in the Y Variables field of the Variables tab. • Subsequent selected nominal variables are automatically entered in the Classification Variables field. • Selected interval variables are automatically entered in the Quantitative Variables field. Chapter 23. Model Fitting: Logistic Regression Any variable in the data table with a Frequency or Weight role is automatically entered in the appropriate field of the Roles tab. 315 316 Chapter 24 Model Fitting: Generalized Linear Models The generalized linear model is a generalization of the traditional linear model. It differs from a linear model in that it assumes that the response distribution is related to the linear predictor through a function called the link function. Specifically, a generalized linear model has a linear component η = η 0 + Xβ and a monotonic differentiable function, g, that links the expected response mean, µ, to the linear predictor η: η = g(µ) The response y is assumed to have a distribution from the exponential family (for example, normal, gamma, Poisson, binomial, etc.). The vector η 0 is called an offset variable. As in least squares regression, X is the design matrix and β is a vector of unknown parameters. The explanatory variables in the Generalized Linear Models analysis can be interval variables or nominal variables (also known as classification variables). You can also specify more complex model terms such as interactions and nested effects. As mentioned in Chapter 21, “Model Fitting: Linear Regression,” the Linear Regression analysis in Stat Studio does not support classification variables. You can use the Generalized Linear Models analysis to fit a linear regression with classification variables by specifying that the response variable is normally distributed and that the link function is the identity function. The first example in this chapter demonstrates this technique. The second example in this chapter fits a Poisson regression model. The link function for this example is the log function. You can run a Generalized Linear Models analysis by selecting Analysis I Model Fitting I Generalized Linear Models from the main menu. The computation of the regression function and related statistics is implemented by calling the GENMOD procedure in SAS/STAT. See the documentation for the GENMOD procedure in the SAS/STAT User’s Guide for additional details. 318 Chapter 24. Model Fitting: Generalized Linear Models Example 1: Linear Regression with Classification Variables In this example you use the Generalized Linear Models analysis to fit a linear regression model with classification variables and an interaction term. In particular, you model how two variables affect the change in blood pressure in a designed experiment. The Drug data set contains results of an experiment carried out to evaluate the effect of four drugs with three experimentally induced diseases. Each drug-by-disease combination was applied to six randomly selected dogs. The response variable, chang– bp, is the increase in systolic blood pressure due to the treatment. The variables drug and disease are classification variables: their values identify distinct levels or groups. =⇒ Open the Drug data set. You need to specify that the drug and disease variables are nominal in order to model them as classification variables. “Context Menus” in Chapter 4, “The Data Table,” describes measure levels for variables. The following steps change the measure level of these variables from interval to nominal: =⇒ Select the drug and disease variables by holding down the CTRL key while you click on the column heading for each variable. =⇒ Right-click on the column heading for either variable and select Nominal from the pop-up menu, as shown in Figure 24.1. Figure 24.1. Changing the Measure Level for Variables =⇒ Clear the selected variables by clicking the blank cell in the upper-left corner of the data table. Chapter 24. Model Fitting: Generalized Linear Models Exploring the Data You can use box plots to explore how blood pressure changes according to the levels of drug and disease. The section “Box Plots” on page 63 describes how to create a box plot. =⇒ Select Graph I Box Plot from the main menu. Create a box plot of chang– bp versus drug. The following steps add an indicator of the mean and standard deviation of each group to the box plot. =⇒ Right-click near the center of the scatter plot, and select Plot Area Properties from the pop-up menu. A dialog box appears, as shown in Figure 24.2. You can use the Boxes tab to change attributes of the box plot. Figure 24.2. The Box Plot Dialog Box =⇒ Select Mean: with one standard deviation. =⇒ Click OK. Note: As a shortcut to the previous three steps, you can press the “m” key while the box plot window is active to toggle the display of means and standard deviations. The box plot is shown in Figure 24.3. The mean change in blood pressure for drug 1 and drug 2 is higher than the mean change for drug 3 and drug 4 (averaged over all three levels of disease). This difference might indicate that the main effect for drug should be included in a model for predicting chang– bp. 319 320 Chapter 24. Model Fitting: Generalized Linear Models Figure 24.3. Blood Pressure Grouped by Drug =⇒ Repeat the previous steps to create a box plot of chang– bp versus disease. Add means and standard deviations to the plot. A box plot that groups the response by disease is shown in Figure 24.4. The means for these groups vary according to the values of disease. The differences between the three disease levels are not as pronounced as those observed for drug. Still, the plot indicates that disease might be a factor in predicting chang– bp. Figure 24.4. Blood Pressure Grouped by Disease Chapter 24. Model Fitting: Generalized Linear Models Creating an Initial Model The two box plots indicate that both drug and disease affect the change in blood pressure in the experimental subjects. Kutner (1974) proposed a two-way analysis of variance model for these data. You can use the Generalized Linear Models analysis to determine which effects are significant and to estimate parameters in the model. However, note that the analysis does not create an ANOVA table, since the GENMOD procedure does not produce ANOVA tables. To begin the analysis, follow these steps: =⇒ Select Analysis I Model Fitting I Generalized Linear Models from the main menu, as shown in Figure 24.5. Figure 24.5. Selecting a Generalized Linear Models Analysis A dialog box appears as in Figure 24.6. =⇒ Select chang– bp, and click Add Y. =⇒ Select drug. While holding down the CTRL key, select disease. Click Add Class. Note: Alternatively, you can double-click on a variable to automatically add it as an explanatory variable. Nominal variables are automatically added as classification variables; interval variables are automatically added as quantitative variables. 321 322 Chapter 24. Model Fitting: Generalized Linear Models Figure 24.6. The Variables Tab When you add explanatory variables to the model by using the Variables tab, the main effects for those variables are automatically added to the Effects tab. It is not clear from the box plots whether drug and disease interact. By adding an interaction term, you can determine whether the level of drug affects the change in blood pressure differently for different levels of disease. The following steps add an interaction term to the model: =⇒ Click the Effects tab. =⇒ Select drug and disease from the Explanatory Variables list. =⇒ Select Cross from the Standard Effects list, if it is not already selected. =⇒ Click Add. The interaction term drug*disease is added to the Effects in Model list, as shown in Figure 24.7. Chapter 24. Model Fitting: Generalized Linear Models Figure 24.7. The Effects Tab =⇒ Click the Method tab. The Method tab (Figure 24.8) enables you to specify aspects of the generalized linear model such as the response distribution and the link function. The default distribution for the response is the normal distribution, and the default link function is the identity function. You do not need to modify this tab since these choices are appropriate for the current analysis. Figure 24.8. The Effects Tab 323 324 Chapter 24. Model Fitting: Generalized Linear Models =⇒ Click the Tables tab. The Tables tab becomes active, as shown in Figure 24.9. This tab controls which tables are produced by the analysis. By default, the analysis displays Type 3 Wald statistics for the significance of effects. The Wald statistics require less computational time than the Type 3 likelihood ratio statistics, but they can be less accurate. For this example, select the more accurate likelihood ratio statistics. =⇒ Clear Wald in the Type 3 Analysis of Contrasts group box. =⇒ Select Likelihood Ratio to request statistics for Type 3 contrasts. Figure 24.9. The Tables Tab =⇒ Click OK. The analysis creates plots, along with output from the GENMOD procedure. Move the plots so that they are arranged as in Figure 24.10. The tables created by the GENMOD procedure appear in the output window. The “LR Statistics For Type 3 Analysis” table indicates which effects in the model are significant. The Type 3 chi-square value for an effect tests the contribution due to that effect, after correcting for the other effects in the model. For example, the chi-square value for the interaction term drug*disease compares the log likelihood for the full model with the log likelihood for the model with only main effects. The value of the Type 3 likelihood ratio statistic for the interaction term is 11.55. The associated p-value indicates that this term is not significant in predicting the change in blood pressure at the 0.05 significance level. The main effects for drug and disease are significant. Since the interaction effect is not significant, the parameter estimates in the “Analysis Of Maximum Likelihood Parameter Estimates” table are not useful. You Chapter 24. Model Fitting: Generalized Linear Models should rerun the model without the interaction effect before examining the parameter estimates. The next section shows you how to delete the interaction effect and rerun the analysis. Figure 24.10. Preliminary Generalized Linear Models Analysis Revising the Model In this section you remove the interaction effect from the previous model and refit the data. =⇒ Select Analysis I Model Fitting I Generalized Linear Models to redisplay the dialog box for this analysis. Note: The items on the Analysis menu are not available if the output window is active. If the menu is not enabled, you should activate a graphical or tabular view of the data before clicking on the Analysis menu. =⇒ Click the Effects tab. =⇒ Select drug * disease from the Effects in Model list. =⇒ Click Remove. The interaction term is removed from the list of effects, as shown in Figure 24.11. 325 326 Chapter 24. Model Fitting: Generalized Linear Models Figure 24.11. Revising the Model =⇒ Click OK. Move the workspace windows so that they are arranged as in Figure 24.12. The “LR Statistics For Type 3 Analysis” table indicates that both main effects are significant. The “Analysis Of Maximum Likelihood Parameter Estimates” table displays parameter estimates for the model. You can use these values to determine the predicted mean response for each experimental group. The interpretation of the parameter estimates depends on the parameterization used to encode the classification variables in the model design matrix. This example used the GLM coding (see Figure 24.8). For this parameterization, the predicted response for a subject is obtained by adding the estimate for the intercept to the parameter estimates for the groups to which the subject belongs. For example, the predicted change in blood pressure in a subject with drug=1 and disease=2 is 8.9861 + 13.4444 + 5.2917 ≈ 27.7. For a given level, the parameter estimate represents the difference between that level and the last level. For example, the estimate of the difference between the parameters for drug 1 and drug 4 is 13.4444, and this estimate is significantly different from zero (as indicated by the p-value in the “Pr > ChiSq” column). In contrast, the difference in the coefficients between drug 3 and drug 4 is –4.1667, but this estimate is not significantly different from zero. Similarly, the estimate of the difference between disease 2 and disease 3 is (marginally) not significant. The parameter estimates table also estimates the scale parameter. For a normally distributed response, the scale parameter is the standard deviation of the response. See the documentation for the GENMOD procedure in the SAS/STAT User’s Guide for additional details. There are three plots in Figure 24.12. The plot of observed values versus predicted values (upper right in Figure 24.12) shows how well the model fits the data. Since Chapter 24. Model Fitting: Generalized Linear Models this model assumes a normally distributed response with an identity link, the plot of chi-square residuals versus predicted values (lower right in Figure 24.12) is just an ordinary residual plot (see the “Residuals” section of the documentation for the GENMOD procedure). The observations fall along vertical lines because all observations with the ith drug and the jth disease have the same predicted value. The scatter plot of Cook’s D (upper left in Figure 24.12) indicates which observations have a large influence on the parameter estimates. Influential observations (that is, those with relatively large values of Cook’s D) are selected in the figure. The selected observations are highlighted in the other plots. Each observation corresponds to a large negative residual, indicating that the observed change in blood pressure for these subjects was substantially less than the model predicts. Figure 24.12. A Revised Generalized Linear Models Analysis Example 2: Poisson Regression In this example, you examine another example of a generalized linear model: Poisson regression. A Poisson regression analysis might be appropriate when the response variable represents counts or rates. If your explanatory variables are all nominal (that is, you can write a contingency table containing the data), then the Poisson model is often called a log-linear model. Counts are always nonnegative, whereas a linear model can predict negative values for the response. Consequently, it is common to choose a logarithmic link function 327 328 Chapter 24. Model Fitting: Generalized Linear Models for the response. That is, if the response variable is Y and the expected value of Y is µ, a Poisson regression finds parameters that best fit the data to the model log(µ) = Xβ. Sometimes the counts represent the number of events that occurred during an observed time period. Some counts might correspond to longer time periods than others do. In this situation, you want to model the rate at which the events occur. When you model a rate, you are modeling the number of events, Y , per unit of time, T . The expected value of the rate is µ/T , where µ is the expected value of Y . In this case, the Poisson model is log(µ/T ) = Xβ. By using the fact that log(µ/T ) = log(µ) − log(T ), this equation can be rewritten as log(µ) = log(T ) + Xβ The term log(T ) is called an offset variable. The example in this section fits a Poisson model to data in the Ship data set. The data and analysis are from McCullagh and Nelder (1989). The response variable, Y, is the number of damage incidents that occurred during the number of months that ship was in service (contained in the months variable). As discussed in the previous paragraph, the quantity log(months) is an offset variable for this model. The three classification variables are as follows: • the ship type (type), which contains five levels, a–e • the year of construction (year), which contains four levels: 1960–64, 1965–69, 1970–74, and 1975–79 • the period of operation (period), which contains two levels: 1960–74 and 1975–79 Exploring the Data =⇒ Open the Ship data set. You can use box plots to explore how the ratio of Y to months varies according to the levels of the classification variables. The section “Box Plots” on page 63 describes how to create a box plot. Figure 24.13 shows plots that indicate how the number of damage incidents per month varies with the explanatory variables. The Variable Transformation Wizard (described in Chapter 32, “Variable Transformations”) was used to create a new variable, IncidentsPerMonth, as the ratio of Y and months. The new variable was created by using the Y / X transformation from the Two Variable family of transformations. The three box plots indicate that the mean of IncidentsPerMonth is as follows: • highest for ships of type e, and low for the other types • highest for ships constructed in the years 1970–74, and lowest for ships constructed in the years 1960–64 Chapter 24. Model Fitting: Generalized Linear Models • highest for ships that operated in the 1975–79 period, and lowest for ships that operated in the 1960–74 period This preliminary analysis indicates that the main effects of type, year, and period are important in predicting IncidentsPerMonth. The next section creates a generalized linear model with these effects. Figure 24.13. Incidents per Month, Grouped by Classification Variables Creating the Offset Variable As discussed earlier in this example, the quantity log(months) is an offset variable for this model. To create this variable, you can use the Variable Transformation Wizard, described in Chapter 32, “Variable Transformations.” =⇒ Select Analysis I Variable Transformation from the main menu. The Variable Transformation Wizard in Figure 24.14 appears. 329 330 Chapter 24. Model Fitting: Generalized Linear Models Figure 24.14. Selecting a Transformation The transformation log(Y+a) is highlighted by default. Since this is the desired transformation, you can proceed to the next page of the wizard. =⇒ Click Next. The wizard displays the page shown in Figure 24.15. Note that the transformation appears in the page’s title bar. Figure 24.15. Selecting a Variable and a Parameter =⇒ Select the months variable, and click Set Y. =⇒ Click Finish. Because there are six observations for which months=0, a warning message appears (Figure 24.16) informing you that the transformed values for these observations are set to missing values. Chapter 24. Model Fitting: Generalized Linear Models Figure 24.16. A Warning Message =⇒ Click OK to dismiss the warning message. The new variable is named Log– months. It contains six missing values. Observations with missing values for the explanatory variables (including the offset variable) or the response variable are not used in fitting the model. Modeling the Data The previous sections describe the Poisson model and create an offset variable for this model. In this section you specify the model. =⇒ Select Analysis I Model Fitting I Generalized Linear Models from the main menu. A dialog box appears as in Figure 24.17. =⇒ Select y, and click Add Y. =⇒ Select type. While holding down the CTRL key, select year, and period. Click Add Class. Figure 24.17. The Variables Tab 331 332 Chapter 24. Model Fitting: Generalized Linear Models Recall that when you add a variable on the Variables tab, the main effect for that variable is added to the Effects tab. This model includes only the main effects, so you do not need to click the Effects tab. There is one more variable to specify. The following steps specify Log– months as the offset variable: =⇒ Click the Roles tab. The Roles tab appears, as shown in Figure 24.18. =⇒ Select Log– months, and click Set Offset. Figure 24.18. The Roles Tab You have specified the variables in the model. The next steps specify the response distribution and the link function for a Poisson regression: =⇒ Click the Method tab. The Method tab appears as in Figure 24.19. =⇒ Select Poisson for Response Distribution. This specifies that the values of y have a probability distribution that is Poisson. (This also implies that the variance of y is proportional to the mean.) When a response distribution is Poisson, the default link function is the natural log. Consequently, you do not need to change the Link function value. Chapter 24. Model Fitting: Generalized Linear Models Figure 24.19. The Method Tab =⇒ Click the Tables tab. The Tables tab becomes active, as shown in Figure 24.9. This tab controls which tables are produced by the analysis. =⇒ Clear Wald in the Type 3 Analysis of Contrasts group box. =⇒ Select Likelihood Ratio to request statistics for Type 3 contrasts. =⇒ Click OK to run the analysis. The results of the analysis are shown in Figure 24.20. Move the workspace windows so that they are arranged as in the figure. The “LR Statistics For Type 3 Analysis” table indicates that all main effects are significant, although period is the weakest of the three. The “Analysis Of Maximum Likelihood Parameter Estimates” table displays parameter estimates for each level of the effects. The Parameter Estimates column indicates that ships of type b and type c have the lowest risk and ships of type e have the highest. The oldest ships (built from 1960 to 1964) have the lowest risk, and ships built from 1965 to 1974 have the highest risk. However, the estimates of the difference between the older ships and the newer ships are not significantly different from zero (as indicated by the Pr > ChiSq column). Ships operated from 1960 to 1974 have a lower risk than ships operated from 1975 to 1979. The GENMOD procedure displays a note indicating that the scale parameter is fixed—that is, not estimated by the iterative fitting process. There are three plots in Figure 24.20. The scatter plot of Cook’s D (upper left in Figure 24.20) indicates which observations have a large influence on the parameter estimates. Influential observations are highlighted in all plots. Note that the influential observations are not necessarily those with the largest residual values. 333 334 Chapter 24. Model Fitting: Generalized Linear Models Figure 24.20. A Poisson Regression Analysis Modeling Overdispersion Overdispersion is a phenomenon that occurs occasionally with binomial and Poisson data. For Poisson data, it occurs when the variance of the response Y exceeds the Poisson variance. (Recall that the Poisson variance equals the response mean: Var(y) = µ.) To account for the overdispersion that might occur in the Ship data, you can specify a method for estimating the overdispersion. =⇒ Select Analysis I Model Fitting I Generalized Linear Models from the main menu. Each tab of the dialog box initializes with the values from the previous analysis of these data. =⇒ Click the Method tab. =⇒ Select Pearson chi-square/DF for the field Estimate scale parameter as (shown in Figure 24.21). =⇒ Click OK. Chapter 24. Model Fitting: Generalized Linear Models Figure 24.21. Modeling Overdispersion Figure 24.22 shows the output for the analysis. A note states that “the scale parameter was estimated by the square root of Pearson’s Chi-Square/DOF.” The scale value reported in the “Analysis Of Maximum Likelihood Parameter Estimates” table is greater than 1, which suggests that overdispersion exists in the model. Note that the parameter estimates are unchanged by the dispersion estimate. However, the estimate does affect the covariance matrix, standard errors, and log likelihoods used in likelihood ratio tests. A comparison of Figure 24.20 with Figure 24.22 shows multiple differences in the output statistics. Although the estimate of the dispersion parameter is often used to indicate overdispersion or underdispersion, this estimate might also indicate other problems, such as an incorrectly specified model or outliers in the data. See the subsection “Generalized Linear Models Theory” in the “Details” section of the documentation for the GENMOD procedure for a discussion of the dispersion parameter and overdispersion. 335 336 Chapter 24. Model Fitting: Generalized Linear Models Figure 24.22. Estimating the Overdispersion Parameter Specifying the Generalized Linear Models Analysis This section describes the dialog box tabs associated with the Generalized Linear Models analysis. The Generalized Linear Models analysis calls the GENMOD procedure in SAS/STAT. See the documentation for the GENMOD procedure in the SAS/STAT User’s Guide for details. The Variables Tab You can use the Variables tab to specify the variables for the Generalized Linear Models analysis. The Variables tab is shown in Figure 24.6. For most response distributions, you only need to specify a single response variable in the Y Variables list. If you specify two numeric variables, the analysis assumes that the variables contain count data for a binomial experiment. The value of the first variable is the number of positive responses (or events). The value of the second variable is the number of trials. In this case, the response distribution is automatically set to binomial. The dialog box supports multiple explanatory variables. You can include nominal variables in the model by adding them to the Classification variables list. You can include interval variables in the model by adding them to the Quantitative variables list. When you add an explanatory variable, that main effect is added to the Effects tab. You can add interaction effects and nested effects by using the Effects tab. Chapter 24. Model Fitting: Generalized Linear Models The Effects Tab You can use the Effects tab to add several different types of effects to your model. All effects appear in the Effects in Model list. The section “The Effects Tab” on page 303 describes how to use the Effects tab to specify effects. The Method Tab You can use the Method tab (Figure 24.8) to specify aspects of the generalized linear model such as the response distribution and the link function. You can specify the following aspects of the model: Response distribution specifies the distribution of the response variable. This corresponds to the DIST= option in the MODEL statement. Predict probability of specifies whether to model the probability of the first or last level of the response variable. This item is available only when the response distribution is binomial or multinomial. This corresponds to the DESCENDING option in the PROC GENMOD statement. Link function specifies the link function. This corresponds to the LINK= option in the MODEL statement. The following table specifies the default link function for each response distribution. Table 24.1. Default Link Functions Distribution binomial gamma inverse gaussian multinomial negative binomial normal Poisson Default Link Function logit inverse (power(–1)) inverse squared (power(–2)) cumulative logit log identity log When the choice of response distribution is multinomial, the choice of link functions is limited to the cumulative logit, the cumulative probit, and the cumulative complementary log-log. Power specifies the number to use for a power link function. This item is available only when the link function is the power function. Order response by specifies how to order the response variable. This corresponds to the RORDER= option in the PROC GENMOD statement. 337 338 Chapter 24. Model Fitting: Generalized Linear Models Classification variables parameterization specifies the parameterization method for the classification variables. This corresponds to the PARAM= option in the CLASS statement. The dialog box supports the GLM, effect, and reference coding schemes. Estimate scale parameter as specifies the method for estimating the dispersion parameter. This corresponds to the SCALE= option in the MODEL statement. The Plots Tab You can use the Plots tab (Figure 24.23) to create plots that graphically display results of the analysis. There are plots that help you to visualize the fit, the residuals, and various influence diagnostics. Creating a plot often adds one or more variables to the data table. For a multinomial response, residuals and influence diagnostics are not available, so the only possible plot for multinomial data is the predicted response plot. The following plots are available: Observed vs. Predicted creates a scatter plot of the Y variables versus the predicted values, overlaid with the diagonal line that represents a perfect fit. Predicted response plot creates a line plot of the predicted probability versus the continuous explanatory variable. This plot is created only if the following conditions are satisfied: • There is exactly one continuous explanatory variable. • There are three or fewer classification variables. • There are 12 or fewer joint levels of the classification variables. If the response distribution is multinomial, there are k − 1 plots, where k is the number of response levels. Pearson chi-square residuals vs. Predicted creates a scatter plot of the residuals versus the predicted probabilities. Deviance residuals vs. Predicted creates a scatter plot of the deviance residuals versus the predicted probabilities. Likelihood residuals vs. Predicted creates a scatter plot of the likelihood residuals versus the predicted probabilities. Cook’s D vs. Observation number creates a scatter plot of Cook’s D statistic for each observation. Chapter 24. Model Fitting: Generalized Linear Models Leverage (H) vs. Observation number creates a scatter plot of the leverage statistic for each observation. Figure 24.23. The Plots Tab The Tables Tab The Tables tab is shown in Figure 24.9. You can use the Tables tab to display the following tables that summarize the results of the analysis: Model fit statistics displays a table of model fit statistics. Parameter estimates displays estimates for the model parameters and the scale parameter. Wald confidence intervals displays estimates of 95% Wald confidence intervals for the model, based on the asymptotic normality of the parameter estimators. This corresponds to the WALDCI option in the MODEL statement. Note: The GENMOD procedure displays the Wald confidence limits by default. Consequently, Wald confidence limits appear in the parameter estimates table even if you clear both of the check boxes for confidence limits in the dialog box. Likelihood ratio confidence intervals displays estimates of 95% confidence intervals for the model parameters, based on the profile likelihood function. This corresponds to the LRCI option in the MODEL statement. Type 1 sequential analysis specifies that a type 1 sequential analysis be displayed. This corresponds to the TYPE1 option in the MODEL statement. 339 340 Chapter 24. Model Fitting: Generalized Linear Models Likelihood ratio specifies that type 3 likelihood statistics be displayed. This corresponds to the TYPE3 option in the MODEL statement. Wald specifies that a type 3 Wald statistics be displayed. This corresponds to the TYPE3WALD option in the MODEL statement. The Output Variables Tab You can use the Output Variables tab (Figure 24.24) to add analysis variables to the data table. If you request a plot that uses one of the output variables, then that variable is automatically created even if you did not explicitly select the variable on the Output Variables tab. For a multinomial response, residuals and influence diagnostics are not available. The following list describes each output variable and indicates how the output variable is named. Y represents the name of the response variable. If you use events/trials syntax, then Y represents the name of the events variable. Proportions for events/trials adds a variable named Proportion– ET , where E is the name of the events variable and T is the name of the trials variable. The value of the variable is the ratio E/T . This variable is added only when you use events/trials syntax. Predicted values adds predicted values. The variable is named GenP– Y . Confidence limits for predicted values adds 95% confidence limits for the predicted values. The variables are named GenLclm– Y and GenUclm– Y . Linear predictor adds the linear predictor values. The variable is named GenXBeta– Y . Raw residuals adds residuals, calculated as observed minus predicted values. The variable is named GenR– Y . Pearson chi-square residuals adds the Pearson chi-square residuals. The variable is named GenChiSqR– Y . Deviance residuals adds the deviance residuals. The variable is named GenDevR– Y . Likelihood residuals adds the likelihood residuals. The variable is named GenLikR– Y . Cook’s D adds Cook’s D influence statistic. The variable is named GenCooksD– Y . Chapter 24. Model Fitting: Generalized Linear Models Leverage (H) adds the leverage statistic. The variable is named GenH– Y . DFBETAS (influence on coefficients) adds p variables, where p is the number of parameters in the model. A classification variable with k levels counts as k parameters. The variables are scaled measures of the change in each parameter estimate and are calculated by deleting the ith observation. Large values of DFBETAS indicate observations that are influential in estimating a given parameter. Belsley, Kuh, √ and Welsch (1980) recommend 2/ n as a size-adjusted cutoff. The variables are named DFBetaj , for j = 1 . . . p. Figure 24.24. The Output Variables Tab The Roles Tab You can use the Roles tab (Figure 24.18) to specify a frequency variable or weight variable for the analysis. You can also specify an offset variable. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. A weight variable is a numeric variable with values that are relative weights for a weighted regression. An offset variable is a variable used as a vector of constants in the regression. Its regression coefficient is set to 1. This corresponds to the OFFSET= option in the MODEL statement. 341 342 Chapter 24. Model Fitting: Generalized Linear Models Analysis of Selected Variables If one or more interval variables are selected in a data table when you run the analysis, then the following occurs: • The first selected nominal variable is automatically entered in the Y Variables field of the Variables tab. • Subsequent selected nominal variables are automatically entered in the Classification Variables field. • Selected interval variables are automatically entered in the Quantitative Variables field. Any variable in the data table with a Frequency or Weight role is automatically entered in the appropriate field of the Roles tab. References Belsley, D. A., Kuh, E., and Welsch, R. E. (1980), Regression Diagnostics, New York: John Wiley & Sons. Kutner, M. H. (1974), “Hypothesis Testing in Linear Models (Eisenhart Model),” American Statistician, 28, 98–100. McCullagh, P. and Nelder, J. A. (1989), Generalized Linear Models, Second Edition, London: Chapman & Hall. Chapter 25 Multivariate Analysis: Correlation Analysis The Correlation analysis can help you to understand and visualize relationships between pairs of variables. You can use correlation coefficients to measure the strength of the linear association between two numerical variables. You can also use prediction ellipses in scatter plots as a visual test for bivariate normality and an indication of the strength of the correlation. You can run the Correlation analysis by selecting Analysis I Multivariate Analysis I Correlation Analysis from the main menu. The analysis is implemented by calling the CORR procedure in Base SAS. See the CORR procedure documentation in the Base SAS Procedures Guide for additional details. Example In this example, you explore correlations and bivariate relationships between variables in the Hurricanes data set. The data are for North Atlantic tropical cyclones from 1988 to 2003. The data set includes information about each storm’s latitude (in the latitude variable), its sustained low-level winds (wind– kts), its central atmospheric pressure (min– pressure), and the size of its eye (radius– eye). A full description of the Hurricanes data set is included in Appendix A, “Sample Data Sets.” =⇒ Open the Hurricanes data set. =⇒ Select Analysis I Multivariate Analysis I Correlation Analysis from the main menu, as shown in Figure 25.1. Figure 25.1. Selecting the Correlation Analysis 344 Chapter 25. Multivariate Analysis: Correlation Analysis A dialog box appears as in Figure 25.2. You can select variables for the analysis by using the Variables tab. =⇒ Select latitude. While holding down the CTRL key, select wind– kts, min– pressure, and radius– eye, and click Add Y. Figure 25.2. The Variables Tab =⇒ Click the Plots tab. The Plots tab (Figure 25.3) becomes active. =⇒ Select Matrix of pairwise scatter plots. =⇒ Click OK. Chapter 25. Multivariate Analysis: Correlation Analysis Figure 25.3. The Plots Tab The analysis calls the CORR procedure, which uses the options specified in the dialog box. The procedure displays tables in the output document, as shown in Figure 25.4. The “Simple Statistics” table (not shown in the figure) displays basic statistics such as the mean, standard deviation, and range of each variable. The “Pearson Correlation Coefficients” table displays the correlation coefficients between pairs of variables. In addition, the table gives the number of nonmissing observations for each pair of variables, and tests the hypothesis that the coefficient is zero. Note that the number of observations used to compute the correlation coefficients can vary. For example, there are no missing values in the latitude of wind– kts variables, so the correlation coefficient for this pair is computed using all 6188 observations in the data set. In contrast, only 745 values for radius– eye are nonmissing, reflecting the fact that not all cyclones have well-defined eyes. For these data, the correlation between min– pressure and wind– kts is strong and negative, with a value near −0.93. This is not surprising, since winds are determined by a pressure gradient. Although not as strong, there is also negative correlation between latitude and min– pressure. In contrast, the correlation between latitude and radius– eye is positive. The correlation between the following pairs of variables is not significantly different from zero: latitude and wind– kts, radius– eye and wind– kts, and radius– eye and min– pressure. These results are graphically summarized in the pairwise correlations plot, shown in the upper-right corner of Figure 25.4. This plot is not linked to the original data set because it has a different number of observations. However, you can view the data table underlying this plot by pressing the F9 key when the plot is active. 345 346 Chapter 25. Multivariate Analysis: Correlation Analysis Figure 25.4. Output from a Correlation Analysis Partly visible in Figure 25.4 is the matrix of pairwise scatter plots between the variables. Some of these plots are hidden by the output window and the pairwise correlation plot. You can use the Workspace Explorer to view all the scatter plots. =⇒ Close the pairwise correlation plot. =⇒ Press ALT+X to open the Workspace Explorer. You can use the Workspace Explorer to manage the display of plots. The Workspace Explorer is described in the section “Workspace Explorer” on page 165 of Chapter 11 . =⇒ Select the entry in the Workspace Explorer labeled Multivariate Correlation Analysis, as shown in Figure 25.5. =⇒ Click View. The scatter plots associated with the analysis appear in front of other windows. =⇒ Click Close to close the Workspace Explorer. Chapter 25. Multivariate Analysis: Correlation Analysis Figure 25.5. Selecting a Group of Plots The workspace is now arranged as shown in Figure 25.6. The ellipses show where the specified percentage of the data should lie, assuming a bivariate normal distribution. Under bivariate normality, the percentage of observations falling inside the ellipse should closely agree with the specified level. The plots also contain a gradient shading that indicates a nested sequence of ellipses. The darkest shading occurs at the bivariate means for each pair of variables. The lightest shading corresponds to 0.9999 probability. Variables that are bivariate normal have most of their observations close to the bivariate mean and have a bivariate density that is proportional to the gradient shading. The plot of wind– kts versus latitude shows that these two variables are not bivariate normal. Similarly, min– pressure and latitude are not bivariate normal. 347 348 Chapter 25. Multivariate Analysis: Correlation Analysis Figure 25.6. A Matrix of Scatter Plots The variables wind– kts and min– pressure are highly correlated and linearly related. In contrast, wind– kts is not correlated with latitude or radius– eye, although you can still notice certain relationships: • Cyclones with high wind speeds occur only at lower latitudes. • Cyclones north of 43 degrees of latitude tend to have wind speeds less than 75 knots. • The size of a cyclone’s eye seems to be unrelated to the speed of its winds. You can observe similar relationships between min– pressure and the latitude and radius– eye variables. The matrix of scatter plots also reveals an aspect of the data that might not be apparent from univariate plots. The plots involving wind– kts and radius– eye show a granular appearance that indicates the data are rounded. Most of the wind speed measurements are rounded to the nearest five knots, whereas the values for the eye radius are rounded to the nearest 2.5 nautical miles. (You can also find observations for these variables that are not rounded.) Figure 25.7 shows another use of the scatter plot matrix. Some observations with extreme values of min– pressure and wind– kts are selected. The marker shape and color for these observations were changed to make them more noticeable. You can use this technique to investigate whether outliers for one pair of variables are, in Chapter 25. Multivariate Analysis: Correlation Analysis fact, multivariate outliers with respect to multivariate normality. Most of the selected data in Figure 25.7 are inside the 80% ellipse for the radius– eye versus latitude scatter plot. This indicates that these data are not far from the mean in those variables. However, a few observations (corresponding to Hurricane Hugo when it was category 5) do appear to be multivariate outliers in these variables. Figure 25.7. Selecting Bivariate Outliers Specifying the Correlation Analysis This section describes the dialog box tabs associated with the Correlation analysis. The Correlation analysis calls the CORR procedure in Base SAS. See the CORR procedure documentation in the Base SAS Procedures Guide for additional details. The Variables Tab You can use the Variables tab to specify the numerical variables for the analysis. The Variables tab is shown in Figure 25.2. The variables in the Y Variables list correspond to variables in the VAR statement of the CORR procedure. The variables in the X Variables (With) list correspond to variables in the WITH statement of the CORR procedure. The simplest way to analyze correlations is to add the variables of interest to the Y Variables list, as in the example earlier in this chapter. 349 350 Chapter 25. Multivariate Analysis: Correlation Analysis If the X Variables (With) list is empty, the correlation matrix is symmetric. If you request a matrix of pairwise scatter plots (on the Plots tab), you will get plots for pairs of variables in the lower triangular portion of the matrix. If the X Variables (With) list is not empty, the correlation matrix is not symmetric. If you specify C1 , . . . Cm as the Y variables and R1 , . . . Rn as the WITH variables, then the ijth cell of the correlation matrix will be the correlation of Ri with Cj . If you request a matrix of pairwise scatter plots, you will get nm plots, arranged in n rows and m columns. The Partial list is rarely used. The variables in this list correspond to variables in the PARTIAL statement of the CORR procedure. A partial correlation measures the strength of a relationship between two variables, while controlling the effect of other variables. The Pearson partial correlation between two variables, after controlling for variables in the PARTIAL statement, is equivalent to the Pearson correlation between the residuals of the two variables after regression on the controlling variables. If there are variables in the Partial list, then the following conditions hold: • You cannot request Hoeffding’s D correlation statistic. • Observations with missing values are excluded from the analysis. The Plots Tab You can use the Plots tab (Figure 25.3) to create plots that graphically display results of the analysis. These plots do not add any variables to the data table. The following plots are available: Pairwise correlation plot creates a bar chart showing the Pearson correlation between pairs of variables. Matrix of pairwise scatter plots creates a matrix of scatter plots showing bivariate data for pairs of variables. If you do not specify any X variables in the X Variables (With) list on the Variables tab, then you will get a lower triangular array of plots. If you do specify X variables, then you will get a rectangular array of plots. The inset added to each plot contains the following: • the Pearson correlation coefficient • the number of nonmissing observations for each pair of variables • the p-value under the null hypothesis of zero correlation Add prediction ellipse adds a prediction ellipse to the scatter plot. The ellipse is calculated under the assumption that the data are bivariate normal. A prediction ellipse is a region for predicting a new observation in the population. It also approximates a region containing a specified percentage of the population. Chapter 25. Multivariate Analysis: Correlation Analysis Confidence level specifies the confidence level for the prediction ellipse. Shade plot background by confidence level specifies that the background of each scatter plot be shaded according to a nested family of prediction ellipses. The Tables Tab The Tables tab is shown in Figure 25.8. You can use the Tables tab to display the following tables that summarize the results of the analysis: Pearson’s product-moment displays a table of Pearson correlation coefficients. Selecting this field corresponds to the PEARSON option in the PROC CORR statement. Clearing this field corresponds to the NOCORR option in the PROC CORR statement. Hoeffding’s D displays a table of Hoeffding’s D statistic. This statistic is not available if you specify variables in the Partial list on the Variables tab. This corresponds to the HOEFFDING option in the PROC CORR statement. Kendall’s tau-b displays a table of Kendall’s tau-b statistic. This corresponds to the KENDALL option in the PROC CORR statement. Spearman’s rho displays a table of Spearman’s rank-order correlation. This corresponds to the SPEARMAN option in the PROC CORR statement. Show significance probabilities for H0: correlation=0 displays p-values under the null hypothesis of zero correlation. Clearing this field corresponds to the NOPROB option in the PROC CORR statement. Simple descriptive statistics displays descriptive statistics for the variables in the analysis. Clearing this field corresponds to the NOSIMPLE option in the PROC CORR statement. Covariances displays the covariance matrix for the variables in the analysis. This corresponds to the COV option in the PROC CORR statement. Cronbach’s coefficient alpha for estimating reliability displays Cronbach’s coefficient alpha for the variables in the analysis. This corresponds to the ALPHA option in the PROC CORR statement. This statistic is not available if you specify variables in the X Variables (With) list on the Variables tab. This statistic is not available unless you select Listwise for Exclude missing values. 351 352 Chapter 25. Multivariate Analysis: Correlation Analysis Exclude missing values specifies how to treat missing values in the analysis. If you select Listwise, then observations with missing values are excluded from the analysis. This corresponds to the NOMISS option in the PROC CORR statement. Otherwise, statistics are computed using all of the nonmissing pairs of variables. Figure 25.8. The Tables Tab The Roles Tab You can use the Roles tab to specify a frequency variable or weight variable for the analysis. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. A weight variable is a numeric variable with values that are relative weights for the analysis. Analysis of Selected Variables If any numeric variables are selected in a data table when you run the analysis, these variables are automatically entered in the Y Variables field of the Variables tab. Any variable in the data table with a Frequency or Weight role is automatically entered in the appropriate field of the Roles tab. Chapter 26 Multivariate Analysis: Principal Component Analysis Principal component analysis is a technique for reducing the complexity of high-dimensional data. You can use principal component analysis to approximate high-dimensional data with fewer dimensions. Each dimension is called a principal component and represents a linear combination of the original variables. The first principal component accounts for as much variation in the data as possible. Each subsequent principal component accounts for as much of the remaining variation as possible and is orthogonal to all of the previous principal components. You can examine principal components to understand the sources of variation in your data. You can also use them in forming predictive models. If most of the variation in your data exists in a low-dimensional subset, you might be able to model your response variable in terms of the principal components. You can use principal components to reduce the number of variables in regression, clustering, and other statistical techniques. You can run the Principal Component analysis by selecting Analysis I Multivariate Analysis I Principal Component Analysis from the main menu. The analysis is implemented by calling the PRINCOMP procedure in SAS/STAT. See the PRINCOMP procedure documentation in the SAS/STAT User’s Guide for additional details. Example In this example, you compute principal components of several variables in the Baseball data set. The Baseball data set contains performance measures for major league baseball players in 1986. A full description of the Baseball data is included in Appendix A, “Sample Data Sets.” Suppose you are interested in exploring the sources of variation in players’ performances during the 1986 season. There are six measures of players’ batting performance: no– atbat, no– hits, no– home, no– runs, no– rbi, and no– bb. There are three measures of players’ fielding performance: no– outs, no– assts, and no– error. These data form a nine-dimensional space. The goal of this example is to use principal component analysis to capture most of the variance of these data in a low-dimensional subspace—preferably in two or three dimensions. The subspace will be formed by the span of the first few principal components. (Recall that the span of a set of vectors is the vector space consisting of all linear combinations of the vectors.) =⇒ Open the Baseball data set. 354 Chapter 26. Multivariate Analysis: Principal Component Analysis =⇒ Select Analysis I Multivariate Analysis I Principal Component Analysis from the main menu, as shown in Figure 26.1. Figure 26.1. Selecting the Principal Component Analysis A dialog box appears as in Figure 26.2. You can select variables for the analysis by using the Variables tab. =⇒ Select no– atbat. While holding down the CTRL key, select no– hits, no– home, no– runs, no– rbi, and no– bb. Click Add Y. Note: Alternately, you can select the variables by using contiguous selection: click on the first item, hold down the SHIFT key, and click on the last item. All items between the first and last item are selected and can be added by clicking Add Y. The three measures of fielding performance are located near the end of the list of variables. =⇒ Scroll to the end of the variable list. Select no– outs. While holding down the CTRL key, select no– assts and no– error. Click Add Y. Figure 26.2. The Variables Tab Chapter 26. Multivariate Analysis: Principal Component Analysis =⇒ Click the Method tab. The Method tab (Figure 26.4) becomes active. You can use the Method tab to set options in the analysis. By default, the analysis is carried out on the correlation matrix. The alternative is to use the covariance matrix. The covariance matrix is recommended only when all the variables are measured in comparable units. For this example, the correlation matrix is appropriate. By default, the analysis computes all p principal components for the p variables selected in the Variables tab. It is often sufficient to compute a smaller number of principal components. =⇒ Set Number of principal components to 4. Figure 26.3. The Method Tab =⇒ Click the Plots tab. The Plots tab (Figure 26.4) becomes active. =⇒ Clear Proportion plot of eigenvalues (scree plot). =⇒ Select Matrix of component score plots. =⇒ Click OK. 355 356 Chapter 26. Multivariate Analysis: Principal Component Analysis Figure 26.4. The Plots Tab The analysis calls the PRINCOMP procedure, which uses the options specified in the dialog box. The procedure displays tables in the output document, as shown in Figure 26.5. The “Simple Statistics” table displays the mean and standard deviation for each variable. (The “Simple Statistics” table is not visible in Figure 26.5. You can scroll through the output window to view it.) The “Correlation Matrix” table (also not shown) displays the correlation between each pair of variables. The “Eigenvalues of the Correlation Matrix” table contains all the eigenvalues of the correlation matrix, differences between successive eigenvalues, the proportion of variance explained by each eigenvalue, and the cumulative proportion of the variance explained. The eigenvalues correspond to the principal components and represent a partitioning of the total variation in the sample. Because correlations are used, the sum of all the eigenvalues is equal to the number of variables. The first row of the table corresponds to the first principal component, the second row to the second principal component, and so on. In this example, the first three principal components account for over 83% of the variation; the first four account for 90%. Chapter 26. Multivariate Analysis: Principal Component Analysis Figure 26.5. Output from a Principal Component Analysis The “Eigenvectors” table contains the first four eigenvectors of the correlation matrix. The eigenvectors are principal component vectors. The first column of the table corresponds to the first principal component, the second column to the second principal component, and so on. Each principal component is a linear combination of the Y variables. For example, the first principal component corresponds to the linear combination PC1 = 0.42798 no– atbat + 0.42066 no– hits + . . . + 0.15736 no– error The first principal component (PC1) appears to be a weighted measure of the players’ overall performance, as seen by the relative magnitudes of the coefficients. More weight is given to batting performance (the batting coefficients are in the range 0.33–0.43) than to fielding performance (the fielding coefficients are in the range 0.14–0.19). The second principal component (PC2) is primarily related to the no– assts and no– error variables. Players with large values of PC2 have many assists, but also relatively many errors. The third component (PC3) is primarily related to the no– outs variable. The fourth component is a contrast between no– home and no– bb (that is, between home runs and walks). This component separates players with many home runs and few walks from the players who often walk and rarely hit a home run. You can use the correlation pattern plot (Figure 26.6) to examine correlations between the principal components and the original variables. For example, the first 357 358 Chapter 26. Multivariate Analysis: Principal Component Analysis principal component (PC1) is positively correlated with all of the original variables. It is correlated more with batting performance than with the fielding variables. Figure 26.6. Correlation Pattern Plot The relationship between the original variables and observations is shown in the biplot, at the lower right of Figure 26.7. The line segments represent the projection of a vector in the direction of each original variable onto a two-dimensional subspace. The points in the biplot are the projection of the observations onto the same two-dimensional subspace. The section “Biplots” on page 362 discusses biplots in further detail. Chapter 26. Multivariate Analysis: Principal Component Analysis Figure 26.7. Graphs from a Principal Component Analysis The plots tiled across the top of Figure 26.7 are called score plots. These are plots of the observations in the coordinate system defined by the principal components. For these data, each observation represents a player. The following steps set the value of the name variable to be the label you see when you click on an observation. =⇒ Click on the score plot of PC2 versus PC1 to activate it. =⇒ Press the F9 key to display the data table associated with this plot. =⇒ Right-click on the variable heading for name to display the Variables menu. Select Label. =⇒ Click in the upper-left cell of the data table to deselect the variable. =⇒ Close the data table. =⇒ Click on some observations in the score plot of PC2 versus PC1, as shown in Figure 26.8. The first principal component measures a player’s hitting performance during the 1986 season. Consequently, players to the right (such as Jesse Barfield) had strong hitting statistics, whereas players to the left (such as Darrell Porter) had weaker statistics. The second principal component primarily measures the number of assists (and errors) for each player. Consequently, players near the top of the plot (such as 359 360 Chapter 26. Multivariate Analysis: Principal Component Analysis Shawon Dunston) have many assists, whereas players near the bottom (such as Jesse Barfield) have few. Figure 26.8. Score Plot of First Two Principal Components The score plot of the second and third principal components is interesting because it compares two different measures of fielding performance. Also, there are few players in the first quadrant of the plot. Recall that the third principal component primarily measures the no– outs variable. This variable records putouts. Common situations leading to a putout include tagging or forcing out a base runner, catching a fly ball, or (for catchers) catching a third strike. The opportunities for a player to get a putout or an assist are highly dependent on the player’s position. Figure 26.9 shows the score plot for the positions of second base, third base, and shortstop. Note that these observations primarily lie in the fourth quadrant. These players have many assists because they often field ground balls and throw to first base, but they have relatively few opportunities to put out runners themselves. In contrast, Figure 26.10 shows the score plot for outfielders and designated hitters. These observations lie in the third quadrant. These players have few assists and relatively few putouts. (The outfielders are credited with a putout when they catch a fly ball, but there are many fewer fly balls than ground balls in a typical game.) Catchers and first basemen (not shown) have scores primarily in the second quadrant of the plot, corresponding to many putouts but few assists. Chapter 26. Multivariate Analysis: Principal Component Analysis Figure 26.9. Fielding Scores for Some Infielders Figure 26.10. Fielding Scores for Outfielders and Designated Hitters In summary, the analysis shows that most of the variation in these data occurs in the first principal component: an overall measure of batting performance. The next two principal components incorporate variation due to fielding performance. Figure 26.9 and Figure 26.10 show that the source of this fielding variation is differences in player positions. Together, these three components account for 83% of the variation in the nine-dimensional space of the original variables. Principal components can also be used as explanatory variables in regression. For example, you could examine how well overall batting performance in 1986 predicts a player’s salary by using PC1 as an explanatory variable in a regression model. 361 362 Chapter 26. Multivariate Analysis: Principal Component Analysis Biplots A biplot is a display that attempts to represent both the observations and variables of multivariate data in the same plot. Stat Studio provides biplots as part of the Principal Component analysis. The computation of biplots in Stat Studio follows the presentation given in Friendly (1991) and Jackson (1991). Detailed discussions of how to compute and interpret biplots are available in Gabriel (1971) and Gower and Hand (1996). The computation of a biplot begins with the data matrix. If you choose to compute principal components from the covariance matrix (on the Method tab; see Figure 26.3), then the data matrix is centered by subtracting the mean of each column. Otherwise, it is standardized so that each variable has zero mean and unit standard deviation. In either case, let X denote the resulting N × p matrix. The singular value decomposition (SVD) of X is the factorization X = U LV 0 = (U Lα ) L1−α V 0 = GH 0 where L is the diagonal matrix of singular values. If you replace G and H with their first two columns, then an approximate relationship exists: X ≈ GH 0 . This is a rank-two approximation of X. In fact, it is the closest rank-two approximation to X in a least squares sense (Golub and Van Loan 1989). In a biplot, the rows of the N × 2 matrix G are plotted as points, which correspond to observations. The rows of the p × 2 matrix H are plotted as vectors, which correspond to variables. The choice of α determines the scaling of the observations and vectors in the biplot. In general, it is impossible to accurately represent the variables and observations in only two dimensions, but you can choose values of α that preserve certain properties of the high-dimensional data. Common choices are α = 0, 1/2, and 1. Stat Studio implements four different versions of the biplot: GH0 This factorization uses α = 0. This biplot attempts to preserve relationships between variables. This biplot has two useful properties: • The length of a vector (a row of H) is proportional to the variance of the corresponding variable. • The Euclidean distance between the ith and jth rows of G is proportional to the Mahalanobis distance between the ith and jth observations in the data set. JK0 This factorization uses α = 1. This biplot attempts to preserve the distance between observations. This biplot has two useful properties: Chapter 26. Multivariate Analysis: Principal Component Analysis • The positions of the points in the biplot are identical to the score plot of first two principal components. • The Euclidean distance between the ith and jth rows of G is equal to the Euclidean distance between the ith and jth observations in the data set. SYM This factorization uses α = 1/2. This biplot treats observations and variables symmetrically. This biplot attempts to preserve the values of observations. √ COV This factorization uses α = 0, but also multiplies G by N − 1 and divides H by the same quantity. This biplot has two useful properties: • The length of a vector (a row of H) is equal to the variance of the corresponding variable. • The Euclidean distance between the ith and jth rows of G is equal to the Mahalanobis distance between the ith and jth observations in the data set. The axes at the bottom and left of the biplot are the coordinate axes for the observations. The axes at the top and right of the biplot are the coordinate axes for the vectors. If the data matrix X is not well approximated by a rank-two matrix, then the visual information in the biplot is not a good approximation to the data. In this case, you should not try to interpret the biplot. However, if X is close to a rank-two matrix, then you can interpret a biplot in the following ways: • The cosine of the angle between a vector and an axis indicates the importance of the contribution of the corresponding variable to the axis dimension. • The cosine of the angle between vectors indicates correlation between variables. Highly correlated variables point in the same direction; uncorrelated variables are at right angles to each other. • Points that are close to each other in the biplot represent observations with similar values. • You can approximate the coordinates of an observation by projecting the point onto the variable vectors within the biplot. For example, in Figure 26.11 the two principal components account for approximately 74% of the variance in the data. This means that the biplot is a fair (but not good) approximation to the data. The footnote in the plot indicates that the biplot is based on the COV factorization and that the data matrix was standardized (STD). The variables are grouped: the hitting variables point primarily in the direction of the horizontal axis; no– assts and no– error point primarily in the direction of the vertical axis. The no– outs vector is much shorter than the other vectors, which often indicates that the vector does not lie near the span of the two biplot dimensions. 363 364 Chapter 26. Multivariate Analysis: Principal Component Analysis The hitting variables are strongly correlated with each other. The variables no– assts and no– error are correlated with each other, but they are not correlated with the hitting variables or with no– outs. Because the biplot is only a moderately good approximation to the data, the following statements are approximately true: • The first and fourth quadrants contain players who tend to be strong hitters. The other quadrants contain weak hitters. • The first and second quadrants contain players who tend to have many assists and errors. The other quadrants contain players with few assists and errors. Figure 26.11. Biplot for Baseball Data Specifying the Principal Component Analysis This section describes the dialog box tabs associated with the Principal Component analysis. The Principal Component analysis calls the PRINCOMP procedure in SAS/STAT. See the PRINCOMP procedure documentation in the SAS/STAT User’s Guide for additional details. The Variables Tab You can use the Variables tab to specify the numerical variables for the analysis. The Variables tab is shown in Figure 26.2. The variables in the Y Variables list correspond to variables in the VAR statement of the PRINCOMP procedure. The Partial list is rarely used. The variables in this list correspond to variables in the PARTIAL statement of the PRINCOMP procedure. The PRINCOMP procedure Chapter 26. Multivariate Analysis: Principal Component Analysis computes the principal components of the residuals from the prediction of the VAR variables by the PARTIAL variables. The Method Tab You can use the Method tab (Figure 26.3) to set options in the analysis. Each of the following options corresponds to an option in the PRINCOMP procedure: Compute principal components from specifies whether the principal components are computed for the correlation matrix or the covariance matrix. This corresponds to the COV option in the PROC PRINCOMP statement. Number of principal components specifies how many principal components to compute. This corresponds to the N= option in the PROC PRINCOMP statement. Note that you can type in this field. If you want five principal components, you can type 5 even though this is not an option in the list. Standardize principal component scores specifies whether to standardize the principal component score. This corresponds to the STANDARD option in the PROC PRINCOMP statement. If you clear this option, the scores have variance equal to the corresponding eigenvalue. The Plots Tab You can use the Plots tab (Figure 26.4) to create plots that graphically display results of the analysis. Creating a plot often adds one or more variables to the data table. The following plots are available: Proportion plot of eigenvalues (scree plot) creates a plot that summarizes the eigenvalues of the correlation or covariance matrix. Show cumulative proportions adds cumulative proportions of eigenvalues to the proportion plot. Matrix of component score plots creates a matrix of scatter plots showing scores for consecutive pairs of principal components. Correlation pattern plot creates a line plot that shows the correlations between principal components and the original variables. 365 366 Chapter 26. Multivariate Analysis: Principal Component Analysis Biplot creates a biplot. A biplot shows relationships between observations and variables in a single plot. Scale factors specifies how to scale and factor the SVD of the data matrix. The scaling determines the values for the biplot. The methods are described in the section “Biplots” on page 362. Extend variable vectors specifies whether to extend the vectors to the edge of the biplot. This is useful for visualizing the direction of short vectors. The Tables Tab The Tables tab is shown in Figure 26.12. You can use the Tables tab to display the following tables that summarize the results of the analysis: Simple descriptive statistics specifies whether to display the mean and standard deviation for each variable. Correlation or covariance matrix specifies whether to display the correlation or covariance matrix, as selected on the Method tab. Eigenvalues specifies whether to display the eigenvalues of the correlation or covariance matrix, as well as the difference between successive eigenvalues, the proportion of variance explained by each eigenvalue, and the cumulative proportion of variance explained. Eigenvectors specifies whether to display the eigenvectors of the correlation or covariance matrix. The eigenvectors are used to form the principal components. Statistics for automatic selection of principal components specifies whether to display statistics that indicate how many principal components are needed to represent the p-dimensional data. This table is displayed only if you request at least as many principal components as there are variables. Chapter 26. Multivariate Analysis: Principal Component Analysis Figure 26.12. The Tables Tab A primary use of principal component analysis is to represent p-dimensional data in k < p dimensions. In practice, it is often difficult to determine the best choice for k. The “Automatic Selection of Principal Components” table, shown in Figure 26.13, is provided to help you choose k. Numerous papers have been written comparing various methods for choosing k, but no method has shown itself to be superior. The following list briefly describes each method reported in the table. Jackson (1991, p. 41–51) gives further details. Parallel Analysis generates random data sets with N observations and p variables. The variables are normally distributed and uncorrelated. The method chooses k to be the largest integer for which the scree plot of the original data lies above the graph of the upper 95 percentiles of the eigenvalues of the random data. Broken Stick retains components that explain more variance than would be expected by randomly dividing the variance into p parts. Average Root keeps components that explain more variance than the mean of the eigenvalues. 0.7 * Average Root keeps components that explain more variance than 0.7 times the mean of the eigenvalues. Imbedded Error chooses k to be the value that minimizes a certain function of the eigenvalues. 367 368 Chapter 26. Multivariate Analysis: Principal Component Analysis Velicer’s MAP chooses k to minimizes a certain function that involves partial correlations. This method is called Velicer’s minimum average partial (MAP) test or Velicer’s partial correlation procedure. Figure 26.13. How Many Principal Components Are Needed? The Output Variables Tab You can use the Output Variables tab (Figure 26.14) to add principal component scores to the data table. The options on the Method tab determine the number of scores and whether the scores are standardized. Figure 26.14. The Output Tab The Roles Tab You can use the Roles tab to specify a frequency variable or weight variable for the analysis. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. A weight variable is a numeric variable with values that are relative weights for the analysis. Chapter 26. Multivariate Analysis: Principal Component Analysis Analysis of Selected Variables If any numeric variables are selected in a data table when you run the analysis, these variables are automatically entered in the Y Variables field of the Variables tab. Any variable in the data table with a Frequency or Weight role is automatically entered in the appropriate field of the Roles tab. References Friendly, M. (1991), SAS System for Statistical Graphics, SAS Series in Statistical Applications, Cary, NC: SAS Institute. Gabriel, K. R. (1971), “The Biplot Graphical Display of Matrices with Applications to Principal Component Analysis,” Biometrika, 58(3), 453–467. Golub, G. H. and Van Loan, C. F. (1989), Matrix Computations, Second Edition, Baltimore: Johns Hopkins University Press. Gower, J. C. and Hand, D. J. (1996), Biplots, London: Chapman & Hall. Jackson, J. E. (1991), A User’s Guide to Principal Components, New York: John Wiley & Sons. 369 370 Chapter 27 Multivariate Analysis: Factor Analysis Like principal component analysis, common factor analysis is a technique for reducing the complexity of high-dimensional data. (For brevity, this chapter refers to common factor analysis as simply “factor analysis.”) However, the techniques differ in how they construct a subspace of reduced dimensionality. Jackson (1981, 1991) provides an excellent comparison of the two methods. Principal component analysis chooses a coordinate system for the vector space spanned by the variables. (Recall that the span of a set of vectors is the vector space consisting of all linear combinations of the vectors.) The first principal component points in the direction of maximum variation in the data. Subsequent components account for as much of the remaining variation as possible while being orthogonal to all of the previous principal components. Each principal component is a linear combination of the original variables. Dimensional reduction is achieved by ignoring dimensions that do not explain much variation. While principal component analysis explains variability, factor analysis explains correlation. Suppose two variables, x1 and x2 , are correlated, but not collinear. Factor analysis assumes the existence of an unobserved variable that is linearly related to x1 and x2 , and explains the correlation between them. The goal of factor analysis is to estimate this unobserved variable from the structure of the original variables. An estimate of the unobserved variable is called a common factor. The geometry of the relationship between the original variables and the common factor is illustrated in Figure 27.1. (The figure is based on a similar figure in Wickens (1995), as is the following description of the geometry.) The correlated variables x1 and x2 are shown schematically in the figure. Each vector is decomposed into a linear combination of a common factor and a unique factor. That is, xi = ci f + di ui , i = 1, 2. The unique factors, u1 and u2 , are uncorrelated with the common factor, f , and with each other. Note that f , u1 , and u2 are mutually orthogonal in the figure. 372 Chapter 27. Multivariate Analysis: Factor Analysis Figure 27.1. The Geometry of Factor Analysis In contrast to principal components, a factor is not, in general, a linear combination of the original variables. Furthermore, a principal component analysis depends only on the data, whereas a factor analysis requires fitting the theoretical structure in the previous paragraph to the observed data. If there are p variables and you postulate the existence of m common factors, then each variable is represented as a linear combination of the m common factors and a single unique factor. Since the unique factors are uncorrelated with the common factors and with each other, factor analysis requires m + p dimensions. (Figure 27.1 illustrates the case p = 2 and m = 1.) However, the orthogonality of the unique factors means that the geometry is readily understood by projecting the original variables onto the span of the m factors (called the factor space). A graph of this projection is called a pattern plot. In Figure 27.1, the pattern plot is the two points on f obtained by projecting x1 and x2 onto f . The length of the projection of an original variable x onto the factor space indicates the proportion of the variability of x that is shared with the other variables. This proportion is called the communality. Consequently, the variance of each original variable is the sum of the common variance (represented by the communality) and the variance of the unique factor for that variable. In a pattern plot, the communality is the squared distance from the origin to a point. In factor analysis, the common factors are not unique. Typically an initial orthonormal set of common factors is computed, but then these factors are rotated so that the factors are more easily interpreted in terms of the original variables. An orthogonal rotation preserves the orthonormality of the factors; an oblique transformation introduces correlations among one or more factors. You can run the Factor analysis in Stat Studio by selecting Analysis I Multivariate Analysis I Factor Analysis from the main menu. The analysis is implemented by calling the FACTOR procedure in SAS/STAT. See the FACTOR procedure documentation in the SAS/STAT User’s Guide for additional details. The FACTOR procedure provides several methods of estimating the common factors Chapter 27. Multivariate Analysis: Factor Analysis and the communalities. Since an (m + p)-dimensional model is fit by using the original p variables, you should interpret the results with caution. The following list describes special issues that can occur: • Some of the eigenvalues of the reduced correlation matrix might be negative. A reduced correlation matrix is the correlation matrix of the original variables, except that the 1’s on the diagonal are replaced by prior communality estimates. These estimates are less than 1, and so the reduced correlation matrix might not be positive definite. In this case, the factors corresponding to the largest eigenvalues might account for more than 100% of the common variance. • The communalities are the proportions of the variance of the original variables that can be attributed to the common factors. As such, the communalities should be in the interval [0, 1]. However, factor analyses that use iterative fitting estimate the communality at each iteration. For some data, the estimate might equal (or exceed) 1 before the analysis has converged to a solution. This is known as a Heywood (or an ultra-Heywood) case, and it implies that one or more unique factor has a nonpositive variance. When this occurs, the factor analysis stops iterating and reports an error. These and other issues are described in the section “Heywood Cases and Other Anomalies about Communality Estimates” in the documentation for the FACTOR procedure. You can use many different methods to perform a factor analysis. Two popular methods are the principal factor method and the maximum likelihood method. The principal factor method is computationally efficient and has similarities to principal component analysis. The maximum likelihood (ML) method is an iterative method that is computationally more demanding and is prone to Heywood cases, nonconvergence, and multiple optimal solutions. However, the ML method also provides statistics such as standard errors and confidence limits that help you to assess how well the model fits the data, and to interpret factors. Consequently, the ML method is often favored by statisticians. In addition to these various methods of factor analysis, you can use Stat Studio to compute various component analyses: principal component analysis, Harris component analysis, and image component analysis. Example This example investigates factors that explain several variables in the Baseball data set. The Baseball data set contains performance measures for major league baseball players in 1986. A full description of the Baseball data is included in Appendix A, “Sample Data Sets.” Suppose you postulate the existence of unobserved variables that explain the hitting and fielding performance of players’ performances during the 1986 season. (An example of an unobserved variable in the context of baseball is “quickness,” which 373 374 Chapter 27. Multivariate Analysis: Factor Analysis could explain correlation between a player’s runs, stolen bases, and fielding statistics.) There are six variables that measure a player’s batting performance: no– atbat, no– hits, no– home, no– runs, no– rbi, and no– bb. There are three variables that measure a player’s fielding performance: no– outs, no– assts, and no– error. The goal of this example is to form a low-dimensional factor space that explains the relationships among these nine variables. =⇒ Open the Baseball data set. =⇒ Select Analysis I Multivariate Analysis I Factor Analysis from the main menu, as shown in Figure 27.2. Figure 27.2. Selecting the Factor Analysis A dialog box appears as in Figure 27.3. You can select variables for the analysis by using the Variables tab. =⇒ Select no– atbat. While holding down the CTRL key, select no– hits, no– home, no– runs, no– rbi, and no– bb. Click Add Y. Note: Alternately, you can select the variables by using contiguous selection: click on the first item, hold down the SHIFT key, and click on the last item. All items between the first and last item are selected and can be added by clicking Add Y. The three measures of fielding performance are located near the end of the list of variables. =⇒ Scroll to the end of the variable list. Select no– outs. While holding down the CTRL key, select no– assts and no– error. Click Add Y. Chapter 27. Multivariate Analysis: Factor Analysis Figure 27.3. The Variables Tab =⇒ Click the Method tab. The Method tab (Figure 27.4) becomes active. You can use the Method tab to set options in the analysis. The default method is principal factor analysis. However, the default method of estimating the prior communalities is to set all prior communalities to 1. This would result in a principal component analysis rather than a factor analysis. =⇒ Set Prior estimates to Squared multiple correlations. The preceding step sets the prior communality estimate for each variable to its squared multiple correlation with all other variables. 375 376 Chapter 27. Multivariate Analysis: Factor Analysis Figure 27.4. The Method Tab =⇒ Click the Rotation tab. The Rotation tab (Figure 27.5) becomes active. The default behavior is to leave factors unrotated. This example requests that an oblique transformation be applied to the factors in order to illustrate how rotated factors can sometimes be more interpretable. =⇒ Select Promax for the Factor rotation option. Figure 27.5. The Rotation Tab Chapter 27. Multivariate Analysis: Factor Analysis =⇒ Click the Tables tab. The Tables tab (Figure 27.6) becomes active. To help determine whether the data are appropriate for the common factor model, you can request Kaiser’s measure of sampling adequacy (MSA). =⇒ Select Kaiser’s measure of sampling adequacy. =⇒ Click OK. Figure 27.6. The Tables Tab The analysis calls the FACTOR procedure, which uses the options specified in the dialog box. The procedure displays tables in the output document, as shown in Figure 27.7. As is discussed subsequently, the Factor analysis extracts three principal factors for these data. Three plots also appear. 377 378 Chapter 27. Multivariate Analysis: Factor Analysis Figure 27.7. Output from a Factor Analysis The eigenvalue plot shows the eigenvalues of the reduced correlation matrix, along with the cumulative proportion of common variance accounted for by the factors. The first two factors account for almost 95% of the common variance, and the first three factors account for 101%. The reduced correlation matrix for these data has negative eigenvalues, which explains why the factors corresponding to the largest eigenvalues account for more than 100% of the common variance. The initial factor pattern plot shows the projection of the original variables onto the subspace spanned by the first two factors. As shown in Figure 27.7, you can click on a point in order to identify the corresponding variable. The points with high values of Factor1 are all hitting variables, including no– hits. The points with the highest values of Factor2 are two of the fielding variables: no– assts and no– error. The third fielding variable (no– outs) is closest to the origin in this plot. The initial factor pattern plot indicates that the first (unrotated) factor correlates highly with the hitting variables, whereas the second correlates with assists and errors. Note: If you want to visualize the third extracted factor, you can color the observations according to the value of the Factor3 variable or create a three-dimensional scatter plot of the three factors. You can view the data table underlying this plot by pressing the F9 key when the plot is active. The rotated factor pattern plot in Figure 27.7 shows the projection of the original variables onto the subspace spanned by the first two rotated factors. A promax transformation is used to transform the original factors (which are orthogonal to Chapter 27. Multivariate Analysis: Factor Analysis each other) to new factors that, in many cases, are easier to interpret in terms of the original variables. Note that this transformation does not change the common factor space or the communality estimates. In the rotated factor pattern plot, the cluster of points with high values of Pattern1 are the variables no– atbat, no– hits, no– runs, and no– bb. (These points are not labeled in Figure 27.7, but they are labeled in Figure 27.8.) Players with high values of these variables get on base often, so you might interpret the first (rotated) factor to be “Getting on Base.” The two points with high values of Pattern2 are the variables no– home and no– rbi. Players who have high values of these variables contribute many runs to their teams’ scores, so you might interpret the second (rotated) factor as “Scoring.” In the rotated factor pattern plot, the fielding variables are positioned near the origin, indicating that these variables are not strongly correlated with the first two rotated factors. Figure 27.8 shows a three-dimensional scatter plot that visualizes the three rotated factors. The plot shows that no– assts and no– error are highly correlated with the third rotated factor, while no– outs is not strongly correlated with any of the first three factors. The third rotated factor identifies players who make many assists and many errors. These are typically infielders who play second base, shortstop, or third base. Consequently, you might interpret the third rotated factor as a “Fielding Position” factor. Figure 27.8. Plot of Obliquely Transformed Factors Figure 27.7 shows part of the partial correlations matrix for the original variables. If the data are appropriate for the common factor model, the partial correlations (controlling the other variables) should be small compared to the original correlations. Recall that the partial correlation between two variables, controlling for the variables X1 , . . . , Xk , is the correlation between the residuals of the two variables after regression on the Xi . 379 380 Chapter 27. Multivariate Analysis: Factor Analysis Figure 27.7 also shows the MSA statistics. Kaiser’s MSA (Kaiser 1970) is a summary, for each variable and for all variables together, of how much smaller the partial correlations are than the original correlations. Values of 0.8 or 0.9 are considered good, while MSAs less than 0.5 are unacceptable. The no– assts and no– error variables have the poorest MSAs. The overall MSA of 0.76 is adequate for proceeding with the factor analysis; an overall MSA lower than 0.6 often indicates that the data are not likely to factor well. Figure 27.9 shows additional output. The prior communality estimates indicate that the variance of no– outs might not be well explained by the three common factors. The table of eigenvalues displays the eigenvalues for the reduced correlation matrix, which is the correlation matrix of the original variables, except that the 1’s on the diagonal are replaced by the prior communality estimates. A note is printed below this table indicating that three factors are retained because they account for (at least) 100% of the common variance. Figure 27.9. Output from a Factor Analysis Figure 27.10 shows additional output from the FACTOR procedure. The “Factor Pattern” table shows the relationship between the unrotated factors and the original Y variables. Each Y variable is a linear combinations of the common factor and a unique factor. For example, no– atbat corresponds to the linear combination no– atbat = 0.95565 Factor1 + 0.13507 Factor2 − 0.12293 Factor3 + u1 If you decide not to rotate the factors, you can attempt to interpret these factors by looking at the relative magnitudes of the coefficients. For example, the first unrotated factor appears to measure a player’s overall performance. More weight is given to getting on base (coefficients in the range 0.89–0.96), less weight is given to scoring runs (coefficients in the range 0.68–0.72), and little weight is given to the fielding statistics. The figure also shows the common variance explained by each factor and the final communality estimates. Chapter 27. Multivariate Analysis: Factor Analysis Figure 27.10. Unrotated Factors Whereas Figure 27.10 displays information about the unrotated factors, Figure 27.11 displays information about the rotated factors. The promax transformation is the composition of two transformations: an orthogonal varimax rotation and an oblique Procrustean transformation. Figure 27.11 displays information about the factors after the orthogonal varimax rotation. You can also visualize the pattern of the rotated factors as follows: view the data table underlying a factor pattern plot by pressing the F9 key when the factor pattern plot is active, and then create scatter plots of the variables named Prerotatn. The Prerotatn variables correspond to the columns of the “Rotated Factor Pattern Table.” 381 382 Chapter 27. Multivariate Analysis: Factor Analysis Figure 27.11. Orthogonally Rotated Factors Figure 27.12 displays information about the obliquely transformed factors. The Procrustean transformation is displayed, followed by the matrix used to transform the unrotated factors into the factors displayed in the “Rotated Factor Pattern (Standardized Regression Coefficients)” table. The factor loadings shown in this table are shown graphically in the rotated factor pattern plot (Figure 27.7). An oblique transformation introduces correlations between the factors, and the “Inter-Factor Correlations” table shows those correlations. You can convert the correlations into angles between the factors by applying the arccosine function. For example, the angle between the first and second factors is cos−1 (0.59222), or approximately 53.7 degrees, whereas the second and third factors are almost orthogonal. The output contains additional tables (not shown) that display further correlations, structures, and variances. The “Displayed Output” section of the FACTOR procedure documentation describes all of the tables. Chapter 27. Multivariate Analysis: Factor Analysis Figure 27.12. Obliquely Rotated Factors Specifying the Factor Analysis This section describes the dialog box tabs associated with the Factor analysis. The Factor analysis calls the FACTOR procedure in SAS/STAT. See the FACTOR procedure documentation in the SAS/STAT User’s Guide for additional details. The Variables Tab You can use the Variables tab to specify the numerical variables for the analysis. The Variables tab is shown in Figure 27.3. The variables in the Y Variables list correspond to variables in the VAR statement of the FACTOR procedure. The Partial list is rarely used. The variables in this list correspond to variables in the PARTIAL statement of the FACTOR procedure. The FACTOR procedure computes the factors for the residuals of the Y variables after regression on the PARTIAL variables. Equivalently, the factors are determined by the partial correlation matrix between the Y variables, controlling for the PARTIAL variables. 383 384 Chapter 27. Multivariate Analysis: Factor Analysis The Method Tab You can use the Method tab (Figure 27.4) to set options in the analysis. Each of the following options corresponds to an option in the FACTOR procedure. Factoring method specifies the method used to extract factors or specifies a component analysis. This corresponds to the METHOD= option in the PROC FACTOR statement. Compute factors from specifies whether the factors are computed for the correlation matrix or the covariance matrix. This corresponds to the COV option in the PROC PRINCOMP statement. Note: Some methods require a correlation matrix. Number of Factors The number of factors retained is determined by the minimum number satisfying the next three criteria. Maximum number specifies how many factors to compute. This corresponds to the N= option in the PROC FACTOR statement. Note that you can type into the field; if you want five factors, you can enter 5 even though this is not an option on the list. Proportion of variance specifies the proportion of common variance in the retained factors. This value is in the range (0, 1]. The option corresponds to the PROPORTION= option in the PROC FACTOR statement. Minimum eigenvalue specifies the smallest eigenvalue for which a factor is retained. This corresponds to the MINEIGEN= option in the PROC FACTOR statement. Prior estimates specifies a method for computing prior communality estimates. This corresponds to the PRIORS= option in the PROC FACTOR statement. Note that the default method for the principal factor method is to set all priors equal to 1. This results in a principal component analysis. If you want a principal factor analysis, you should select a different method for estimating the prior communalities, as illustrated in the section “Example” on page 373. Heywood Conditions specifies how the factor analysis behaves if a communality is greater than 1. The section “Heywood Cases and Other Anomalies about Communality Estimates” in the documentation for the FACTOR procedure describes why this situation might occur. Do not allow communalities greater than one specifies that an analysis should stop processing if it encounters a communality greater than one. Chapter 27. Multivariate Analysis: Factor Analysis Set any communality greater than one to one specifies that an analysis should set any communality greater than one to one, and then continue. This corresponds to the HEYWOOD option in the PROC FACTOR statement. Allow communalities greater than one specifies that an analysis should allow any communality. This corresponds to the ULTRAHEYWOOD option in the PROC FACTOR statement. The Rotation Tab You can use the Rotation tab (Figure 27.5) to transform the factors by orthogonal or oblique rotations. Orthogonal rotations rigidly rotate the factors; oblique transformations introduce correlations between the factors. Transformed factors are often more interpretable in terms of the original variables. Factor rotation specifies the rotation method. You can select from a set of common orthogonal or oblique transformations. This corresponds to the ROTATE= option in the PROC FACTOR statement. Harris-Kaiser power specifies the power of the square roots of the eigenvalues used to rescale the eigenvectors for Harris-Kaiser orthoblique transformation. This corresponds to the HKPOWER= option in the PROC FACTOR statement. Promax power specifies the power for forming the target Procrustean matrix. This corresponds to the POWER= option in the PROC FACTOR statement. Factor pattern normalization specifies the method for normalizing the rows of the factor pattern for rotation. This corresponds to the NORM= option in the PROC FACTOR statement. The Plots Tab You can use the Plots tab (Figure 27.13) to create plots that display results of the analysis. The plots for the Factor analysis are not linked to the original data table. The scree plot has its own data table; the two factor pattern plots (also called factor loading plots) are linked to each other. You can view the data table underlying a plot by pressing the F9 key when the plot is active. The following plots are available: Proportion plot of eigenvalues (scree plot) creates a plot that summarizes the eigenvalues of the reduced correlation or reduced covariance matrix. 385 386 Chapter 27. Multivariate Analysis: Factor Analysis Show cumulative proportions adds cumulative proportions of eigenvalues to the proportion plot. Initial factor pattern (unrotated) creates a plot showing the relationships between the initial (unrotated) factors and the original variables. Rotated factor pattern creates a plot showing the relationships between the final rotated factors and the original variables. This plot is created only if you specify a rotation on the Rotation tab. Figure 27.13. The Plots Tab The Tables Tab The Tables tab is shown in Figure 27.6. You can use the Tables tab to display the following tables that summarize the results of the analysis: Simple descriptive statistics specifies whether to display the mean and standard deviation for each variable. This corresponds to the SIMPLE option in the PROC FACTOR statement. Correlation matrix specifies whether to display the correlation matrix. This corresponds to the CORR option in the PROC FACTOR statement. Eigenvectors specifies whether to display the eigenvectors of the reduced correlation matrix. This corresponds to the EIGENVECTORS option in the PROC FACTOR statement. Kaiser’s measure of sampling adequacy specifies whether to display partial correlations between each pair of variables (controlling for all other variables), and Kaiser’s measure of sampling adequacy. This corresponds to the MSA option in the PROC FACTOR statement. Factor scoring coefficients specifies whether to display the factor scoring coefficients. This corresponds to the SCORE option in the PROC FACTOR statement. Chapter 27. Multivariate Analysis: Factor Analysis Residual and partial correlations specifies whether to display the residual correlation matrix and the associated partial correlation matrix. This corresponds to the RESIDUALS option in the PROC FACTOR statement. The Output Variables Tab You can use the Output Variables tab (Figure 27.14) to add estimated factor scores to the data table. Each estimated factor score is computed as a linear combination of the standardized values of the variables that are factored. The names of the variables are of the form FACi, where i = 1 . . . k, and k is the number of retained factors. Figure 27.14. The Output Tab The Roles Tab You can use the Roles tab to specify a frequency variable or weight variable for the analysis. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. A weight variable is a numeric variable with values that are relative weights for the analysis. Analysis of Selected Variables If any numeric variables are selected in a data table when you run the analysis, these variables are automatically entered in the Y Variables field of the Variables tab. Any variable in the data table with a Frequency or Weight role is automatically entered in the appropriate field of the Roles tab. 387 388 Chapter 27. Multivariate Analysis: Factor Analysis References Jackson, J. E. (1981), “Principal Components and Factor Analysis: Part III—What Is Factor Analysis?” Journal of Quality Technology, 13(2), 125–130. Jackson, J. E. (1991), A User’s Guide to Principal Components, New York: John Wiley & Sons. Kaiser, H. F. (1970), “A Second Generation Little Jiffy,” Psychometrika, 35, 401–415. Wickens, T. D. (1995), The Geometry of Multivariate Statistics, Hillsdale, NJ: Lawrence Erlbaum Associates. Chapter 28 Multivariate Analysis: Canonical Correlation Analysis Canonical correlation analysis is a technique for analyzing the relationship between two sets (or groups) of variables. Each set can contain multiple variables. Given two sets of variables, canonical correlation analysis finds a linear combination from each set, called a canonical variable, such that the correlation between the two canonical variables is maximized. This correlation between the two canonical variables is the first canonical correlation. The first canonical correlation is at least as large as the multiple correlation between any variable and the opposite set of variables. The coefficients of the linear combinations are canonical coefficients or canonical weights. It is customary to normalize the canonical coefficients so that each canonical variable has a variance of 1. Canonical correlation analysis continues by finding a second set of canonical variables, uncorrelated with the first pair, that produces the second-highest correlation coefficient. The process of constructing canonical variables continues until the number of pairs of canonical variables equals the number of variables in the smaller group. Each canonical variable is uncorrelated with all the other canonical variables of either set except for the one corresponding canonical variable in the opposite set. However, the canonical variables do not represent jointly perpendicular directions through the space of the original variables. You can run the Canonical Correlation analysis by selecting Analysis I Multivariate Analysis I Canonical Correlation Analysis from the main menu. The analysis is implemented by calling the CANCORR procedure in SAS/STAT. See the CANCORR procedure documentation in the SAS/STAT User’s Guide for additional details. Example In this example, you examine canonical correlations between sets of variables in the GPA data set. The GPA data set contains average high school grades in mathematics, science, and English for students applying to a university computer science program. The data also contains the students’ scores on the mathematics and verbal sections of the SAT, which is a standardized test to measure aptitude. Suppose you are interested in the relationship between the variables that represent analytical thinking and those that represent verbal thinking. You can group the following variables into the analytical set: hsm (high school math average), hss (high school science average), and satm (SAT math score). You can group the 390 Chapter 28. Multivariate Analysis: Canonical Correlation Analysis following variables into the verbal set: hse (high school English average) and satv (SAT verbal score). =⇒ Open the GPA data set. =⇒ Select Analysis I Multivariate Analysis I Canonical Correlation Analysis from the main menu, as shown in Figure 28.1. Figure 28.1. Selecting the Canonical Correlation Analysis A dialog box appears as in Figure 28.2. You can select variables for the analysis by using the Variables tab. =⇒ Select hsm. While holding down the CTRL key, select hss and satm. Click Add Y. =⇒ Select hse. While holding down the CTRL key, select satv. Click Add X. Figure 28.2. The Variables Tab Chapter 28. Multivariate Analysis: Canonical Correlation Analysis =⇒ Click the Tables tab. The Tables tab (Figure 28.3) becomes active. You can use the Tables tab to display statistics associated with the analysis, and to specify labels that identify the two sets of variables. For this example, you can label the first set of variables as the “Analytical” set and the second set as the “Verbal” set. =⇒ Type Analytical into the Y variables field. =⇒ Type Verbal into the X variables field. Figure 28.3. The Tables Tab =⇒ Click OK. The analysis calls the CANCORR procedure, which uses the options specified in the dialog box. The procedure displays tables in the output document, as shown in Figure 28.4. Two plots are also created. The plot of the first canonical variables shows the strength of the relationship between the set of analytical variables and the set of verbal variables. The second plot shows the second canonical variables. The footnote of these plots displays the canonical correlations. Note that the correlation between the second pair of canonical variables is less than the correlation between the first pair. The output window in Figure 28.4 displays the canonical correlation, which is the correlation between the first pair of canonical variables. The value 0.6106 represents the highest possible correlation between any linear combination of the analytical variables and any linear combination of the verbal variables. 391 392 Chapter 28. Multivariate Analysis: Canonical Correlation Analysis Figure 28.4. Output from a Canonical Correlation Analysis The output window contains additional tables, as shown in Figure 28.5. The figure displays the likelihood ratios and associated statistics for testing the hypothesis that the canonical correlations in the current row and all that follow are zero. The first approximate F value of 30.80 corresponds to the test that all canonical correlations are zero. Since the p-value is small, you can reject the null hypothesis at the 95% level. Similarly, the second approximate F value of 29.50 corresponds to the test that the second canonical correlation is zero. This test also rejects the hypothesis. Several multivariate statistics and F test approximations are also provided. These statistics test the null hypothesis that all canonical correlations are zero. The small p-values for these tests (< 0.0001) are evidence for rejecting the null hypothesis. Chapter 28. Multivariate Analysis: Canonical Correlation Analysis Figure 28.5. Testing Whether Canonical Correlations Are Zero The analysis creates canonical variables and adds them to the data table. The canonical variables for the analytical group are named CCAY1 and CCAY2. The canonical variables for the verbal group are named CCAX1 and CCAX2. The canonical variables are linear combinations of the original variables, so you can sometimes interpret the meaning of the canonical variables in terms of the original variables. To interpret the variables, inspect the standardized coefficients of the canonical variables and the correlations between the canonical variables and their original variables. These statistics are shown in Figure 28.6. For example, the first canonical variables are represented by CCAY1 = 0.0249 satm + 0.8009 hss + 0.2836 hsm CCAX1 = 0.9129 hse + 0.2424 satv The standardized canonical coefficients show that the first canonical variable for the analytical group is a weighted sum of the variables hss (with coefficient 0.8009) and hsm (0.2836), with the emphasis on the science grade. The coefficient for the variable satm is close to zero. The second canonical variable for the analytical group is a contrast between the variables satm (1.1208) and hsm (–0.4752), with the SAT math score receiving the most weight. The coefficients for the verbal variables show that hse contributes heavily to the CCAX1 canonical variable (0.9129), whereas CCAX2 is heavily influenced by satv (1.0022). 393 394 Chapter 28. Multivariate Analysis: Canonical Correlation Analysis Figure 28.6. Canonical Coefficients Figure 28.7 displays the table of correlations between the canonical variables and the original variables. These univariate correlations must be interpreted with caution, since they do not indicate how the original variables contribute jointly to the canonical analysis. However, they are often useful in the interpretation of the canonical variables. The first canonical variable for the analytical group is strongly correlated with hsm and hss, with correlations 0.7560 and 0.9702, respectively. The second canonical variable for the analytical group is strongly correlated with satm, with a correlation of 0.8982. The first canonical variable for the verbal group is strongly correlated with hse, with a correlation of 0.9720. The second canonical variable for the verbal group is strongly correlated with satv, with a correlation of 0.8854. Figure 28.7. Correlations between Canonical and Original Variables In summary, the analytical and verbal variables are moderately correlated with each other, with a canonical correlation of 0.6106. The first canonical variables are close to the linear subspace spanned by the variables that measure a student’s high school grades. The second canonical variables are close to the linear subspace spanned by the SAT variables. (Recall that the span of a set of vectors is the vector space consisting of all linear combinations of the vectors.) Chapter 28. Multivariate Analysis: Canonical Correlation Analysis Specifying the Canonical Correlation Analysis This section describes the dialog box tabs associated with the Canonical Correlation analysis. The Canonical Correlation analysis calls the CANCORR procedure in SAS/STAT. See the CANCORR procedure documentation in the SAS/STAT User’s Guide for additional details. The Variables Tab You can use the Variables tab to specify the numerical variables for the analysis. The Variables tab is shown in Figure 28.2. The variables in the Y Variables list correspond to variables in the VAR statement of the CANCORR procedure. The variables in the X Variables (With) list correspond to variables in the WITH statement of the CANCORR procedure. The Partial list is rarely used. The variables in this list correspond to variables in the PARTIAL statement of the CANCORR procedure. The CANCORR procedure computes the canonical correlations of the residuals from the prediction of the VAR and WITH variables by the PARTIAL variables. The Method Tab You can use the Method tab (Figure 28.8) to set options in the analysis. You can use the Number of canonical variables option to specify the number of canonical variables displayed in the output window. This option corresponds to the NCAN= option in the PROC CANCORR statement. Figure 28.8. The Method Tab The Plots Tab You can use the Plots tab (Figure 28.9) to create plots that graphically display results of the analysis. Creating a plot adds canonical variables to the data table. The following plots are available: Matrix of canonical score plots creates a plot for each pair of canonical variables that summarizes the strength of the relationship between the variables. 395 396 Chapter 28. Multivariate Analysis: Canonical Correlation Analysis Add regression line adds a least squares regression line to each score plot. The regression line predicts the ith canonical variable in the second group from the ith canonical variable in the first group. Figure 28.9. The Plots Tab The Tables Tab The Tables tab is shown in Figure 28.3. You can use the Tables tab to display the following tables that summarize the results of the analysis: Simple statistics specifies whether to display the mean and standard deviation for each variable. This option corresponds to the SIMPLE option in the PROC CANCORR statement. Correlation matrices specifies whether to display the correlation matrices for each set of variables. This option corresponds to the CORR option in the PROC CANCORR statement. Multivariate statistics specifies whether to display a table of multivariate statistics and F approximations. Canonical coefficients specifies whether to display the raw and standardized canonical coefficients for each set of variables. Canonical structures specifies whether to display correlations between the canonical variables and the original variables. Canonical redundancy analysis specifies whether to display a canonical redundancy analysis. This option corresponds to the REDUNDANCY option in the PROC CANCORR statement. The Output Variables Tab You can use the Output Variables tab (Figure 28.10) to add canonical variables (also called canonical scores) to the data table. The option on the Method tab determines how many variables are added. Chapter 28. Multivariate Analysis: Canonical Correlation Analysis Figure 28.10. The Output Tab The Roles Tab You can use the Roles tab to specify a frequency variable or weight variable for the analysis. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. A weight variable is a numeric variable with values that are relative weights for the analysis. Analysis of Selected Variables If any numeric variables are selected in a data table when you run the analysis, these variables are automatically entered in the Y Variables field of the Variables tab. Any variable in the data table with a Frequency or Weight role is automatically entered in the appropriate field of the Roles tab. 397 398 Chapter 29 Multivariate Analysis: Canonical Discriminant Analysis Canonical discriminant analysis is a dimension-reduction technique related to principal component analysis and canonical correlation. Given a nominal classification variable and several interval variables, canonical discriminant analysis derives canonical variables (linear combinations of the interval variables) that summarize between-class variation in much the same way that principal components summarize total variation. Canonical discriminant analysis is equivalent to canonical correlation analysis between the quantitative variables and a set of dummy variables coded from the classification variable. Given two or more groups of observations with measurements on several interval variables, canonical discriminant analysis derives a linear combination of the variables that has the highest possible multiple correlation with the groups. This maximum multiple correlation is called the first canonical correlation. The coefficients of the linear combination are the canonical coefficients. The variable defined by the linear combination is the first canonical variable. The second canonical correlation is obtained by finding the linear combination uncorrelated with the first canonical variable that has the highest possible multiple correlation with the groups. The process of extracting canonical variables can be repeated until the number of canonical variables equals the number of original variables or the number of classes minus one, whichever is smaller. Canonical variables are also called canonical components. You can run the Canonical Discriminant analysis by selecting Analysis I Multivariate Analysis I Canonical Discriminant Analysis from the main menu. The analysis is implemented by calling the DISCRIM procedure with the CANONICAL option in SAS/STAT. See the documentation for the DISCRIM and CANDISC procedures in the SAS/STAT User’s Guide for additional details. The analysis calls the DISCRIM procedure (rather than the CANDISC procedure) because the DISCRIM procedure produces a discriminant function that can be used to classify current or future observations. 400 Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis Example In this example, you examine measurements of 159 fish caught in Finland’s Lake Laengelmavesi. The fish are one of seven species: bream, parkki, perch, pike, roach, smelt, and whitefish. Associated with each fish are physical measurements of weight, length, height, and width. A full description of the Fish data is included in Appendix A, “Sample Data Sets.” The goal of this example is to use canonical discriminant analysis to construct linear combinations of the size and weight variables that best discriminate between the species. By looking at the coefficients of the linear combinations, you can determine which physical measurements are most important in discriminating between groups. You can also determine whether there are two or more groups that cannot be discriminated using these measurements. =⇒ Open the Fish data set. =⇒ Select Analysis I Multivariate Analysis I Canonical Discriminant Analysis from the main menu, as shown in Figure 29.1. Figure 29.1. Selecting the Canonical Discriminant Analysis A dialog box appears as in Figure 29.2. You can select variables for the analysis by using the Variables tab. =⇒ Select Species and click Set Y. =⇒ Select Weight. While holding down the CTRL key, select Length1, Length2, Length3, Height, and Width. Click Add X. Note: Alternately, you can select the variables by using contiguous selection: click on the first variable (Weight), hold down the SHIFT key, and click on the last variable (Width). All variables between the first and last item are selected and can be added by clicking Add X. Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis Figure 29.2. The Variables Tab =⇒ Click the Method tab. The Method tab (Figure 29.3) becomes active. You can use the Method tab to set options in the analysis. =⇒ Select 3 for Number of canonical variables. The number of fish in any lake varies by species. That is, there is no reason to suspect that the number of whitefish in the lake is the same as the number of perch or bream. In the absence of prior knowledge about the distribution of fish species, you can assume that the number of fish of each species in the lake is proportional to the number in the sample. =⇒ Select Proportional to group sizes for Prior probability of group membership. 401 402 Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis Figure 29.3. The Method Tab =⇒ Click OK. The analysis calls the DISCRIM procedure with the CANONICAL option. The procedure uses the options specified in the dialog box. The procedure displays tables in the output document, as shown in Figure 29.4. Two plots are also created. The plot of the first two canonical components shows how well the first two canonical variables discriminate between the species of fish. The first canonical component differentiates among four groups: the pike-perch-smelt group, the roach-whitefish group, the parkki group, and the bream group. The second canonical component differentiates the pike groups from the other groups. Thus, the first two canonical components cannot differentiate between perch and smelt, nor between roach and whitefish. In Figure 29.4, a cloud of observations is selected. You can see from the linked bar chart that these observations consist of perch and smelt. The location of the multivariate means for each species is indicated in the plot of the first two canonical components, along with an 80% confidence ellipse for the mean. The means of the perch and smelt groups are close to each other, as are the means of the roach and whitefish. Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis Figure 29.4. Output from a Canonical Discriminant Analysis Note: The third canonical component helps to differentiate between perch and smelt, and between roach and whitefish. The canonical variables were added to the data table by the analysis, so you can create a scatter plot of the second and third canonical variables (CDA– 3 versus CDA– 2) or create a rotating plot of all three canonical components, as shown in Figure 29.5. 403 404 Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis Figure 29.5. A Rotating Plot of the Canonical Components The output window contains many tables of statistics. Figure 29.4 shows a summary of the model, as well as the frequency and proportion of each species. Recall that canonical discriminant analysis is equivalent to canonical correlation analysis between the quantitative variables and a set of dummy variables coded from the classification variable (in this case, Species). Figure 29.6 displays statistics related to the canonical correlations. The multivariate statistics and F approximations test the null hypothesis that all canonical correlations are zero. The small p-values for these tests (< 0.0001) are evidence for rejecting the null hypothesis that all canonical correlations are zero. The table of canonical correlations shows that the first three canonical components are all highly correlated with the classification variable. Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis Figure 29.6. Canonical Correlations The portion of the output window shown in Figure 29.7 shows the canonical structure. These are tables of correlations between the canonical variables and the original variables. The canonical variables are linear combinations of the original variables, so you can sometimes interpret the canonical variables in terms of the original variables. The “Total Canonical Structure” table displays the correlations without regard for group membership. Since these correlations do not account for the groups, they can sometimes be misleading. The “Between Canonical Structure” table removes the within-class variability before computing the correlations. For each variable X, define the group mean vector of X to be the vector whose ith element is the mean of all values of X that belong to the same group as Xi . The values in the “Between Canonical Structure” table are the correlations between the group mean vectors of the canonical variables and the group mean vectors of the original variables. The “Pooled Within Canonical Structure” table removes the between-class variability before computing the correlations. The values in this table are the correlations between the residuals of the original and canonical variables, after regressing them onto the group variable. For this example, the “Total Canonical Structure” table and the “Between Canonical Structure” table have similar interpretations: the first canonical component is strongly correlated with Height. The second canonical variable is strongly correlated with the length variables, and also with Weight. The third canonical component is a weighted average of all the variables, with slightly more weight given to Width. 405 406 Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis Figure 29.7. Canonical Structure The first canonical variable separates the species most effectively. An examination of the “Raw Canonical Coefficients” table (Figure 29.8) shows that the first canonical variable is the following linear combination of the centered variables: Can1 = −0.0006 Weight − 0.328 Length1 + . . . − 1.44 Width The coefficients are standardized so that the canonical variables have zero mean and a pooled within-class variance equal to one. The second canonical variable provides the greatest difference between group means while being uncorrelated with the first canonical variable. Figure 29.8 also shows the coordinates of the group means in terms of the canonical variables. For example, the mean of the bream species projected onto the span of the first two canonical components is (10.91, 0.51). (Recall that the span of a set of vectors is the vector space consisting of all linear combinations of the vectors.) This agrees with the graph shown in Figure 29.4. The means of the perch and smelt groups are close to each other when projected onto the span of the first two canonical components. However, the third canonical component separates these means. Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis Figure 29.8. Canonical Coefficients and Group Means Figure 29.9 displays a table that summarizes how many fish are classified (or misclassified) into each species. If the canonical components capture most of the between-class variation of the data, then the elements on the table’s main diagonal are large, compared to the off-diagonal elements. For these data, two smelt are misclassified as perch, but no other fish are misclassified. This indicates that the first three canonical components are good discriminators for Species. Note: If you choose different options on the Method tab, the classification of observations will be different. Figure 29.9. Classification of Observations into Groups 407 408 Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis In summary, it is possible to use canonical discriminant analysis to discriminate between these species of fish by using three canonical components that are linear combinations of physical measurements. Trying to discriminate by using only two canonical components leads to classification errors, because the projection onto the span of the first two canonical components does not separate the perch group from the smelt group, nor does it separate the roach group from the whitefish group. Specifying the Canonical Discriminant Analysis This section describes the dialog box tabs associated with the Canonical Discriminant analysis. The Canonical Discriminant analysis calls the DISCRIM procedure with the CANONICAL option. See the DISCRIM procedure documentation in the SAS/STAT User’s Guide for additional details. The Variables Tab You can use the Variables tab to specify the variables for the analysis. The Variables tab is shown in Figure 29.2. The variable in the Y Variable (Classification) list corresponds to the variable in the CLASS statement of the DISCRIM procedure. This variable must be nominal. The variables in the X Variables list correspond to variables in the VAR statement of the DISCRIM procedure. The Method Tab You can use the Method tab (Figure 29.3) to set options in the analysis. The tab supports the following options: Number of canonical variables specifies the number of canonical variables. This option corresponds to the NCAN= option in the PROC DISCRIM statement. Covariance within groups specifies assumptions about the homogeneity of within-group covariances. This option corresponds to the POOL= option in the PROC DISCRIM statement. Prior probability of group membership specifies assumptions about the prior probabilities of group membership. This option corresponds to choosing either the EQUAL or PROPORTIONAL option in the PRIORS statement. Classify observations by specifies a method of classifying observations based on their canonical scores. This option corresponds to the CROSSVALIDATE option in the PROC DISCRIM statement. Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis The Plots Tab You can use the Plots tab (Figure 29.10) to create plots that graphically display results of the analysis. Figure 29.10. The Plots Tab Creating a plot often adds one or more variables to the data table. The following plots are available: Observed groups creates a spine plot (a one-dimensional mosaic plot) of the groups for the Y variable. Observed vs. Predicted groups creates a mosaic plot of the groups for the Y variable versus the group as classified by a discriminant function. Each observation is placed in the group that minimizes the generalized squared distance between the observation and the group mean. Classification fit plot creates a plot that indicates how well each observation is classified by the discriminant function. This plot is shown in Figure 29.11. Observations that are close to two or more group means are selected in the plot. For each observation, PROC DISCRIM computes posterior probabilities for membership in each group. Let mi be the maximum posterior probability for the ith observation. The classification fit plot is a plot of − log(mi ) versus i. 409 410 Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis Figure 29.11. A Classification Fit Plot Canonical score plot creates a plot of the first two canonical variables. (If there is only one canonical variable, then a histogram of that variable is created instead.) Show group means displays the mean of each group in the score plot. Add confidence ellipses for means displays a confidence ellipse for the mean of each group in the score plot. Confidence level specifies the probability level for the confidence ellipse. Shade plot background by confidence level specifies that the background of each scatter plot be shaded according to a nested family of confidence ellipses. Add prediction ellipses displays a prediction ellipse for the mean of each group in the score plot, assuming multivariate normality within each group. Prediction level specifies the probability level for the prediction ellipse. The Tables Tab The Tables tab is shown in Figure 29.12. You can use the Tables tab to display the following tables that summarize the results of the analysis. For more information, Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis see the “Displayed Output” subsection of the “Details” section in the documentation for the DISCRIM procedure. Simple statistics specifies whether to display descriptive statistics for the total sample and within each group. This option corresponds to the SIMPLE option in the PROC DISCRIM statement. Univariate ANOVA specifies whether to display univariate statistics for testing the hypothesis that the population group means are equal for each variable. This option corresponds to the ANOVA option in the PROC DISCRIM statement. Multivariate ANOVA specifies whether to display multivariate statistics for testing the hypothesis that the population group means are equal for each variable. This option corresponds to the MANOVA option in the PROC DISCRIM statement. Squared distances between group means specifies whether to display the squared Mahalanobis distances (and associated statistics) between the group means. This option corresponds to the DISTANCE option in the PROC DISCRIM statement. Standardized group means specifies whether to display total-sample and pooled within-group standardized group means. This option corresponds to the STDMEAN option in the PROC DISCRIM statement. Covariance matrices specifies whether to display the correlation matrices for each set of variables. This option corresponds to the BCOV, PCOV, TCOV, and WCOV options in the PROC DISCRIM statement. Correlation matrices specifies whether to display the correlation matrices for each set of variables. This option corresponds to the BCORR, PCORR, TCORR, and WCORR options in the PROC DISCRIM statement. Canonical structures specifies whether to display correlations between the canonical variables and the original variables. Canonical coefficients specifies whether to display the raw and standardized canonical coefficients for each set of variables. 411 412 Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis Figure 29.12. The Tables Tab In addition to the previous optional tables, the Canonical Discriminant analysis always creates the following tables. The name of the table refers to the ODS table name. Counts corresponds to the Counts table. Class level information corresponds to the Levels table. Canonical correlations corresponds to the CanCorr table. Note: This table looks like three tables: canonical correlations, eigenvalues of E −1 H, and tests for hypothesis that the canonical coefficients equal zero. Class means on canonical variables corresponds to the CanonicalMeans table. Linear discriminant function corresponds to the LinearDiscFunc table. This table is displayed only for the linear parametric classification method. Number of observations and percent classified corresponds to the ClassifiedResub or ClassifiedCrossVal table. Error count estimates corresponds to the ErrorResub or ErrorCrossVal table. Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis The Output Variables Tab You can use the Output Variables tab (Figure 29.13) to add analysis variables to the data table. If you request a plot that uses one of the output variables, then that variable is automatically created even if you did not explicitly select the variable on the Output Variables tab. The following list describes each output variable added to the data table and indicates how the output variable is named. Y represents the name of the classification variable. Posterior probabilities of group membership adds variables named CDAProb– X, where X is the name of an X variable. Predicted groups adds a variable named CDAPred– Y that contains the name of the group to which each observation is assigned. Canonical scores adds variables named CDA– 1 through CDA– k, where k is the number of canonical components. If a classification fit plot is requested on the Plots tab, then a variable named CDALogProb– Y is created, as described in the section “The Plots Tab” on page 409. Figure 29.13. The Output Tab The Roles Tab You can use the Roles tab to specify a frequency variable or weight variable for the analysis. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. A weight variable is a numeric variable with values that are relative weights for the analysis. 413 414 Chapter 29. Multivariate Analysis: Canonical Discriminant Analysis Analysis of Selected Variables If a nominal variable is selected in a data table when you run the analysis, this variable is automatically entered in the Y Variable (Classification) field of the Variables tab. Any selected interval variables are automatically entered in the X Variables field of the Variables tab. Any variable in the data table with a Frequency or Weight role is automatically entered in the appropriate field of the Roles tab. Chapter 30 Multivariate Analysis: Discriminant Analysis For a set of observations containing one or more interval variables and a classification variable defining groups of observations, discriminant analysis derives a discriminant criterion function to classify each observation into one of the groups. When the distribution within each group is assumed to be multivariate normal, a parametric method can be used to develop a discriminant function. The discriminant function, also known as a classification criterion, is determined by a generalized squared distance. The classification criterion can be based on either the individual within-group covariance matrices (yielding a quadratic function) or the pooled covariance matrix (yielding a linear function). It also takes into account the prior probabilities of the groups. When no assumptions can be made about the distribution within each group, or when the distribution is not assumed to be multivariate normal, nonparametric methods can be used to estimate the group-specific densities. These methods include the kernel and k-nearest-neighbor methods. You can run the Discriminant analysis by selecting Analysis I Multivariate Analysis I Discriminant Analysis from the main menu. The analysis is implemented by calling the DISCRIM procedure in SAS/STAT. See the documentation for the DISCRIM procedure in the SAS/STAT User’s Guide for additional details. Example In this example, you examine measurements of 159 fish caught in Finland’s Lake Laengelmavesi. The fish are one of seven species: bream, parkki, perch, pike, roach, smelt, and whitefish. Associated with each fish are physical measurements of weight, length, height, and width. The goal of this example is to construct a discriminant function that classifies species based on physical measurements. =⇒ Open the Fish data set. =⇒ Select Analysis I Multivariate Analysis I Discriminant Analysis from the main menu, as shown in Figure 30.1. 416 Chapter 30. Multivariate Analysis: Discriminant Analysis Figure 30.1. Selecting the Discriminant Analysis A dialog box appears as in Figure 30.2. You can select variables for the analysis by using the Variables tab. =⇒ Select Species and click Set Y. =⇒ Select Weight. While holding down the CTRL key, select Length1, Length2, Length3, Height, and Width. Click Add X. Note: Alternately, you can select the variables by using contiguous selection: click on the first variable (Weight), hold down the SHIFT key, and click on the last variable (Width). All variables between the first and last item are selected and can be added by clicking Add X. Figure 30.2. The Variables Tab =⇒ Click the Method tab. The Method tab (Figure 30.3) becomes active. You can use the Method tab to set options in the analysis. Chapter 30. Multivariate Analysis: Discriminant Analysis =⇒ Select Kernel density for Classification method. The options associated with the kernel density classification method become active. =⇒ Select Normal for Kernel. The number of fish in the lake probably varies by species. That is, there is no reason to suspect that the number of whitefish in the lake is the same as the number of perch or bream. In the absence of prior knowledge about the distribution of fish species, you can assume that the number of fish of each species in the lake is proportional to the number in the sample. =⇒ Select Proportional to group sizes for Prior probability of group membership. Figure 30.3. The Method Tab =⇒ Click the Plots tab. The Plots tab (Figure 30.4) becomes active. =⇒ Select Classification fit plot. Figure 30.4. The Plots Tab =⇒ Click OK. The analysis calls the DISCRIM procedure. The procedure uses the options specified in the dialog box. The procedure displays tables in the output document, as 417 418 Chapter 30. Multivariate Analysis: Discriminant Analysis shown in Figure 30.5. Two plots are also created. Move the classification fit plot so that the workspace is arranged as in Figure 30.5. The classification fit plot indicates how well each observation is classified by the discriminant function. For each observation, PROC DISCRIM computes posterior probabilities for membership in each group. Let mi be the maximum posterior probability for the ith observation. The classification fit plot is a plot of − log(mi ) versus i. In Figure 30.5, the selected observations are those with − log(mi ) ≥ 0.1. Equivalently, the maximum posterior probability for membership for the selected observations is less than exp(−0.1) ≈ 0.9. The selected fish are those with relatively large probabilities of misclassification. Conversely, selecting the bream, parkki, and pike species in the spine plot (the upper-right plot in Figure 30.5) shows that the classification criterion discriminates between these species quite well. A spine plot is a one-dimensional mosaic plot in which the width of a bar represent the number of observations in a category. Note: If there are k groups, then the maximum posterior probability of membership is at least 1/k, so the vertical axis of the classification fit plot is bounded above by log(k). Figure 30.5. Output from a Discriminant Analysis The output window contains many tables of statistics. The first table in Figure 30.5 is produced by Stat Studio. It is associated with a heuristic method of choosing the bandwidth for the kernel density classification method. This table is described in the section “The Method Tab” on page 420. Chapter 30. Multivariate Analysis: Discriminant Analysis Figure 30.6 displays a table that summarizes how many fish are classified (or misclassified) into each species. If the discriminant function correctly classifies most observations, then the elements on the table’s main diagonal are large compared to the off-diagonal elements. For this example, the nonparametric discriminant function correctly classified all fish into the species to which they belong. Note: The classification in this example was performed using resubstitution. This estimate of the error rate is optimistically biased. You can obtain a less biased estimate by using cross validation. You can select cross validation for the Classify observations by option on the Method tab. Figure 30.6. Classification of Observations into Groups In summary, the nonparametric discriminant function in this example does an excellent job of discriminating among these species of fish. Specifying the Discriminant Analysis This section describes the dialog box tabs associated with the Discriminant analysis. The Discriminant analysis calls the DISCRIM procedure option. See the DISCRIM procedure documentation in the SAS/STAT User’s Guide for additional details. 419 420 Chapter 30. Multivariate Analysis: Discriminant Analysis The Variables Tab You can use the Variables tab to specify the variables for the analysis. The Variables tab is shown in Figure 30.2. The variable in the Y Variable (Classification) list corresponds to the variable in the CLASS statement of the DISCRIM procedure. This variable must be nominal. The variables in the X Variables list correspond to variables in the VAR statement of the DISCRIM procedure. The Method Tab You can use the Method tab (Figure 30.3) to set options in the analysis. The tab supports the following options. Classification method specifies the method used to construct the discriminant function. Parametric specifies that a parametric method based on a multivariate normal distribution within each group be used to derive a linear or quadratic discriminant function. This corresponds to the METHOD=NORMAL option in the PROC DISCRIM statement. k nearest neighbors specifies that a nonparametric classification method be used. An observation is classified into a group based on the information from the k nearest neighbors of the observation. This corresponds to the METHOD=NPAR K= option in the PROC DISCRIM statement. Kernel density specifies that a nonparametric classification method be used. An observation is classified into a group based on the information from observations within a given radius of the observation. This corresponds to the METHOD=NPAR R= option in the PROC DISCRIM statement. k specifies the number of nearest neighbors for the k nearest neighbors method. You can select a fixed number of observations, or a proportion of the total number of observations. You can type a value in this field or choose from a set of standard values. This option corresponds to the K= or KPROP= option in the PROC DISCRIM statement. Kernel specifies the shape of the kernel function for the Kernel density method. You can specify a uniform, Epanechnikov (quadratic), or normal kernel function. This corresponds to the KERNEL= option in the PROC DISCRIM statement. Bandwidth specifies the bandwidth for the kernel density classification method. This corresponds to the R= option in the PROC DISCRIM statement. There are two options for choosing the bandwidth: Chapter 30. Multivariate Analysis: Discriminant Analysis Maximum of radii that minimizes AMISE of group densities This option uses a heuristic to automatically choose a bandwidth. The “Background” subsection of the “Details” section in the documentation for the DISCRIM procedure presents formulas for the bandwidths that minimize an approximate mean integrated square error of the estimated density within each group. The formulas assume the data within each group are multivariate normal. The optimal radius for each group is determined for each group, as shown in Figure 30.5. Descriptive statistics of the radii are also displayed, including the mean of the radii weighted by the number of observations in each group. The bandwidth used for the R= option in the PROC DISCRIM statement is the maximum of the radii. Manual sets the kernel bandwidth to the value in the Value field. Covariance within groups specifies assumptions about the homogeneity of within-group covariances. This option corresponds to the POOL= option in the PROC DISCRIM statement. For the parametric classification method, the assumption of equal covariances results in a linear discriminant function. The assumption of unequal covariances results in a quadratic discriminant function. Prior probability of group membership specifies assumptions about the prior probabilities of group membership. This option corresponds to the EQUAL and PROPORTIONAL options in the PRIORS statement. Classify observations by specifies a method of classifying observations based on their canonical scores. This option corresponds to the CROSSVALIDATE option in the PROC DISCRIM statement. The Plots Tab You can use the Plots tab (Figure 30.4) to create plots that graphically display results of the analysis. Creating a plot often adds one or more variables to the data table. The following plots are available: Observed groups creates a spine plot (a one-dimensional mosaic plot) of the groups for the Y variable. Observed vs. Predicted groups creates a mosaic plot of the groups for the Y variable versus the group as classified by a discriminant function. Each observation is placed in the group that minimizes the generalized squared distance between the observation and the group mean. 421 422 Chapter 30. Multivariate Analysis: Discriminant Analysis Classification fit plot creates a plot that indicates how well each observation is classified by the discriminant function. This plot is shown in Figure 30.5. The observations selected in the plot have a low posterior probability of group membership. For each observation, PROC DISCRIM computes posterior probabilities for membership in each group. Let mi be the maximum posterior probability for the ith observation. The classification fit plot is a plot of − log(mi ) versus i. The Tables Tab The Tables tab is shown in Figure 30.7. You can use the Tables tab to display the following tables that summarize the results of the analysis. For more information, see the “Displayed Output” subsection of the “Details” section in the documentation for the DISCRIM procedure. Simple statistics specifies whether to display descriptive statistics for the total sample and within each group. This option corresponds to the SIMPLE option in the PROC DISCRIM statement. Univariate ANOVA specifies whether to display univariate statistics for testing the hypothesis that the population group means are equal for each variable. This option corresponds to the ANOVA option in the PROC DISCRIM statement. Multivariate ANOVA specifies whether to display multivariate statistics for testing the hypothesis that the population group means are equal for each variable. This option corresponds to the MANOVA option in the PROC DISCRIM statement. Squared distances between group means specifies whether to display the squared Mahalanobis distances (and associated statistics) between the group means. This option corresponds to the DISTANCE option in the PROC DISCRIM statement. Standardized group means specifies whether to display total-sample and pooled within-group standardized group means. This option corresponds to the STDMEAN option in the PROC DISCRIM statement. Covariance matrices specifies whether to display the correlation matrices for each set of variables. This option corresponds to the BCOV, PCOV, TCOV, and WCOV options in the PROC DISCRIM statement. Correlation matrices specifies whether to display the correlation matrices for each set of variables. This option corresponds to the BCORR, PCORR, TCORR, and WCORR options in the PROC DISCRIM statement. Chapter 30. Multivariate Analysis: Discriminant Analysis Figure 30.7. The Tables Tab In addition to the previous optional tables, the Discriminant analysis always creates the following tables. The name of the table refers to the ODS table name. Counts corresponds to the Counts table. Class level information corresponds to the Levels table. Linear discriminant function corresponds to the LinearDiscFunc table. This table is displayed only for the linear parametric classification method. Number of observations and percent classified corresponds to the ClassifiedResub or ClassifiedCrossVal table. Error count estimates corresponds to the ErrorResub or ErrorCrossVal table. The Output Variables Tab You can use the Output Variables tab (Figure 30.8) to add analysis variables to the data table. If you request a plot that uses one of the output variables, then that variable is automatically created even if you did not explicitly select the variable on the Output Variables tab. The following list describes each output variable and indicates how the output variable is named. Y represents the name of the classification variable. 423 424 Chapter 30. Multivariate Analysis: Discriminant Analysis Posterior probabilities of group membership adds variables named DiscProb– X, where X is the name of an X variable. Predicted groups adds a variable named DiscPred– Y that contains the name of the group to which each observation is assigned. If a classification fit plot is requested on the Plots tab, then a variable named DiscLogProb– Y is created, as described in the section “The Plots Tab” on page 421. Figure 30.8. The Output Tab The Roles Tab You can use the Roles tab to specify a frequency variable or weight variable for the analysis. A frequency variable is a numeric variable whose value represents the frequency of the observation. If you use a frequency variable, the underlying procedure assumes that each observation represents n observations, where n is the value of the frequency variable. A weight variable is a numeric variable with values that are relative weights for the analysis. Analysis of Selected Variables If a nominal variable is selected in a data table when you run the analysis, this variable is automatically entered in the Y Variable (Classification) field of the Variables tab. Any selected interval variables are automatically entered in the X Variables field of the Variables tab. Any variable in the data table with a Frequency or Weight role is automatically entered in the appropriate field of the Roles tab. Chapter 31 Multivariate Analysis: Correspondence Analysis The Correspondence analysis performs simple correspondence analysis, which you can use to analyze frequency data and associations between two or more nominal variables. The correspondence analysis finds a low-dimensional representation of the rows and columns of a contingency table consisting of the counts for the variables. While principal component analysis constructs directions in the space of variables that explain variance, correspondence analysis constructs directions (sometimes called principal coordinates) that explain inertia. Inertia is the total chi-square statistic divided by the total number of observations. Correspondence analysis computes directions that best explain deviations from expected values (assuming no association). The analysis graphically represents each row and column by a point in a configuration plot. You can run the Correspondence analysis by selecting Analysis I Multivariate Analysis I Correspondence Analysis from the main menu. The analysis is implemented by calling the CORRESP procedure in SAS/STAT. See the documentation for the CORRESP procedure in the SAS/STAT User’s Guide for additional details. For a general introduction to correspondence analysis, see Friendly (2000). Example In this example, you examine data from 1991 about 127 companies from five nations in four industries. The companies are from Britain, France, Germany, Japan, and the United States. The companies are in the following industries: automobiles, electronics, food, and oil. =⇒ Open the Business data set. Table 31.1 shows a contingency table of the number of companies in each Industry for each Nation. The goal of this example is to use correspondence analysis to examine relationships between and among the Nation and Industry variables. Table 31.1. Contingency Table of Industry and Nation Industry Automobiles Electronics Food Oil Britain 2 1 11 2 France 3 3 2 2 Nation Germany 5 1 0 1 Japan 14 12 11 5 U.S. 7 11 19 13 426 Chapter 31. Multivariate Analysis: Correspondence Analysis =⇒ Select Analysis I Multivariate Analysis I Correspondence Analysis from the main menu, as shown in Figure 31.1. Figure 31.1. Selecting the Correspondence Analysis A dialog box appears as in Figure 31.2. You can select variables for the analysis by using the Variables tab. In Table 31.1, the levels of Industry specify the rows of the table and are displayed along the vertical dimension of the table. Thus Industry is the Y variable whose values determine the rows. Similarly, Nation is the X variable whose values determine the columns. =⇒ Select Industry and click Add Y. =⇒ Select Nation and click Add X. Figure 31.2. The Variables Tab =⇒ Click the Plots tab. The Plots tab (Figure 31.3) becomes active. =⇒ Select Mosaic plot (single Y only). Chapter 31. Multivariate Analysis: Correspondence Analysis Figure 31.3. The Plots Tab =⇒ Click the Tables tab. The Tables tab (Figure 31.3) becomes active. For this example, it is informative to see how each cell, column, and row of Table 31.1 contributes to the chi-square association statistic for the table. =⇒ Select Contributions to chi-square statistic. Figure 31.4. The Tables Tab =⇒ Click OK. The analysis calls the CORRESP procedure. The procedure uses the options specified in the dialog box. The procedure displays tables in the output document, as shown in Figure 31.5. Two plots are also created. The mosaic plot indicates the frequency count for each cell in the contingency table. You can add labels to the cells of the mosaic plot to make the frequency count more evident. =⇒ Activate the mosaic plot. Press the “l” key (lowercase “L”) to toggle labels. 427 428 Chapter 31. Multivariate Analysis: Correspondence Analysis Figure 31.5. Output from a Correspondence Analysis The mosaic plot shows several interesting facts. The British companies are not evenly divided among industries; many British companies in these data are food companies. Similarly, the lack of German food companies is evident, as is the preponderance of German automobile companies. The United States has the largest proportion of oil companies. Correspondence analysis plots all the categories in a Euclidean space. The first two dimensions of this space are plotted in a configuration plot, shown in the upper-right corner of Figure 31.5. As indicated by the labels for the axes, the first principal coordinate accounts for 75% of the inertia, while the second accounts for almost 20%. Thus, these two principal coordinates account for almost 95% of the inertia in this example. The plot should be thought of as two different overlaid plots, one for each categorical variable. Distances between points within a variable have meaning, but distances between points from different variables do not. The configuration plot summarizes association between categories, and indicates the contribution to the chi-square statistic from each cell. To interpret the plot, start by interpreting the row points: the categories of Industry. The points for food and automobiles are farthest from the origin, so these industries contribute the most to the chi-square statistic. Oil and electronics contribute relatively less to the chi-square statistic. For the column points, the points for the United States, France, and Japan are near the origin, so these countries contribute a relatively small amount to the chi-square Chapter 31. Multivariate Analysis: Correspondence Analysis statistic. The points for Britain and Germany are far from the origin; they make relatively large contributions to the chi-square statistic. The “Contributions to the Total Chi-Square Statistic” table in Figure 31.6 displays the contributions to the chi-square statistic for each industry and country. The last column summarizes the contributions for industry. Automobiles (12.4) and food (10.15) contribute the most, a fact apparent from the configuration plot. Similarly, the last row summarizes the contributions for countries. Britain and Germany make the largest contributions. The “Inertia and Chi-Square Decomposition” table summarizes the chi-square decomposition. The first two components account for almost 95% of the chi-square association. Figure 31.6. Contributions to the Chi-Square Statistic The next series of tables summarize the correspondence analysis for the row variable (Industry). These tables are shown in Figure 31.7. The “Row Coordinates” table displays the coordinates of the various industries in the configuration plot. The “Summary Statistics” table displays various statistics, including the so-called quality of the representation. Categories with low quality values (for example, oil) are not well represented by the two principal coordinates. The quality statistic is equal to the sum of the squared cosines, which are displayed in the last table of Figure 31.7. The squared cosines are the square of the cosines of the angles between each axis and a vector from the origin to the point. Thus, points with a squared cosine near 1 are located near a principal coordinate axis, and so have high quality. The “Partial Contributions to Inertia” table indicates how much of the total inertia is accounted for by each category in each dimension. This table corresponds to the spread of the points in the configuration plot in the horizontal and vertical dimensions. In the first principal coordinate, automobiles and food contribute the most. In the second principal coordinate, electronics contributes the most, although the contributions are more evenly spread across categories. 429 430 Chapter 31. Multivariate Analysis: Correspondence Analysis For further details, see the “Algorithm and Notation” and “Displayed Output” sections of the documentation for the CORRESP procedure. Figure 31.7. Output from a Correspondence Analysis The analysis of the countries is similar. Figure 31.8 shows a partial view of the related statistics. Note that the quality statistic helps explain a seeming discrepancy in the configuration plot (Figure 31.5). From the configuration plot (and from the “Column Coordinates” table), it is apparent that the point representing Japan is closer to the origin than the point representing France. It is tempting to conclude that Japan contributes less to the chi-square statistic than France. But the “Contributions to the Total Chi-Square Statistic” table in Figure 31.6 and the “Partial Contributions to Inertia” table in Figure 31.8 show that the opposite is true. The contradictory evidence can be resolved by noticing that the quality statistic for Japan is only 0.787. That value is the sum of the squared cosines for each dimension. The squared cosine for the second dimension is nearly zero, indicating that Japan’s position is almost completely determined by the first dimension. Chapter 31. Multivariate Analysis: Correspondence Analysis Figure 31.8. Output from a Correspondence Analysis Note that you cannot compare row points with column points in the configuration plot. For example, you cannot compare the distance from the origin for electronics to the distance for Japan and draw any meaningful conclusions. However, you can interpret associations between rows and columns. For example, the first principal coordinate shows a greater association with being British and being a food company than would be expected if these two categories were independent. Similarly, the association between being German and being an automobile company is greater than expected under the assumption of independence. Specifying the Correspondence Analysis This section describes the dialog box tabs associated with the Correspondence analysis. The Correspondence analysis calls the CORRESP procedure option. See the CORRESP procedure documentation in the SAS/STAT User’s Guide for additional details. The Variables Tab You can use the Variables tab to specify the variables for the analysis. The Variables tab is shown in Figure 31.2. The variables in the Y Variables (Row) list corresponds to the row variables in the TABLE statement of the CORRESP procedure. These variables must be nominal. The variables in the X Variables (Col) list corresponds to the column variables in the TABLE statement of the CORRESP procedure. These variables must be nominal. 431 432 Chapter 31. Multivariate Analysis: Correspondence Analysis These variables are used to construct the rows and columns of a contingency table. You can specify a Weight variable on the Roles tab to read category frequencies. The Method Tab You can use the Method tab (Figure 31.9) to set options in the analysis. The tab supports the following options: Cross levels of row variables specifies that each combination of levels for all row variables become a row label. Otherwise, each level of every row variable becomes a row label. This corresponds to the CROSS=ROW option in the PROC CORRESP statement. Cross levels of column variables specifies that each combination of levels for all column variables become a column label. Otherwise, each level of every column variable becomes a column label. This corresponds to the CROSS=COL option in the PROC CORRESP statement. Selecting both of the previous options corresponds to specifying the CROSS=BOTH option in the PROC CORRESP statement; clearing both of the previous options corresponds to specifying the CROSS=NONE option. Missing values specifies whether to include observations with missing values in the analysis. Exclude from analysis specifies that observations with missing values be excluded from the analysis. Use as category levels specifies that missing values be treated as a distinct level of each categorical variable. This corresponds to the MISSING option in the PROC CORRESP statement. Number of dimensions specifies the number of principal coordinates to use for the analysis. You can type a value in this field. If your contingency table is an R × C table, the number of dimensions in the correspondence analysis is at most min(R − 1, C − 1). This corresponds to the DIMENS= option in the PROC CORRESP statement. Standardize coordinates from specifies the standardization for the row and column coordinates. This corresponds to the PROFILE= option in the PROC CORRESP statement. Chapter 31. Multivariate Analysis: Correspondence Analysis Figure 31.9. The Method Tab The Plots Tab You can use the Plots tab (Figure 31.3) to create plots that graphically display results of the analysis. The following plots are available: Mosaic plot (single Y only) creates a mosaic plot of a single Y variable versus the X variables. The mosaic plot is a graphical representation of the contingency table for the data. Configuration plot creates a plot of the first two principal coordinates. These directions account for the greatest deviation from independence. The row and column categories are plotted in these coordinates. Note: The configuration plot is not linked to the original data set because it has a different number of observations. However, you can view the data table underlying this plot by pressing the F9 key when the plot is active. The data are created by the combination of the SOURCE and OUTC= options in the PROC CORRESP statement. 433 434 Chapter 31. Multivariate Analysis: Correspondence Analysis The Tables Tab The Tables tab is shown in Figure 31.4. You can use the Tables tab to display the following tables that summarize the results of the analysis. For more information, see the “Displayed Output” subsection of the “Details” section in the documentation for the CORRESP procedure. Observed frequencies specifies whether to display the contingency table of observed frequencies. This option corresponds to the OBSERVED option in the PROC CORRESP statement. Chi-square expected values specifies whether to display the expected frequencies for the contingency table. This option corresponds to the EXPECTED option in the PROC CORRESP statement. Deviations (observed minus expected) specifies whether to display the difference between the observed and expected frequencies for the contingency table. This option corresponds to the DEVIATIONS option in the PROC CORRESP statement. Contributions to chi-square statistic specifies whether to display contributions to the total chi-square test statistic, including the row and column marginals and the total chi-square statistic. This option corresponds to the CELLCHI2 option in the PROC CORRESP statement. Row and column profiles specifies whether to display row and column profiles. The row profile is the matrix of row-conditional probabilities. The column profile is the matrix of column-conditional probabilities. This option corresponds to the RP and CP options in the PROC CORRESP statement. Inertias and squared cosines specifies whether to display statistics related to inertia and squared cosines. The names of the ODS tables displayed by this option are Inertias, ColBest, ColContr, ColQualMassIn, ColSqCos, RowBest, RowContr, RowQualMassIn, and RowSqCos. In addition to the previous optional tables, the Correspondence analysis always creates the following tables: Row coordinates corresponds to the RowCoors table. Column coordinates corresponds to the ColCoors table. Chapter 31. Multivariate Analysis: Correspondence Analysis The Roles Tab You can use the Roles tab (Figure 31.10) to specify a weight variable or supplementary variables for the analysis. A weight variable is a numeric variable representing category frequencies. In the absence of a weight variable, each observation contributes a value of 1 to the frequency count for its category. That is, each observation represents one subject. When you specify a weight variable, each observation contributes the value of the weighting variable for that observation. For example, a weight of 3 means that the observation represents three subjects. Supplementary variables are displayed as points in the configuration plot, but these variables are not used in computing the correspondence analysis. In other words, a supplementary variable is projected onto the principal coordinate directions, but it is not used to compute the principal coordinates. Note: In the CORRESP procedure, supplementary variables must be listed in the TABLE statement in addition to being listed in the SUPPLEMENTARY statement. In Stat Studio, you should not specify supplementary variables on the Variables tab. As an example of using supplementary variables, suppose you use the Variable Transformation Wizard to create a nominal variable that indicates whether a company is profitable. You can display the levels of this variable in the configuration plot by adding the variable to a supplementary variable list, as shown in Figure 31.10. Figure 31.10. The Roles Tab 435 436 Chapter 31. Multivariate Analysis: Correspondence Analysis Analysis of Selected Variables If a nominal variable is selected in a data table when you run the analysis, this variable is automatically entered in the Y Variables (Row) field of the Variables tab. Any variable in the data table with a Weight role is automatically entered in the appropriate field of the Roles tab. References Friendly, M. (2000), Visualizing Categorical Data, Cary, NC: SAS Institute Inc. Chapter 32 Variable Transformations Transforming data is an important technique in exploratory data analysis. Centering and scaling are simple examples of transforming data. More complex transformations are useful for a variety of purposes. A variable that violates the assumptions of a statistical technique can sometimes be transformed to fit the assumptions better. For example, a variable that is not normally distributed can be transformed in an attempt to improve normality; a variable with nonhomogeneous variance can be transformed in an attempt to improve homogeneity of variance. You can create new variables in a data set by transforming existing variables. Stat Studio provides a Variable Transformation Wizard that enables you to quickly apply standard transformations to your data. These include normalizing transformations (such as logarithmic and power transformations), logit and probit transformations, affine transformations (including centering and standardizing), and rank transformations. You can create your own transformations within the Variable Transformation Wizard by using SAS DATA step syntax and functions. These enable you to recode variables, to create variables with simulated values from known distributions, and to use arbitrarily complex formulas and logical statements to define new variables. Most Stat Studio transformations create a new numerical variable from an existing numerical variable. You can define custom DATA step transformations that use and create variables of any type. You can apply transformations to all observations, or you can apply the transformation only to observations that are included in analyses. Example: A Logarithmic Transformation Many statistical analyses assume that the data are normally distributed. If a variable is not normally distributed, it is often possible to improve normality by using an appropriate transformation of the variable. The three transformations used most often for this purpose are the logarithmic, square root, and inverse transformations. In this example, you apply a logarithmic transformation to the driltime variable of the Miningx data set. Note that the driltime variable is nonnegative, so a logarithmic transformation is well-defined. =⇒ Open the Miningx data set. =⇒ Create a histogram of the driltime variable. 438 Chapter 32. Variable Transformations The histogram is shown in Figure 32.1. Figure 32.1. Histogram of Drilling Time Clearly, the driltime variable is not normally distributed. You might explore whether some transformation of driltime is approximately normal. To begin, you might try a logarithmic transformation. =⇒ Select Analysis I Variable Transformation from the main menu. The Variable Transformation Wizard in Figure 32.2 appears. Figure 32.2. Selecting a Transformation The first page of the wizard enables you to select a transformation family and a Chapter 32. Variable Transformations specific transformation within that family. The logarithmic transformation is available from several items in the Family list, including the Common family. This transformation is of the form log(y + a), so you need to specify the variable y and the parameter a. The transformation log(Y+a) is highlighted by default. Since this is the desired transformation, you can proceed to the next page of the wizard. =⇒ Click Next. The wizard displays the page shown in Figure 32.3. Note that the transformation appears on the page’s title bar. Figure 32.3. Selecting a Variable and a Parameter =⇒ Select the driltime variable, and click Set Y. The parameter a is an offset that is useful if your variable contains nonpositive values. For these data, you can accept the default value of 0. =⇒ Click Next. The wizard displays the page shown in Figure 32.4. You can use this page to specify a variable name (and, optionally, a label) for the new variable. 439 440 Chapter 32. Variable Transformations Figure 32.4. Specifying the Variable Name and Label For this example, you can accept the default variable name. =⇒ Click Finish. Stat Studio adds the new variable, Log– driltime, as the last variable in the data set. You can horizontally scroll through the data table to see the variable. To complete this example, you can visualize the distribution of the new variable. =⇒ Create a histogram of the Log– driltime variable. The histogram (Figure 32.5) shows improved normality, but the transformed data distribution is still skewed to the right. Chapter 32. Variable Transformations Figure 32.5. A Histogram of the Transformed Data Example: A Box-Cox Transformation This example is a continuation of the previous example. The goal is the same: to normalize the driltime variable in the Miningx data set. In the previous example, you tried a logarithmic transformation. Unfortunately, it is often not clear which transformation most improves normality. One strategy is to consider a family of transformations, and to select the transformation within the family for which the transformed data are “most normal.” The Box-Cox family (Box and Cox 1964) is a family of power transformations that includes the logarithmic transformation as a limiting case: ( λ y −1 if λ 6= 0 λ BC(y; λ) = log y if λ = 0 The parameter λ can be chosen by maximizing a log-likelihood function. For details see the section “Normalizing Transformations” on page 446. Note: The Box-Cox parameter is traditionally denoted by λ, as in the previous formula and in the plot in Figure 32.8. However, the Variable Transformation Wizard uses a as a generic notation for a transformation parameter, as shown in Figure 32.7. =⇒ Open the Miningx data set , if it is not already open. =⇒ Select Analysis I Variable Transformation from the main menu. =⇒ Select Normalizing from the Family list. =⇒ Select the Box-Cox(Y;a) transformation from the Transformations list, as shown in Figure 32.6. 441 442 Chapter 32. Variable Transformations Figure 32.6. Selecting a Box-Cox Transformation =⇒ Click Next. The wizard displays the page shown in Figure 32.7. Figure 32.7. Selecting a Variable and Parameters Chapter 32. Variable Transformations =⇒ Select the driltime variable, and click Set Y. By default, the Box-Cox parameter is estimated by maximum likelihood estimation. Alternatively, you can manually specify the parameter. For this example, accept the default method. You could proceed to the next page of the wizard if you wanted to change the default name for the new variable. (The default name is BC– driltime.) For this example, accept the default name and skip the last page of the wizard. =⇒ Click Finish. A graph appears (Figure 32.8) that plots the log-likelihood function as a function of the parameter. An inset gives the lower and upper 95% confidence limits for the maximum log-likelihood estimate, the maximum likelihood estimate (MLE), and a convenient estimate. A convenient estimate is a fraction with a small denominator (such as an integer, a half integer, or an integer multiple of 1/3 or 1/4) that is within the 95% confidence limits about the MLE. Using a convenient estimate sometimes results in a Box-Cox transformation that is more interpretable in terms of the original variable. Note: If there is no convenient estimate within the 95% confidence limits, then the inset does not include this information. Figure 32.8. Plot of Log Likelihood A dialog box (Figure 32.9) also appears that prompts you for a parameter value to use for the Box-Cox transformation. For this example, you are prompted to accept the convenient estimate of –1, even though the MLE estimate is approximately –1.2. 443 444 Chapter 32. Variable Transformations Figure 32.9. Setting the Box-Cox Parameter =⇒ Click OK to accept the value of –1. The parameter –1 specifies the Box-Cox transformation as BC(y, −1) = 1 − y −1 , which is essentially an inverse transformation followed by a reflection and translation. To complete this example, you can visualize the distribution of the new variable. =⇒ Create a histogram of the BC– driltime variable. The histogram is shown in Figure 32.10. The transformed data show improved normality: the distribution is more symmetric and the tails are not as long. Figure 32.10. Histogram of the Box-Cox Transformed Data Chapter 32. Variable Transformations Common Transformations Figure 32.11 shows the transformations that are available when you select Common from the Family list. Equations for these transformations are given in Table 32.1. Figure 32.11. Common Transformations Table 32.1. Description of Common Transformations Transformation log(Y+a) log10(Y+a) sqrt(Y+a) exp(Y) 1/Y Y*Y Default Parameter a=0 a=0 a=0 Name of New Variable Log– Y Log10– Y Sqrt– Y Exp– Y Inv– Y Squared– Y Equation log(Y + a), Y + a > 0 log √ 10 (Y + a), Y + a > 0 Y + a, Y + a > 0 exp(Y ) 1/Y, Y 6= 0 Y2 The logarithmic transformations are often used when the scale of the data range exceeds an order of magnitude. The square root transformation is often used when your data are counts. The inverse transformation is often used to transform waiting times. 445 446 Chapter 32. Variable Transformations Normalizing Transformations Figure 32.12 shows the transformations that are available when you select Normalizing from the Family list. These transformations are often used to improve the normality of a variable. Equations for these transformations are given in Table 32.2. Figure 32.12. Normalizing Transformations Table 32.2. Description of Normalizing Transformations Transformation log(Y+a) log10(Y+a) sqrt(Y+a) exp(Y) power(Y;a) arcsinh(Y) Box-Cox(Y;a) Default Parameter a=0 a=0 a=0 a=1 MLE Name of New Variable Log– Y Log10– Y Sqrt– Y Exp– Y Pow– Y Arcsinh– Y BC– Y Equation log(Y + a), Y + a > 0 log √ 10 (Y + a), Y + a > 0 Y + a, Y + a > 0 exp(Y ) Y a, Y √ > 0 if a is not integral log(Y + Y 2 + 1) See text. The Box-Cox transformation (Box and Cox 1964) is a one-parameter family of power transformations that includes the logarithmic transformation as a limiting case. For Y > 0, ( λ y −1 if λ 6= 0 λ BC(y; λ) = log y if λ = 0 You can specify the parameter, λ, for the Box-Cox transformation, but typically you choose a value for λ that maximizes (or nearly maximizes) a log-likelihood function. Chapter 32. Variable Transformations Stat Studio plots the log-likelihood function versus the parameter, as shown in Figure 32.8. An inset gives the lower and upper 95% confidence limits for the maximum log-likelihood estimate, the MLE estimate, and a convenient estimate. A convenient estimate is a fraction with a small denominator (such as an integer, a half integer, or an integer multiple of 1/3 or 1/4) that is within the 95% confidence limits about the MLE. Although the value of the parameter is not bounded, Stat Studio graphs the log-likelihood function restricted to the interval [−2, 2]. A dialog box (Figure 32.9) also appears that prompts you to enter the parameter value to use for the Box-Cox transformation. The log-likelihood function for the Box-Cox transformation is defined as follows. Write the normalized Box-Cox transformation, z, as ( z(λ; y) = y λ −1 λẏ λ−1 if λ 6= 0 ẏ log y if λ = 0 where ẏ is the geometric mean of y. Let N be the number of nonmissing values, and define R(λ; z) = z0 z − (Σzi )2 /N The log-likelihood function is (Atkinson 1985, p. 87) L(λ; z) = −(N/2) log(R(λ; z)/(N − 1)) Variance Stabilizing Transformations Figure 32.13 shows the transformations that are available when you select Variance stabilizing from the Family list. Variance stabilizing transformations are often used to transform a variable whose variance depends on the value of the variable. For example, the variability of a variable Y might increase as Y increases. Equations for these transformations are given in Table 32.3. 447 448 Chapter 32. Variable Transformations Figure 32.13. Variance Stabilizing Transformations Table 32.3. Description of Variance Stabilizing Transformations Transformation log(Y+a) log10(Y+a) sqrt(Y+a) 1/Y arcsinh(Y) generalized log(Y;a) log-linear hybrid(Y;a) Default Parameter a=0 a=0 a=0 a=0 a=1 Name of New Variable Log– Y Log10– Y Sqrt– Y Inv– Y Arcsinh– Y GLog– Y LogLin– Y Equation log(Y + a), Y + a > 0 log √ 10 (Y + a), Y + a > 0 Y + a, Y + a > 0 1/Y, Y√6= 0 log(Y + √Y 2 + 1) log((Y + Y 2 + a2 )/2) See text. The log-linear hybrid transformation is defined for a > 0 as follows: H(y; a) = y/a + log(a) − 1 if y < a log y if y ≥ a The function is linear for y < a, logarithmic for y > a, and continuously differentiable. The generalized log and the log-linear hybrid transformations were introduced in the context of gene-expression microarray data by Rocke and Durbin (2003). Chapter 32. Variable Transformations Transformations for Proportion Variables Figure 32.14 shows the transformations that are available when you select For proportions from the Family list. These transformations are intended for variables that represent proportions. That is, the Y variable must take values between 0 and 1. You can also use these transformations for percentages if you first divide the percentages by 100. Chapter 7 of Atkinson (1985) is devoted to transformations of proportions. Equations for these transformations are given in Table 32.4. Figure 32.14. Transformations for Proportions Table 32.4. Description of Transformations for Proportions Y ∈ [0, 1) Transformation odds(Y) logit(Y) probit(Y) arcsin(Y) arcsin(sqrt(Y)) folded power(Y;a) Guerrero-Johnson(Y;a) Aranda-Ordaz(Y;a) Default Parameter MLE MLE MLE Name of New Variable Odds– Y Logit– Y Probit– Y Arcsin– Y Angular– Y FPow– Y GJ– Y AO– Y Equation Y /(1 − Y ) log(Y /(1 − Y )) probit(Y ) arcsin(Y √) arcsin( Y ) See text. See text. See text. The probit function is the quantile function of the standard normal distribution. The last three transformations in the list are similar to the Box-Cox transformation described in the section “Normalizing Transformations” on page 446. The 449 450 Chapter 32. Variable Transformations parameter for each transformation is in the unit interval: a ∈ [0, 1]. Typically, you choose a parameter that maximizes (or nearly maximizes) a log-likelihood function. The log-likelihood function is defined as follows. Let N be the number of nonmissing values, and let G(·) be the geometric mean function. Each transformation has a corresponding normalized transformation z(λ; y), to be defined later. Define R(λ; z) = z0 z − (Σzi )2 /N and define the log-likelihood function as L(λ; z) = −(N/2) log(R(λ; z)/(N − 1)) The following sections define the normalized transformation for the folded power, Guerrero-Johnson, and Aranda-Ordaz transformations. In each section, p = y/(1 − y). The Folded Power Transformation The folded power transformation is defined as ( λ y −(1−y)λ if λ 6= 0 λ f(y; λ) = log(p) if λ = 0 The normalized folded power transformation is defined as (Atkinson 1985, p. 139) ( λ y −(1−y)λ if λ 6= 0 λGf (λ) zf (λ; y) = log(p)G(y(1 − y)) if λ = 0 where Gf (λ) = G(y λ−1 + (1 − y)λ−1 ). When you select the folded power transformation, a plot of L(λ; zf ) appears. You should choose a value close to the MLE value. The Guerrero-Johnson Transformation The Guerrero-Johnson transformation is defined as ( λ p −1 if λ 6= 0 λ GJ(y; λ) = log(p) if λ = 0 The normalized Guerrero-Johnson transformation is defined as (Atkinson 1985, p. 145) ( λ p −1 if λ 6= 0 λGGJ (λ) zGJ (λ; y) = log(p)G(y(1 − y)) if λ = 0 where GGJ (λ) = G(y λ−1 /(1 − y)λ+1 ). When you select the Guerrero-Johnson transformation, a plot of L(λ; zGJ ) appears. You should choose a value close to the MLE value. Chapter 32. Variable Transformations The Aranda-Ordaz Transformation The Aranda-Ordaz transformation is defined as ( 2(pλ −1) if λ 6= 0 λ(pλ +1) AO(y; λ) = log(p) if λ = 0 The normalized Aranda-Ordaz transformation is defined as (Atkinson 1985, p. 149) ( pλ −1 if λ 6= 0 λ +1)G (λ) λ(p AO zAO (λ; y) = log(p)G(y(1 − y)) if λ = 0 where GAO (λ) = G(2pλ−1 (1 + p)2 /(pλ + 1)2 ). When you select the Aranda-Ordaz transformation, a plot of L(λ; zAO ) appears. You should choose a value close to the MLE value. Scaling and Translation Transformations Figure 32.15 shows the transformations that are available when you select Scaling/Translation from the Family list. These transformations are used to center and scale a variable. Equations for these transformations are given in Table 32.5. Figure 32.15. Scaling and Translation Transformations Table 32.5. Description of Scaling and Translation Transformations Transformation center(Y) stdize(Y) reflect(Y;a) a+b*Y Default Parameter a=0 a = 0, b = 1 Name of New Variable Center– Y Stdize– Y Reflect– Y Linear– Y Equation Y − mean(Y ) See text. 2a − Y a + bY 451 452 Chapter 32. Variable Transformations The stdize(Y) transformation transforms the data to have zero mean and unit variance. The reflect(Y) transformation reflects the data about the value Y = a. Rank Transformations Figure 32.16 shows the transformations that are available when you select Rank from the Family list. The rank transformation of a variable Y is a new variable containing the ranks of the corresponding values of Y . Figure 32.16. Rank Transformations There are actually four different rank functions, depending on the options you select on the second page of the wizard (Figure 32.17). If you select Assign arbitrarily as the Rank of Ties option, then the SAS/IML RANK function is used to compute ranks. If you select Assign to average, then the SAS/IML RANKTIE function is used. This is summarized in Table 32.6. Chapter 32. Variable Transformations Figure 32.17. Rank Transformations Table 32.6. Description of Rank Transformations Transformation rank(Y) Order Ascending Descending Ascending Descending Rank of Ties Arbitrary Arbitrary Average Average Name of New Variable Rank– Y Rank– Y Rank– Y Rank– Y Equation rank(Y) rank(-Y) ranktie(Y) ranktie(-Y) Lag Transformations Figure 32.18 shows the transformations that are available when you select Lag from the Family list. These transformations are used to compute lagged transformations of a variable’s value. Equations for these transformations are given in Table 32.7. 453 454 Chapter 32. Variable Transformations Figure 32.18. Lag Transformations Table 32.7. Description of Lag Transformations Transformation lag(Y;a) dif(Y;a) Default Parameter a=1 a=1 Name of New Variable Lag– Y Dif– Y Equation lag(Y, a) dif(Y, a) The lag(Y;a) transformation creates a new variable whose ith value is equal to Yi−a for i > a. For i ≤ a, the new variable contains missing values. See the documentation for the LAG function in Base SAS for further details. The dif(Y;a) transformation creates a new variable whose ith value is equal to Yi − Yi−a for i > a. For i ≤ a, the new variable contains missing values. If either Yi or Yi−a is missing, then so is their difference. See the documentation for the DIF function in Base SAS for further details. If some observations are excluded from analyses and you select Only observations in analysis, shown in Figure 32.18, then the lag transformations use only the observations included in analyses. Figure 32.19 presents an example of how these transformations behave when some observations are excluded. In the data table, Y has values 1–5, but observations 3 and 4 are excluded from analyses. The Lag– Y variable is the result of the lag(Y;1) transformation. The third and fourth values are missing because these observations are excluded from analyses. The fifth value of Lag– Y is 2, the previous value of Y that is included in analyses. The Dif– Y variable is the result of the dif(Y;1) transformation. The values are the difference between the first and second columns. Chapter 32. Variable Transformations Figure 32.19. Transformations with Excluded Observations Two-Variable Transformations Figure 32.20 shows the transformations that are available when you select Two Variable from the Family list. The two-variable transformations are used to compute a new variable from standard arithmetic operations on two variables. The arithmetic is performed for each observation. Equations for these transformations are given in Table 32.8. Figure 32.20. Two-Variable Transformations Table 32.8. Description of Two-Variable Transformations Transformation Y*X Y/X a*X+b*Y Default Parameter a = 1, b = 1 Name of New Variable Mult– Y– X Div– Y– X Linear– Y– X Equation YX Y /X aX + bY 455 456 Chapter 32. Variable Transformations Custom Transformations While Stat Studio provides many standard transformations, the most powerful feature of the Variable Transformation Wizard is that you can use the SAS DATA step to create new variables defined by arbitrarily complex formulas. You can define custom transformations after selecting Custom from the Family list in the Variable Transformation Wizard (Figure 32.21). Figure 32.21. Selecting a Custom Transformation The second page of the wizard provides a window for you to enter DATA step code. The wizard displays the page shown in Figure 32.22. Chapter 32. Variable Transformations Figure 32.22. A Window for Entering DATA Step Code You can enter any valid DATA step code into this window, with the following conditions: • The code must begin with a DATA statement. • The code must include a SET statement. • The code must end with a RUN statement. • The code must create an output data set containing the same number of observations as the data table (or the same number as are included in analyses). The data set specified in the SET statement is called the input data set. The data set specified in the DATA statement is called the output data set. Note that the dialog box shown in Figure 32.22 contains a DATA step template satisfying the first three conditions in the previous list. It is up to you to satisfy the last condition by inserting code before the RUN statement. The name of the output data set defaults to – VarTransform; the name of the input data set is automatically generated based on the name of your data table. You can accept these default data set names, or you can enter different names. When you click Finish, the following steps occur: 1. Stat Studio scans the text in the window. If the names of any variables in the current data table are found in the text, then these variables are written to the input data set on the SAS server. 2. The DATA step is executed on the server. This creates the output data set. 457 458 Chapter 32. Variable Transformations 3. The variables in the output data set are compared with the variables in the input data set. (a) Any variables in the output data set that are not in the input data set are copied from the server and added to the current data table. (b) Any variables common to the input and output data sets are compared. If the DATA step changed any values, the new values are copied to the current data table. 4. The input and output data sets are deleted from the server. Each workspace remembers the last custom transformation you entered. If there is an error in your DATA step code, you can again select Analysis I Variable Transformation from the main menu and attempt to correct your error. Custom transformations are not remembered between Stat Studio sessions. Example This example illustrates how to define a custom transformation by using the Variable Transformation Wizard. Note: This example is intended for SAS programmers who are comfortable writing DATA step code. Kimball and Mulekar (2004) analyze the intensification tendency of Atlantic cyclones. This example is based on their analysis and graphics. In this example, you use the Variable Transformation Wizard to write DATA step code that creates a character variable, Tendency, that encodes whether a storm is strengthening or weakening. The Tendency variable is computed by transforming a numeric variable for wind speed. For each observation of each storm, the Tendency variable has the value “Intensifying” when the wind speed is stronger than it was for the previous observation, “Steady” when the wind speed stays the same, and “Weakening” when the wind speed is less than it was for the previous observation. =⇒ Open the Hurricanes data set. The wind speed is contained in the wind– kts variable. Note that the values of the wind– kts variable are rounded to the nearest 5 knots. The name of each storm is contained in the name variable. The data are grouped according to storm name, so an algorithm for creating the Tendency variable is as follows. For each named storm: Compute the difference between the current wind speed and the previous wind speed by using the DIF function in Base SAS. Specify a value for the tendency variable according to whether the difference in wind speed is less than zero, exactly zero, or greater than zero. Chapter 32. Variable Transformations If you were to write a DATA step to create the Tendency variable in a data set, you might write code like the following. The DATA step creates two new variables: a numeric variable called dif– wind– kts and a character variable of length 12 called Tendency. The BY statement is used to loop through the names of cyclones; the NOTSORTED option specifies that the Name variable in the input data set is not sorted in alphabetic order. data WindTendency; set Hurricanes; by name notsorted; length Tendency $12; dif_wind_kts = dif(wind_kts); if first.name then do; Tendency = "Intensifying"; dif_wind_kts = .; end; else do; if dif_wind_kts < 0 then Tendency = "Weakening"; else if dif_wind_kts > 0 then Tendency = "Intensifying"; else Tendency = "Steady"; end; run; The Tendency variable is assigned to “Intensifying” for the first observation of each storm because the storm system was weaker six hours earlier. The dif– wind– kts variable is assigned a missing value for the first observation of each storm because the previous wind speed is unknown. For subsequent storm observations, the dif– wind– kts variable is assigned the results of the DIF function, which computes the difference between the current and previous values of wind– kts. Submitting this DATA step in the Variable Transformation Wizard is easy. No changes are required. =⇒ Select Analysis I Variable Transformation from the main menu. =⇒ Select Custom from the Family list on the left side of the page, as shown in Figure 32.21. =⇒ Click Next. The wizard displays the page shown in Figure 32.22. =⇒ Type the DATA step into the Variable Transformation Wizard, as shown in Figure 32.23. 459 460 Chapter 32. Variable Transformations Figure 32.23. A Custom Transformation =⇒ Click Finish. Stat Studio scans the contents of the window and determines that the name and wind– kts variables are needed by the DATA step. The input data set, Hurricanes, is created in the work library. The input data set contains the name and wind– kts variables. Next, the DATA step executes on the SAS server. The DATA step creates the output data set, WindTendency, which contains the dif– wind– kts and Tendency variables. The dif– wind– kts and Tendency variables are copied from the output data set to the Stat Studio data table. =⇒ Scroll the data table to the extreme right to see the newly created variables. You can now investigate the relationship between the Tendency variable and other variables of interest. =⇒ Create a box plot of latitude versus Tendency. The box plot in Figure 32.24 shows the distribution of latitudes for intensifying, steady, and weakening storms. Intensifying storms tend to occur at more southerly latitudes, whereas weakening storms tend to occur at more northerly latitudes. Chapter 32. Variable Transformations Figure 32.24. Latitude Stratified by Intensification Tendency Applying Normalizing Transformations This section describes some issues to consider when you are applying normalizing transformations. Translating Data The logarithmic and square root transformations are typically most effective at normalizing data that have a minimum value near 1 and have a range that is at most a few orders of magnitude. If a variable consists entirely of large positive values, the transformed data do not show improved normality. For example, if the minimum value of your data is m, you might want to subtract m − 1 from your data as a first step so that the new minimum value is 1. You can translate (and scale) data by using the a+b*Y transformation in the Scaling/Translation family. Alternatively, the square root and logarithmic transformations are defined as log(Y+a) and sqrt(Y+a), so you can specify negative values for the a parameter in these transformations. An example of this is presented in the next section. Skewness Data can be positively or negatively skewed. The transformations commonly used to improve normality compress the right side of the distribution more than the left side. Consequently, they improve the normality of positively skewed distributions. For example, look at the histogram of the min– pressure variable in the Hurricanes data, shown in Figure 32.25. The data are negatively skewed. 461 462 Chapter 32. Variable Transformations Figure 32.25. A Negatively Skewed Variable To improve the normality of these data, you first need to reflect the distribution to make it positively skewed. You can reflect data by using the Reflect(Y;a) transformation in the Scaling/Translation family. Reflecting the data about any point accomplishes the goal of reversing the sign of the skewness. The transformation shown in Figure 32.26 uses a = 1025. Figure 32.26. Defining a Reflection Transformation A histogram of the reflected data is shown in Figure 32.27. Chapter 32. Variable Transformations Figure 32.27. A Histogram of Reflected Data You can now apply a normalizing transformation to the Reflect– min– pressure variable. The minimum value of this variable is 1026. As described in the section “Translating Data” on page 461, you can translate and apply a logarithmic transformation in a single step: select the log(Y+a) transformation with a = −1025. A histogram for the logarithmically transformed variable shows improved normality (Figure 32.28), but it is still far from normal. Figure 32.28. A Histogram of the Logarithm of Reflected Data 463 464 Chapter 32. Variable Transformations Alternatively, you could transform the Reflect– min– pressure variable in two steps: use the a+b*Y transformation with a = −1025 and b = 1, and then apply a normalizing transformation. This technique is recommended for transformations (such as the Box-Cox family) that do not have a built-in translation parameter. References Atkinson, A. C. (1985), Plots, Transformations, and Regression, New York: Oxford University Press. Box, G. E. P. and Cox, D. R. (1964), “An Analysis of Transformations,” Journal of the Royal Statistics Society, Series B, 26, 211–234. Kimball, S. K. and Mulekar, M. S. (2004), “A 15-year Climatology of North Atlantic Tropical Cyclones. Part I: Size Parameters,” Journal of Climatology, 3555–3575. Rocke, D. M. and Durbin, B. P. (2003), “Approximate Variance-Stabilizing Transformations for Gene-Expression Microarray Data,” Bioinformatics, 19(8), 966–972. Chapter 33 Running Custom Analyses The programming language in Stat Studio, which is called IMLPlus, is an enhanced version of the IML programming language. The “Plus” part of the name refers to new features that extend the IML language, including the ability to create and manipulate statistical graphics and to call SAS procedures. You can write programs in IMLPlus to perform analyses not included in Stat Studio. The analyses can be quite complex. In fact, when you use the Stat Studio GUI to select an analysis from the Analysis menu, Stat Studio actually calls an IMLPlus program, so you have already seen examples of what you can accomplish by running IMLPlus programs. Sample Programs Stat Studio is distributed with samples of programs written in IMLPlus. To open these programs, do the following: 1. Select File I Open I File from the main menu. 2. Click Go to Installation directory near the bottom of the dialog box. 3. Double-click the Programs folder. 4. Double-click one of the subfolders: Demos, Doc, or Samples. Navigate additional subfolders as necessary. 5. Select a file with an .sx extension. 6. Click Open. The Demos folder contains advanced programs that demonstrate some of the capabilities of the IMLPlus language. The Doc folder contains introductory programs that are described in Stat Studio for SAS/STAT Users. The Samples folder contains elementary programs that demonstrate how to perform simple tasks in IMLPlus. You can refer to these sample programs as you write more sophisticated programs. 466 Chapter 33. Running Custom Analyses Running a User Analysis from the Main Menu You can create your own custom analyses by writing an IMLPlus program. An introduction to IMLPlus programming is described in Stat Studio for SAS/STAT Users and in the Stat Studio online Help. You can display the online Help by selecting Help I Help Topics from the main menu. When you select Analysis I User Analysis from the main menu, Stat Studio calls a module called UserAnalysis. Stat Studio distributes a sample UserAnalysis module as an example of the sort of analyses that you can write. You can copy and modify the UserAnalysis module to execute your own IMLPlus programs. The following steps run the sample UserAnalysis module. =⇒ Open the Baseball data set. =⇒ Select Analysis I User Analysis from the main menu, as shown in Figure 33.1. Figure 33.1. Running a User Analysis The sample UserAnalysis module displays a simple dialog box (Figure 33.2) containing a list of analyses that you can run on the data. The dialog box displays a list of two analyses. Chapter 33. Running Custom Analyses Figure 33.2. Selecting from a List of Analyses =⇒ Select Create Plot Matrix and click OK. The Create Plot Matrix analysis demonstrates one way to query information from the person running the analysis. In this case, the program prompts you to select several variables to plot. If you select n variables from this list, the variables will be plotted against each other in an (n − 1) × (n − 1) lower-triangular array of plots of the pairwise combination of variables. =⇒ Hold down the CTRL key and select yr– major, cr– atbat, league, and division, as shown in Figure 33.3. Click OK. Figure 33.3. Selecting Variables 467 468 Chapter 33. Running Custom Analyses These four variables are plotted in pairwise combinations, as shown in Figure 33.4. Three different plots are created. Mosaic plots display the relationship between pairs of nominal variables. Box plots are used to plot an interval variable against a nominal variable. Scatter plots display the relationship between pairs of interval variables. Windows along the diagonal display variable names and values of each axis. Figure 33.4. The Results of the Analysis You can modify the UserAnalysis module to call your own analyses from the Analysis I User Analysis menu item. To create your own UserAnalysis module, do the following: 1. Copy the UserAnalysis.sxs file distributed with Stat Studio to your personal modules directory. The UserAnalysis.sxs file is distributed in the Modules\System subdirectory of the Stat Studio installation directory. Your personal modules directory is usually the Modules subdirectory of your personal files directory. (See “The Personal Files Directory” for more information about the personal files directory.) 2. Edit your personal copy of the UserAnalysis.sxs file. Modify the body of the UserAnalysis module so that it performs an analysis of your choosing. 3. Save the file. 4. Select Program I Run to store the module. 5. Open any data set, and choose Analysis I User Analysis to run the module. Chapter 33. Running Custom Analyses The UserAnalysis module must take a DataObject variable as its single argument. When you select Analysis I User Analysis, the module is called. The currently active DataObject is used as the argument to the module. Table 33.1 lists a few of the methods in the DataObject class. You might find these methods useful in writing your analyses. These and other IMLPlus class methods are documented in the Stat Studio online Help, in the “DataObject” section of the “IMLPlus Class Reference” chapter. Table 33.1. Frequently Used DataObject Methods Method AddAnalysisVar GetNumObs GetSelectedObsNumbers GetSelectedVarNames GetVarData IsNominal IsNumeric SelectObs SetMarkerColor SetMarkerShape Description Adds a new variable to the DataObject. Returns the number of observations in the DataObject. Gets the index of observations selected in the DataObject. Gets the names of variables selected in the DataObject. Gets the data for a variable in the DataObject. Returns true if the named variable is nominal. Returns true if the named variable is numeric. Selects observations in the DataObject. Sets the color of observation markers. Sets the shape of observation markers. For example, you could modify the body of the UserAnalysis module to include the following statements. If you select a nominal variable from a data table and then select Analysis I User Analysis, these statements assign a distinct marker shape to each unique value of the nominal variable. (If there are more unique values than marker shapes, the shapes are reused.) The NCOL, UNIQUE, LOC, and MOD functions are all part of SAS/IML, as are the IF and DO statements. start UserAnalysis(DataObject dobj); dobj.GetSelectedVarNames(VarName); /* get selected var name */ if ncol(VarName) = 0 then return; /* return if no selected variable */ if dobj.IsNominal(VarName) then do;/* if it is nominal... */ shapes = MARKER_SQUARE || MARKER_PLUS || MARKER_CIRCLE || MARKER_DIAMOND ||MARKER_X || MARKER_TRIANGLE || MARKER_INVTRIANGLE || MARKER_STAR; dobj.GetVarData(VarName, x); /* get the data */ ux = unique(x); /* find the unique values */ do i = 1 to ncol(ux); /* for each unique value... */ idx = loc(x = ux[i]); /* find obs with that value */ iShape = 1 + mod(i-1, 8); /* choose next shape (mod 8) */ /* set the shape of the relevant observations */ dobj.SetMarkerShape(idx, shapes[iShape]); end; end; finish; store module=UserAnalysis; 469 470 Chapter 33. Running Custom Analyses Action Menus You can create a custom menu for a plot and associate one or more IMLPlus statements with each item on the menu. Such a menu is referred to as an action menu. To display a plot’s action menu, press F11 while the plot’s window is active. Selecting an item on the menu executes the IMLPlus statements associated with that item. Several previous chapters use action menus to run an analysis on a plot. For example, Figure 12.14 and Figure 18.9 show action menus attached to plots. Action menus are described in the Stat Studio online Help, in the section called “The Action Menu” in the chapter titled “The Plots.” As an example, the following statements create a histogram and attach an action menu to the plot. When the menu item is selected, the module PrintMean is executed. If the X variable is numeric, then the PrintMean module gets the data associated with the X variable of the plot and computes the mean value of these data. x = normal( j(100,1,1) ); declare Histogram plot; plot = Histogram.Create("Histogram", x); plot.AppendActionMenuItem("Print Mean", "run PrintMean();"); /* Press F11 in the plot window and select the menu item. */ /* module to run when menu item is selected */ start PrintMean(); declare Plot plot; plot = DataView.GetInitiator(); /* get the active plot */ plot.GetVars(ROLE_X, VarName); /* get the X var name */ declare DataObject dobj; dobj = plot.GetDataObject(); /* get the DataObject */ if dobj.IsNumeric(VarName) then do; dobj.GetVarData(VarName, x); /* get the X values */ mean = x[:]; /* compute the mean */ print "The mean X value is " mean; end; finish; Chapter 34 Configuring the Stat Studio Interface You can configure many aspects of Stat Studio, including the following: • the appearance of GUI items, such as toolbars • the behavior of the program editor • the default SAS server • the default positions of Stat Studio windows, such as graphs, data tables, and output documents • the directories that Stat Studio searches when trying to locate Java classes, data files, matrices, and modules • the location of your personal files directory This chapter describes configuring Stat Studio by using the Options dialog box. You can open the Options dialog box by selecting Tools I Options from the main menu. If you change options in the Options dialog box, the changes apply to all workspaces. Some changes affect only new workspaces. Stat Studio Window Types Stat Studio provides the following different types of windows. Program Window A program window is an editor for IMLPlus programs. For each program window, Stat Studio creates a workspace. There is always a one-to-one correspondence between a program window and a workspace. It is not possible to have two program windows share a single workspace, nor is it possible to have a single program window connected to more than one workspace. Program windows provide the following features: • color coding of IMLPlus keywords, string literals, comments, and constants • automatic indention of program statements • drag-and-drop text editing • positioning the cursor at the source of a program error • following errors into IML modules • multilevel undo and redo • bookmarks 472 Chapter 34. Configuring the Stat Studio Interface • finding and replacing text Error Log Window An error log window reports warnings and errors from analyses, and programming errors that occur when you run a program. Output Document Window An output document window displays output from analyses, output from the SAS/IML PRINT statement, and output from programs that you run. The output window supports the Microsoft rich text format (RTF), so you can paste graphical objects (including Stat Studio graphics) from the Windows clipboard into the output document. Data View Window A data view is a generic name for a data table or a plot. Data views that display common data are linked together, meaning that selections made in one view are displayed in all views of the same data. Auxiliary Input Window An auxiliary input window is a secondary programming window that is linked to the main program window. The IMLPlus PAUSE statement pauses the main program, creates an auxiliary input window, and waits until you click Resume. You can use the auxiliary input window as a debugging tool or to prompt for user input. For example, the following program prompts for user input: pause "Enter starting value in x. Example: x=10;"; do while ( x > 0 ); print x; x = x - 1; end; When this program is executed, the auxiliary input window appears with the PAUSE statement’s message displayed, as shown in Figure 34.1. You can then type the statement x=10; into the Input box, and click Resume. IMLPlus executes the statement to create the matrix x, and then resumes execution of the main program from the line following the PAUSE statement. Chapter 34. Configuring the Stat Studio Interface Figure 34.1. The Auxiliary Input Window General Options You can configure aspects of the Stat Studio GUI. If you select Tools I Options from the main menu, the Options dialog box appears. By default, the General tab is active, as shown in Figure 34.2. Figure 34.2. The General Tab The General tab has the following fields: Show toolbar specifies whether to display the toolbar below the main menu. You can use the toolbar to initiate commonly used actions. 473 474 Chapter 34. Configuring the Stat Studio Interface Show workspace bar specifies whether to display the workspace bar at the bottom of Stat Studio’s main window. You can use the workspace bar to switch between different Stat Studio workspaces. Show status bar specifies whether to display the status bar at the bottom of Stat Studio’s main window. The status bar displays a short message, such as an error message or a description of a menu item. Show icons in main menu specifies whether to display icons on the main Stat Studio menus (File, Edit, View, etc.). Large buttons specifies whether to display the buttons on the main toolbar in a large size. Show welcome dialog box specifies whether to display the Welcome dialog box, shown in Figure 34.3, when you start Stat Studio. Configure Java Runtime Environment enables you to select the Java runtime environment for Stat Studio. If you click this button, the dialog box in Figure 34.4 appears. Figure 34.3. The Welcome Dialog Box Chapter 34. Configuring the Stat Studio Interface Figure 34.4. Configuring the Java Runtime Environment Program Editor Options You can configure aspects of the Stat Studio program editor. The program editor is used to write and debug IMLPlus programs. IMLPlus programming is described in Stat Studio for SAS/STAT Users and in the Stat Studio online Help. You can display the online Help by selecting Help I Help Topics from the main menu. To display the Program Editor tab (shown in Figure 34.5), select Tools I Options from the main menu, and click Program Editor. Figure 34.5. The Program Editor Tab 475 476 Chapter 34. Configuring the Stat Studio Interface The Program Editor tab has the following fields: Enable automatic indention specifies whether the program editor automatically indents new lines to match the indention of the previous line. Enable syntax color coding specifies whether the program editor color-codes keywords, string literals, comments, and predefined IMLPlus constants. Allow cursor movement past end of line specifies whether you can move the cursor beyond an end-of-line character in the program editor. Insert tabs/spaces specifies whether the program editor inserts a tab character or space characters when you press the TAB key, and when the program editor automatically indents a line. Tab width specifies the width (in characters) of the tab positions. Display tabs as visible characters specifies whether the program editor displays each tab character as the symbol . Output Options You can configure aspects of the way that Stat Studio displays output in the output document. Output from SAS procedures is sent to the output document when you run analyses. To display the Output tab (shown in Figure 34.6), select Tools I Options from the main menu, and click Output. Chapter 34. Configuring the Stat Studio Interface Figure 34.6. The Output Tab The Output tab has a single option. If you select Always append program output to end of output document, then output from SAS procedures and IMLPlus programs is always added at the bottom of the output document. If you clear this option, then output is inserted into the output document at the current cursor position. Runtime Options You can configure aspects of the Stat Studio programming environment. To configure default options for new program windows, select Tools I Options from the main menu, and click the Runtime tab. This tab is shown in Figure 34.7. 477 478 Chapter 34. Configuring the Stat Studio Interface Figure 34.7. The Runtime Tab The Runtime tab has the following fields: Statement Mode specifies that the program environment defaults to Statement Mode. For information about Statement Mode, see the Stat Studio online Help. You can display the online Help by selecting Help I Help Topics from the main menu. Warn if running an out-of-date stored module specifies that a warning message is printed to the error log when an IMLPlus program executes an out-of-date module. An out-of-date module is one whose source code has been changed since the module was last stored by using the IML STORE statement. To change these options for a currently open workspace, select Program I Configure from the main menu, and click the Runtime tab. Server Options The PC running Stat Studio is called the client. The computer running SAS is called the SAS server. You can specify the default SAS server that Stat Studio should use. Different workspaces can be connected to different servers. To configure the default server for new workspaces, select Tools I Options from the main menu, and click the Server tab. This tab is shown in Figure 34.8. Chapter 34. Configuring the Stat Studio Interface Figure 34.8. The Server Tab The Server tab enables you to specify which SAS server is the default server for new workspaces. After you select a server, click the Set button. To change the SAS server for a currently open workspace, select Program I Configure from the main menu, and click the Server tab. Windows Options You can configure the default positioning of each Stat Studio window type. Stat Studio provides the following types of windows: • program windows • error log windows • output document windows • data view windows (plots and data tables) Stat Studio assigns two properties to each type of window. These properties are as follows: Auto Position specifies a default window position. Auto Hide specifies that the window is hidden when not attached to the active workspace. Error log windows always have the Auto Hide property. 479 480 Chapter 34. Configuring the Stat Studio Interface In addition, output document windows have a third property: Auto Close specifies that an output document window is automatically closed when the last associated workspace is closed. (Note that output document windows can be attached to multiple workspaces.) To change a property for an existing window, click on the icon in the window’s title bar. This displays the Control menu, as shown in Figure 34.9. (You can also display the Control menu for the active window by pressing ALT+HYPHEN.) You can use this menu to toggle the Auto Position, Auto Hide, and (for an output document window) Auto Close properties. Figure 34.9. A Control Menu You can configure the default window properties for each type of window. Select Tools I Options from the main menu, and click the Windows tab. This tab is shown in Figure 34.10. Chapter 34. Configuring the Stat Studio Interface Figure 34.10. The Windows Tab You can select an Auto Position location for all window types. This specifies the default location for a window. Caution: If you create multiple windows of the same type (for example, two graphs), then the second window is positioned on top of the first. Move the topmost window to reveal the window hidden beneath. You can select Auto hide for all window types except error log windows. A window with this property is hidden when it is not attached to the current workspace. This means that if you change to a different workspace, the windows associated with the previous workspace disappear from view. Error log windows always have this property; they appear only in the workspace to which they are attached. You can select Auto activate for error log windows. This causes the error log window to open and become the active window when an error occurs. You can select Auto close for output document windows. This causes the output document window to close when you close the last workspace to which it is attached. (Note that output document windows can be attached to multiple workspaces.) Directory and Search Path Options You can configure the directories that Stat Studio searches when trying to locate Java classes, data files, matrices, and modules. Select Tools I Options from the main menu, and click the Directories tab. This tab is shown in Figure 34.11. 481 482 Chapter 34. Configuring the Stat Studio Interface Figure 34.11. The Directories Tab The Directories tab has the following fields: Show directories for specifies the type of file (Java classes, data files, matrices, or modules) that the search path applies to. Search Path specifies the directories to search when Stat Studio tries to find the indicated type of file. The directories are searched in the order listed. Add opens the Browse for Folder dialog box (Figure 34.12). When you select a directory, the directory name is added to the Search Path list. Remove removes the selected directory from the Search Path list. Move Up moves the selected directory up one position in the Search Path list. The directories in the list are searched in order, from top to bottom, so to reduce search time you should position frequently used directories near the top of the list. Caution: Do not change the relative positions of the four standard entries. Move Down moves the selected directory down one position in the Search Path list. Set Default copies the selected directory into the Default Storage Directory field. Chapter 34. Configuring the Stat Studio Interface Default Storage Directory specifies the directory in which to store modules or matrices when an IMLPlus program executes a STORE statement. To change this field, click Change or Set Default. Personal Files Directory specifies the personal files directory. To change this field, click Change. The personal files directory is described in the section “The Personal Files Directory” on page 485. Figure 34.12. The Browse for Folder Dialog Box Example: Changing the Search Path for Data Files In this section, you add a new directory to the search path for data files. Data files include SAS data sets (with extensions sd6 or sas7bdat) and Microsoft Excel files (with extension xls). When you try to load an IMLPlus matrix (with extension imx), Stat Studio searches the directories in the search path for matrices. If the file is not found, Stat Studio searches the directories in the search path for data files. Assume that you have SAS data sets in a directory on your PC. The following steps add this directory to the beginning of the search path for data sets. =⇒ Select Tools I Options from the main menu, and click the Directories tab. The Directories tab is shown in Figure 34.11. =⇒ Select Data Files from the Show directories for list. =⇒ Click Add. The Browse for Folder dialog box appears. =⇒ Navigate to the directory containing your data, as shown in Figure 34.13. Click OK. 483 484 Chapter 34. Configuring the Stat Studio Interface Figure 34.13. Changing the Search Path The directory is appended to the end of the Search Path list, as shown in Figure 34.14. Figure 34.14. Adding a New Directory =⇒ Click Move Up twice. The directory is now at the beginning of the Search Path list, as shown in Figure 34.15. =⇒ Click OK to apply the changes. Chapter 34. Configuring the Stat Studio Interface Figure 34.15. The New Search Path The Personal Files Directory The first time you run Stat Studio, a personal files directory called My Stat Studio Files is created. By default, the personal files directory corresponds to the Windows directory shown in Table 34.1. Table 34.1. The Personal Files Directory Windows XP Windows Vista C:\Documents and Settings\userid\My Documents\ My Stat Studio Files C:\Users\userid\Documents\My Stat Studio Files It is recommended that you store the files you create with Stat Studio in subdirectories of the personal files directory. This provides the following advantages: • Each person who logs on to the computer has a unique personal files directory. • The personal files directory keeps your files separate from files distributed with Stat Studio. • If all your Stat Studio files are in subdirectories of the personal files directory, it is easier for you to back up your files. • When you open a file by selecting File I Open I File from the main menu, the dialog box contains a button that lets you navigate directly to the personal files directory. 485 486 Chapter 34. Configuring the Stat Studio Interface In the personal files directory, Stat Studio creates the following subdirectories: Classes directory for user-written Java classes Data Sets directory for SAS data sets Matrices directory for IMLPlus matrices stored on the client computer Modules directory for IMLPlus modules Programs directory for IMLPlus programs Example: Changing the Personal Files Directory If you want to change the location of your personal files directory, follow the steps in this section. =⇒ Select Tools I Options from the main menu, and click the Directories tab. The Directories tab is shown in Figure 34.11. =⇒ Click Change next to the Personal Files Directory field. The Browse for Folder dialog box appears, as shown in Figure 34.12. =⇒ Select the directory you want to become your new personal files directory, and click OK. A message box appears, as shown in Figure 34.16. You are asked whether you want to create the standard subdirectories in this new personal files directory. Figure 34.16. A Message Box =⇒ Usually, you will want to respond to this prompt by clicking Yes. =⇒ Click OK to close the Options dialog box. Note: When you change the location of the personal files directory, Stat Studio does not move files from the previous personal files directory location. You must move the files yourself. Appendix A Sample Data Sets Stat Studio is distributed with several data sets. These data sets are used in this documentation to demonstrate the capabilities and features of Stat Studio. To open any data sets described in this section, do the following: 1. Select File I Open I File from the main menu. A dialog box appears. 2. Click Go to Installation directory near the bottom of the dialog box. 3. Double-click on the Data Sets folder. 4. Select a data set. 5. Click Open. The following sections describe the Stat Studio sample data sets. Air Data The Air data set contains measurements of pollutant concentrations from a city in Germany during a week in November 1989. The following list describes each variable. datetime date and hour, in SAS datetime format day day of the week hour hour of the day co carbon monoxide concentration o3 ozone concentration so2 sulfur dioxide concentration no nitrogen oxide concentration dust dust concentration wind wind speed, in knots 488 Appendix A. Sample Data Sets Baseball Data The Baseball data set contains performance measures and salary levels for regular hitters and leading substitute hitters in Major League Baseball for the year 1986 (Reichler 1987). There is one observation per hitter. The following list describes each variable. name player’s name no– atbat no– hits number of times at bat (in 1986) no– home no– runs number of home runs (in 1986) no– rbi no– bb number of runs batted in (in 1986) yr– major years in the major leagues cr– atbat career at-bats cr– hits career hits cr– home career home runs cr– runs career runs cr– rbi career runs batted in cr– bb career bases on balls league player’s league at the end of 1986 division player’s division at the end of 1986 team player’s team at the end of 1986 position positions played (in 1986) no– outs no– assts number of putouts (in 1986) no– error salary number of errors (in 1986) number of hits (in 1986) number of runs (in 1986) number of bases on balls (in 1986) number of assists (in 1986) salary, in thousands of dollars (in 1986) Appendix A. Sample Data Sets The position variable in the Baseball data set is encoded as follows: 13 1B 1O 23 2B 2S 32 3B 3O 3S C CD CF first base, third base first base first base, outfield second base, third base second base second base, shortstop third base, second base third base third base, outfield third base, shortstop catcher center field, designated hitter center field CS DH DO LF O1 OD OF OS RF S3 SS UT center field, shortstop designated hitter designated hitter, outfield left field outfield, first base outfield, designated hitter outfield outfield, shortstop right field shortstop, third base shortstop utility Business Data The Business data set contains information about publicly held German, Japanese, and U.S. companies in the automotive, chemical, electronics, and oil refining industries in 1991. There is one observation for each company. The following list describes each variable. nation nationality of the company industry principal business of the company employs number of employees sales sales for 1991, in millions of dollars profits profits for 1991, in millions of dollars Caribbean Data The Caribbean data set contains geographical data for countries in the western Atlantic Ocean. The data are used to create a map of the Caribbean islands. To create a map, plot lat versus lon, and select ID and segment as ID (grouping) variables. The following list describes each variable. ID country code identifier segment segment code identifier for a country lon longitude of each point of a country segment lat latitude of each point of a country segment 489 490 Appendix A. Sample Data Sets Central America Data The CentralAmerica data set contains geographical data for countries in Central America. The data are used to create a map of Central America. To create a map, plot lat versus lon, and select ID and segment as ID (grouping) variables. The following list describes each variable. ID country code identifier segment segment code identifier for a country lon longitude of each point of a country segment lat latitude of each point of a country segment Climate Data The Climate data set contains geographical and meteorological data for certain cities in the 48 contiguous states of the United States. The following list describes each variable. station name of city containing the weather station longitude longitude of city latitude latitude of city elevationFeet elevation of city, in feet above mean sea level JanMaxF average maximum temperature in January, in degrees Fahrenheit JanMinF average minimum temperature in January, in degrees Fahrenheit AprMaxF average maximum temperature in April, in degrees Fahrenheit AprMinF average minimum temperature in April, in degrees Fahrenheit JulMaxF average maximum temperature in July, in degrees Fahrenheit JulMinF average minimum temperature in July, in degrees Fahrenheit OctMaxF average maximum temperature in October, in degrees Fahrenheit OctMinF average minimum temperature in October, in degrees Fahrenheit extremeMaxF highest recorded temperature, in degrees Fahrenheit extremeMinF lowest recorded temperature, in degrees Fahrenheit JanAvePrecipIn average precipitation in January, in inches FebAvePrecipIn average precipitation in February, in inches Appendix A. Sample Data Sets MarAvePrecipIn average precipitation in March, in inches AprAvePrecipIn average precipitation in April, in inches MayAvePrecipIn average precipitation in May, in inches JunAvePrecipIn average precipitation in June, in inches JulAvePrecipIn average precipitation in July, in inches AugAvePrecipIn average precipitation in August, in inches SepAvePrecipIn average precipitation in September, in inches OctAvePrecipIn average precipitation in October, in inches NovAvePrecipIn average precipitation in November, in inches DecAvePrecipIn average precipitation in December, in inches totalAvePrecipIn average total annual precipitation, in inches Drug Data The Drug data set contains results of an experiment to evaluate drug effectiveness (Afifi and Azen 1972). Four drugs were tested against three diseases on six subjects; there is one observation for each test. The following list describes each variable. drug drug used in treatment disease disease identifier chang– bp change in systolic blood pressure due to treatment Fish Data The Fish data set contains measurements of 159 fish caught in Finland’s Lake Laengelmavesi (Journal of Statistics Education Data Archive 2006). The following list describes each variable. species species of fish weight weight of the fish, in grams length1 length of the fish from the nose to the beginning of the tail, in centimeters length2 length of the fish from the nose to the notch of the tail, in centimeters length3 length of the fish from the nose to the end of the tail, in centimeters height maximum height of the fish, in centimeters 491 492 Appendix A. Sample Data Sets width maximum width of the fish, in centimeters In addition to these variables, the data set contains the following transformed variables. cubeRootWeight cube root of the weight scaledLength1 the ratio length1 / cubeRootWeight scaledLength2 the ratio length2 / cubeRootWeight scaledLength3 the ratio length3 / cubeRootWeight scaledHeight the ratio height / cubeRootWeight scaledWidth the ratio width / cubeRootWeight logLengthRatio logarithm of the ratio length3 / length1 GPA Data The GPA data set contains data collected to determine which applicants at a large midwestern university were likely to succeed in its computer science program (Campbell and McCabe 1984). There is one observation per student. The following list describes each variable. gpa grade point average of students in the computer science program hsm average high school grade in mathematics hse average high school grade in English hss average high school grade in science satm score on the mathematics section of the SAT satv score on the verbal section of the SAT sex student’s gender Hurricanes Data The U.S. National Hurricane Center records intensity and track information for tropical cyclones at six-hour intervals. The Hurricanes data set is an “extended best-track” (EBT) data set that adds six measured size parameters to the best-track data. The data were prepared by DeMaria, Pennington, and Williams (2004). The cyclones from 1988 to 2003 are included. The version distributed with Stat Studio is Version 1.6, released February 2004. An earlier version of the EBT data was analyzed in Mulekar and Kimball (2004) and Kimball and Mulekar (2004). The data as assembled by DeMaria include the following variables. Appendix A. Sample Data Sets name storm name date date of observation, in SAS date format hms time of observation (UTC), in SAS time format latitude latitude of observation, in degrees north latitude longitude longitude of observation. Note: DeMaria encodes this variable as degrees west longitude. For ease of plotting, this variable is recoded as a (usually) negative value in degrees east longitude. wind– kts min– pressure maximum low-level sustained wind speed, in knots minimum central sea-level pressure, in hPa pressure– outer– isobar pressure of outer closed isobar, in hPa radius– eye radius of eye (if an eye exists), in nautical miles. Note: A nautical mile is one minute of latitude, or approximately 1.15 statute miles. radius– max– wind radius at which maximum wind speed was measured, in nautical miles radius– 64kt average radius of 64-knot (hurricane strength) winds, in nautical miles radius– 50kt radius– 34kt average radius of 50-knot winds, in nautical miles radius– outer– isobar storm– type radius of outer closed isobar, in nautical miles month month day day of the month time time of day (UTC) year year ID storm identification number radius– 34kt– ne radius of 34-knot (tropical storm strength) winds northeast of the storm’s center, in nautical miles radius– 34kt– se radius of 34-knot (tropical storm strength) winds southeast of the storm’s center, in nautical miles radius– 34kt– sw radius of 34-knot (tropical storm strength) winds southwest of the storm’s center, in nautical miles radius– 34kt– nw radius of 34-knot (tropical storm strength) winds northwest of the storm’s center, in nautical miles radius– 50kt– ne radius of 50-knot winds northeast of the storm’s center, in nautical miles average radius of 34-knot (tropical storm strength) winds, in nautical miles indicator of whether the system was purely tropical, subtropical, or extra-tropical 493 494 Appendix A. Sample Data Sets radius– 50kt– se radius of 50-knot winds southeast of the storm’s center, in nautical miles radius– 50kt– sw radius of 50-knot winds southwest of the storm’s center, in nautical miles radius– 50kt– nw radius of 50-knot winds northwest of the storm’s center, in nautical miles radius– 64kt– ne radius of 64-knot (hurricane strength) winds northeast of the storm’s center, in nautical miles radius– 64kt– se radius of 64-knot (hurricane strength) winds southeast of the storm’s center, in nautical miles radius– 64kt– sw radius of 64-knot (hurricane strength) winds southwest of the storm’s center, in nautical miles radius– 64kt– nw radius of 64-knot (hurricane strength) winds northwest of the storm’s center, in nautical miles The storm– type variable is encoded as follows: ∗ W D S E L Tropical system Tropical wave Tropical disturbance Subtropical storm Extra-tropical storm Remnant low In addition to these variables, the data set contains the following variables, suggested in the analyses of Mulekar and Kimball (2004) and Kimball and Mulekar (2004). Missing values were converted to the SAS missing value. category indicator variable corresponding to the Saffir-Simpson wind intensity scale wind– mph maximum low-level sustained wind speed, in miles per hour. This variable is computed as wind– kts times 1.15. radius– 64kt average of nonmissing values of the 64-knot radii in the northeast, southeast, southwest, and northwest directions radius– 50kt average of nonmissing values of the 50-knot radii in the northeast, southeast, southwest, and northwest directions radius– 34kt average of nonmissing values of the 34-knot radii in the northeast, southeast, southwest, and northwest directions The category variable is encoded according to the value of wind– kts (wind speed) as in Table A.1. Appendix A. Sample Data Sets Table A.1. The Saffir-Simpson Intensity Scale Category TD TS Cat1 Cat2 Cat3 Cat4 Cat5 Description Tropical Depression Tropical Storm Category 1 Hurricane Category 2 Hurricane Category 3 Hurricane Category 4 Hurricane Category 5 Hurricane Wind Speed (knots) 22–33 34–63 64–82 83–95 96–113 114–134 135 or greater Iris Data The Iris data set is Fisher’s iris data (Fisher 1936). Sepal and petal size were measured for 50 specimens from each of three species of iris. There is one observation per specimen. The following list describes each variable. sepallen sepal length, in millimeters sepalwid sepal width, in millimeters petallen petal length, in millimeters petalwid petal width, in millimeters species species of iris Mining Data The Mining data set contains the results of an experiment to determine whether drilling time was faster for wet drilling or dry drilling (Penner and Watts 1991). Tests were replicated three times for each method at different test holes. There is one observation per five-foot interval for each replication. The following list describes each variable. depth depth of the hole, in feet driltime time to drill the last five feet of the current depth, in minutes method drilling method, wet or dry rep replicate number Miningx Data The Miningx data set is a subset of the Mining data set. It contains data from only one of the test holes. 495 496 Appendix A. Sample Data Sets Neuralgia Data Neuralgia is pain that follows the path of specific nerves. Neuralgia is most common in elderly persons, but it can occur at any age. The Neuralgia data set contains data on 60 patients. These data are hypothetical, but they are similar to data reported by Layman, Agyras, and Glynn (1986). Two test treatments and a placebo are compared. The response variable is Pain, which has the value “No” if the patient reports no pain or a substantial lessening of pain, and the value “Yes” if the patient still experienced pain after treatment. The explanatory variables are as follows: Treatment treatment administered. “A” and “B” represent the two test treatments. “P” represents the placebo treatment. Sex gender of the patient Age age of the patient, in years, when treatment began Duration duration of complaint, in months, before the treatment began Patient Data The Patient data set contains data collected on cancer patients (Lee 1974). There is one observation per patient. The response variable is remiss, which has the value 1 if the patient experienced cancer remission, and 0 otherwise. The explanatory variables are the results from blood tests and physiological measurements on each patient. The variables are rescaled. The explanatory variables are cell, smear, infil, li, blast, and temp. PRDSALE Data The PRDSALE data set is also distributed in the SASHELP library. The data are artificial; the data set is typically used for resolving technical support issues. The following list describes each variable. actual revenue from the sale of an item of furniture, in dollars predict predicted revenue from the sale, in dollars country country in which the item was sold region region in which the item was sold prodtype product type product item of furniture quarter quarter of year in which the item was sold Appendix A. Sample Data Sets year year in which the item was sold month month in which the item was sold Ship Data The Ship data set contains data from an investigation of wave damage to cargo ships (McCullagh and Nelder 1989). The purpose of the investigation was to set standards for hull construction. There is one observation per ship. The following list describes each variable. type type of ship year year of construction period period of operation months aggregate months of service y number of damage incidents States48 Data The States48 data set contains geographical data for the 48 contiguous states in the United States. The data are used to create a map of the continental United States. To create a map, plot lat versus lon, and select state and segment as ID (grouping) variables. The following list describes each variable. state state code identifier segment segment code identifier for a state postal postal code identifier for a state lon longitude of each point of a state segment, in degrees west longitude lat latitude of each point of a state segment, in degrees north latitude References Afifi, A. A. and Azen, S. P. (1972), Statistical Analysis: A Computer-Oriented Approach, New York: Academic Press. Campbell, P. F. and McCabe, G. P. (1984), “Predicting the Success of Freshmen in a Computer Science Major,” Communications of the ACM, 27, 1108–1113. DeMaria, M., Pennington, J., and Williams, K. (2004), “Description of the Extended Best Track File,” Version 1.6, ftp://ftp.cira.colostate.edu/demaria/ebtrk/ (accessed March 1, 2004). 497 498 Appendix A. Sample Data Sets Fisher, R. A. (1936), “The Use of Multiple Measurements in Taxonomic Problems,” Annals of Eugenics, 7, 179–188. Journal of Statistics Education Data Archive (2006), “Fish Catch data set (1917),” http://www.amstat.org/publications/jse/jse_data_ archive.html. Kimball, S. K. and Mulekar, M. S. (2004), “A 15-year Climatology of North Atlantic Tropical Cyclones. Part I: Size Parameters,” Journal of Climatology, 3555–3575. Layman, P. R., Agyras, E., and Glynn, C. J. (1986), “Iontophoresis of Vincristine versus Saline in Post-herpetic Neuralgia: A Controlled Trial,” Pain, 25, 165–170. Lee, E. T. (1974), “A Computer Program for Linear Logistic Regression Analysis,” Computer Programs in Biomedicine, 80–92. McCullagh, P. and Nelder, J. A. (1989), Generalized Linear Models, Second Edition, London: Chapman & Hall. Mulekar, M. S. and Kimball, S. K. (2004), “The Statistics of Hurricanes,” STATS, 39, 3–8. Penner, R. and Watts, D. G. (1991), “Mining Information,” The American Statistician, 45(1), 4–9. Reichler, J. L., ed. (1987), The 1987 Baseball Encyclopedia Update, New York: Macmillan. Appendix B SAS/INSIGHT Features Not Available in Stat Studio The following list presents general features of SAS/INSIGHT that are not included in Stat Studio. • SAS/INSIGHT can be launched from SAS DMS mode (from the SAS program editor, from the Solutions I Analysis menu, or from the SAS command line). • SAS/INSIGHT shares the libraries and catalogs defined in DMS mode. • SAS/INSIGHT automatically recomputes analyses (including curves on graphs) and statistics if data are changed. • SAS/INSIGHT supports recording an interactive session for later playback. The following list presents features of SAS/INSIGHT data views (tables and plots) that are not included in Stat Studio. • SAS/INSIGHT supports multiple plots in a single window. • SAS/INSIGHT supports “renewing” a plot or analysis. • SAS/INSIGHT provides GUI support for animation. • SAS/INSIGHT supports changing the orientation of plots. • SAS/INSIGHT supports changing the formats of table cells after the table is created. • SAS/INSIGHT supports saving tables to data sets after they are created. • SAS/INSIGHT supports changing the attributes of a curve after it is created. • SAS/INSIGHT supports user-defined formats. • SAS/INSIGHT provides a “Tools window” for rapidly changing attributes of markers and curves. • SAS/INSIGHT provides a mechanism to set a common view range for all plots that display a given variable. • SAS/INSIGHT can put multiple plots (for example, BY-group plots and scatter plot matrices) into a single window. The following list presents features of SAS/INSIGHT analyses that are not included in Stat Studio. 500 Appendix B. SAS/INSIGHT Features Not Available in Stat Studio • SAS/INSIGHT supports adding or deleting curves, graphs, variables, and tables from existing analyses without explicitly rerunning the analysis. • SAS/INSIGHT supports “group” variables for the analysis of BY-groups. • SAS/INSIGHT supports “freezing” an analysis for easy comparison with subsequent analyses. • SAS/INSIGHT provides sliders for interactively varying parameters in models. • SAS/INSIGHT supports creating a parametric CDF. • SAS/INSIGHT supports a kernel smoother for scatter plot smoothing. • SAS/INSIGHT supports maximum redundancy analysis. • SAS/INSIGHT supports biplots for many multivariate analyses. Index A action menu, 182, 240 action menus, 470 active window, 21 AddAnalysisVar method, 469 adding observations, 30 variables, 28 aggregate, 310 Air data set, 79, 487 Akaike information criterion, 242 analysis menu, 187 not enabled, 325 animation, 499 annotations deleting, 122 inserting, 120 properties, 122 ANOVA, 280, 411, 422 AppendActionMenuItem method, 470 ASCII order, 45, 155 aspect ratio, 123, 126, 144 auto close property, 480 auto hide property, 479 auto position property, 479 auxiliary input window, 472 axes changing range, 145 changing tick marks, 145 labels, 148 location, 102 properties, 147 setting common view range, 185 axis area, 129 axis label area, 129 B bar charts, 12, 53 properties, 55 Baseball data set, 267, 285, 353, 374, 466, 487 bin tool, 62, 119 biplots, 362, 366, 500 box plots, 18, 63 displaying means, 126 displaying notches, 126 displaying serifs, 126 properties, 65 Business data set, 69, 425, 489 BY groups, 155, 173, 174 BY variables, 173 BY-group analysis, 500 BY-group plots, 182 copying to output doc, 184 layout, 184 not linked to original data, 182 writing to files, 184 C CANCORR procedure, 389 CANDISC procedure, 399 canonical components, 399 canonical correlation analysis, 389 canonical discriminant analysis, 399 canonical variables, 389 Caribbean data set, 489 CDF plot parametric, 500 CDF plots, 208, 213, 214 CentralAmerica data set, 489 changing contours, 109 chi-square residuals, 310 chi-squared (χ2 ) symbol, 153 classification criterion, 415 classification fit plots, 409, 422 classification variables, 297, 303, 317, 336 client, 478 Climate data set, 99, 106, 490 closing windows, 170 color blend, 76, 125 colors of lines, 80 of markers, 41, 76, 133 predefined, 125 column headings, 31 column variables, 431 common factors, 371 communality, 372 comparing smoothers, 237 complement of selected observations, 125 confidence ellipses, 410 confidence interval displacement diagnostic, 311 confidence intervals, 200, 339 confidence levels, 351 confidence limits for means, 243, 252, 262 confidence limits for parameters, 280 configuration plots, 428, 433 configuring Stat Studio, 471 confirmatory data analysis, 3 502 Index context areas, 129 context menus, 31, 129 contiguous selection, 354, 374, 400, 416 contingency tables, 69 contour plots, 105 properties, 113 contours changing, 109 levels, 114 styles, 114 control menu, 480 convenient estimate, 443 Cook’s D statistic, 272, 279, 327, 338 copying data, 47 plots, 124, 172 CORR procedure, 343 correlation, 21, 75 pairwise, 350 partial, 350 correlation analysis, 343 correlation matrix in correlation analysis, 351 in factor analysis, 384 in principal component analysis, 355 reduced, 373 correlation pattern plots, 357, 365 CORRESP procedure, 425 correspondence analysis, 425 covariance matrix in correlation analysis, 351 in factor analysis, 384 in principal component analysis, 355 covariance ratio, 277, 279 creating data, 25 curve attributes, 499 custom analysis, 466 cyclones, 11 D data copying, 47 creating, 25 editing, 25 saving, 28, 48 subsetting, 47 data smoothing loess, 233 polynomial regression, 257 thin-plate spline, 247 data tables, 31 creating new from selected data, 151 properties, 49 data views, 16 DataObject methods AddAnalysisVar, 469 GetNumObs, 469 GetSelectedObsNumbers, 469 GetSelectedVarNames, 469 GetVarData, 469 IsNominal, 469 IsNumeric, 469 SelectObs, 469 SetMarkerColor, 469 SetMarkerShape, 469 DataObject.SetVarValueOrder method, 159 DataView methods AppendActionMenuItem , 470 GetDataObject, 470 GetInitiator, 470 default label variables, 139 delete annotations, 122 design points, 251 deviance residuals, 310 DFBETAS, 313, 341 DFFIT statistic, 279 DIFCHISQ statistic, 310 DIFDEV statistic, 311 DISCRIM procedure, 399, 415 discriminant analysis, 415 discriminant function, 420 dispersion, 338 distribution analysis descriptive statistics, 187 distributional modeling, 203 frequency counts, 217 location and scale statistics, 195 outlier detection, 225 dmm file, 48 Drug data set, 318, 491 dynamically linked, 2, 16 E editing data, 25 observations, 30 effects, 297, 303, 337 crossed, 304 factorial, 306 main, 304 multivariate polynomial, 308 nested, 305 polynomial, 307 reordering, 309 eigenvalues, 356, 365, 366, 378, 384 eigenvectors, 357, 366, 386 error log window, 472 events, 303, 336 events/trials syntax, 303 examining selected observations, 47, 228 exclude from analyses, 39, 42, 125 exclude from plots, 14, 39, 42, 125 excluding observations, 153 analyses not rerun, 154 plots recomputed, 154 explanatory variables, 267 exploratory data analysis, 2, 3, 11 extended selection, 14, 65 Index F factor analysis, 371 factor plots, 372 FACTOR procedure, 372 factor spaces, 372 finding observations, 43 Fish data set, 400, 415, 491 font, 141 footnote, 144 format, 27, 46 freezing an analysis, 500 FREQ procedure, 217 frequency role, 32 frequency variables, 33 G generalized cross validation, 242, 251 generalized squared distance, 415 GENMOD procedure, 317 GetDataObject method, 470 GetInitiator method, 470 GetNumObs method, 469 GetSelectedObsNumbers method, 469 GetSelectedVarNames method, 469 GetVarData method, 469 GetVars method, 470 Gini’s mean difference, 230 global selection mode, 160, 164 goodness-of-fit test, 223 GPA data set, 390, 492 gradient colormap, 89 graph area, 129 margins, 123, 144 properties, 143 graphical filtering, 165 group mean vector, 405 group variables, 79, 82 H hat matrix, 277 Help I Help Topic, 2 Heywood case, 373, 384 hiding windows, 170 high leverage points, 277 HISTOGRAM statement, 212 histograms, 15, 57 anchor, 60 bin tool, 62 bin width, 60 binning, 60, 62 properties, 59 Hurricanes data set, 11, 53, 57, 63, 74, 93, 174, 187, 195, 203, 217, 226, 258, 343, 492 I IMLPlus, 2, 465 include in analyses, 39, 42, 125 include in plots, 14, 39, 42, 125 including observations, 155 inertia, 425 influence diagnostics, 277 informat, 45 input data set, 457 insert annotations, 120 interaction tools, 117 interquartile range, 230 Iris data set, 495 IsNominal method, 469 IsNumeric method, 469 iterative reweighting, 243 K kernel bandwidth, 191 kernel density estimate, 191 kernel smoother, 500 keyboard shortcuts in data tables, 51 in plots, 125 kurtosis, 192 L label role, 32 label variables, 138 labeling observations, 138 labels, 125 large left arrow, 85, 110–112, 114 layout, 176, 184 level tool, 119 leverage points, 285 leverage statistic, 277, 279, 311 line plots, 78 changing line properties, 127 properties, 85 selecting line, 127 setting line color, 127 lines colors, 80 selecting, 82 styles, 80 link function, 317, 337 local regression, 233 local selection mode, 160, 164 local sorting, 49 location estimates, 200, 229 location parameter, 225 LOESS procedure, 233 log-linear model, 327 LOGISTIC procedure, 297 M MAD, See median absolute deviation Mahalanobis distance, 294, 363 markers attributes, 177 changing size, 126, 133 changing size difference, 126, 133 coloring, 125 503 504 Index colors, 41, 76, 133 properties, 41 shapes, 41, 76, 130 sizes, 76 maximum likelihood estimate, 443 maximum likelihood estimation, 212, 297 maximum redundancy analysis, 500 mean, 192 measure level, 29, 33 median absolute deviation, 195, 230 metadata, 48 Mining data set, 495 Miningx data set, 233, 247, 437, 441, 495 missing values, 45, 193, 220, 331, 345, 432 in bar charts, 14, 55 in box plots, 19, 65 MLE, See maximum likelihood estimation model fitting generalized linear models, 317 linear regression, 267 logistic regression, 297 robust regression, 285 modes, 200 mosaic plots, 69 properties, 72 multivariate analysis canonical correlation analysis, 389 canonical discriminat analysis, 399 correlation analysis, 343 correspondence analysis, 425 discriminant analysis, 415 factor analysis, 371 principal component analysis, 353 N Neuralgia data set, 298 normal density, 207 normalizing transformations, 437 notches, 65 O oblique rotations, 385 observation inspector, 123 multiple observations, 124 scrolling, 124 observation inspector mode, 123 observations adding, 30 editing, 30 excluding, 153 finding, 43 including, 155 labeling, 138 labels, 41, 141 properties, 38 selecting, 39 sorting, 37 observations menu, 38 observer view, 160 of the intersection, 160 of the union, 160 offset variables, 314, 317, 328, 329, 341 online Help, 2 ordering, 155 by data, 156, 158 by frequency count, 156, 157 missing values, 156 nominal variables, 33 ordinary least squares regression, 267 orientation of plots, 499 orthogonal rotations, 385 Other threshold, 56, 72 Others category, 126 outliers, 225, 285 output data set, 457 output document, 184, 476 output document window, 472 overdispersion, 334 overplotting, 95, 135, 174 P pairwise correlation, 350 pan tool, 118 parameter estimates, 280, 294, 339 parameterization, 310, 338 parametric distributions, 212, 213 partial correlation, 350 partial leverage, 279 partial leverage plots, 273 partial variables, 350, 364, 383, 395 pasting plots, 124 Patient data set, 496 pattern plots, 386 PAUSE statement, 472 personal files directory, 483, 485 changing the location, 486 players, 487 plot area, 129 margins, 142, 143 properties, 142 values at edges, 143 Plot methods GetVars, 470 plots copying, 124, 172 not linked to original data, 345, 433 pasting, 124 regions, 129 Poisson regression, 327 pollutants, 487 polygon plots, 87 coloring regions, 88 filling polygons, 127 properties, 90 power transformations, 441 PRDSALE data set, 496 prediction ellipses, 347, 350, 410 Index RANK function, 452 RANKTIE function, 452 RD plots, 289 rebinning, 119 reduced correlation matrix, 373 reference lines, 125, 140 REG procedure, 257, 267 removing smoothers, 240 renewing a plot, 499 reset plot view, 119 residual plots, 243, 252, 263, 274, 279, 293, 310, 338 response distribution, 337 response variables, 267 robust distance, 294 robust regression algorithm, 292 ROBUSTREG procedure, 285 ROC curve, 310 role frequency, 32 label, 32 weight, 32 rotating buttons, 94 rotating plots, 93 properties, 101, 127 rotating, 127 row headings, 31 row variables, 431 comparing, 237 loess, 237 removing, 240 scatter plots, 20, 74 matrix, 346, 350 properties, 76 score plots, 359, 365, 395, 410 scree plots, 365, 385 scrolling selected observations into view, 50 search path, 482 select tool, 117 selecting lines, 82 observations, 39 selection rectangle, 16, 65 SelectObs method, 469 selector view, 160, 164 limit, 165 serifs, 65 server, 7, 478 SetMarkerColor method, 469 SetMarkerShape method, 469 shape parameter, 210 Ship data set, 328, 497 show only selected observations, 76, 126, 135, 174 single-trial syntax, 303 singular value decomposition, 362 skewness, 192 slicing, 135 sliders, 500 smoothing criterion, 244 sorting observations, 37 span, 353, 371, 394, 406 spin tool, 119 spine plots, 409, 418, 421 standard deviation, 230 statement mode, 478 States48 data set, 497 status bar, 474 STORE statement, 483 studentized residuals, 277, 279, 294 subsetting data, 47, 151 supplementary variables, 435 surface drawing modes, 102 surface plots, 99 S T Saffir-Simpson Intensity Scale, 12, 53 sample programs, 465 SAS servers, 7, 478 SAS/INSIGHT, 5, 35 saving data, 28, 48 plots, 184 saving tables, 499 scale estimates, 200, 230 scale multiplier, 225, 230 scale parameter, 210, 225 scatter plot smoothers TABLES statement, 223 testing for normality, 208 threshold parameters, 210 ticks adjusting, 60 anchor, 147 major, 147 minor, 147 range of, 148 title, 144 tolerance, 44 tool bar, 473 prediction limits, 262 PRESS residuals, 277, 279 principal component analysis, 353 principal components, 353 automatic selection, 366 principal coordinates, 425 PRINCOMP procedure, 353 prior probability, 408 program editor, 475 program window, 471 programming language, 465 Q Q-Q plots, 208, 213, 214, 243, 253, 263, 275, 279, 293 quantiles, 192 R 505 506 Index tools window, 499 TPSPLINE procedure, 247 transformations Aranda-Ordaz, 451 Box-Cox, 441 common, 445 custom, 456 folded power, 450 for proportion variables, 449 Guerrero-Johnson, 450 inverse, 445 issues to consider, 461 lag, 453 logarithmic, 437 normalizing, 437, 446 rank, 452 scaling and translation, 451 square root, 445 two-variable, 455 variance stabilizing, 447 trials, 303, 336 trimmed mean, 200 Type 1 sequential analysis, 339 Type 3 statistic, 340 U unicode characters, v unique factors, 371 UNIVARIATE procedure, 187, 195, 203, 225 user analysis, 466 user-defined formats, 499 UserAnalysis module, 466 V variable transformation wizard, 437 variables adding, 28 BY, 173 canonical, 389 classification, 297, 303, 317, 336 explanatory, 267 frequency, 33 group, 79, 82 label, 138 offset, 314, 317, 328, 329, 341 partial, 350, 364, 383, 395 properties, 32 response, 267 roles, 32 supplementary, 435 weight, 33 WITH, 350, 395 variables menu, 32 variance, 192 W weight role, 32 weight variables, 33 welcome dialog, 474 whiskers, 63, 65 windows clipboard, 124, 172 Windows Device Independent Bitmap Format (BMP), 172 Windows Enhanced 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