# The PLM Procedure SAS/STAT User’s Guide (Book Excerpt)

® SAS/STAT 9.22 User’s Guide The PLM Procedure (Book Excerpt) SAS® Documentation This document is an individual chapter from SAS/STAT® 9.22 User’s Guide. The correct bibliographic citation for the complete manual is as follows: SAS Institute Inc. 2010. SAS/STAT® 9.22 User’s Guide. Cary, NC: SAS Institute Inc. Copyright © 2010, SAS Institute Inc., Cary, NC, USA All rights reserved. Produced in the United States of America. 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). 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Chapter 66 The PLM Procedure Contents Overview: PLM Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5408 Basic Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5408 PROC PLM Contrasted with Other SAS Procedures . . . . . . . . . . . . . 5409 Getting Started: PLM Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410 Syntax: PLM Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5417 PROC PLM Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418 EFFECTPLOT Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421 ESTIMATE Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5422 FILTER Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5423 LSMEANS Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5425 LSMESTIMATE Statement . . . . . . . . . . . . . . . . . . . . . . . . . . 5426 SCORE Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5427 SHOW Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5429 SLICE Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5431 TEST Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5431 WHERE Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5432 Details: PLM Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5434 BY Processing and the PLM Procedure . . . . . . . . . . . . . . . . . . . . 5434 Analysis Based on Posterior Estimates . . . . . . . . . . . . . . . . . . . . 5434 User-Defined Formats and the PLM Procedure . . . . . . . . . . . . . . . . 5436 ODS Table Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5437 ODS Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5438 Examples: PLM Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5438 Example 66.1: Scoring with PROC PLM . . . . . . . . . . . . . . . . . . . 5438 Example 66.2: Working with Item Stores . . . . . . . . . . . . . . . . . . . 5440 Example 66.3: Group Comparisons in Ordinal Model . . . . . . . . . . . . 5442 Example 66.4: Posterior Inference for Binomial Data . . . . . . . . . . . . 5444 Example 66.5: By-Group Processing . . . . . . . . . . . . . . . . . . . . . 5449 Example 66.6: Comparing Multiple B-Splines . . . . . . . . . . . . . . . . 5454 Example 66.7: Linear Inference with Arbitrary Estimates . . . . . . . . . . 5460 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5463 5408 F Chapter 66: The PLM Procedure Overview: PLM Procedure The PLM procedure performs postfitting statistical analyses for the contents of a SAS item store that was previously created with the STORE statement in some other SAS/STAT procedure. An item store is a special SAS-defined binary file format used to store and restore information with a hierarchical structure. The statements available in the PLM procedure are designed to reveal the contents of the source item store via the Output Delivery System (ODS) and to perform postfitting tasks such as the following: testing hypotheses computing confidence intervals producing prediction plots scoring a new data set The use of item stores and PROC PLM enables you to separate common postprocessing tasks, such as testing for treatment differences and predicting new observations under a fitted model, from the process of model building and fitting. A numerically expensive model fitting technique can be applied once to produce a source item store. The PLM procedure can then be called multiple times and the results of the fitted model analyzed without incurring the model fitting expenditure again. The PLM procedure offers the most advanced postprocessing techniques available in SAS/STAT software. These techniques include step-down multiplicity adjustments for p-values, F tests with order restrictions, analysis of means (ANOM), and sampling-based linear inference based on Bayes posterior estimates. The following procedures support the STORE statement for the generation of item stores that can be processed with the PLM procedure: GENMOD, GLIMMIX, GLM, LOGISTIC, MIXED, ORTHOREG, PHREG, SURVEYLOGISTIC, SURVEYPHREG, and SURVEYREG. For details about the STORE statement, see the section “STORE Statement” on page 529 of Chapter 19, “Shared Concepts and Topics.” Basic Features The PLM procedure, unlike most SAS/STAT procedures, does not operate primarily on an input data set. Instead, the procedure requires you to specify an item store with the SOURCE= option in the PROC PLM statement. The item store contains the necessary information and context about the statistical model that was fit when the store was created. SAS data sets are used only to provide input information in some circumstances. For example, when scoring a data set or when computing least squares means with specially defined population margins. In other words, instead of reading raw data and fitting a model, the PLM procedure reads the results of a model having been fit. PROC PLM Contrasted with Other SAS Procedures F 5409 In order to interact with the item store and to reveal its contents, the PLM procedure supports the SHOW statement which converts item store information into standard ODS tables for viewing and further processing. The PLM procedure is sensitive to the contents of the item store. For example, if a BAYES statement was in effect when the item store was created, the posterior parameter estimates are saved to the item store so that the PLM procedure can perform postprocessing tasks by taking the posterior distribution of estimable functions into account. As another example, for item stores that are generated by a mixed model procedure using the Satterthwaite or Kenward-Roger (Kenward and Roger 1997) degrees-of-freedom method, these methods continue to be available when the item store contents are processed with the PLM procedure. Because the PLM procedure does not read data and does not fit a model, the processing time of this procedure is usually considerably less than the processing time of the procedure that generates the item store. PROC PLM Contrasted with Other SAS Procedures In contrast to other analytic procedures in SAS/STAT software, the PLM procedure does not use an input data set. Instead, it retrieves information from an item store. Some of the statements in the PLM procedure are also available as postprocessing statements in other procedures. Table 66.1 lists SAS/STAT procedures that support the same postprocessing statements as PROC PLM does. Table 66.1 SAS/STAT Procedures with Postprocessing Statements Similar to PROC PLM GENMOD GLIMMIX GLM LOGISTIC MIXED ORTHOREG PHREG SURVEYLOGISTIC SURVEYPHREG SURVEYREG EFFECTPLOT p p p ESTIMATE p p p p p p p p p p LSMEANS p p p p p p p p p p LSMESTIMATE p p SLICE p p p p p p p p p p p p p p p p p Table entries marked with indicate procedures that support statements with the same functionality p as in PROC PLM. Those entries marked with indicate procedures that support statements with same names but different syntaxes from PROC PLM. You can find the most comprehensive set of features for these statements in the PLM procedure. For example, the LSMEANS statement is available in all of the listed procedures. For example, the ESTIMATE statement available in the GENMOD, GLIMMIX, GLM and MIXED procedures does not support all options that PROC PLM supports, such as multiple rows and multiplicity adjustments. TEST p p p p p p p 5410 F Chapter 66: The PLM Procedure The WHERE statement in other procedures enables you to conditionally select a subset of the observations from the input data set so that the procedure processes only the observations that meet the specified conditions. Since the PLM procedure does not use an input data set, the WHERE statement in the PLM procedure has different functionality. If the item store contains information about By groups—that is, a BY statement was in effect when the item store was created—you can use the WHERE statement to select specific BY groups for the analysis. You can also use the FILTER statement in the PLM procedure to filter results from the ODS output and output data sets. Getting Started: PLM Procedure The following DATA step creates a data set from a randomized block experiment with a factorial treatment structure of factors A and B: data BlockDesign; input block a b y datalines; 1 1 1 56 1 1 2 1 2 1 50 1 2 2 1 3 1 39 1 3 2 2 1 1 30 2 1 2 2 2 1 36 2 2 2 2 3 1 33 2 3 2 3 1 1 32 3 1 2 3 2 1 31 3 2 2 3 3 1 15 3 3 2 4 1 1 30 4 1 2 4 2 1 35 4 2 2 4 3 1 17 4 3 2 ; @@; 41 36 35 25 28 30 24 27 19 25 30 18 The GLM procedure is used in the following statements to fit the model and to create a source item store for the PLM procedure: proc glm class model store run; data=BlockDesign; block a b; y = block a b a*b / solution; sasuser.BlockAnalysis / label='PLM: Getting Started'; The CLASS statement identifies the variables Block, A, and B as classification variables. The MODEL statement specifies the response variable and the model effects. The block effect models the design effect, and the a, b, and a*b effects model the factorial treatment structure. The STORE statement requests that the context and results of this analysis be saved to an item store named sasuser.BlockAnalysis. Because the SASUSER library is specified as the library name of the item store, the store will be available after the SAS session completes. The optional label in the STORE statement identifies the store in subsequent analyses with the PLM procedure. Getting Started: PLM Procedure F 5411 Note that having BlockDesign as the name of the output store would not create a conflict with the input data set name, because data sets and item stores are saved as files of different types. Figure 66.1 displays the results from the GLM procedure. The “Class Level Information” table shows the number of levels and their values for the three classification variables. The “Parameter Estimates” table shows the estimates and their standard errors along with t tests. Figure 66.1 Class Variable Information, Fit Statistics, and Parameter Estimates The GLM Procedure Class Level Information Levels Values block 4 1 2 3 4 a 3 1 2 3 b 2 1 2 R-Square Coeff Var Root MSE y Mean 0.848966 15.05578 4.654747 30.91667 Parameter Intercept block block block block a a a b b a*b a*b a*b a*b a*b a*b Class Estimate 1 2 3 4 1 2 3 1 2 1 1 2 2 3 3 1 2 1 2 1 2 20.41666667 17.00000000 4.50000000 -1.16666667 0.00000000 3.25000000 4.75000000 0.00000000 0.50000000 0.00000000 7.75000000 0.00000000 7.25000000 0.00000000 0.00000000 0.00000000 B B B B B B B B B B B B B B B B Standard Error t Value Pr > |t| 2.85043856 2.68741925 2.68741925 2.68741925 . 3.29140294 3.29140294 . 3.29140294 . 4.65474668 . 4.65474668 . . . 7.16 6.33 1.67 -0.43 . 0.99 1.44 . 0.15 . 1.66 . 1.56 . . . <.0001 <.0001 0.1148 0.6704 . 0.3391 0.1695 . 0.8813 . 0.1167 . 0.1402 . . . The following statements invoke the PLM procedure and use sasuser.BlockAnalysis as the source item store: proc plm source=sasuser.BlockAnalysis; run; 5412 F Chapter 66: The PLM Procedure These statements produce Figure 66.2. The “Store Information” table displays information that is gleaned from the source item store. For example, the store was created by the GLM procedure at the indicated time and date, and the input data set for the analysis was WORK.BLOCKDESIGN. The label used earlier in the STORE statement of the GLM procedure also appears as a descriptor in Figure 66.2. Figure 66.2 Default Information The PLM Procedure Store Information Item Store Label Data Set Created From Created By Date Created Response Variable Class Variables Model Effects SASUSER.BLOCKANALYSIS PLM: Getting Started WORK.BLOCKDESIGN PROC GLM 13JAN10:12:58:48 y block a b Intercept block a b a*b Class Level Information Class Levels block a b 4 3 2 Values 1 2 3 4 1 2 3 1 2 The “Store Information” table also echoes partial information about the variables and model effects that are used in the analysis. The “Class Level Information” table is produced by the PLM procedure by default whenever the model contains effects that depend on CLASS variables. The following statements request a display of the fit statistics and the parameter estimates from the source item store and a test of the treatment main effects and their interactions: proc plm source=sasuser.BlockAnalysis; show fit parms; test a b a*b; run; Getting Started: PLM Procedure F 5413 The statements produce Figure 66.3. Notice that the estimates and standard errors in the “Parameter Estimates” table agree with the results displayed earlier by the GLM procedure, except for small differences in formatting. Figure 66.3 Fit Statistics, Parameter Estimates, and Tests of Effects The PLM Procedure Fit Statistics MSE Error df 21.66667 15 Parameter Estimates Effect Intercept block block block block a a a b b a*b a*b a*b a*b a*b a*b block a b Estimate Standard Error 1 2 1 2 1 2 1 2 20.4167 17.0000 4.5000 -1.1667 0 3.2500 4.7500 0 0.5000 0 7.7500 0 7.2500 0 0 0 2.8504 2.6874 2.6874 2.6874 . 3.2914 3.2914 . 3.2914 . 4.6547 . 4.6547 . . . 1 2 3 4 1 2 3 1 1 2 2 3 3 Type III Tests of Model Effects Effect a b a*b Num DF Den DF F Value Pr > F 2 1 2 15 15 15 7.54 8.38 1.74 0.0054 0.0111 0.2097 Since the main effects, but not the interaction are significant in this experiment, the subsequent analysis focuses on the main effects, in particular on the effect of variable A. The following statements request the least squares means of the A effect along with their pairwise differences: proc plm source=sasuser.BlockAnalysis seed=3; lsmeans a / diff; lsmestimate a -1 1, 1 1 -2 / uppertailed ftest; run; 5414 F Chapter 66: The PLM Procedure The LSMESTIMATE statement tests two linear combinations of the A least squares means: equality of the first two levels and whether the sum of the first two level effects equals twice the effect of the third level. The FTEST option in the LSMESTIMATE statement requests a joint F tests for this two-row contrast. The UPPERTAILED option requests that the F test also be carried out under one-sided order restrictions. Since F tests under order restrictions (chi-bar-square statistic) require a simulation-based approach for the calculation of p-values, the random number stream is initialized with a known seed value through the SEED= option in the PROC PLM statement. The results of the LSMEANS and the LSMESTIMATE statement are shown in Figure 66.4. Figure 66.4 LS-Means Related Inference for A Effect The PLM Procedure a Least Squares Means a Estimate Standard Error DF t Value Pr > |t| 1 2 3 32.8750 34.1250 25.7500 1.6457 1.6457 1.6457 15 15 15 19.98 20.74 15.65 <.0001 <.0001 <.0001 Differences of a Least Squares Means a _a 1 1 2 2 3 3 Estimate Standard Error DF t Value Pr > |t| -1.2500 7.1250 8.3750 2.3274 2.3274 2.3274 15 15 15 -0.54 3.06 3.60 0.5991 0.0079 0.0026 Least Squares Means Estimates Effect Label Estimate Standard Error DF t Value Tails Pr > t a a Row 1 Row 2 1.2500 15.5000 2.3274 4.0311 15 15 0.54 3.85 Upper Upper 0.2995 0.0008 F Test for Least Squares Means Estimates Effect a Num DF Den DF F Value 2 15 7.54 Pr > F ChiBar Sq Value Pr > ChiBarSq 0.0054 15.07 0.0001 The least squares means for the three levels of variable A are 32:875, 34:125, and 25:75. The differences between the third level and the first and second levels are statistically significant at the 5% level (p-values of 0:0079 and 0:0026, respectively). There is no significant difference between the first two levels. The first row of the “Least Squares Means Estimates” table also displays the Getting Started: PLM Procedure F 5415 difference between the first two levels of factor A. Although the (absolute value of the) estimate and its standard error are identical to those in the “Differences of a Least Squares Means” table, the p-values do not agree because one-sided tests were requested in the LSMESTIMATE statement. The “F Test” table in Figure 66.4 shows the two degree-of-freedom test for the linear combinations of the LS-means. The F value of 7:54 with p-value of 0:0054 represents the usual (two-sided) F test. Under the one-sided right-tailed order restriction imposed by the UPPERTAILED option, the ChiBarSq value of 15:07 represents the observed value of the chi-bar-square statistic of Silvapulle and Sen (2004). The associated p-value of 0:0001 was obtained by simulation. Now suppose that you are interested in analyzing the relationship of the interaction cell means. (Typically this would not be the case in this example since the a*b interaction is not significant; see Figure 66.3.) The SLICE statement in the following PROC PLM run produces an F test of equality and all pair-wise differences of the interaction means for the subset (partition) where variable B is at level ‘1’. With the ODS GRAPHICS ON statement, the pairwise differences are also visualized in a diffogram by default. ods graphics on; proc plm source=sasuser.BlockAnalysis; slice a*b / sliceby(b='1') diff; run; ods graphics off; The results are shown in Figure 66.5. Since variable A has three levels, the test of equality of the A means at level ‘1’ of B is a two-degree comparison. This comparison is statistically significant (p-value equals 0:0040). You can conclude that the three levels of A are not the same for the first level of B. Figure 66.5 Results from Analyzing an Interaction Partition The PLM Procedure F Test for a*b Least Squares Means Slice Slice b 1 Num DF Den DF F Value Pr > F 2 15 8.18 0.0040 Simple Differences of a*b Least Squares Means Slice a _a Estimate Standard Error DF t Value Pr > |t| b 1 b 1 b 1 1 1 2 2 3 3 -1.0000 11.0000 12.0000 3.2914 3.2914 3.2914 15 15 15 -0.30 3.34 3.65 0.7654 0.0045 0.0024 5416 F Chapter 66: The PLM Procedure The table of “Simple Differences” was produced by the DIFF option in the SLICE statement. As is the case with the marginal comparisons in Figure 66.4, there are significant differences against the third level of A if variable B is held fixed at ‘1’. Figure 66.6 shows the diffogram that displays the three pairwise least squares mean differences and their significance. Each line segment corresponds to a comparison. It centers at the least squares means in the pair with its length corresponding to the projected width of a confidence interval for the difference. If the variable B is held fixed at ‘1’, both the first two levels are significantly different from the third level, but the difference between the first and the second level is not significant. Figure 66.6 LS-Means Difference Diffogram Syntax: PLM Procedure F 5417 Syntax: PLM Procedure You can specify the following statements in the PLM procedure: PROC PLM SOURCE=item-store-specification < options > ; EFFECTPLOT < plot-type < (plot-definition-options) > > < / options > ; ESTIMATE < ‘label’ > estimate-specification < (divisor =n) > < , . . . < ‘label’ > estimate-specification < (divisor =n) > > < / options > ; FILTER expression ; LSMEANS < model-effects > < / options > ; LSMESTIMATE model-effect < ‘label’ > values < divisor =n > < , . . . < ‘label’ > values < divisor =n > > < / options > ; SCORE DATA=SAS-data-set < OUT=SAS-data-set > < keyword< =name > >. . . < keyword< =name > > < / options > ; SHOW options ; SLICE model-effect < / options > ; TEST < model-effects > < / options > ; WHERE expression ; With the exception of the PROC PLM statement and the FILTER statement, any statement can appear multiple times and in any order. The default order in which the statements are processed by the PLM procedure depends on the specification in the item store and can be modified with the STMTORDER= option in the PROC PLM statement. In contrast to many other SAS/STAT modeling procedures, the PLM procedure does not have common modeling statements such as the CLASS and MODEL statements. This is because the information about classification variables and model effects is contained in the source item store that is passed to the procedure in the PROC PLM statement. All subsequent statements are checked for consistency with the stored model. For example, the statement lsmeans c / diff; is detected as not valid unless one of the following conditions was true at the time when the source store was created: The effect C was used in the model. C was specified in the CLASS statement. The CLASS variables in the model had a GLM parameterization. The FILTER, SCORE, SHOW, and WHERE statements are described in full after the PROC PLM statement in alphabetical order. The EFFECTPLOT, ESTIMATE, LSMEANS, LSMESTIMATE, SLICE, and TEST statements are also used by many other procedures. Summary descriptions of functionality and syntax for these statements are also given after the PROC PLM statement in alphabetical order, but full documentation about them is available in Chapter 19, “Shared Concepts and Topics.” 5418 F Chapter 66: The PLM Procedure PROC PLM Statement PROC PLM SOURCE=item-store-specification < options > ; The PROC PLM statement invokes the procedure. The SOURCE= option with item-storespecification is required. You can specify the following options: ALPHA=˛ specifies the nominal significance level for multiplicity corrections and for the construction of confidence intervals. The value of ˛ must be between 0 and 1. The default is the value specified in the source item store, or 0.05 if the item store does not provide a value. The confidence level based on ˛ is 1 ˛. DDFMETHOD=RESIDUAL | RES | ERROR DDFMETHOD=NONE DDFMETHOD=KENROG | KR | KENWARDROGER DDFMETHOD=SATTERTH | SAT | SATTERTHWAITE specifies the method for determining denominator degrees of freedom for tests and confidence intervals. The default degree-of-freedom method is determined by the contents of the item store. You can override the default to some extent with the DDFMETHOD= option. If you choose DDFMETHOD=NONE, then infinite denominator degrees of freedom are assumed for tests and confidence intervals. This essentially produces z tests and intervals instead of t tests and intervals and chi-square tests instead of F tests. The KENWARDROGER and SATTERTHWAITE methods require that the source item store contain information about these methods. This information is currently available for item stores that were created with the MIXED or GLIMMIX procedures when the appropriate DDFM= option was in effect. ESTEPS= specifies the tolerance value used in determining the estimability of linear functions. The default value is determined by the contents of the source item store; it is usually 1E 4. FORMAT=NOLOAD | RELOAD specifies how the PLM procedure handles user-defined formats, which are not permanent. When the item store is created, user-defined formats are stored. When the PLM procedure opens an item store, these formats are loaded by default. If the format already exists in your SAS session, this operation amounts to a reloading of the format (FORMAT=RELOAD) that replaces the existing format. With FORMAT=NOLOAD, you prevent the PLM procedure from reloading the format from the item store. As a consequence, PLM statements might fail if a format was present at the item store creation and is not available in your SAS session. Also, if you modify the format that was used in the item store creation and use FORMAT=NOLOAD, you might obtain unexpected results because levels of classification variables are remapped. PROC PLM Statement F 5419 The “Class Level Information” table always displays the formatted values of classification variables that were used in fitting the model, regardless of the FORMAT= option. For more details about using formats with the PLM procedure, see “User-Defined Formats and the PLM Procedure” on page 5436. MAXLEN=n determines the maximum length of informational strings in the “Store Information” table. This table displays, for example, lists of classification or BY variables and lists of model effects. The value of n determines the truncation length for these strings. The minimum and maximum values for n are 20 and 256, respectively. The default is n D 100. NOCLPRINT< =number > suppresses the display of the “Class Level Information” table if you do not specify number. If you specify number, only levels with totals that are less than number are listed in the table. The PLM procedure produces the “Class Level Information” table by default when the model contains effects that depend on classification variables. NOINFO suppresses the display of the “Store Information” table. NOPRINT suppresses the generation of tabular and graphical output. When the NOPRINT option is in effect, ODS tables are also not produced. PERCENTILES=value-list PERCENTILE=value-list supplies a list of percentiles for the construction of highest posterior density (HPD) intervals when the PLM procedure performs a sampling-based analysis (for example, when processing an item store that contains posterior parameter estimates from a Bayesian analysis). The default set of percentiles depends on the contents of the source item store; it is typically PERCENTILES=25, 50, 75. The entries in value-list must be strictly between 0 and 100. PLOTS < (global-plot-option) > < =specific-plot-options > requests that the PLM procedure produce statistical graphics via the Output Delivery System, provided that the ODS GRAPHICS ON statement has been specified. For general information about ODS Graphics, see Chapter 21, “Statistical Graphics Using ODS.” You can request statistical graphics in the EFFECTPLOT, ESTIMATE, LSMEANS, LSMESTIMATE, and SLICE statements. For information about these plots, see the corresponding sections of Chapter 19, “Shared Concepts and Topics.” Global Plot Option The following global-plot-option applies to all plots produced by PROC PLM. UNPACKPANEL UNPACK breaks a graphic that is otherwise paneled into individual component plots. 5420 F Chapter 66: The PLM Procedure Specific Plot Options You can specify the following specific-plot-options: ALL requests that all the appropriate plots be produced. NONE suppresses all plots. SEED=number specifies the random number seed for analyses that depend on a random number stream. You can also specify the random number seed through some PLM statements (for example, through the SEED= options in the ESTIMATE, LSMEANS, and LSMESTIMATE statements). However, note that there is only a single random number stream per procedure run. Specifying the SEED= option in the PROC PLM statement initializes the stream for all subsequent statements. If you do not specify a random number seed, the source item store might supply one for you. If a seed is in effect when the PLM procedure opens the source store, the “Store Information” table displays its value. If the random number seed is less than or equal to zero, the seed is generated from reading the time of day from the computer clock and a log message indicates the chosen seed value. SINGCHOL=number tunes the singularity criterion in Cholesky decompositions. The default value depends on the contents of the source item store. The default value is typically 1E4 times the machine epsilon; this product is approximately 1E 12 on most computers. SINGRES=number sets the tolerance for which the residual variance or scale parameter is considered to be zero. The default value depends on the contents of the source item store. The default value is typically 1E4 times the machine epsilon; this product is approximately 1E 12 on most computers. SINGULAR=number tunes the general singularity criterion applied by the PLM procedure in divisions and inversions. The default value used by the PLM procedure depends on the contents of the item store. The default value is typically 1E4 times the machine epsilon; this product is approximately 1E 12 on most computers. SOURCE=item-store-specification RESTORE=item-store-specification specifies the source item store for processing. This option is required because, in contrast to SAS data sets, there is no default item store. An item-store-specification consists of a one- or two-level name as with SAS data sets. As with data sets, the default library association of an item store is with the WORK library, and any stores created in this library are deleted when the SAS session concludes. EFFECTPLOT Statement F 5421 STMTORDER=SYNTAX | GROUP STMT=SYNTAX | GROUP affects the order in which statements are grouped during processing. The default behavior depends on the contents of the source item store and can be modified with the STMTORDER= option. If STMTORDER=SYNTAX is in effect, the statements are processed in the order in which they appear. Note that this precludes the hierarchical grouping of ODS objects. If STMTORDER=GROUP is in effect, the statements are processed in groups and in the following order: SHOW, TEST, LSMEANS, SLICE, LSMESTIMATE, ESTIMATE, and SCORE. WHEREFORMAT specifies that the constants (literals) specified in WHERE expressions for group selection are in terms of the formatted values of the BY variables. By default, WHERE expressions are specified in terms of the unformatted (raw) values of the BY variables, as in the SAS DATA step. ZETA=number tunes the sensitivity in forming Type III functions. Any element in the estimable function basis with an absolute value less than number is set to 0. The default depends on the contents of the source item store; it usually is 1E 8. EFFECTPLOT Statement EFFECTPLOT < plot-type < (plot-definition-options) > > < / options > ; The EFFECTPLOT statement produces a display of the fitted model and provides options for changing and enhancing the displays. Table 66.2 describes the available plot-types and their plotdefinition-options. Table 66.2 Plot-Types and Plot-Definition-Options Description Plot-Definition-Options BOX plot-type Displays a box plot of continuous response data at each level of a CLASS effect, with predicted values superimposed and connected by a line. This is an alternative to the INTERACTION plot-type. PLOTBY= variable or CLASS effect X= CLASS variable or effect CONTOUR plot-type Displays a contour plot of predicted values against two continuous covariates. PLOTBY= variable or CLASS effect X= continuous variable Y= continuous variable FIT plot-type Displays a curve of predicted values versus a continuous variable. PLOTBY= variable or CLASS effect X= continuous variable 5422 F Chapter 66: The PLM Procedure Table 66.2 continued Description Plot-Definition-Options INTERACTION plot-type Displays a plot of predicted values (possibly with error bars) versus the levels of a CLASS effect. The predicted values are connected with lines and can be grouped by the levels of another CLASS effect. PLOTBY= variable or CLASS effect SLICEBY= variable or CLASS effect X= CLASS variable or effect SLICEFIT plot-type Displays a curve of predicted values versus a continuous variable grouped by the levels of a CLASS effect. PLOTBY= variable or CLASS effect SLICEBY= variable or CLASS effect X= continuous variable For full details about the syntax and options of the EFFECTPLOT statement, see the section “EFFECTPLOT Statement” on page 436 of Chapter 19, “Shared Concepts and Topics.” ESTIMATE Statement ESTIMATE < ‘label’ > estimate-specification < (divisor =n) > < , . . . < ‘label’ > estimate-specification < (divisor =n) > > < / options > ; The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. Estimates are formed as linear estimable functions of the form Lˇ. You can perform hypothesis tests for the estimable functions, construct confidence limits, and obtain specific nonlinear transformations. Table 66.3 summarizes important options in the ESTIMATE statement. Table 66.3 Option Important ESTIMATE Statement Options Description Construction and Computation of Estimable Functions DIVISOR= Specifies a list of values to divide the coefficients NOFILL Suppresses the automatic fill-in of coefficients for higher-order effects SINGULAR= Tunes the estimability checking difference FILTER Statement F 5423 Table 66.3 continued Option Description Degrees of Freedom and p-values ADJUST= Determines the method for multiple comparison adjustment of estimates ALPHA=˛ Determines the confidence level (1 ˛) LOWER Performs one-sided, lower-tailed inference STEPDOWN Adjusts multiplicity-corrected p-values further in a step-down fashion TESTVALUE= Specifies values under the null hypothesis for tests UPPER Performs one-sided, upper-tailed inference Statistical Output CL CORR COV E JOINT PLOTS= SEED= Constructs confidence limits Displays the correlation matrix of estimates Displays the covariance matrix of estimates Prints the L matrix Produces a joint F or chi-square test for the estimable functions Requests ODS statistical graphics if the analysis is sampling-based Specifies the seed for computations that depend on random numbers Generalized Linear Modeling CATEGORY= Specifies how to construct estimable functions with multinomial data EXP Exponentiates and displays estimates ILINK Computes and displays estimates and standard errors on the inverse linked scale For details about the syntax of the ESTIMATE statement, see the section “ESTIMATE Statement” on page 462 of Chapter 19, “Shared Concepts and Topics.” FILTER Statement FILTER expression ; The FILTER statement enables you to filter the results of the PLM procedure, specifically the contents of ODS tables and the output data sets. There can be at most one FILTER statement per PROC PLM run, and the filter is applied to all BY groups and to all queries generated through WHERE expressions. A filter expression follows the same pattern as a where-expression in the WHERE statement. The expressions consist of operands and operators. For more information about specifying whereexpressions, see the WHERE statement for the PLM procedure and SAS Language Reference: Concepts. 5424 F Chapter 66: The PLM Procedure Valid keywords for the formation of operands in the FILTER statement are shown in Table 66.4. Table 66.4 Keywords for Filtering Results Keyword Description Prob ProbChi ProbF ProbT AdjP Estimate Pred Resid Std Mu Regular (unadjusted) p-values from t, F, or chi-square tests Regular (unadjusted) p-values from chi-square tests Regular (unadjusted) p-values from F tests Regular (unadjusted) p-values from t tests Adjusted p-values Results displayed in “Estimates” column of ODS tables Predicted values in SCORE output data sets Residuals in SCORE output data sets. Standard errors in ODS tables and in SCORE results Results displayed in the “Mean” column of ODS tables (this column is typically produced by the ILINK option) The value of the usual t statistic The value of the usual F statistic The value of the chi-square statistic The value of the test statistic (a generic keyword for the ‘tValue’, ‘FValue’, and ‘Chisq’ tokens) The lower confidence limit displayed in ODS tables The upper confidence limit displayed in ODS tables The adjusted lower confidence limit displayed in ODS tables The adjusted upper confidence limit displayed in ODS tables The lower confidence limit for the mean displayed in ODS tables The upper confidence limit for the mean displayed in ODS tables The adjusted lower confidence limit for the mean displayed in ODS tables The adjusted upper confidence limit for the mean displayed in ODS tables tValue FValue Chisq testStat Lower Upper AdjLower AdjUpper LowerMu UpperMu AdjLowerMu AdjUpperMu When you write filtering expressions, be advised that filtering variables that are not used in the results are typically set to missing values. For example, the following statements select all results (filter nothing) because no adjusted p-values are computed: proc plm source=MyStore; lsmeans a / diff; filter adjp < 0.05; run; If the adjusted p-values are set to missing values, the condition < 0:05 is true in each case (missing values always compare smaller than the smallest nonmissing value). See “Example 66.6: Comparing Multiple B-Splines” on page 5454 for an example of using the FILTER statement. Filtering results has no affect on the item store contents that are displayed with the SHOW statement. However, BY-group selection with the WHERE statement can limit the amount of information that is displayed by the SHOW statements. LSMEANS Statement F 5425 LSMEANS Statement LSMEANS < model-effects > < / options > ; The LSMEANS statement computes and compares least squares means (LS-means) of fixed effects. LS-means are predicted population margins—that is, they estimate the marginal means over a balanced population. In a sense, LS-means are to unbalanced designs as class and subclass arithmetic means are to balanced designs. Table 66.5 summarizes important options in the LSMEANS statement. Table 66.5 Important LSMEANS Statement Options Option Description Construction and Computation of LS-Means AT Modifies the covariate value in computing LS-means BYLEVEL Computes separate margins DIFF Requests differences of LS-means Specifies the weighting scheme for LS-means computation as deOM= termined by the input data set SINGULAR= Tunes estimability checking Degrees of Freedom and p-values ADJUST= Determines the method for multiple comparison adjustment of LSmeans differences ALPHA=˛ Determines the confidence level (1 ˛) STEPDOWN Adjusts multiple comparison p-values further in a step-down fashion Statistical Output CL CORR COV E LINES MEANS PLOTS= SEED= Constructs confidence limits for means and mean differences Displays the correlation matrix of LS-means Displays the covariance matrix of LS-means Prints the L matrix Produces a “Lines” display for pairwise LS-means differences Prints the LS-means Requests ODS statistical graphics of means and mean comparisons Specifies the seed for computations that depend on random numbers Generalized Linear Modeling EXP Exponentiates and displays estimates of LS-means or LS-means differences ILINK Computes and displays estimates and standard errors of LS-means (but not differences) on the inverse linked scale ODDSRATIO Reports (simple) differences of least squares means in terms of odds ratios if permitted by the link function 5426 F Chapter 66: The PLM Procedure For details about the syntax of the LSMEANS statement, see the section “LSMEANS Statement” on page 479 of Chapter 19, “Shared Concepts and Topics.” LSMESTIMATE Statement LSMESTIMATE model-effect < ‘label’ > values < divisor =n > < , . . . < ‘label’ > values < divisor =n > > < / options > ; The LSMESTIMATE statement provides a mechanism for obtaining custom hypothesis tests among least squares means. Table 66.6 summarizes important options in the LSMESTIMATE statement. Table 66.6 Important LSMESTIMATE Statement Options Option Description Construction and Computation of LS-Means AT Modifies covariate values in computing LS-means BYLEVEL Computes separate margins DIVISOR= Specifies a list of values to divide the coefficients OM= Specifies the weighting scheme for LS-means computation as determined by a data set SINGULAR= Tunes estimability checking Degrees of Freedom and p-values ADJUST= Determines the method for multiple comparison adjustment of LSmeans differences ALPHA=˛ Determines the confidence level (1 ˛) LOWER Performs one-sided, lower-tailed inference STEPDOWN Adjusts multiple comparison p-values further in a step-down fashion TESTVALUE= Specifies values under the null hypothesis for tests UPPER Performs one-sided, upper-tailed inference Statistical Output CL CORR COV E ELSM JOINT PLOTS= SEED= Constructs confidence limits for means and mean differences Displays the correlation matrix of LS-means Displays the covariance matrix of LS-means Prints the L matrix Prints the K matrix Produces a joint F or chi-square test for the LS-means and LSmeans differences Requests ODS statistical graphics of means and mean comparisons Specifies the seed for computations that depend on random numbers SCORE Statement F 5427 Table 66.6 continued Option Description Generalized Linear Modeling CATEGORY= Specifies how to construct estimable functions with multinomial data EXP Exponentiates and displays LS-means estimates ILINK Computes and displays estimates and standard errors of LS-means (but not differences) on the inverse linked scale For details about the syntax of the LSMESTIMATE statement, see the section “LSMESTIMATE Statement” on page 496 of Chapter 19, “Shared Concepts and Topics.” SCORE Statement SCORE DATA=SAS-data-set < OUT=SAS-data-set > < keyword< =name > >. . . < keyword< =name > > < / options > ; The SCORE statement applies the contents of the source item store to compute predicted values and other observation-wise statistics for a SAS data set. You can specify the following syntax elements in the SCORE statement before the option slash (/): DATA=SAS-data-set specifies the input data set for scoring. This option is required, and the data set is examined for congruity with the previously fitted (and stored) model. For example, all necessary variables to form a row of the X matrix must be present in the input data set and must be of the correct type and format. The following variables do not have to be present in the input data set: the response variable the events and trials variables used in the events/trials syntax for binomial data variables used in WEIGHT or FREQ statements OUT=SAS-data-set specifies the name of the output data set. If you do not specify an output data set with the OUT= option, the PLM procedure uses the DATAn convention to name the output data set. keyword< =name > specifies a statistic to be included in the OUT= data set and optionally assigns the statistic the variable name name. Table 66.7 lists the keywords and the default names assigned by the PLM procedure if you do not specify a name. 5428 F Chapter 66: The PLM Procedure Table 66.7 Keywords for Output Statistics Keyword Description PREDICTED Linear predictor STDERR Standard deviation of linear predictor Residual RESIDUAL LCLM UCLM LCL UCL Lower confidence limit for the linear predictor Upper confidence limit for the linear predictor Lower prediction limit for the linear predictor Upper prediction limit for the linear predictor Expression b D xb̌ p Var.b / y b Name Predicted StdErr Resid LCLM UCLM LCL UCL Prediction limits (LCL, UCL) are available only for statistical models that allow such limits, typically regression-type models for normally distributed data with an identity link function. You can specify the following options in the SCORE statement after a slash (/): ALPHA=number determines the coverage probability for two-sided confidence and prediction intervals. The coverage probability is computed as 1 number. The value of number must be between 0 and 1; the default is 0.05. DF=number specifies the degrees of freedom to use in the construction of prediction and confidence limits. ILINK requests that predicted values be inversely linked to produce predictions on the data scale. By default, predictions are produced on the linear scale where covariate effects are additive. NOUNIQUE requests that names not be made unique in the case of naming conflicts. By default, the PLM procedure avoids naming conflicts by assigning a unique name to each output variable. If you specify the NOUNIQUE option, variables with conflicting names are not renamed. In that case, the first variable added to the output data set takes precedence. NOVAR requests that variables from the input data set not be added to the output data set. OBSCAT requests that statistics in models for multinomial data be written to the output data set only for the response level that corresponds to the observed level of the observation. SHOW Statement F 5429 SAMPLE requests that the sample of parameter estimates in the item store be used to form scoring statistics. This option is useful when the item store contains the results of a Bayesian analysis and a posterior sample of parameter estimates. The predicted value is then computed as the average predicted value across the posterior estimates, and the standard error measures the standard deviation of these estimates. For example, let b̌1 ; : : : ; b̌k denote the k posterior sample estimates of ˇ, and let xi denote the x-vector for the i th observation in the scoring data set. If the SAMPLE option is in effect, the output statistics for the predicted value, the standard error, and the residual of the i th observation are computed as ij D xi b̌j PREDi STDERRi k 1X D i D ij k j D1 0 k 1 X D @ ij k 1 11=2 2 i A j D1 RESIDUALi where g 1 ./ D yi g 1 .i / denotes the inverse link function. If, in addition, the ILINK option is in effect, the calculations are as follows: ij D xi b̌j k PREDi STDERRi 1X 1 g ij D k j D1 0 k 1 X g 1 .ij / D @ k 1 11=2 2 PREDi A j D1 RESIDUALi D yi PREDi The LCL and UCL statistics are not available with the SAMPLE option. When the LCLM and UCLM statistics are requested, the SAMPLE option yields the lower 100 ˛=2% and upper 100 .1 ˛=2/% percentiles of the predicted values under the sample (posterior) distribution. When you request residuals with the SAMPLE option, the calculation depends on whether the ILINK option is specified. SHOW Statement SHOW options ; The SHOW statement uses the Output Delivery System to display contents of the item store. This statement is useful for verifying that the contents of the item store apply to the analysis and for generating ODS tables. You can specify the following options after the SHOW statement: 5430 F Chapter 66: The PLM Procedure ALL | _ALL_ displays all applicable contents. BYVAR | BY displays information about the BY variables in the source item store. If a BY statement was present when the item store was created, the PLM procedure performs the analysis separately for each BY group. CLASSLEVELS | CLASS displays the “Class Level Information” table. This table is produced by the PLM procedure by default if the model contains effects that depend on classification variables. CORRELATION | CORR | CORRB produces the correlation matrix of the parameter estimates. If the source item store contains a posterior sample of parameter estimates, the computed matrix is the correlation matrix of the sample covariance matrix. COVARIANCE | COV | COVB produces the covariance matrix of the parameter estimates. If the source item store contains a posterior sample of parameter estimates, the PLM procedure computes the empirical sample covariance matrix from the posterior estimates. You can convert this matrix into a sample correlation matrix with the CORRELATION option in the SHOW statement. EFFECTS displays information about the constructed effects in the model. Constructed effects are those that were created with the EFFECT statement in the procedure run that generated the source item store. FITSTATS | FIT displays the fit statistics from the item store. HESSIAN | HESS displays the Hessian matrix. HERMITE | HERM generates the Hermite matrix H D .X0 X/ .X0 X/. The PLM procedure chooses a reflexive, g2 -inverse for the generalized inverse of the crossproduct matrix X0 X. See “Important Linear Algebra Concepts” on page 47 of Chapter 3, “Introduction to Statistical Modeling with SAS/STAT Software,” for information about generalized inverses and the sweep operator. PARAMETERS< =n > PARMS< =n > displays the parameter estimates. The structure of the display depends on whether a posterior sample of parameter estimates is available in the source item store. If such a sample is present, up to the first 20 parameter vectors are shown in wide format. You can modify this number with the n argument. If no posterior sample is present, the single vector of parameter estimates is shown in narrow format. If the store contains information about the covariance matrix of the parameter estimates, then standard errors are added. SLICE Statement F 5431 PROGRAM< (WIDTH=n) > PROG< (WIDTH=n) > displays the SAS program that generated the item store, provided that this was stored at store generation time. The program does not include comments, titles, or some other global statements. The optional width parameter n determines the display width of the source code. XPX | CROSSPRODUCT displays the crossproduct matrix X0 X. XPXI displays the generalized inverse of the crossproduct matrix X0 X. The PLM procedure obtains a reflexive g2 -inverse by sweeping. See “Important Linear Algebra Concepts” on page 47 of Chapter 3, “Introduction to Statistical Modeling with SAS/STAT Software,” for information about generalized inverses and the sweep operator. SLICE Statement SLICE model-effect < / options > ; The SLICE statement provides a general mechanism for performing a partitioned analysis of the LS-means for an interaction. This analysis is also known as an analysis of simple effects. The SLICE statement uses the same options as the LSMEANS statement, which are summarized in Table 19.19. For details about the syntax of the SLICE statement, see the section “SLICE Statement” on page 526 of Chapter 19, “Shared Concepts and Topics.” TEST Statement TEST < model-effects > < / options > ; The TEST statement enables you to perform F tests for model effects that test Type I, II, or Type III hypotheses. See Chapter 15, “The Four Types of Estimable Functions,” for details about the construction of Type I, II, and III estimable functions. Table 66.8 summarizes options in the TEST statement. 5432 F Chapter 66: The PLM Procedure Table 66.8 TEST Statement Options Option Description CHISQ DDF= E E1 E2 E3 HTYPE= INTERCEPT Requests chi-square tests Specifies denominator degrees of freedom for fixed effects Requests Type I, Type II, and Type III coefficients Requests Type I coefficients Requests Type II coefficients Requests Type III coefficients Indicates the type of hypothesis test to perform Adds a row that corresponds to the overall intercept For details about the syntax of the TEST statement, see the section “TEST Statement” on page 530 of Chapter 19, “Shared Concepts and Topics.” WHERE Statement WHERE expression ; The WHERE statement in the PLM procedure is helpful when the item store contains BY-variable information and you want to apply the PROC PLM statements to only a subset of the BY groups. A WHERE expression is a type of SAS expression that defines a condition. In the DATA step and in procedures that use SAS data sets as input source, the WHERE expression is used to select observations for inclusion in the DATA step or in the analysis. In the PLM procedure, which does not accept a SAS data set but rather takes an item store that was created by a qualifying SAS/STAT procedure, the WHERE statement is also used to specify conditions. The conditional selection does not apply to observations in PROC PLM, however. Instead, you use the WHERE statement in the PLM procedure to select a subset of BY groups from the item store to which to apply the PROC PLM statements. The general syntax of the WHERE statement is WHERE operand < operator > < operand > ; where operand is something to be operated on. The operand can be the name of a BY variable in the item store, a SAS function, a constant, or a predefined name to identify columns in result tables. operator is a symbol that requests a comparison, logical operation, or arithmetic calculation. All SAS expression operators are valid for a WHERE expression. WHERE Statement F 5433 For more details about how to specify general WHERE expressions, see SAS Language Reference: Concepts. Notice that the FILTER statement accepts similar expressions that are specified in terms of predefined keywords. Expressions in the WHERE statement of the PLM procedure are written in terms of BY variables. There is no limit to the number of WHERE statements in the PLM procedure. When you specify multiple WHERE statements, the statements are not cumulative. Each WHERE statement is executed separately. You can think of each selection WHERE statement as one analytic query to the item store: the WHERE statement defines the query, and the PLM procedure is the querying engine. For example, suppose that the item store contains results for the numeric BY variables A and B. The following statements define two separate queries of the item store: WHERE a = 4; WHERE (b < 3) and (a > 4); The PLM procedure first applies the requested analysis to all BY groups where a equals 4 (irrespective of the value of variable b). The analysis is then repeated for all BY groups where b is less than 3 and a is greater than 4. Group selection with WHERE statements is possible only if the item store contains BY variables. You can use the BYVAR option in the SHOW statement to display the BY variables in the item store. Note that WHERE expressions in the SAS DATA step and in many procedures are specified in terms of the unformatted values of data set variables, even if a format was applied to the variable. If you specify the WHEREFORMAT option in the PROC PLM statement, the PLM procedure evaluates WHERE expressions for BY variables in terms of the formatted values. For example, assume that the following format was applied to the variable tx when the item store was created: proc format; value bf 1 = 'Control' 2 = 'Treated'; run; Then the following two PROC PLM runs are equivalent: proc plm source=MyStore; show parms; where b = 2; run; proc plm source=MyStore whereformat; show parms; where b = 'Treated'; run; 5434 F Chapter 66: The PLM Procedure Details: PLM Procedure BY Processing and the PLM Procedure When a BY statement is in effect for the anlaysis that creates an item store, the information about BY variables and BY-group-specific modeling results are transferred to the item store. In this case, the PLM procedure automatically assumes a processing mode for the item store that is akin to BY processing, with the PLM statements being applied in turn for each of the BY groups. Also, you can then obtain a table of BY groups with the BYVAR option in the SHOW statement. The “Source Information” table also displays the variable names of the BY variables if BY groups are present. The WHERE statement can be used to restrict the analysis to specific BY groups that meet the conditions of the WHERE expression. See Example 66.4 for an example that uses BY-group-specific information in the source item store. As with procedures that operate on input data sets, the BY variable information is added automatically to any output data sets and ODS tables produced by the PLM procedure. When you score a data set with the SCORE statement and the item store contains BY variables, three situations can arise: None of the BY variables are present in the scoring data set. In this situation the results of the BY groups in the item store are applied in turn to the entire scoring data set. For example, if the scoring data set contains 50 observations and no BY-variable information, the number of observations in the output data set of the SCORE statement equals 50 times the number of BY groups. The scoring data set contains only a part of the BY variables, or the variables have different type or format. The PLM procedure does not process such an incompatible scoring data set. All BY variables are in the scoring data set in the same type and format as when the item store was created. The BY-group-specific results are applied to each observation in the scoring data set. The scoring data set does not have to be sorted or grouped by the BY variables. However, it is computationally more efficient if the scoring data set is arranged by groups of the BY variables. Analysis Based on Posterior Estimates If an item store that are saved from a Bayesian analysis (by PROC GENMOD or PROC PHREG), then PROC PLM can perform sampling-based inference based on Bayes posterior estimates that are saved in the item store. For example, the following statements request that a Bayesian analysis and results be saved to an item store named sasuser.gmd. For the Bayesian analysis, the random number Analysis Based on Posterior Estimates F 5435 generator seed is set to 1. By default, a noninformative distribution is set as the prior distribution for the regression coefficients and the posterior sample size is 10,000. proc genmod data=gs; class a b; model y = a b; bayes seed=1; store sasuser.gmd / label='Bayesian Analysis'; run; When the PLM procedure opens the item store sasuser.gmd, it detects that the results were saved from a Bayesian analysis. The posterior sample of regression coefficient estimates are then loaded to perform statistical inference tasks. The majority of postprocessing tasks involve inference based on an estimable linear function Lb̌, which often requires its mean and variance. When the standard frequentist analyses are performed, the mean and variance have explicit forms because the parameter estimate b̌ is analytically tractable. However, explicit forms are not usually available when Bayesian models are fitted. Instead, empirical means and variance-covariance matrices for the estimable function are constructed from the posterior sample. Let b̌i ; i D 1; : : : ; Np denote the Np vectors of posterior sample estimates of ˇ saved in sasuser.gmd. Use these vectors to construct the posterior sample of estimable functions Lˇi . The posterior mean of the estimable function is thus Np X 1 Lb̌ D Lb̌i Np i D1 and the posterior variance of the estimable function is V Lb̌ D 1 Np 1 Np X Lb̌i 2 Lb̌ i D1 Sometimes statistical inference on a transformation of Lb̌ is requested. For example, the EXP option for the ESTIMATE and LSMESTIMATE statements requests analysis based on exp.Lb̌/, exponentiation of the estimable function. If this type of analysis is requested, the posterior sample of transformed estimable functions is constructed by transforming each of the estimable function evaluated at the posterior sample: f .Lb̌i /; i D 1; : : : ; Np . The posterior mean and variance for f .Lb̌/ are then computed from the constructed sample to make the inference: Np 1 X b̌ f .L / D f .Lb̌i / Np i D1 V f .Lb̌/ D 1 Np 1 Np X f .Lb̌i / 2 f .Lb̌/ i D1 After obtaining the posterior mean and variance, the PLM procedure proceeds to perform statistical inference tasks based on them. 5436 F Chapter 66: The PLM Procedure User-Defined Formats and the PLM Procedure The PLM procedure does not support a FORMAT statement because it operates without an input data set, and also because changing the format properties of variables could alter the interpretation of parameter estimates, thus creating a dissonance with variable properties in effect when the item store was created. Instead, user-defined formats that are applied to classification variables when the item store is created are saved to the store and are by default reloaded by the PLM procedure. When the PLM procedure loads a format, notes are issued to the log. You can change the load behavior for formats with the user-defined FORMAT= option in the PROC PLM statement. User-defined formats do not need to be supplied in a new SAS session. However, when a user-defined format with the same name as a stored format exists and the default FORMAT=RELOAD option is in effect, the format definition loaded from the item store replaces the format currently in effect. In the following statements, the format AFORM is created and applied to the variable a in the PROC GLM step. This format definition is transferred to the item store sasuser.glm through the STORE statement. proc format; value aform 1='One' 2='Two' 3='Three'; run; proc glm data=sp; format a aform.; class block a b; model y = block a b x; store sasuser.glm; weight x; run; The following statements replace the format definition for aform in the PROC FORMAT step. The PLM step then reloads the AFORM format and thereby restores its original state. proc format; value aform 1='Un' 2='Deux' 3='Trois'; run; proc plm source=sasuser.glm; show class; score data=sp out=plmout lcl lclm ucl uclm; run; The following notes, issued by the PLM procedure, inform you that the procedure loaded the format, the format already existed, and the existing format was replaced: NOTE: The format AFORM was loaded from item store SASUSER.GLM. NOTE: Format AFORM is already on the library. NOTE: Format AFORM has been output. ODS Table Names F 5437 After the PROC PLM run, the definition that is in effect for the format AFORM corresponds to the following SAS statements: proc format; value aform 1='One' 2='Two' 3='Three'; run; ODS Table Names PROC PLM assigns a name to each table it creates. You can use these names to refer to the table when you use the Output Delivery System (ODS) to select tables and create output data sets. These names are listed in Table 66.9. For more information about ODS, see Chapter 20, “Using the Output Delivery System.” Each of the EFFECTPLOT, ESTIMATE, LSMEANS, LSMESTIMATE, and SLICE statements also creates tables, which are not listed in Table 66.9. For information about these tables, see the corresponding sections of Chapter 19, “Shared Concepts and Topics.” Table 66.9 ODS Tables Produced by PROC PLM Table Name Description Required Option ByVarInfo Information about BY variables in source item store (if present) Level information from the CLASS statement SHOW BYVAR ClassLevels Corr Cov FitStatistics Hessian Hermite ParameterEstimates ParameterSample Program StoreInfo XpX XpXI Default output when model effects depend on CLASS variables Correlation matrix of parameter esti- SHOW CORR mates Covariance matrix of parameter esti- SHOW COV mates Fit statistics SHOW FIT Hessian matrix SHOW HESSIAN Hermite matrix SHOW HERMITE Parameter estimates SHOW PARMS Sampled (posterior) parameter esti- SHOW PARMS mates Originating source code SHOW PROGRAM Information about source item store Default X0 X matrix SHOW XPX 0 .X X/ matrix SHOW XPXI 5438 F Chapter 66: The PLM Procedure ODS Graphics When the ODS Graphics are in effect, then each of the EFFECTPLOT, ESTIMATE, LSMEANS, LSMESTIMATE, and SLICE statements can produce plots associated with their analyses. For information about these plots, see the corresponding sections of Chapter 19, “Shared Concepts and Topics.” Examples: PLM Procedure Example 66.1: Scoring with PROC PLM Logistic regression with model selection is often used to extract useful information and build interpretable models for classification problems with many variables. This example demonstrates how you can use PROC LOGISTIC to build a spline model on a simulated data set and how you can later use the fitted model to classify new observations. The following DATA step creates a data set named SimuData, which contains 5,000 observations and 100 continuous variables: %let nObs = 5000; %let nVars = 100; data SimuData; array x{&nVars}; do obsNum=1 to &nObs; do j=1 to &nVars; x{j}=ranuni(1); end; linp = 10 + 11*x1 - 10*sqrt(x2) + 2/x3 - 8*exp(x4) + 7*x5*x5 - 6*x6**1.5 + 5*log(x7) - 4*sin(3.14*x8) + 3*x9 - 2*x10; TrueProb = 1/(1+exp(-linp)); if ranuni(1) < TrueProb then y=1; else y=0; output; end; run; The response is binary based on the inversely transformed logit values. The true logit is a function of only 10 of the 100 variables, including nonlinear transformations of seven variables, as follows: p 2 8 exp.x4 /C7x52 6x61:5 C5 log.x7 / 4 sin.3:14x8 /C3x9 2x10 logit.p/ D 10C11x1 10 x2 C x3 Example 66.1: Scoring with PROC PLM F 5439 Now suppose the true model is not known. With some exploratory data analysis, you determine that the dependency of the logit on some variables is nonlinear. Therefore, you decide to use splines to model this nonlinear dependence. Also, you want to use stepwise regression to remove unimportant variable transformations. The following statements perform the task: proc logistic data=SimuData; effect splines = spline(x1-x&nVars/separate); model y = splines/selection=stepwise; store sasuser.SimuModel; run; By default, PROC LOGISTIC models the probability that y D 0. The EFFECT statement requests an effect named splines constructed by all predictors in the data. The SEPARATE option specifies that the spline basis for each variable be treated as a separate set so that model selection applies to each individual set. The SELECTION=STEPWISE specifies the stepwise regression as the model selection technique. The STORE statement requests that the fitted model be saved to an item store sasuser.SimuModel. See “Example 66.2: Working with Item Stores” on page 5440 for an example with more details about working with item stores. The spline effect for each predictor produces seven columns in the design matrix, making stepwise regression computationally intensive. For example, a typical Pentium 4 workstation takes around ten minutes to run the preceding statements. Real data sets for classification can be much larger. See examples at UCI Machine Learning Repository (Asuncion and Newman 2007). If new observations about which you want to make predictions are available at model fitting time, you can add the SCORE statement in the LOGISTIC procedure. However, if observations to predict become available after fitting the model, you must use the LOGISTIC procedure to refit the model to make predictions for new observations. With PROC PLM, you do not have to repeat the intimidating model-fitting processes multiple times. You can use the SCORE statement in the PLM procedure to score new observations based on the item store sasuser.SimuModel that was created during the initial model building. For example, to compute the probability of y D 0 for one new observation with all predictor values equal to 0.15 in the data set test, you can use the following statements: data test; array x{&nVars}; do j=1 to &nVars; x{j}=0.15; end; drop j; output; run; proc plm source=sasuser.SimuModel; score data=test out=testout predicted / ilink; run; The ILINK option in the SCORE statement requestes that predicted values be inversely transformed to the response scale. In this case, it is the predicted probability of y D 0. Output 66.1.1 shows the predicted probability for the new observation. 5440 F Chapter 66: The PLM Procedure Output 66.1.1 Predicted Probability for One New Observation Obs Predicted 1 0.56649 Example 66.2: Working with Item Stores This example demonstrates how procedures save statistical analysis context and results into item stores and how you can use PROC PLM to make post hoc inference based on saved item stores. The data are taken from McCullagh and Nelder (1989) and concern the effects on taste of various cheese additives. Four cheese additives were tested, and 52 response ratings for each additive were obtained. The response was measured on a scale of nine categories that range from strong dislike (1) to excellent taste (9). The following program saves the data in the data set Cheese. The variable y contains the taste rating, the variable Additive contains cheese additive types, and the variable freq contains the frequencies with which each additive received each rating. data Cheese; do Additive = 1 to 4; do y = 1 to 9; input freq @@; output; end; end; label y='Taste Rating'; datalines; 0 0 1 7 8 8 19 8 1 6 9 12 11 7 6 1 0 0 1 1 6 8 23 7 5 1 0 0 0 0 1 3 7 14 16 11 ; The response y is a categorical variable that contains nine ordered levels. You can use PROC LOGISTIC to fit an ordinal model to investigate the effects of the cheese additive types on taste ratings. Suppose you also want to save the ordinal model into an item store so that you can make statistical inference later. You can use the following statements to perform the tasks: proc logistic data=cheese; freq freq; class additive y / param=glm; model y=additive; store sasuser.cheese; title 'Ordinal Model on Cheese Additives'; run; By default, PROC LOGISTIC uses the cumulative logit model for the ordered categorical response. The STORE statement requests that the fitted model be saved to a SAS item store named sasuser.cheese. The name is a two-level SAS name of the form libname.membername. If libname Example 66.2: Working with Item Stores F 5441 is not specified in the STORE statement, the fitted results are saved in work.membername and the item store is deleted after the current SAS session ends. With this example, the fitted model is saved to an item store named sasuser.cheese in the SASUSER library. It is not deleted after the current SAS session ends. You can use PROC PLM to restore the results later. The following statements use PROC PLM to load the saved model context and results by specifying SOURCE= with the target item store sasuser.cheese. Then they use two SHOW statements to display separate information saved in the item store. The first SHOW statement with the PROGRAM option displays the program that was used to generate the item store sasuser.cheese. The second SHOW statement with the PARMS option displays parameter estimates and associated statistics of the fitted ordinal model. proc plm source=sasuser.cheese; show program; show parms; run; Output 66.2.1 displays the program that generated the item store sasuser.cheese. Except for the title information, it matches the original program. Output 66.2.1 Program Information from sasuser.cheese Ordinal Model on Cheese Additives The PLM Procedure SAS Program Information proc logistic data=cheese; freq freq; class additive y / param=glm; model y=additive; store sasuser.cheese; run; Output 66.2.2 displays estimates of the intercept terms and covariates and associated statistics. The intercept terms correspond to eight cumulative logits defined on taste ratings; that is, the i th intercept for i th logit is ! P j i pj P log 1 j i pj 5442 F Chapter 66: The PLM Procedure Output 66.2.2 Parameter Estimates of the Ordinal Model Parameter Estimates Parameter Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Additive 1 Additive 2 Additive 3 Additive 4 Taste Rating 1 2 3 4 5 6 7 8 Estimate Standard Error -7.0801 -6.0249 -4.9254 -3.8568 -2.5205 -1.5685 -0.06688 1.4930 1.6128 4.9645 3.3227 0 0.5624 0.4755 0.4272 0.3902 0.3431 0.3086 0.2658 0.3310 0.3778 0.4741 0.4251 . You can perform various statistical inference tasks from a saved item store, as long as the task is applicable under the model context. For example, you can perform group comparisons between different cheese additive types. See the next example for details. Example 66.3: Group Comparisons in Ordinal Model This example continues the study of the effects on taste of various cheese additives. You have finished fitting an ordinal logistic model and saved it to an item store named sasuser.cheese in the previous example. Suppose you want to make comparisons between any pair of cheese additives. You can conduct the analysis by using the ESTIMATE statement and constructing an appropriate L matrix, or by using the LSMEANS statement to compute least squares means differences. For an ordinal logistic model with the cumulative logit link, the least squares means are predicted population margins of the cumulative logits. The following statements compute and display differences between least squares means of cheese additive: ods graphics on; proc plm source=sasuser.cheese; lsmeans additive / cl diff oddsratio plot=diff; run; ods graphics off; There are four options specified for the LSMEANS statement in the preceding statements. The DIFF option requests least squares means differences for cheese additives. Since the fitted model is an ordinal logistic model with the cumulative logit link, the least squares means differences represent log cumulative odds ratios. The ODDSRATIO option requests exponentiation of the LSmeans differences which produces cumulative odds ratios. The CL option requests that confidence limits be constructed for the LS-means differences. When ODS GRAPHICS ON is specified, the PLOTS=DIFF option requests a display of all pairwise least squares means differences and their significance. Example 66.3: Group Comparisons in Ordinal Model F 5443 Output 66.3.1 displays the LS-means differences. The reported log odds ratios indicate the relative difference among the cheese additives. A negative log odds ratio indicates that the first category (displayed in the “Additive” column) having a lower taste rating is less likely than the second category (displayed in the “_Additive” column) having a lower taste rating. For example, the log odds ratio between cheese additive 1 and 2 is 3:3517 and the corresponding odds ratio is 0:035. This means the odds of cheese additive 1 receiving a poor rating is 0:035 times the odds of cheese additive 2 receiving a poor rating. In addition to the highly significant p-value (< 0:0001), the confidence limits for both the log odds ratio and the odds ratio indicate that you can reject the null hypothesis that the odds of cheese additive 1 having a lower taste rating is the same as that of cheese additive 2 having a lower rating. Similarly, the odds of cheese additive 2 having a lower rating is 143:241 (with 95% confidence limits .56:558; 362:777/) times the odds of cheese additive 4 having a lower rating. With the same logic, you can conclude that the preference order for the four cheese types from the most favorable to the least favorable is: 4, 1, 3 and 2. Output 66.3.1 LS-Means Differences of Additive Ordinal Model on Cheese Additives The PLM Procedure Differences of Additive Least Squares Means Additive _Additive 1 1 1 2 2 3 2 3 4 3 4 4 Estimate Standard Error z Value Pr > |z| Alpha -3.3517 -1.7098 1.6128 1.6419 4.9645 3.3227 0.4235 0.3731 0.3778 0.3738 0.4741 0.4251 -7.91 -4.58 4.27 4.39 10.47 7.82 <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 0.05 0.05 0.05 0.05 0.05 0.05 Differences of Additive Least Squares Means Additive _Additive 1 1 1 2 2 3 2 3 4 3 4 4 Lower Upper Odds Ratio Lower Confidence Limit for Odds Ratio -4.1818 -2.4410 0.8724 0.9092 4.0353 2.4895 -2.5216 -0.9787 2.3532 2.3746 5.8938 4.1558 0.035 0.181 5.017 5.165 143.241 27.734 0.015 0.087 2.393 2.482 56.558 12.055 Upper Confidence Limit for Odds Ratio 0.080 0.376 10.520 10.746 362.777 63.805 Output 66.3.2 displays the DiffPlot. This shows that all pairs of LS-means differences, equivalent to log odds ratios in this case, are significant at the level of ˛ D 0:05. This means that the preference between any pair of the four cheese additive types are statistically significantly different. 5444 F Chapter 66: The PLM Procedure Output 66.3.2 LS-Means Plot of Pairwise Differences Example 66.4: Posterior Inference for Binomial Data This example demonstrates how you can use PROC PLM to perform posterior inference from a Bayesian analysis. The data for this example are taken from Weisberg (1985) and concern the effect of small electrical currents on farm animals. The ultimate goal of the experiment was to understand the effects of high-voltage power lines on livestock and to better protect farm animals. Seven cows and six shock intensities were used in two experiments. In one experiment, each cow was given 30 electrical shocks with five at each shock intensity in random order. The number of shock responses was recorded for each cow at each shock level. The experiment was then repeated to investigate whether the response diminished due to fatigue of cows, or due to learning. So each cow received a total of 60 shocks. For the following analysis, the cow difference is ignored. The following DATA step lists the data where the variable current represents the shock level, the variable response represents the number of shock responses, the variable trial represents the total number of trials at each shock level, and the variable experiment represents the experiment number (1 for the initial experiment and 2 for the repeated one): Example 66.4: Posterior Inference for Binomial Data F 5445 data cow; input current response trial experiment; datalines; 0 0 35 1 0 0 35 2 1 6 35 1 1 3 35 2 2 13 35 1 2 8 35 2 3 26 35 1 3 21 35 2 4 33 35 1 4 27 35 2 5 34 35 1 5 29 35 2 ; Suppose you are interested in modeling the distribution of the shock response based on the level of the current and the experiment number. You can use the GENMOD procedure to fit a frequentist logistic model for the data. However, if you have some prior information about parameter estimates, you can fit a Bayesian logistic regression model to take this prior information into account. In this case, suppose you believe the logit of response has a positive association with the shock level but you are uncertain about the ranges of other regression coefficients. To incorporate this prior information in the regression model, you can use the following statements: data prior; input _type_$ current; datalines; mean 100 var 50 ; proc genmod data=cow; class experiment; bayes coeffprior=normal(input=prior) seed=1; model response/trial = current|experiment / dist=binomial; store cowgmd; title 'Bayesian Logistic Model on Cow'; run; The DATA step before the GENMOD procedure creates a data set prior that specifies the prior distribution information for current, which in this case is a normal distribution with mean 100 and variance 50. This reflects a rough belief in a positive coefficient in a moderate range for current. The prior distribution parameters are not specified for experiment and the interaction between experiment and current, and so PROC GENMOD assigns a default prior for them, which is a normal distribution with mean 0 and variance 1E6. After the DATA step, the BAYES statement in PROC GENMOD specifies that the regression coefficients follow a normal distribution with mean and variance specified in the input data set named prior. It also specifies 1 as the seed for the random number generator in the simulation of the posterior sample. The MODEL statement requests a logistic regression model with a logit link. The STORE statement requests that the fitted results be saved into an item store named cowgmd. 5446 F Chapter 66: The PLM Procedure The convergence diagnostics in the output of PROC GENMOD indicate that the Markov chain has converged. Output 66.4.1 displays summaries on the posterior sample of the regression coefficients. The posterior mean for the intercept is 3:5857 with a 95% HPD interval . 4:5226; 2:6303/. The posterior mean of the coefficient for current is 1:1893 with a 95% HPD interval .0:8950; 1:4946/, which indicates a positive association between the logit of response and the shock level. Further investigation about whether shock reaction was different between two experiment is warranted. Output 66.4.1 Posterior Summaries on the Bayesian Logistic Model Bayesian Logistic Model on Cow The GENMOD Procedure Bayesian Analysis Posterior Summaries Parameter Intercept current experiment1 experiment1current N Mean Standard Deviation 25% 10000 10000 10000 10000 -3.5857 1.1893 0.00727 0.3695 0.4822 0.1536 0.7025 0.2529 -3.9014 1.0833 -0.4483 0.1977 Percentiles 50% -3.5704 1.1843 0.00849 0.3651 75% -3.2553 1.2893 0.4879 0.5332 Posterior Intervals Parameter Alpha Equal-Tail Interval Intercept current experiment1 experiment1current 0.050 0.050 0.050 0.050 -4.5814 0.9016 -1.4347 -0.1134 -2.6799 1.5047 1.3517 0.8802 HPD Interval -4.5226 0.8950 -1.4390 -0.1105 -2.6303 1.4946 1.3439 0.8809 Bayesian model fitting typically involves a large amount of simulation. Using the item store and PROC PLM, you do not need to refit the model to perform further posterior inference. Suppose you want to determine whether the shock reaction for the current level is different between the two experiments. You can use PROC PLM with the ESTIMATE statement in the following statements: proc plm source=cowgmd; estimate 'Diff at current 0' experiment 'Diff at current 1' experiment 'Diff at current 2' experiment 'Diff at current 3' experiment 'Diff at current 4' experiment 'Diff at current 5' experiment / exp cl; run; 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 current*experiment current*experiment current*experiment current*experiment current*experiment current*experiment [1, [1, [1, [1, [1, [1, 0 1 2 3 4 5 1] 1] 1] 1] 1] 1] [-1, [-1, [-1, [-1, [-1, [-1, 0 1 2 3 4 5 2], 2], 2], 2], 2], 2] Example 66.4: Posterior Inference for Binomial Data F 5447 Each line in the ESTIMATE statement compares the fits between the two groups at each current level. The nonpositional syntax is used for the interaction effect current*experiment. For example, the first line requests coefficient 1 for the interaction effect at current level 0 for the initial experiment, and coefficient –1 for the effect at current level 0 for the repeated experiment. The two terms are then added to derive the difference. For more details about the nonpositional syntax, see “Positional and Nonpositional Syntax for Coefficients in Linear Functions” on page 473 of Chapter 19, “Shared Concepts and Topics.” The EXP option exponentiates log odds ratios to produce odds ratios. The CL option requests that confidence limits be constructed for both log odds ratios and odds ratios. Output 66.4.2 lists the posterior sample estimates for differences between experiments at different current levels. Output 66.4.2 Comparisons between Experiments at Different Current Levels Bayesian Logistic Model on Cow The PLM Procedure Sample Estimates Label Diff Diff Diff Diff Diff Diff at at at at at at current current current current current current 0 1 2 3 4 5 N Estimate Standard Deviation 10000 10000 10000 10000 10000 10000 0.007272 0.3767 0.7462 1.1156 1.4851 1.8546 0.7025 0.4840 0.3207 0.3151 0.4729 0.6899 ---------Percentiles-------25th 50th 75th -0.4483 0.0590 0.5316 0.9023 1.1681 1.3925 0.00849 0.3802 0.7500 1.1113 1.4739 1.8382 0.4879 0.7051 0.9581 1.3253 1.7943 2.3004 Sample Estimates Label Diff Diff Diff Diff Diff Diff at at at at at at current current current current current current 0 1 2 3 4 5 Alpha Lower HPD Upper HPD Exponentiated Standard Deviation of Exponentiated 0.05 0.05 0.05 0.05 0.05 0.05 -1.4390 -0.6113 0.1091 0.5205 0.5601 0.4712 1.3439 1.3007 1.3665 1.7407 2.4287 3.1885 1.2811 1.6362 2.2202 3.2082 4.9514 8.1917 0.974564 0.824141 0.730518 1.052458 2.602392 7.099208 Sample Estimates -------Percentiles for Exponentiated-------25th 50th 75th Label Diff Diff Diff Diff Diff Diff at at at at at at current current current current current current 0 1 2 3 4 5 0.6387 1.0608 1.7017 2.4652 3.2157 4.0250 1.0085 1.4626 2.1170 3.0383 4.3661 6.2849 1.6289 2.0240 2.6066 3.7632 6.0152 9.9777 Lower HPD of Exponentiated Upper HPD of Exponentiated 0.07387 0.4184 0.9713 1.4772 1.3250 0.8604 3.1001 3.2783 3.6418 5.3149 9.9922 20.2432 5448 F Chapter 66: The PLM Procedure The sample statistics are constructed from the posterior sample saved in the item store cowgmd. From the output, the odds of a cow showing shock reaction at level 0 in the initial experiment is 1.2811 (with a 95% HPD interval (0:07387, 3:1001)) times the odds in the repeated experiment. The HPD interval for the odds ratio is constructed based on the mean and variance of the sample of the exponentiated log odds ratios, instead of based on the exponentiated mean and variance of the posterior sample of log odds ratios. The HPD interval suggests that there is not much evidence that the cows responded differently at current level 0 between the two experiments. Similar conclusions can be drawn for current level 1, 2, and 5. However, there is strong evidence that cows responded differently at current level 3 and 4 between the two experiments. The possible explanation is that, if the current level is so small that cows could hardly feel it or the current level is so strong that cows could hardly bear it, cows would respond consistently in the two experiment. If the current level is moderate, cows might get used to it and their response diminished in the repeated experiment. You can visualize the distribution of the posterior sample of log odds ratios by specifying the PLOTS= option in the ESTIMATE statement. In the following statements, ODS Graphics is enabled by the ODS GRAPHICS ON statement, the PLOTS=BOXPLOT option requests a box plot of posterior distribution of log odds ratios. The suboption ORIENT=HORIZONTAL specifies a horizontal orientation of the boxes. ods graphics on; proc plm source=cowgmd; estimate 'Diff at current 0' experiment 1 -1 current*experiment 'Diff at current 1' experiment 1 -1 current*experiment 'Diff at current 2' experiment 1 -1 current*experiment 'Diff at current 3' experiment 1 -1 current*experiment 'Diff at current 4' experiment 1 -1 current*experiment 'Diff at current 5' experiment 1 -1 current*experiment / plots=boxplot(orient=horizontal); run; ods graphics off; [1, [1, [1, [1, [1, [1, 0 1 2 3 4 5 1] 1] 1] 1] 1] 1] [-1, [-1, [-1, [-1, [-1, [-1, 0 1 2 3 4 5 Output 66.4.3 displays the box plot of the posterior sample of log odds ratios. The two boxes for differences at current level 3 and 4 show that the corresponding log odds ratios are significantly larger than the reference value x D 0. This indicate that there is obvious evidence that the probability of cow response is larger in the initial experiment than in the repeated one at the two current levels. The other four boxes show that the corresponding log odds ratios are not significantly different from 0, which suggests that there is no obvious reaction difference at current level 0, 1, 2, and 5 between the two experiments. 2], 2], 2], 2], 2], 2] Example 66.5: By-Group Processing F 5449 Output 66.4.3 Box Plot of Difference between Two Experiments Example 66.5: By-Group Processing This example uses a data set on a study of the analgesic effects of treatments on elderly patients with neuralgia. The purpose of this example is to show how PROC PLM behaves under different situations when By-group processing is present. Two test treatments and a placebo are compared to test whether the patient reported pain or not. For each patient, the information of age, gender, and the duration of complaint before the treatment began were recorded. The following DATA step creates the data set named Neuralgia: 5450 F Chapter 66: The PLM Procedure Data Neuralgia; input Treatment $ datalines; P F 68 1 No B M 74 P M 66 26 Yes B F 67 A F 71 12 No B F 72 A M 71 17 Yes A F 63 B F 66 12 No A M 62 A F 64 17 No P M 74 P M 70 1 Yes B M 66 A F 64 30 No A M 70 B F 78 1 No P M 83 B M 75 30 Yes P M 77 A M 70 12 No A F 69 B M 70 1 No B M 67 P M 78 12 Yes B M 77 P M 66 4 Yes P F 65 A M 78 15 Yes B M 75 P F 72 27 No P F 70 B F 65 7 No P F 68 P M 67 17 Yes B M 70 P F 67 1 Yes A M 67 A F 74 1 No B M 80 ; Sex $ Age Duration Pain $ @@; 16 28 50 27 42 4 19 28 1 29 12 23 1 29 21 13 27 22 10 21 No No No No No No No No Yes Yes No No Yes No Yes Yes Yes No No Yes P B B A P A B A B P B A B P A A P A P A F F F F F F M M F F F M F M F M M M F F 67 77 76 69 64 72 59 69 69 79 65 76 69 60 67 75 68 65 72 69 30 16 9 18 1 25 29 1 42 20 14 25 24 26 11 6 11 15 11 3 No No Yes Yes Yes No No No No Yes No Yes No Yes No Yes Yes No Yes No The data set contains five variables. Treatment is a classification variable that has three levels: A and B represent the two test treatments, and P represents the placebo treatment. Sex is a classification variable that indicates each patient’s gender. Age is a continuous variable that indicates the age in years of each patient when a treatment began. Duration is a continuous variable that indicates the duration of complaint in months. The last variable Pain is the response variable with two levels: ‘Yes’ if pain was reported, ‘No’ if no pain was reported. Suppose there is some preliminary belief that the dependency of pain on the explanatory variables is different for male and female patients, leading to separate models between genders. You believe there might be redundant information for predicting the probability of Pain. Thus, you want to perform model selection to eliminate unnecessary effects. You can use the following statements: proc sort data=Neuralgia; by sex; run; proc logistic data=Neuralgia; class Treatment / param=glm; model pain = Treatment Age Duration / selection=backward; by sex; store painmodel; title 'Logistic Model on Neuralgia'; run; Example 66.5: By-Group Processing F 5451 PROC SORT is called to sort the data by variable Sex. The LOGISTIC procedure is then called to fit the probability of no pain. Three variables are specified for the full model: Treatment, Age, and Duration. The backward elimination is used as the model selection method. The BY statement specifies that separate models be fitted for male and female patients. Finally, the STORE statement specifies that the fitted results be saved to an item store named painmodel. Output 66.5.1 lists parameter estimates from the two models after backward elimination is performed. From the model for female patients, Treatment is the only factor that affects the probability of no pain, and Treatment A and B have the same positive effect in predicting the probability of no pain. From the model for male patients, both Treatment and Age are included in the selected model. Treatment A and B have different positive effects, while Age has a negative effect in predicting the probability of no pain. Output 66.5.1 Parameter Estimates for Male and Female Patients Logistic Model on Neuralgia ------------------------------------ Sex=F ------------------------------------The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates Parameter Intercept Treatment A Treatment B Treatment P DF Estimate Standard Error Wald Chi-Square Pr > ChiSq 1 1 1 0 -0.4055 2.6027 2.6027 0 0.6455 1.2360 1.2360 . 0.3946 4.4339 4.4339 . 0.5299 0.0352 0.0352 . ------------------------------------ Sex=M ------------------------------------Analysis of Maximum Likelihood Estimates Parameter Intercept Treatment A Treatment B Treatment P Age DF Estimate Standard Error Wald Chi-Square Pr > ChiSq 1 1 1 0 1 20.6178 3.9982 4.5556 0 -0.3416 9.1638 1.7333 1.9252 . 0.1408 5.0621 5.3208 5.5993 . 5.8869 0.0245 0.0211 0.0180 . 0.0153 5452 F Chapter 66: The PLM Procedure Now the fitted models are saved to the item store painmodel. Suppose you want to use it to score several new observations. The following DATA steps create three data sets for scoring: data score1; input Treatment $ Sex $ Age; datalines; A F 20 B F 30 P F 40 A M 20 B M 30 P M 40 ; data score2; set score1(drop=sex); run; data score3; set score2(drop=Age); run; The first score data set score1 contains six observations and all the variables that are specified in the full model. The second score data set score2 is a duplicate of score1 except that Sex is dropped. The third score data set score3 is a duplicate of score2 except that Age is dropped. You can use the following statements to score the three data sets: proc plm score score score run; source=painmodel; data=score1 out=score1out predicted; data=score2 out=score2out predicted; data=score3 out=score3out predicted; Output 66.5.2 lists the store information that PROC PLM reads from the item store painmodel. The “Model Effects” entry lists all three variables that are specified in the full model before the By-group processing. Output 66.5.2 Item Store Information for painmodel Logistic Model on Neuralgia The PLM Procedure Store Information Item Store Data Set Created From Created By Date Created By Variable Response Variable Link Function Distribution Class Variables Model Effects WORK.PAINMODEL WORK.NEURALGIA PROC LOGISTIC 13JAN10:13:02:41 Sex Pain Logit Binary Treatment Pain Intercept Treatment Age Duration Example 66.5: By-Group Processing F 5453 With the three SCORE statements, three data sets are thus produced: score1out, score2out, and score3out. They contain the linear predictors in addition to all original variables. The data set score1out contains the values shown in Output 66.5.3: Output 66.5.3 Values of Data Set score1out Logistic Model on Neuralgia Obs Treatment Sex 1 2 3 4 5 6 A B P A B P F F F M M M Age Predicted 20 30 40 20 30 40 2.1972 2.1972 -0.4055 17.7850 14.9269 6.9557 Linear predictors are computed for all six observations. Because the BY variable Sex is available in score1, PROC PLM uses separate models to score observations of male and female patients. So an observation with the same Treatment and Age has different linear predictors for different genders. The data set score2out contains the values shown in Output 66.5.4: Output 66.5.4 Values of Data Set score2out Logistic Model on Neuralgia Obs 1 2 3 4 5 6 7 8 9 10 11 12 Sex Treatment F F F F F F M M M M M M A B P A B P A B P A B P Age Predicted 20 30 40 20 30 40 20 30 40 20 30 40 2.1972 2.1972 -0.4055 2.1972 2.1972 -0.4055 17.7850 14.9269 6.9557 17.7850 14.9269 6.9557 The second score data set score2 does not contain the BY variable Sex. PROC PLM continues to score the full data set two times. Each time the scoring is based on the fitted model for each corresponding By-group. In the output data set, Sex is added at the first column as the By-group indicator. The first six entries correspond to the model for female patients, and the next six entries correspond to the model for male patients. Age is not included in the first model, and Treatment A and B have the same parameter estimates, so observations 1, 2, 4, and 5 have the same linear predicted value. 5454 F Chapter 66: The PLM Procedure The data set score3out contains the values shown in Output 66.5.5: Output 66.5.5 Values of Data Set score3out Logistic Model on Neuralgia Obs 1 2 3 4 5 6 7 8 9 10 11 12 Sex Treatment F F F F F F M M M M M M A B P A B P A B P A B P Predicted 2.19722 2.19722 -0.40547 2.19722 2.19722 -0.40547 . . . . . . The third score data set score3 does not contain the BY variable Sex. PROC PLM scores the full data twice with separate models. Furthermore, it does not contain the variable Age, which is a selected variable for predicting the probability of no pain for male patients. Thus, PROC PLM computes linear predictor values for score3 by using the first model for female patients, and sets the linear predictor to missing when using the second model for male patients to score the data set. Example 66.6: Comparing Multiple B-Splines This example conducts an analysis similar to Example 15 in Chapter 38.33, “Examples: GLIMMIX Procedure.” It uses simulated data to perform multiple comparisons among predicted values in a model with group-specific trends that are modeled through regression splines. The estimable functions are formed using nonpositional syntax with constructed effects. Consider the data in the following DATA step. Each of the 100 observations for the continuous response variable y is associated with one of two groups. Example 66.6: Comparing Multiple B-Splines F 5455 data spline; input group y @@; x = _n_; datalines; 1 -.020 1 0.199 2 -.397 1 0.065 1 0.253 2 -.460 1 0.379 1 0.971 2 0.574 2 0.755 2 1.088 2 0.607 1 0.629 2 1.237 2 1.002 2 1.201 1 1.329 1 1.580 2 1.052 2 1.108 2 1.726 2 1.179 2 2.105 2 1.828 1 1.984 2 1.867 2 1.522 2 2.200 1 2.769 1 2.534 1 2.873 1 2.678 1 2.893 1 3.023 2 2.549 1 2.836 1 3.727 1 3.806 1 2.948 2 1.954 1 3.744 2 2.431 2 1.996 2 2.028 2 2.337 1 4.516 2 2.474 2 2.221 1 5.253 2 3.024 ; 2 2 2 1 1 2 2 1 2 2 2 2 1 1 2 1 1 2 1 2 2 2 2 1 2 -1.36 -.861 0.195 0.712 0.316 0.959 0.734 1.520 1.098 1.257 1.338 1.368 2.771 2.562 1.969 3.135 3.050 2.375 3.269 2.326 2.040 2.321 2.326 4.867 2.403 1 1 2 2 2 1 2 1 1 2 1 1 1 1 1 2 2 2 1 2 1 2 2 2 1 -.026 0.251 -.108 0.811 0.961 0.653 0.299 1.105 1.613 2.005 1.707 2.252 2.052 2.517 2.460 1.705 2.273 1.841 3.533 2.017 3.995 2.479 2.144 2.453 5.498 The following statements fit a model with separate trends for the two groups; the trends are modeled as B-splines. proc orthoreg data=spline; class group; effect spl = spline(x); model y = group spl*group / noint; store ortho_spline; title 'B-splines Comparisons'; run; Results from this analysis are shown in Output 66.6.1. The “Parameter Estimates” table shows the estimates for the spline coefficients in the two groups. 5456 F Chapter 66: The PLM Procedure Output 66.6.1 Results for Group-Specific Spline Model B-splines Comparisons The ORTHOREG Procedure Dependent Variable: y Source Model Error Uncorrected Total DF Sum of Squares 14 86 100 481.92117059 6.3223804119 488.243551 Root MSE R-Square Parameter (group='1') (group='2') spl_group_1_1 spl_group_1_2 spl_group_2_1 spl_group_2_2 spl_group_3_1 spl_group_3_2 spl_group_4_1 spl_group_4_2 spl_group_5_1 spl_group_5_2 spl_group_6_1 spl_group_6_2 spl_group_7_1 spl_group_7_2 Mean Square F Value Pr > F 34.422940756 0.0735160513 468.24 <.0001 0.2711384357 0.9603214326 DF Parameter Estimate Standard Error t Value Pr > |t| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 9.70265463962039 6.30619220563569 -11.1786451718041 -20.1946092746139 -9.53273697995301 -5.85652496534967 -8.96118371893294 -5.55671605245205 -7.26153231478755 -4.36778889738236 -6.44615256510896 -4.03801618914902 -4.63816959094139 -4.30290104395061 0 0 3.1341899987 2.6299147768 3.7008097395 3.9765046236 3.2575832048 2.7906116773 3.0717508806 2.5716715573 3.243690314 2.7246809593 2.9616955361 2.4588839125 3.7094636319 3.0478540171 . . 3.10 2.40 -3.02 -5.08 -2.93 -2.10 -2.92 -2.16 -2.24 -1.60 -2.18 -1.64 -1.25 -1.41 . . 0.0026 0.0187 0.0033 <.0001 0.0044 0.0388 0.0045 0.0335 0.0278 0.1126 0.0323 0.1042 0.2146 0.1616 . . By default, the ORTHOREG procedure constructs B-splines with seven knots. Since B-spline coefficients satisfy a sum-to-one constraint and since the model contains group-specific intercepts, the last spline coefficient for each group is redundant and estimated as 0. The following statements make a prediction for the input data set by using the SCORE statement with PROC PLM and graph the observed and predicted values in the two groups: proc plm source=ortho_spline; score data=spline out=ortho_pred predicted=p; run; proc sgplot data=ortho_pred; series y=p x=x / group=group name="fit"; scatter y=y x=x / group=group; keylegend "fit" / title="Group"; run; Example 66.6: Comparing Multiple B-Splines F 5457 The prediction plot in Output 66.6.2 suggests that there is some separation of the group trends for small values of x and for values that exceed about x D 40. Output 66.6.2 Observed Data and Predicted Values by Group In order to determine the range on which the trends separate significantly, the PLM procedure is executed in the following statements with an ESTIMATE statement that applies group comparisons at a number of values for the spline variable x: %macro GroupDiff; %do x=0 %to 75 %by 5; "Diff at x=&x" group 1 -1 group*spl [1,1 &x] [-1,2 %end; 'Diff at x=80' group 1 -1 group*spl [1,1 80] [-1,2 80] %mend; proc plm source=ortho_spline; show effects; estimate %GroupDiff / adjust=simulate seed=1 stepdown; run; &x], 5458 F Chapter 66: The PLM Procedure For example, the following ESTIMATE statement compares the trends between the two groups at x D 25: estimate 'Diff at x=25' group 1 -1 group*spl [1,1 25] [-1,2 25]; The nonpositional syntax is used for the group*spl effect. For example, the specification Œ 1; 2 25 requests that the spline be computed at x D 25 for the second level of variable group. The resulting coefficients are added to the bL vector for the estimate after being multiplied with 1. Because comparisons are made at a large number of values for x, a multiplicity correction is in order to adjust the p-values to reflect familywise error control. Simulated p-values with step-down adjustment are used here. Output 66.6.3 displays the “Store Information” for the item store and information about the spline effect (the result of the SHOW statement). Output 66.6.3 Spline Details B-splines Comparisons The PLM Procedure Store Information Item Store Data Set Created From Created By Date Created Response Variable Class Variable Constructed Effect Model Effects WORK.ORTHO_SPLINE WORK.SPLINE PROC ORTHOREG 13JAN10:13:03:14 y group spl group spl*group B-splines Comparisons The PLM Procedure Knots for Spline Effect spl Knot Number 1 2 3 4 5 6 7 8 9 Boundary * * * * * * x -48.50000 -23.75000 1.00000 25.75000 50.50000 75.25000 100.00000 124.75000 149.50000 Example 66.6: Comparing Multiple B-Splines F 5459 Output 66.6.3 continued B-splines Comparisons The PLM Procedure Basis Details for Spline Effect spl Column 1 2 3 4 5 6 7 Support Knots --------Support-------48.50000 -48.50000 -23.75000 1.00000 25.75000 50.50000 75.25000 25.75000 50.50000 75.25000 100.00000 124.75000 149.50000 149.50000 1-4 1-5 2-6 3-7 4-8 5-9 6-9 Output 66.6.4 displays the results from the ESTIMATE statement. Output 66.6.4 Estimate Results with Multiplicity Correction Estimates Adjustment for Multiplicity: Holm-Simulated Label Diff Diff Diff Diff Diff Diff Diff Diff Diff Diff Diff Diff Diff Diff Diff Diff Diff at at at at at at at at at at at at at at at at at x=0 x=5 x=10 x=15 x=20 x=25 x=30 x=35 x=40 x=45 x=50 x=55 x=60 x=65 x=70 x=75 x=80 Estimate Standard Error DF t Value Pr > |t| Adj P 12.4124 1.0376 0.3778 0.05822 -0.02602 0.02014 0.1023 0.1924 0.2883 0.3877 0.4885 0.5903 0.7031 0.8401 1.0147 1.2400 1.5237 4.2130 0.1759 0.1540 0.1481 0.1243 0.1312 0.1378 0.1236 0.1114 0.1195 0.1308 0.1231 0.1125 0.1203 0.1348 0.1326 0.1281 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 2.95 5.90 2.45 0.39 -0.21 0.15 0.74 1.56 2.59 3.24 3.74 4.79 6.25 6.99 7.52 9.35 11.89 0.0041 <.0001 0.0162 0.6952 0.8346 0.8783 0.4600 0.1231 0.0113 0.0017 0.0003 <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 0.0206 <.0001 0.0545 0.9101 0.9565 0.9565 0.7418 0.2925 0.0450 0.0096 0.0024 <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 Notice that the “Store Information” in Output 66.6.3 displays the classification variables (from the CLASS statement in PROC ORTHOREG), the constructed effects (from the EFFECT statement in PROC ORTHOREG), and the model effects (from the MODEL statement in PROC ORTHOREG). Output 66.6.4 shows that at the 5% significance level the trends are significantly different for x 10 and for x 40. Between those values you cannot reject the hypothesis of trend congruity. 5460 F Chapter 66: The PLM Procedure To see this effect more clearly, you can filter the results by adding the following filtering statement to the previous PROC PLM run: filter adjp > 0.05; This produces Output 66.6.5, which displays the subset of the results in Output 66.6.4 that meets the condition in the FILTER expression. Output 66.6.5 Filtered Estimate Results B-splines Comparisons The PLM Procedure Estimates Adjustment for Multiplicity: Holm-Simulated Label Diff Diff Diff Diff Diff Diff at at at at at at x=10 x=15 x=20 x=25 x=30 x=35 Estimate Standard Error DF t Value Pr > |t| Adj P 0.3778 0.05822 -0.02602 0.02014 0.1023 0.1924 0.1540 0.1481 0.1243 0.1312 0.1378 0.1236 86 86 86 86 86 86 2.45 0.39 -0.21 0.15 0.74 1.56 0.0162 0.6952 0.8346 0.8783 0.4600 0.1231 0.0545 0.9101 0.9565 0.9565 0.7418 0.2925 Example 66.7: Linear Inference with Arbitrary Estimates Suppose that you have calculated a vector of parameter estimates of dimension .p 1/ and its associated variance-covariance matrix by some statistical method. You are now interested in using these results to perform linear inference, or perhaps to score a data set and to calculate predicted values and their standard errors. The following DATA steps create two SAS data sets. The first, called parms, contains six estimates that represent two uncorrelated groups. The data set cov contains the covariance matrix of the estimates. The lack of correlation between the two sets of three parameters is evident in the blockdiagonal structure of the covariance matrix. data parms; length name $6; input Name$ Value; datalines; alpha1 -3.5671 beta1 0.4421 gamma1 -2.6230 alpha2 -3.0111 beta2 0.3977 gamma2 -2.4442 ; Example 66.7: Linear Inference with Arbitrary Estimates F 5461 data cov; input Parm row col1-col6; datalines; 1 1 0.007462 -0.005222 0.010234 0.000000 0.000000 0.000000 1 2 -0.005222 0.048197 -0.010590 0.000000 0.000000 0.000000 1 3 0.010234 -0.010590 0.215999 0.000000 0.000000 0.000000 1 4 0.000000 0.000000 0.000000 0.031261 -0.009096 0.015785 1 5 0.000000 0.000000 0.000000 -0.009096 0.039487 -0.019996 1 6 0.000000 0.000000 0.000000 0.015785 -0.019996 0.126172 ; Suppose that you are interested in testing whether the parameters are homogeneous across groups— that is, whether ˛1 D ˛2 ; ˇ1 D ˇ2 ; 1 D 2. You are interested in testing the hypothesis jointly and separately with multiplicity adjustment. In order to use the facilities of the PLM procedure, you first need to create an item store that contains the necessary information as if the preceding parameter vector and covariance matrix were the result of a statistical modeling procedure. The following statements use the multivariate facilities of the GLIMMIX procedure to create such an item store, by fitting a saturated linear model with the GLIMMIX procedure where the data set that contains the parameter estimates serves as the input data set: proc glimmix data=parms order=data; class Name; model Value = Name / noint ddfm=none s; random _residual_ / type=lin(1) ldata=cov v; parms (1) / noiter; store ArtificialModel; title 'Linear Inference'; run; The RANDOM statement is used to form the covariance structure for the estimates. The PARMS statement prevents iterative updates of the covariance parameters. The resulting marginal covariance matrix of the “data” is thus identical to the covariance matrix in the data set cov. The ORDER=DATA option in the PROC GLIMMIX statement is used to arrange the levels of the classification variable Name in the order in which they appear in the data set so that the order of the parameters matches that of the covariance matrix. The results of this analysis are shown in Output 66.7.1. Note that the parameter estimates are identical to the values passed in the input data set and their standard errors equal the square root of the diagonal elements of the cov data set. 5462 F Chapter 66: The PLM Procedure Output 66.7.1 “Fitted” Parameter Estimates and Covariance Matrix Linear Inference The GLIMMIX Procedure Estimated V Matrix for Subject 1 Row Col1 Col2 Col3 1 2 3 4 5 6 0.007462 -0.00522 0.01023 -0.00522 0.04820 -0.01059 0.01023 -0.01059 0.2160 Col4 Col5 Col6 0.03126 -0.00910 0.01579 -0.00910 0.03949 -0.02000 0.01579 -0.02000 0.1262 Solutions for Fixed Effects Effect name name name name name name name alpha1 beta1 gamma1 alpha2 beta2 gamma2 Estimate Standard Error DF t Value Pr > |t| -3.5671 0.4421 -2.6230 -3.0111 0.3977 -2.4442 0.08638 0.2195 0.4648 0.1768 0.1987 0.3552 Infty Infty Infty Infty Infty Infty -41.29 2.01 -5.64 -17.03 2.00 -6.88 <.0001 0.0440 <.0001 <.0001 0.0454 <.0001 There are other ways to fit a saturated model with the GLIMMIX procedure. For example, you can use the TYPE=UN covariance structure in the RANDOM statement with a properly prepared input data set for the PDATA= option in the PARMS statement. See Example 17 in Chapter 38.33, “Examples: GLIMMIX Procedure,” for details. Once the item store exists, you can apply the linear inference capabilities of the PLM procedure. For example, the ESTIMATE statement in the following statements test the hypothesis of parameter homogeneity across groups: proc plm source=ArtificialModel; estimate 'alpha1 = alpha2' Name 1 0 'beta1 = beta2 ' Name 0 1 'gamma1 = gamma2' Name 0 0 adjust=bon stepdown run; 0 -1 0 0, 0 0 -1 0, 1 0 0 -1 / ftest(label='Homogeneity'); References F 5463 Output 66.7.2 Results from the PLM Procedure Linear Inference The PLM Procedure Estimates Adjustment for Multiplicity: Holm Label alpha1 = alpha2 beta1 = beta2 gamma1 = gamma2 Estimate Standard Error DF t Value Pr > |t| Adj P -0.5560 0.04440 -0.1788 0.1968 0.2961 0.5850 Infty Infty Infty -2.83 0.15 -0.31 0.0047 0.8808 0.7599 0.0142 1.0000 1.0000 F Test for Estimates Label Homogeneity Num DF Den DF F Value Pr > F 3 Infty 2.79 0.0389 The F test in Output 66.7.2 shows that the joint test of homogeneity is rejected. The individual tests with familywise control of the Type I error show that the overall difference is due to a significant change in the ˛ parameters. The hypothesis of homogeneity across the two groups cannot be rejected for the ˇ and parameters. References Asuncion, A. and Newman, D. J. (2007). UCI Machine Learning Repository, Irvine, CA: University of California, School of Information and Computer Science. Kenward, M. G. and Roger, J. H. (1997), “Small Sample Inference for Fixed Effects from Restricted Maximum Likelihood,” Biometrics, 53, 983–997. McCullagh, P. and Nelder, J. A. (1989), Generalized Linear Models, Second Edition, London: Chapman & Hall. Silvapulle, M. J. and Sen, P. K. (2004), Constrained Statistical Inference: Order, Inequality, and Shape Constraints, New York: John Wiley & Sons. Weisberg, S. (1985), Applied Linear Regression, Second Edition. New York: John Wiley & Sons. Subject Index alpha level PLM procedure, 5428 degrees of freedom PLM procedure, 5428 options summary ESTIMATE statement, 5422 PLM procedure alpha level, 5428 BY processing, 5434 common postprocessing statements, 5409 degrees of freedom, 5428 filter PLM results, 5423 item store, 5408 least squares means, 5429 ODS graph names, 5438 ODS Graphics, 5419 ODS table names, 5437 posterior inference, 5434 scoring statistics, 5429 user-defined formats, 5436 scoring statistics PLM procedure, 5429 Syntax Index procedure, EFFECTPLOT statement, 5421 procedure, ESTIMATE statement, 5422 procedure, LSMEANS statement, 5425 procedure, LSMESTIMATE statement, 5426 procedure, SLICE statement, 5431 procedure, TEST statement, 5431 ALL option SHOW statement (PLM), 5430 ALPHA= option PROC PLM statement (PLM), 5418 SCORE statement (PLM), 5428 BYVAR option SHOW statement (PLM), 5430 CLASS option SHOW statement (PLM), 5430 CORRELATION option SHOW statement (PLM), 5430 COVARIANCE option SHOW statement (PLM), 5430 DDFMETHOD= option PROC PLM statement (PLM), 5418 DF= option SCORE statement (PLM), 5428 EFFECTPLOT statement procedure, 5421 EFFECTS option SHOW statement (PLM), 5430 ESTEPS= option PROC PLM statement (PLM), 5418 ESTIMATE statement procedure, 5422 FILTER statement PLM procedure, 5423 FITSTATS option SHOW statement (PLM), 5430 FORMAT= option PROC PLM statement (PLM), 5418 HERMITE option SHOW statement (PLM), 5430 HESSIAN option SHOW statement (PLM), 5430 ILINK option SCORE statement (PLM), 5428 LSMEANS statement procedure, 5425 LSMESTIMATE statement procedure, 5426 MAXLEN= option PROC PLM statement (PLM), 5419 NOCLPRINT option PROC PLM statement (PLM), 5419 NOINFO option PROC PLM statement (PLM), 5419 NOPRINT option PROC PLM statement (PLM), 5419 NOUNIQUE option SCORE statement (PLM), 5428 NOVAR option SCORE statement (PLM), 5428 OBSCAT option SCORE statement (PLM), 5428 PARAMETERS option SHOW statement (PLM), 5430 PERCENTILES= option PROC PLM statement (PLM), 5419 PLM procedure, 5417 FILTER statement, 5423 PROC PLM statement, 5418 SHOW statement, 5429 syntax, 5417 WHERE statement, 5432 PLM procedure, FILTER statement, 5423 PLM procedure, PROC PLM statement, 5418 ALPHA= option, 5418 DDFMETHOD= option, 5418 ESTEPS= option, 5418 FORMAT= option, 5418 MAXLEN= option, 5419 NOCLPRINT option, 5419 NOINFO option, 5419 PERCENTILES= option, 5419 PLOT option, 5419 PLOTS option, 5419 SEED= option, 5420 SINGCHOL= option, 5420 SINGRES= option, 5420 SINGULAR= option, 5420 SOURCE= option, 5420 STMTORDER= option, 5421 WHEREFORMAT option, 5421 ZETA= option, 5421 PLM procedure, SCORE statement ALPHA= option, 5428 DF= option, 5428 ILINK option, 5428 NOUNIQUE option, 5428 NOVAR option, 5428 OBSCAT option, 5428 SAMPLE option, 5429 PLM procedure, SHOW statement, 5429 ALL option, 5430 BYVAR option, 5430 CLASS option, 5430 CORREATION option, 5430 COVARIANCE option, 5430 EFFECTS option, 5430 FITSTATS option, 5430 HERMITE option, 5430 HESSIAN option, 5430 PARAMETERS option, 5430 PROGRAM option, 5431 XPX option, 5431 XPXI option, 5431 PLM procedure, WHERE statement, 5432 PLOT option PROC PLM statement, 5419 PLOTS option PROC PLM statement, 5419 PROC PLM statement, see PLM procedure PLM procedure, 5418 PROGRAM option SHOW statement (PLM), 5431 SAMPLE option SCORE statement (PLM), 5429 SEED= option PROC PLM statement (PLM), 5420 SHOW statement PLM procedure, 5429 SINGCHOL= option PROC PLM statement (PLM), 5420 SINGRES= option PROC PLM statement (PLM), 5420 SINGULAR= option PROC PLM statement (PLM), 5420 SLICE statement procedure, 5431 SOURCE= option PROC PLM statement (PLM), 5420 STMTORDER= option PROC PLM statement (PLM), 5421 TEST statement procedure, 5431 WHERE statement PLM procedure, 5432 WHEREFORMAT option PROC PLM statement (PLM), 5421 XPX option SHOW statement (PLM), 5431 XPXPI option SHOW statement (PLM), 5431 ZETA= option PROC PLM statement (PLM), 5421 Your Turn We welcome your feedback. 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