Version 13
Basic Analysis
“The real voyage of discovery consists not in seeking new
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13.1
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JMP® 13 Basic Analysis. Cary, NC: SAS Institute Inc.
JMP® 13 Basic Analysis
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Contents
Basic Analysis
1
Learn about JMP
Documentation and Additional Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Formatting Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
JMP Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
JMP Documentation Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
JMP Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Resources for Learning JMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tutorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sample Data Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Learn about Statistical and JSL Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Learn JMP Tips and Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tooltips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
JMP User Community . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
JMPer Cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
JMP Books by Users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The JMP Starter Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Technical Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
20
21
21
27
27
28
28
28
28
29
29
29
30
30
30
Introduction to Basic Analysis
Overview of Fundamental Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3
Distributions
Using the Distribution Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Overview of the Distribution Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Categorical Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Continuous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Distribution Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Launch the Distribution Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Distribution Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Frequencies Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Quantiles Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Summary Statistics Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Distribution Platform Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Options for Categorical Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
34
34
34
36
37
39
42
42
42
45
46
10
Basic Analysis
Display Options Submenu for Categorical Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Histogram Options Submenu for Categorical Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Save Submenu for Categorical Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Options for Continuous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Display Options Submenu for Continuous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Histogram Options Submenu for Continuous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Normal Quantile Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Outlier Box Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quantile Box Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stem and Leaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CDF Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Test Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Test Std Dev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Test Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Confidence Intervals for Continuous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Save Commands for Continuous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Prediction Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tolerance Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Capability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Continuous Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discrete Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Distribution Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Examples of the Distribution Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Selecting Data in Multiple Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Examples of the Test Probabilities Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Prediction Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Tolerance Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Capability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Statistical Details for the Distribution Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Standard Error Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Normal Quantile Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wilcoxon Signed Rank Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Standard Deviation Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Normal Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Saving Standardized Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Prediction Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tolerance Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Capability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Continuous Fit Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discrete Fit Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
46
47
47
47
48
50
51
52
52
53
53
54
55
56
57
58
59
59
61
61
62
63
66
66
67
69
70
71
72
72
73
73
74
75
77
77
77
78
78
79
83
90
Basic Analysis
11
Fitted Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Fit Distribution Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4
Introduction to Fit Y by X
Examine Relationships Between Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Overview of the Fit Y by X Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Launch the Fit Y by X Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Launch Specific Analyses from the JMP Starter Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5
Bivariate Analysis
Examine Relationships between Two Continuous Variables . . . . . . . . . . . . . . . . . . . . . . . . 99
Example of Bivariate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Launch the Bivariate Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Bivariate Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fitting Commands and Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Mean Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fitting Command Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit the Same Command Multiple Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Histogram Borders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Mean Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Line and Fit Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linear Fit and Polynomial Fit Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Special . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Special Reports and Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flexible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kernel Smoother . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Each Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Orthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Orthogonal Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Orthogonal Regression Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Robust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Robust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Density Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Correlation Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonpar Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonparametric Bivariate Density Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Group By . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fitting Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fitting Menu Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Diagnostics Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
100
101
102
103
104
105
105
105
106
106
107
113
113
114
115
116
117
117
117
118
118
118
119
119
120
120
121
121
122
122
125
12
Basic Analysis
Additional Examples of the Bivariate Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Example of the Fit Special Command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Example Using the Fit Orthogonal Command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Example Using the Fit Robust Command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Example of Group By Using Density Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Example of Group By Using Regression Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Statistical Details for the Bivariate Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Fit Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Fit Spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Fit Orthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Summary of Fit Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Lack of Fit Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Parameter Estimates Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Smoothing Fit Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Correlation Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6
Oneway Analysis
Examine Relationships between a Continuous Y and a Categorical X Variable . . . . 137
Overview of Oneway Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Oneway Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Launch the Oneway Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Oneway Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Oneway Platform Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Display Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Outlier Box Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Means/Anova and Means/Anova/Pooled t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Summary of Fit Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The t-test Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Analysis of Variance Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Means for Oneway Anova Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Block Means Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mean Diamonds and X-Axis Proportional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mean Lines, Error Bars, and Standard Deviation Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of Means Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of Means for Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of Means for Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of Means Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of Means Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Compare Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Using Comparison Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Each Pair, Student’s t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
All Pairs, Tukey HSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
138
140
140
141
144
146
146
147
147
148
149
150
150
150
151
152
152
153
154
155
155
157
158
158
Basic Analysis
With Best, Hsu MCB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
With Control, Dunnett’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Compare Means Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonparametric Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Wilcoxon, Median, and Van der Waerden Test Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kolmogorov-Smirnov Two-Sample Test Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonparametric Multiple Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Unequal Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tests That the Variances Are Equal Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equivalence Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Robust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Robust Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cauchy Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power Details Window and Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Normal Quantile Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CDF Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Matching Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Examples of the Oneway Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of an Analysis of Means Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of an Analysis of Means for Variances Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Each Pair, Student’s t Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the All Pairs, Tukey HSD Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the With Best, Hsu MCB Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the With Control, Dunnett’s Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example Contrasting All of the Compare Means Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Nonparametric Wilcoxon Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Unequal Variances Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of an Equivalence Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Robust Fit Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Power Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of a Normal Quantile Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of a CDF Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Densities Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Matching Column Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Stacking Data for a Oneway Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Statistical Details for the Oneway Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of Fit Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tests That the Variances Are Equal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonparametric Test Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
159
160
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165
167
168
170
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176
178
180
181
182
183
186
187
188
190
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192
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195
202
202
203
204
204
206
14
7
Basic Analysis
Contingency Analysis
Examine Relationships between Two Categorical Variables . . . . . . . . . . . . . . . . . . . . . . 209
Example of Contingency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Launch the Contingency Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Contingency Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contingency Platform Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mosaic Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pop-Up Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contingency Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Description of the Contingency Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Description of the Tests Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fisher’s Exact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of Means for Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Correspondence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Understanding Correspondence Analysis Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Correspondence Analysis Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Details Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cochran-Mantel-Haenszel Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Agreement Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Relative Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two Sample Test for Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measures of Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cochran Armitage Trend Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Examples of the Contingency Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Analysis of Means for Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Correspondence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of a Cochran Mantel Haenszel Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Agreement Statistic Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Relative Risk Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of a Two Sample Test for Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Measures of Association Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Cochran Armitage Trend Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Statistical Details for the Contingency Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Agreement Statistic Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Odds Ratio Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tests Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Details Report in Correspondence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
210
211
212
213
214
215
216
218
218
219
219
219
220
220
220
221
221
222
222
223
223
225
225
226
226
227
229
231
232
233
234
235
236
236
236
237
238
Logistic Analysis
Examine Relationships between a Categorical Y and a Continuous X Variable . . . . 239
Overview of Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
240
Basic Analysis
Nominal Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ordinal Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Nominal Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Launch the Logistic Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Logistic Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Logistic Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Whole Model Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Logistic Platform Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ROC Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Save Probability Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inverse Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Examples of Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Ordinal Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Example of a Logistic Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of ROC Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Inverse Prediction Using the Crosshair Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Inverse Prediction Using the Inverse Prediction Option . . . . . . . . . . . . . . . . . . . . . .
Statistical Details for the Logistic Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Whole Model Test Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
15
240
240
241
242
243
243
244
245
245
247
247
249
250
250
250
250
252
254
255
256
258
258
Tabulate
Create Summary Tables Interactively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Example of the Tabulate Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Launch the Tabulate Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Use the Dialog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Add Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Tabulate Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Grouping Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Column and Row Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Edit Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tabulate Platform Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Show Test Build Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Right-Click Menu for Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Examples of the Tabulate Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Combining Columns into a Single Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example Using a Page Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
260
263
265
266
269
271
271
272
273
273
274
275
275
279
281
16
Basic Analysis
10 Simulate
Answer Challenging Questions with Parametric Resampling . . . . . . . . . . . . . . . . . . . . . . 285
Overview of Simulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Examples That Use Simulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Construct an Accurate Confidence Interval for Variance Components . . . . . . . . . . . . . . . . . . . . .
Conduct a Permutation Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explore Retaining a Factor in Generalized Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conduct Prospective Power Analysis for a Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Launch the Simulate Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Simulate Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Simulate Results Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation Results Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulated Power Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
286
286
287
292
295
300
308
308
309
310
310
11 Bootstrapping
Approximate the Distribution of a Statistic through Resampling . . . . . . . . . . . . . . . . . . 311
Overview of Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bootstrapping Window Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stacked Results Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Unstacked Bootstrap Results Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of Bootstrap Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Example of Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Statistical Details for Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculation of Fractional Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bias-Corrected Percentile Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
312
313
315
316
317
318
319
324
324
324
12 Text Explorer
Explore Unstructured Text in Your Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Text Explorer Platform Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Text Processing Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the Text Explorer Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Launch the Text Explorer Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Customize Regex: Regular Expression Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Text Explorer Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary Counts Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Term and Phrase Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Text Explorer Platform Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Text Preparation Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Text Analysis Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Save Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Report Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Latent Class Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
328
329
330
333
335
340
340
341
343
343
349
351
353
354
Basic Analysis
Latent Semantic Analysis (SVD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SVD Plots Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Topic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Topic Words Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Topic Scores Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Example of the Text Explorer Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A
17
355
356
356
356
357
357
Statistical Details
Basic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Platforms That Support Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
B
References
Index
Basic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
18
Basic Analysis
Chapter 1
Learn about JMP
Documentation and Additional Resources
This chapter includes the following information:
•
book conventions
•
JMP documentation
•
JMP Help
•
additional resources, such as the following:
– other JMP documentation
– tutorials
– indexes
– Web resources
– technical support options
20
Learn about JMP
Formatting Conventions
Chapter 1
Basic Analysis
Formatting Conventions
The following conventions help you relate written material to information that you see on
your screen:
•
Sample data table names, column names, pathnames, filenames, file extensions, and
folders appear in Helvetica font.
•
Code appears in Lucida Sans Typewriter font.
•
Code output appears in Lucida Sans Typewriter italic font and is indented farther than
the preceding code.
•
Helvetica bold formatting indicates items that you select to complete a task:
– buttons
– check boxes
– commands
– list names that are selectable
– menus
– options
– tab names
– text boxes
•
The following items appear in italics:
– words or phrases that are important or have definitions specific to JMP
– book titles
– variables
– script output
•
Features that are for JMP Pro only are noted with the JMP Pro icon
of JMP Pro features, visit http://www.jmp.com/software/pro/.
Note: Special information and limitations appear within a Note.
Tip: Helpful information appears within a Tip.
. For an overview
Chapter 1
Basic Analysis
Learn about JMP
JMP Documentation
21
JMP Documentation
JMP offers documentation in various formats, from print books and Portable Document
Format (PDF) to electronic books (e-books).
•
Open the PDF versions from the Help > Books menu.
•
All books are also combined into one PDF file, called JMP Documentation Library, for
convenient searching. Open the JMP Documentation Library PDF file from the Help > Books
menu.
•
You can also purchase printed documentation and e-books on the SAS website:
http://www.sas.com/store/search.ep?keyWords=JMP
JMP Documentation Library
The following table describes the purpose and content of each book in the JMP library.
Document Title
Document Purpose
Document Content
Discovering JMP
If you are not familiar
with JMP, start here.
Introduces you to JMP and gets you
started creating and analyzing data.
Using JMP
Learn about JMP data
tables and how to
perform basic
operations.
Covers general JMP concepts and
features that span across all of JMP,
including importing data, modifying
columns properties, sorting data, and
connecting to SAS.
Basic Analysis
Perform basic analysis
using this document.
Describes these Analyze menu platforms:
•
Distribution
•
Fit Y by X
•
Tabulate
•
Text Explorer
Covers how to perform bivariate,
one-way ANOVA, and contingency
analyses through Analyze > Fit Y by X.
How to approximate sampling
distributions using bootstrapping and
how to perform parametric resampling
with the Simulate platform are also
included.
22
Learn about JMP
JMP Documentation
Chapter 1
Basic Analysis
Document Title
Document Purpose
Document Content
Essential Graphing
Find the ideal graph
for your data.
Describes these Graph menu platforms:
•
Graph Builder
•
Overlay Plot
•
Scatterplot 3D
•
Contour Plot
•
Bubble Plot
•
Parallel Plot
•
Cell Plot
•
Treemap
•
Scatterplot Matrix
•
Ternary Plot
•
Chart
The book also covers how to create
background and custom maps.
Profilers
Learn how to use
interactive profiling
tools, which enable you
to view cross-sections
of any response
surface.
Covers all profilers listed in the Graph
menu. Analyzing noise factors is
included along with running simulations
using random inputs.
Design of
Experiments Guide
Learn how to design
experiments and
determine appropriate
sample sizes.
Covers all topics in the DOE menu and
the Specialized DOE Models menu item
in the Analyze > Specialized Modeling
menu.
Chapter 1
Basic Analysis
Learn about JMP
JMP Documentation
Document Title
Document Purpose
Document Content
Fitting Linear Models
Learn about Fit Model
platform and many of
its personalities.
Describes these personalities, all
available within the Analyze menu Fit
Model platform:
•
Standard Least Squares
•
Stepwise
•
Generalized Regression
•
Mixed Model
•
MANOVA
•
Loglinear Variance
•
Nominal Logistic
•
Ordinal Logistic
•
Generalized Linear Model
23
24
Learn about JMP
JMP Documentation
Chapter 1
Basic Analysis
Document Title
Document Purpose
Document Content
Predictive and
Specialized Modeling
Learn about additional
modeling techniques.
Describes these Analyze > Predictive
Modeling menu platforms:
•
Modeling Utilities
•
Neural
•
Partition
•
Bootstrap Forest
•
Boosted Tree
•
K Nearest Neighbors
•
Naive Bayes
•
Model Comparison
•
Formula Depot
Describes these Analyze > Specialized
Modeling menu platforms:
•
Fit Curve
•
Nonlinear
•
Gaussian Process
•
Time Series
•
Matched Pairs
Describes these Analyze > Screening
menu platforms:
•
Response Screening
•
Process Screening
•
Predictor Screening
•
Association Analysis
The platforms in the Analyze >
Specialized Modeling > Specialized DOE
Models menu are described in Design of
Experiments Guide.
Chapter 1
Basic Analysis
Learn about JMP
JMP Documentation
Document Title
Document Purpose
Document Content
Multivariate
Methods
Read about techniques
for analyzing several
variables
simultaneously.
Describes these Analyze > Multivariate
Methods menu platforms:
•
Multivariate
•
Principal Components
•
Discriminant
•
Partial Least Squares
Describes these Analyze > Clustering
menu platforms:
Quality and Process
Methods
Read about tools for
evaluating and
improving processes.
•
Hierarchical Cluster
•
K Means Cluster
•
Normal Mixtures
•
Latent Class Analysis
•
Cluster Variables
Describes these Analyze > Quality and
Process menu platforms:
•
Control Chart Builder and individual
control charts
•
Measurement Systems Analysis
•
Variability / Attribute Gauge Charts
•
Process Capability
•
Pareto Plot
•
Diagram
25
26
Learn about JMP
JMP Documentation
Chapter 1
Basic Analysis
Document Title
Document Purpose
Document Content
Reliability and
Survival Methods
Learn to evaluate and
improve reliability in a
product or system and
analyze survival data
for people and
products.
Describes these Analyze > Reliability and
Survival menu platforms:
Consumer Research
Scripting Guide
Learn about methods
for studying consumer
preferences and using
that insight to create
better products and
services.
Learn about taking
advantage of the
powerful JMP
Scripting Language
(JSL).
•
Life Distribution
•
Fit Life by X
•
Cumulative Damage
•
Recurrence Analysis
•
Degradation and Destructive
Degradation
•
Reliability Forecast
•
Reliability Growth
•
Reliability Block Diagram
•
Repairable Systems Simulation
•
Survival
•
Fit Parametric Survival
•
Fit Proportional Hazards
Describes these Analyze > Consumer
Research menu platforms:
•
Categorical
•
Multiple Correspondence Analysis
•
Multidimensional Scaling
•
Factor Analysis
•
Choice
•
MaxDiff
•
Uplift
•
Item Analysis
Covers a variety of topics, such as writing
and debugging scripts, manipulating
data tables, constructing display boxes,
and creating JMP applications.
Chapter 1
Basic Analysis
Learn about JMP
Additional Resources for Learning JMP
Document Title
Document Purpose
Document Content
JSL Syntax Reference
Read about many JSL
functions on functions
and their arguments,
and messages that you
send to objects and
display boxes.
Includes syntax, examples, and notes for
JSL commands.
27
Note: The Books menu also contains two reference cards that can be printed: The Menu Card
describes JMP menus, and the Quick Reference describes JMP keyboard shortcuts.
JMP Help
JMP Help is an abbreviated version of the documentation library that provides targeted
information. You can open JMP Help in several ways:
•
On Windows, press the F1 key to open the Help system window.
•
Get help on a specific part of a data table or report window. Select the Help tool
from
the Tools menu and then click anywhere in a data table or report window to see the Help
for that area.
•
Within a JMP window, click the Help button.
•
Search and view JMP Help on Windows using the Help > Help Contents, Search Help, and
Help Index options. On Mac, select Help > JMP Help.
•
Search the Help at http://jmp.com/support/help/ (English only).
Additional Resources for Learning JMP
In addition to JMP documentation and JMP Help, you can also learn about JMP using the
following resources:
•
Tutorials (see “Tutorials” on page 28)
•
Sample data (see “Sample Data Tables” on page 28)
•
Indexes (see “Learn about Statistical and JSL Terms” on page 28)
•
Tip of the Day (see “Learn JMP Tips and Tricks” on page 28)
•
Web resources (see “JMP User Community” on page 29)
•
JMPer Cable technical publication (see “JMPer Cable” on page 29)
•
Books about JMP (see “JMP Books by Users” on page 30)
•
JMP Starter (see “The JMP Starter Window” on page 30)
28
Learn about JMP
Additional Resources for Learning JMP
•
Chapter 1
Basic Analysis
Teaching Resources (see “Sample Data Tables” on page 28)
Tutorials
You can access JMP tutorials by selecting Help > Tutorials. The first item on the Tutorials menu
is Tutorials Directory. This opens a new window with all the tutorials grouped by category.
If you are not familiar with JMP, then start with the Beginners Tutorial. It steps you through the
JMP interface and explains the basics of using JMP.
The rest of the tutorials help you with specific aspects of JMP, such as designing an experiment
and comparing a sample mean to a constant.
Sample Data Tables
All of the examples in the JMP documentation suite use sample data. Select Help > Sample
Data Library to open the sample data directory.
To view an alphabetized list of sample data tables or view sample data within categories,
select Help > Sample Data.
Sample data tables are installed in the following directory:
On Windows: C:\Program Files\SAS\JMP\13\Samples\Data
On Macintosh: \Library\Application Support\JMP\13\Samples\Data
In JMP Pro, sample data is installed in the JMPPRO (rather than JMP) directory. In JMP
Shrinkwrap, sample data is installed in the JMPSW directory.
To view examples using sample data, select Help > Sample Data and navigate to the Teaching
Resources section. To learn more about the teaching resources, visit http://jmp.com/tools.
Learn about Statistical and JSL Terms
The Help menu contains the following indexes:
Statistics Index Provides definitions of statistical terms.
Lets you search for information about JSL functions, objects, and display
boxes. You can also edit and run sample scripts from the Scripting Index.
Scripting Index
Learn JMP Tips and Tricks
When you first start JMP, you see the Tip of the Day window. This window provides tips for
using JMP.
Chapter 1
Basic Analysis
Learn about JMP
Additional Resources for Learning JMP
29
To turn off the Tip of the Day, clear the Show tips at startup check box. To view it again, select
Help > Tip of the Day. Or, you can turn it off using the Preferences window. See the Using JMP
book for details.
Tooltips
JMP provides descriptive tooltips when you place your cursor over items, such as the
following:
•
Menu or toolbar options
•
Labels in graphs
•
Text results in the report window (move your cursor in a circle to reveal)
•
Files or windows in the Home Window
•
Code in the Script Editor
Tip: On Windows, you can hide tooltips in the JMP Preferences. Select File > Preferences >
General and then deselect Show menu tips. This option is not available on Macintosh.
JMP User Community
The JMP User Community provides a range of options to help you learn more about JMP and
connect with other JMP users. The learning library of one-page guides, tutorials, and demos is
a good place to start. And you can continue your education by registering for a variety of JMP
training courses.
Other resources include a discussion forum, sample data and script file exchange, webcasts,
and social networking groups.
To access JMP resources on the website, select Help > JMP User Community or visit
https://community.jmp.com/.
JMPer Cable
The JMPer Cable is a yearly technical publication targeted to users of JMP. The JMPer Cable is
available on the JMP website:
http://www.jmp.com/about/newsletters/jmpercable/
30
Learn about JMP
Technical Support
Chapter 1
Basic Analysis
JMP Books by Users
Additional books about using JMP that are written by JMP users are available on the JMP
website:
http://www.jmp.com/en_us/software/books.html
The JMP Starter Window
The JMP Starter window is a good place to begin if you are not familiar with JMP or data
analysis. Options are categorized and described, and you launch them by clicking a button.
The JMP Starter window covers many of the options found in the Analyze, Graph, Tables, and
File menus. The window also lists JMP Pro features and platforms.
•
To open the JMP Starter window, select View (Window on the Macintosh) > JMP Starter.
•
To display the JMP Starter automatically when you open JMP on Windows, select File >
Preferences > General, and then select JMP Starter from the Initial JMP Window list. On
Macintosh, select JMP > Preferences > Initial JMP Starter Window.
Technical Support
JMP technical support is provided by statisticians and engineers educated in SAS and JMP,
many of whom have graduate degrees in statistics or other technical disciplines.
Many technical support options are provided at http://www.jmp.com/support, including the
technical support phone number.
Chapter 2
Introduction to Basic Analysis
Overview of Fundamental Analysis Methods
This book describes the initial types of analyses that you often perform in JMP:
•
The Distribution platform illustrates the distribution of a single variable using histograms,
additional graphs, and reports. Once you know how your data is distributed, you can plan
the appropriate type of analysis going forward. See Chapter 3, “Distributions”.
•
The Fit Y by X platform analyzes the pair of X and Y variables that you specify, by context,
based on modeling type. See Chapter 4, “Introduction to Fit Y by X”. The four types of
analyses include:
– The Bivariate platform, which analyzes the relationship between two continuous X
variables. See Chapter 5, “Bivariate Analysis”.
– The Oneway platform, which analyzes how the distribution of a continuous Y variable
differs across groups defined by a categorical X variable. See Chapter 6, “Oneway
Analysis”.
– The Contingency platform, which analyzes the distribution of a categorical response
variable Y as conditioned by the values of a categorical X factor. See Chapter 7,
“Contingency Analysis”.
– The Logistic platform, which fits the probabilities for response categories (Y) to a
continuous X predictor. See Chapter 8, “Logistic Analysis”.
•
The Tabulate platform interactively constructs tables of descriptive statistics. See Chapter
9, “Tabulate”.
•
The Simulate feature provides parametric and nonparametric simulation capability. See
Chapter 10, “Simulate”.
•
Bootstrap analysis approximates the sampling distribution of a statistic. The data is
re-sampled with replacement and the statistic is computed. This process is repeated to
produce a distribution of values for the statistic. See Chapter 11, “Bootstrapping”.
•
The Text Explorer platform enables you to categorize and analyze unformatted text data.
You can use regular expressions to clean up the data before you proceed to analysis. See
Chapter 12, “Text Explorer”.
32
Introduction to Basic Analysis
Chapter 2
Basic Analysis
Chapter 3
Distributions
Using the Distribution Platform
The Distribution platform illustrates the distribution of a single variable using histograms,
additional graphs, and reports. The word univariate simply means involving one variable
instead of two (bivariate) or many (multivariate). However, you can examine the distribution
of several individual variables within a report. The report content for each variable changes
depending on whether the variable is categorical (nominal or ordinal) or continuous.
Once you know how your data is distributed, you can plan the appropriate type of analysis
going forward.
The Distribution report window is interactive. Clicking on a histogram bar highlights the
corresponding data in any other histograms and in the data table. See Figure 3.1.
Figure 3.1 Example of the Distribution Platform
34
Distributions
Overview of the Distribution Platform
Chapter 3
Basic Analysis
Overview of the Distribution Platform
The treatment of variables in the Distribution platform is different, depending on the
modeling type of the variable, which can be categorical (nominal or ordinal) or continuous.
Categorical Variables
For categorical variables, the initial graph that appears is a histogram. The histogram shows a
bar for each level of the ordinal or nominal variable. You can also add a divided (mosaic) bar
chart.
The reports show counts and proportions. You can add confidence intervals and test the
probabilities.
Continuous Variables
For numeric continuous variables, the initial graphs show a histogram and an outlier box plot.
The histogram shows a bar for grouped values of the continuous variable. The following
options are also available:
•
quantile box plot
•
normal quantile plot
•
stem and leaf plot
•
CDF plot
The reports show selected quantiles and summary statistics. Report options are available for
the following:
•
saving ranks, probability scores, normal quantile values, and so on, as new columns in the
data table
•
testing the mean and standard deviation of the column against a constant you specify
•
fitting various distributions and nonparametric smoothing curves
•
performing a capability analysis for a quality control application
•
confidence intervals, prediction intervals, and tolerance intervals
Example of the Distribution Platform
Suppose that you have data on 40 students, and you want to see the distribution of age and
height among the students.
1. Select Help > Sample Data Library and open Big Class.jmp.
Chapter 3
Basic Analysis
Distributions
Example of the Distribution Platform
2. Select Analyze > Distribution.
3. Select age and height and click Y, Columns.
4. Click OK.
Figure 3.2 Example of the Distribution Platform
From the histograms, you notice the following:
•
The ages are not uniformly distributed.
•
For height, there are two points with extreme values (that might be outliers).
Click on the bar for 50 in the height histogram to take a closer look at the potential outliers.
•
The corresponding ages are highlighted in the age histogram. The potential outliers are
age 12.
•
The corresponding rows are highlighted in the data table. The names of the potential
outliers are Lillie and Robert.
35
36
Distributions
Launch the Distribution Platform
Chapter 3
Basic Analysis
Add labels to the potential outliers in the height histogram.
1. Select both outliers.
2. Right-click on one of the outliers and select Row Label.
Label icons are added to these rows in the data table.
3. Resize the box plot wider to see the full labels.
Figure 3.3 Potential Outliers Labeled
Launch the Distribution Platform
Launch the Distribution platform by selecting Analyze > Distribution.
Figure 3.4 The Distribution Launch Window
Y, Columns Assigns the variables that you want to analyze. A histogram and associated
reports appear for each variable.
Weight Assigns a variable that specifies weights for observations on continuous Ys. For
categorical Ys, the Weight column is ignored. Any statistic that is based on the sum of the
weights is affected by weights.
Chapter 3
Basic Analysis
Distributions
The Distribution Report
37
Freq Assigns a frequency variable to this role. This is useful if you have summarized data. In
this instance, you have one column for the Y values and another column for the frequency
of occurrence of the Y values. The sum of this variable is included in the overall count
appearing in the Summary Statistics report (represented by N). All other moment statistics
(mean, standard deviation, and so on) are also affected by the Freq variable.
By
Produces a separate report for each level of the By variable. If more than one By variable
is assigned, a separate report is produced for each possible combination of the levels of the
By variables.
Histograms Only Removes everything except the histograms from the report window.
For general information about launch windows, see the Get Started chapter in the Using JMP
book.
After you click OK, the Distribution report window appears. See “The Distribution Report” on
page 37.
The Distribution Report
Follow the instructions in “Example of the Distribution Platform” on page 34 to produce the
report shown in Figure 3.5.
38
Distributions
The Distribution Report
Chapter 3
Basic Analysis
Figure 3.5 The Initial Distribution Report Window
Note: When you apply only the Hidden row state to rows in the data table, the
corresponding points do not appear in plots that show points. However, histograms are
constructed using the hidden rows. If you want to exclude rows from the construction of
the histograms and from analysis results, apply the Exclude row state and select Redo >
Redo Analysis from the red triangle menu next to Distributions.
The initial Distribution report contains a histogram and reports for each variable. Note the
following:
•
To replace a variable in a report, from the Columns panel of the associated data table, drag
and drop the variable into the axis of the histogram.
•
To insert a new variable into a report, creating a new histogram, drag and drop the
variable outside of an existing histogram. The new variable can be placed before, between,
or after the existing histograms.
Note: To remove a variable, select Remove from the red triangle menu.
Chapter 3
Basic Analysis
Distributions
The Distribution Report
39
•
The red triangle menu next to Distributions contains options that affect all of the variables.
See “Distribution Platform Options” on page 45.
•
The red triangle menu next to each variable contains options that affect only that variable.
See “Options for Categorical Variables” on page 46 or “Options for Continuous Variables”
on page 47. If you hold down the Control key and select a variable option, the option
applies to all of the variables that have the same modeling type.
•
Histograms visually display your data. See “Histograms” on page 39.
•
The initial report for a categorical variable contains a Frequencies report. See “The
Frequencies Report” on page 42.
•
The initial report for a continuous variable contains a Quantiles and a Summary Statistics
report. See “The Quantiles Report” on page 42 and “The Summary Statistics Report” on
page 42.
Histograms
Histograms visually display your data. For categorical (nominal or ordinal) variables, the
histogram shows a bar for each level of the ordinal or nominal variable. For continuous
variables, the histogram shows a bar for grouped values of the continuous variable.
Highlighting data Click on a histogram bar or an outlying point in the graph. The
corresponding rows are highlighted in the data table, and corresponding sections of other
histograms are also highlighted, if applicable. See “Highlight Bars and Select Rows” on
page 41.
Creating a subset Double-click on a histogram bar, or right-click on a histogram bar and
select Subset. A new data table is created that contains only the selected data.
Resizing the entire histogram Hover over the histogram borders until you see a double-sided
arrow. Then click and drag the borders.
Rescaling the axis (Continuous variables only) Click and drag on an axis to rescale it.
Alternatively, hover over the axis until you see a hand. Then, double-click on the axis and
set the parameters in the Axis Specification window.
Resizing histogram bars (Continuous variables only) There are multiple options to resize
histogram bars. See “Resize Histogram Bars for Continuous Variables” on page 40.
Specify the data that you select in multiple histograms. See “Specify
Your Selection in Multiple Histograms” on page 41.
Specifying your selection
To see additional options for the histogram or the associated data table:
•
Right-click on a histogram. For details, see the Using JMP book.
•
Right-click on an axis. You can add a label or modify the axis. For details, see the JMP
Reports chapter in the Using JMP book.
40
Distributions
The Distribution Report
•
Chapter 3
Basic Analysis
Click on the red triangle next to the variable, and select Histogram Options. Options are
slightly different depending on the variable modeling type. See “Options for Categorical
Variables” on page 46 or “Options for Continuous Variables” on page 47.
Resize Histogram Bars for Continuous Variables
Resize histogram bars for continuous variables by using the following:
•
the Grabber (hand) tool
•
the Set Bin Width option
•
the Increment option
Use the Grabber Tool
The Grabber tool is a quick way to explore your data.
1. Select Tools > Grabber.
Note: (Windows only) To see the menu bar, you might need to hover over the bar below
the window title. You can also change this setting in File > Preferences > Windows
Specific.
2. Place the grabber tool anywhere in the histogram.
3. Click and drag the histogram bars.
Think of each bar as a bin that holds a number of observations:
•
Moving the hand to the left increases the bin width and combines intervals. The number of
bars decreases as the bar size increases.
•
Moving the hand to the right decreases the bin width, producing more bars.
•
Moving the hand up or down shifts the bin locations on the axis, which changes the
contents and size of each bin.
Use the Set Bin Width Option
The Set Bin Width option is a more precise way to set the width for all bars in a histogram. To
use the Set Bin Width option, from the red triangle menu for the variable, select Histogram
Options > Set Bin Width. Change the bin width value.
Use the Increment Option
The Increment option is another precise way to set the bar width. To use the Increment option,
double-click on the axis, and change the Increment value.
Chapter 3
Basic Analysis
Distributions
The Distribution Report
41
Highlight Bars and Select Rows
Clicking on a histogram bar highlights the bar and selects the corresponding rows in the data
table. The appropriate portions of all other graphical displays also highlight the selection.
Figure 3.6 shows the results of highlighting a bar in the height histogram. The corresponding
rows are selected in the data table.
Tip: To deselect specific histogram bars, press the Control key and click the highlighted bars.
Figure 3.6 Highlighting Bars and Rows
Select a bar to highlight rows
and parts of other output.
Specify Your Selection in Multiple Histograms
Extend or narrow your selection in histograms as follows:
•
To extend your selection, hold down the Shift key and select another bar. This is the
equivalent of using an or operator.
•
To narrow your selection, hold down the Control and Alt keys (Windows) or Command
and Option keys (Macintosh) and select another bar. This is the equivalent of using an and
operator.
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The Distribution Report
Chapter 3
Basic Analysis
Related Information
•
“Example of Selecting Data in Multiple Histograms” on page 66
The Frequencies Report
For nominal and ordinal variables, the Frequencies report lists the levels of the variables,
along with the associated frequency of occurrence and probabilities.
For each level of a categorical (nominal or ordinal) variable, the Frequencies report contains
the information described in the following list. Missing values are omitted from the analysis.
Tip: Click a value in the Frequencies report to select the corresponding data in the histogram
and data table.
Level
Lists each value found for a response variable.
Lists the number of rows found for each level of a response variable. If you use a Freq
variable, the Count is the sum of the Freq variables for each level of the response variable.
Count
Prob Lists the probability (or proportion) of occurrence for each level of a response variable.
The probability is computed as the count divided by the total frequency of the variable,
shown at the bottom of the table.
Lists the standard error of the probabilities. This column might be hidden. To
show the column, right-click in the table and select Columns > StdErr Prob.
StdErr Prob
Cum Prob Contains the cumulative sum of the column of probabilities. This column might be
hidden. To show the column, right-click in the table and select Columns > Cum Prob.
The Quantiles Report
For continuous variables, the Quantiles report lists the values of selected quantiles (sometimes
called percentiles).
Related Information
•
“Quantiles” on page 73
The Summary Statistics Report
For continuous variables, the Summary Statistics report displays the mean, standard
deviation, and other summary statistics. You can control which statistics appear in this report
by selecting Customize Summary Statistics from the red triangle menu next to Summary
Statistics.
Chapter 3
Basic Analysis
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The Distribution Report
43
Tip: To specify which summary statistics show in the report each time you run a Distribution
analysis for a continuous variable, select File > Preferences > Platforms > Distribution
Summary Statistics, and select the ones you want to appear.
•
“Description of the Summary Statistics Report” describes the statistics that appear by
default.
•
“Additional Summary Statistics” describes additional statistics that you can add to the
report using the Customize Summary Statistics window.
Description of the Summary Statistics Report
Mean Estimates the expected value of the underlying distribution for the response variable,
which is the arithmetic average of the column’s values. It is the sum of the non-missing
values divided by the number of non-missing values.
The normal distribution is mainly defined by the mean and standard deviation.
These parameters provide an easy way to summarize data as the sample becomes large:
Std Dev
– 68% of the values are within one standard deviation of the mean
– 95% of the values are within two standard deviations of the mean
– 99.7% of the values are within three standard deviations of the mean
Std Err Mean The standard error of the mean, which estimates the standard deviation of the
distribution of the mean.
Are 95% confidence limits about the mean. They
define an interval that is very likely to contain the true population mean.
Upper 95% Mean and Lower 95% Mean
N
Is the total number of nonmissing values.
Additional Summary Statistics
Sum Weight The sum of a column assigned to the role of Weight (in the launch window).
Sum Wgt is used in the denominator for computations of the mean instead of N.
Sum The sum of the response values.
Variance The sample variance, and the square of the sample standard deviation.
Skewness Measures sidedness or symmetry.
Measures peakedness or heaviness of tails. See “Kurtosis” on page 74 for formula
details.
Kurtosis
CV The percent coefficient of variation. It is computed as the standard deviation divided by
the mean and multiplied by 100. The coefficient of variation can be used to assess relative
variation, for example when comparing the variation in data measured in different units
or with different magnitudes.
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The Distribution Report
N Missing
Chapter 3
Basic Analysis
The number of missing observations.
N Zero The number of zero values.
N Unique The number of unique values.
Uncorrected SS The uncorrected sum of squares or sum of values squared.
Corrected SS The corrected sum of squares or sum of squares of deviations from the mean.
(Appears only if you have not specified a Frequency variable.) First
autocorrelation that tests if the residuals are correlated across the rows. This test helps
detect non-randomness in the data.
Autocorrelation
Minimum Represents the 0 percentile of the data.
Maximum
Represents the 100 percentile of the data.
Median Represents the 50th percentile of the data.
Mode The value that occurs most often in the data. If there are multiple modes, the smallest
mode appears.
Trimmed Mean The mean calculated after removing the smallest p% and the largest p% of the
data. The value of p is entered in the Enter trimmed mean percent text box at the bottom of
the window. The Trimmed Mean option is not available if you have specified a Weight
variable.
Geometric Mean The nth root of the product of the data. Zero and negative numbers are
treated like missing values. For example, you might want to compare two companies
based on varying metrics that come from different ranges. The statistic is also helpful
when the data contains a large value in a skewed distribution.
Range The difference between the maximum and minimum of the data.
Interquartile Range The difference between the 3rd and 1st quartiles.
(Does not appear if you have specified a Weight variable.) The
median of the absolute deviations from the median.
Median Absolute Deviation
Robust Mean The robust mean, calculated in a way that is resistant to outliers, using Huberʹs
M-estimation. See Huber and Ronchetti (2009).
The robust standard deviation, calculated in a way that is resistant to
outliers, using Huberʹs M-estimation. See Huber and Ronchetti (2009).
Robust Std Dev
Enter (1-alpha) for mean confidence interval
Specify the alpha level for the mean confidence
interval.
Enter trimmed mean percent Specify the trimmed mean percentage. The percentage is
trimmed off each side of the data.
Chapter 3
Basic Analysis
Distributions
Distribution Platform Options
45
Summary Statistics Options
The red triangle menu next to Summary Statistics contains these options:
Select which statistics you want to appear from the list. You
can select or deselect all summary statistics.
Customize Summary Statistics
Show All Modes
Shows all of the modes if there are multiple modes.
Related Information
•
“Summary Statistics” on page 73
Distribution Platform Options
The red triangle menu next to Distributions contains options that affect all of the reports and
graphs in the Distribution platform.
Uniform Scaling Scales all axes with the same minimum, maximum, and intervals so that the
distributions can be easily compared.
Stack Changes the orientation of the histogram and the reports to horizontal and stacks the
individual distribution reports vertically. Deselect this option to return the report window
to its original layout.
Arrange in Rows Enter the number of plots that appear in a row. This option helps you view
plots vertically rather than in one wide row.
Saves the histograms as .swf files that are Adobe Flash
player compatible. Use these files in presentations and in Web pages. An HTML page is
also saved that shows you the correct code for using the resulting .swf file.
Save for Adobe Flash platform (.SWF)
For more information about this option, go to http://www.jmp.com/support/swfhelp/en.
See the JMP Reports chapter in the Using JMP book for more information about the following
options:
Shows or hides the local data filter that enables you to filter the data used in
a specific report.
Local Data Filter
Contains options that enable you to repeat or relaunch the analysis. In platforms that
support the feature, the Automatic Recalc option immediately reflects the changes that
you make to the data table in the corresponding report window.
Redo
Contains options that enable you to save a script that reproduces the report to
several destinations.
Save Script
Save By-Group Script Contains options that enable you to save a script that reproduces the
platform report for all levels of a By variable to several destinations. Available only when a
By variable is specified in the launch window.
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Options for Categorical Variables
Chapter 3
Basic Analysis
Options for Categorical Variables
The red triangle menus next to each variable in the report window contain additional options
that apply to the variable. This section describes the options that are available for categorical
(nominal or ordinal) variables.
To see the options that are available for continuous variables, see “Options for Continuous
Variables” on page 47.
Display Options Submenu for Categorical Variables
Frequencies Shows or hides the Frequencies report. See “The Frequencies Report” on
page 42.
Horizontal Layout
Changes the orientation of the histogram and the reports to vertical or
horizontal.
Axes on Left Moves the Count, Prob, and Density axes to the left instead of the right.
This option is applicable only if Horizontal Layout is selected.
Histogram Options Submenu for Categorical Variables
Histogram Shows or hides the histogram. See “Histograms” on page 39.
Vertical Changes the orientation of the histogram from a vertical to a horizontal orientation.
Std Error Bars Draws the standard error bar on each level of the histogram.
Separate Bars Separates the histogram bars.
Histogram Color Changes the color of the histogram bars.
Count Axis Adds an axis that shows the frequency of column values represented by the
histogram bars.
Adds an axis that shows the proportion of column values represented by
histogram bars.
Prob Axis
Density Axis Adds an axis that shows the length of the bars in the histogram.
The count and probability axes are based on the following calculations:
prob = (bar width)*density
count = (bar width)*density*(total count)
Show Percents Labels the percent of column values represented by each histogram bar.
Show Counts Labels the frequency of column values represented by each histogram bar.
Chapter 3
Basic Analysis
Distributions
Options for Continuous Variables
47
Mosaic Plot Displays a mosaic bar chart for each nominal or ordinal response variable. A
mosaic plot is a stacked bar chart where each segment is proportional to its group’s
frequency count.
Order By Reorders the histogram, mosaic plot, and Frequencies report in ascending or
descending order, by count. To save the new order as a column property, use the Save >
Value Ordering option.
Test Probabilities Displays a report that tests hypothesized probabilities. See “Examples of
the Test Probabilities Option” on page 67 for more details.
Confidence Interval This menu contains confidence levels. Select a value that is listed, or
select Other to enter your own. JMP computes score confidence intervals.
Save Submenu for Categorical Variables
Level Numbers Creates a new column in the data table called Level <colname>. The level
number of each observation corresponds to the histogram bar that contains the
observation.
(Use with the Order By option) Creates a new value ordering column
property in the data table, reflecting the new order.
Value Ordering
Script to log Displays the script commands to generate the current report in the log window.
Select View > Log to see the log window.
Remove Permanently removes the variable and all its reports from the Distribution report.
Options for Continuous Variables
The red triangle menus next to each variable in the report window contain additional options
that apply to the variable. This section describes the options that are available for continuous
variables.
To see the options that are available for categorical (nominal and ordinal) variables, see
“Options for Categorical Variables” on page 46.
Display Options Submenu for Continuous Variables
Quantiles Shows or hides the Quantiles report. See “The Quantiles Report” on page 42.
Set Quantile Increment
Changes the quantile increment or revert back to the default quantile
increment.
Custom Quantiles Sets custom quantiles by values or by increments. You can specify the
confidence level and choose whether to compute smoothed empirical likelihood quantiles
(for large data sets, this can take some time).
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Options for Continuous Variables
Chapter 3
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– For details about how the weighted average quantiles are estimated, see “Quantiles”
on page 73.
– For details about distribution-free confidence limits for the weighted average quantiles,
see section 5.2 in Hahn and Meeker (1991).
– Smoothed empirical likelihood quantiles are based on a kernel density estimate. For
more details about how these quantiles and their confidence limits are estimated, see
Chen and Hall (1993).
Summary Statistics Shows or hides the Summary Statistics report. See “The Summary
Statistics Report” on page 42.
Adds or removes statistics from the Summary Statistics
report. See “The Summary Statistics Report” on page 42.
Customize Summary Statistics
Horizontal Layout
Changes the orientation of the histogram and the reports to vertical or
horizontal.
Axes on Left Moves the Count, Prob, Density, and Normal Quantile Plot axes to the left instead
of the right.
This option is applicable only if Horizontal Layout is selected.
Histogram Options Submenu for Continuous Variables
Histogram Shows or hides the histogram. See “Histograms” on page 39.
Shadowgram Replaces the histogram with a shadowgram. To understand a shadowgram,
consider that if the bin width of a histogram is changed, the appearance of the histogram
changes. A shadowgram overlays histograms with different bin widths. Dominant
features of a distribution are less transparent on the shadowgram.
Note that the following options are not available for shadowgrams:
– Std Error Bars
– Show Counts
– Show Percents
Vertical Changes the orientation of the histogram from a vertical to a horizontal orientation.
Std Error Bars Draws the standard error bar on each level of the histogram using the
standard error. The standard error bar adjusts automatically when you adjust the number
of bars with the hand tool. See “Resize Histogram Bars for Continuous Variables” on
page 40, and “Standard Error Bars” on page 72.
Set Bin Width Changes the bin width of the histogram bars. See “Resize Histogram Bars for
Continuous Variables” on page 40.
Histogram Color Changes the color of the histogram bars.
Chapter 3
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Distributions
Options for Continuous Variables
49
Count Axis Adds an axis that shows the frequency of column values represented by the
histogram bars.
Note: If you resize the histogram bars, the count axis also resizes.
Adds an axis that shows the proportion of column values represented by
histogram bars.
Prob Axis
Note: If you resize the histogram bars, the probability axis also resizes.
Density Axis The density is the length of the bars in the histogram. Both the count and
probability are based on the following calculations:
prob = (bar width)*density
count = (bar width)*density*(total count)
When looking at density curves that are added by the Fit Distribution option, the density
axis shows the point estimates of the curves.
Note: If you resize the histogram bars, the density axis also resizes.
Show Percents Labels the proportion of column values represented by each histogram bar.
Show Counts Labels the frequency of column values represented by each histogram bar.
Normal Quantile Plot Adds a normal quantile plot that shows the extent to which the variable
is normally distributed. See “Normal Quantile Plot” on page 50.
Adds an outlier box plot that shows the outliers in your data. See “Outlier
Box Plot” on page 51.
Outlier Box Plot
Quantile Box Plot Adds a quantile box plot that shows specific quantiles from the Quantiles
report. See “Quantile Box Plot” on page 52.
Adds a stem and leaf report, which is a variation of the histogram. See “Stem
and Leaf” on page 52.
Stem and Leaf
Adds a plot of the empirical cumulative distribution function. See “CDF Plot” on
page 53.
CDF Plot
Test Mean Performs a one-sample test for the mean. See “Test Mean” on page 53.
Test Std Dev
Performs a one-sample test for the standard deviation. See “Test Std Dev” on
page 54.
Test Equivalence Performs an equivalence test. See “Test Equivalence” on page 55.
Confidence Interval Choose confidence intervals for the mean and standard deviation. See
“Confidence Intervals for Continuous Variables” on page 56.
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Options for Continuous Variables
Chapter 3
Basic Analysis
Choose prediction intervals for a single observation, or for the mean and
standard deviation of the next randomly selected sample. See “Prediction Intervals” on
page 58.
Prediction Interval
Computes an interval to contain at least a specified proportion of the
population. See “Tolerance Intervals” on page 59.
Tolerance Interval
Capability Analysis Measures the conformance of a process to given specification limits. See
“Capability Analysis” on page 59.
Continuous Fit Fits distributions to continuous variables. See “Fit Distributions” on page 61.
Discrete Fit
Fits distributions to discrete variables. See “Fit Distributions” on page 61.
Save Saves information about continuous or categorical variables. See “Save Commands for
Continuous Variables” on page 57.
Remove Permanently removes the variable and all its reports from the Distribution report.
Normal Quantile Plot
Use the Normal Quantile Plot option to visualize the extent to which the variable is normally
distributed. If a variable is normally distributed, the normal quantile plot approximates a
diagonal straight line. This type of plot is also called a quantile-quantile plot, or Q-Q plot.
The normal quantile plot also shows Lilliefors confidence bounds (Conover 1980) and
probability and normal quantile scales.
Figure 3.7 Normal Quantile Plot
normal quantile scale
Lilliefors confidence
bounds
probability scale
Note the following information:
•
The y-axis shows the column values.
•
The x-axis shows the empirical cumulative probability for each value.
Related Information
•
“Normal Quantile Plot” on page 74
Chapter 3
Basic Analysis
Distributions
Options for Continuous Variables
51
Outlier Box Plot
Use the outlier box plot (also called a Tukey outlier box plot) to see the distribution and
identify possible outliers. Generally, box plots show selected quantiles of continuous
distributions.
Figure 3.8 Outlier Box Plot
whisker
3rd quartile
shortest half
median sample value
confidence diamond
1st quartile
whisker
Note the following aspects about outlier box plots:
•
The horizontal line within the box represents the median sample value.
•
The confidence diamond contains the mean and the upper and lower 95% of the mean. If
you drew a line through the middle of the diamond, you would have the mean. The top
and bottom points of the diamond represent the upper and lower 95% of the mean.
•
The ends of the box represent the 25th and 75th quantiles, also expressed as the 1st and 3rd
quartile, respectively.
•
The difference between the 1st and 3rd quartiles is called the interquartile range.
•
The box has lines that extend from each end, sometimes called whiskers. The whiskers
extend from the ends of the box to the outermost data point that falls within the distances
computed as follows:
1st quartile - 1.5*(interquartile range)
3rd quartile + 1.5*(interquartile range)
If the data points do not reach the computed ranges, then the whiskers are determined by
the upper and lower data point values (not including outliers).
•
The bracket outside of the box identifies the shortest half, which is the most dense 50% of
the observations (Rousseuw and Leroy 1987).
Remove Objects from the Outlier Box Plot
To remove the confidence diamond or the shortest half, proceed as follows:
1. Right-click on the outlier box plot and select Customize.
2. Click Box Plot.
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Options for Continuous Variables
Chapter 3
Basic Analysis
3. Deselect the check box next to Confidence Diamond or Shortest Half.
For more details about the Customize Graph window, see the JMP Reports chapter in the
Using JMP book.
Quantile Box Plot
The Quantile Box Plot displays specific quantiles from the Quantiles report. If the distribution
is symmetric, the quantiles in the box plot are approximately equidistant from each other. At a
glance, you can see whether the distribution is symmetric. For example, if the quantile marks
are grouped closely at one end, but have greater spacing at the other end, the distribution is
skewed toward the end with more spacing. See Figure 3.9.
Figure 3.9 Quantile Box Plot
90% quantile
10% quantile
Quantiles are values where the pth quantile is larger than p% of the values. For example, 10%
of the data lies below the 10th quantile, and 90% of the data lies below the 90th quantile.
Stem and Leaf
Each line of the plot has a Stem value that is the leading digit of a range of column values. The
Leaf values are made from the next-in-line digits of the values. You can see the data point by
joining the stem and leaf. In some cases, the numbers on the stem and leaf plot are rounded
versions of the actual data in the table. The stem-and-leaf plot actively responds to clicking
and the brush tool.
Note: The stem-and-leaf plot does not support fractional frequencies.
Chapter 3
Basic Analysis
Distributions
Options for Continuous Variables
53
CDF Plot
The CDF plot creates a plot of the empirical cumulative distribution function. Use the CDF
plot to determine the percent of data that is at or below a given value on the x-axis.
Figure 3.10 CDF Plot
For example, in this CDF plot, approximately 30% of the data is less than a total fat value of 10
grams.
Test Mean
Use the Test Mean window to specify options for and perform a one-sample test for the mean.
If you specify a value for the standard deviation, a z-test is performed. Otherwise, the sample
standard deviation is used to perform a t-test. You can also request the nonparametric
Wilcoxon Signed-Rank test.
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Chapter 3
Basic Analysis
Use the Test Mean option repeatedly to test different values. Each time you test the mean, a
new Test Mean report appears.
Description of the Test Mean Report
Statistics that are calculated for Test Mean
t Test (or z Test) Lists the value of the test statistic and the p-values for the two-sided and
one-sided alternatives.
Signed-Rank (Only appears if the Wilcoxon Signed-Rank test is selected.) Lists the value of
the Wilcoxon signed-rank statistic followed by p-values for the two-sided and one-sided
alternatives. The test uses the Pratt method to address zero values. This is a nonparametric
test whose null hypothesis is that the median equals the postulated value. It assumes that
the distribution is symmetric. See “Wilcoxon Signed Rank Test” on page 75.
Probability values
Prob>|t| The probability of obtaining an absolute t-value by chance alone that is greater than
the observed t-value when the population mean is equal to the hypothesized value. This is
the p-value for observed significance of the two-tailed t-test.
The probability of obtaining a t-value greater than the computed sample t ratio by
chance alone when the population mean is not different from the hypothesized value. This
is the p-value for an upper-tailed test.
Prob>t
The probability of obtaining a t-value less than the computed sample t ratio by
chance alone when the population mean is not different from the hypothesized value. This
is the p-value for a lower-tailed test.
Prob<t
Descriptions of the Test Mean Options
Starts an interactive visual representation of the p-value. Enables you to
change the hypothesized mean value while watching how the change affects the p-value.
PValue animation
Power animation Starts an interactive visual representation of power and beta. You can
change the hypothesized mean and sample mean while watching how the changes affect
power and beta.
Remove Test Removes the mean test.
Test Std Dev
Use the Test Std Dev option to perform a one-sample test for the standard deviation (details in
Neter, Wasserman, and Kutner 1990). Use the Test Std Dev option repeatedly to test different
values. Each time you test the standard deviation, a new Test Standard Deviation report
appears.
Chapter 3
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Distributions
Options for Continuous Variables
55
Test Statistic Provides the value of the Chi-square test statistic. See “Standard Deviation
Test” on page 77.
Min PValue The probability of obtaining a more extreme Chi-square value by chance alone
when the population standard deviation does not differ from the hypothesized value. See
“Standard Deviation Test” on page 77.
Prob>ChiSq The probability of obtaining a Chi-square value greater than the computed
sample Chi-square by chance alone when the population standard deviation is not
different from the hypothesized value. This is the p-value for observed significance of a
one-tailed t-test.
Prob<ChiSq The probability of obtaining a Chi-square value less than the computed sample
Chi-square by chance alone when the population standard deviation is not different from
the hypothesized value. This is the p-value for observed significance of a one-tailed t-test.
Test Equivalence
The equivalence test assesses whether a population mean is equivalent to a hypothesized
value. You must define a threshold difference that is considered equivalent to no difference.
The Test Equivalence option uses the Two One-Sided Tests (TOST) approach. Two one-sided
t-tests are constructed for the null hypotheses that the difference between the true mean and
the hypothesized value exceeds the threshold. If both null hypotheses are rejected, this implies
that the true difference does not exceed the threshold. You conclude that the mean can be
considered practically equivalent to the hypothesized value.
When you select the Test Equivalence option, you specify the Hypothesized Mean, the
threshold difference (Difference Considered Practically Zero), and the Confidence Level. The
Confidence Level is 1 - alpha, where alpha is the significance level for each one-sided test.
The Test Equivalence report in Figure 3.11 is for the variable BMI in the Diabetes.jmp sample
data table. The Hypothesized Mean is 26.5 and the Difference Considered Practically Zero is
specified as 0.5.
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Chapter 3
Basic Analysis
Figure 3.11 Equivalence Test Report
The report shows the following:
•
A plot of your defined equivalence region that shows the Target and boundaries, defined
by vertical lines labeled Lower and Upper.
•
A confidence interval for the hypothesized mean. This confidence interval is a 1 - 2*alpha
level interval.
•
A table that shows the calculated mean, the specified lower and upper bounds, and a
(1 - 2*alpha) level confidence interval for the mean.
•
A table that shows the results of the two one-sided tests.
•
A note that summarizes the results, and states whether the mean can be considered
equivalent to the Target value.
Confidence Intervals for Continuous Variables
The Confidence Interval options display confidence intervals for the mean and standard
deviation. The 0.90, 0.95, and 0.99 options compute two-sided confidence intervals for the
mean and standard deviation. Use the Confidence Interval > Other option to select a
confidence level, and select one-sided or two-sided confidence intervals. You can also type a
known sigma. If you use a known sigma, the confidence interval for the mean is based on
z-values rather than t-values.
The Confidence Intervals report shows the mean and standard deviation parameter estimates
with upper and lower confidence limits for 1 - α.
Chapter 3
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Distributions
Options for Continuous Variables
57
Save Commands for Continuous Variables
Use the Save menu commands to save information about continuous variables. Each Save
command generates a new column in the current data table. The new column is named by
appending the variable name (denoted <colname> in the following definitions) to the Save
command name. See Table 3.1.
Select the Save commands repeatedly to save the same information multiple times under
different circumstances, such as before and after combining histogram bars. If you use a Save
command multiple times, the column name is numbered (name1, name2, and so on) to ensure
unique column names.
Table 3.1 Descriptions of Save Commands
Command
Column Added
to Data Table
Description
Level Numbers
Level
<colname>
The level number of each observation corresponds to
the histogram bar that contains the observation. The
histogram bars are numbered from low to high,
beginning with 1.
Note: To maintain source information, value labels
are added to the new column, but they are turned off
by default.
Level Midpoints
Midpoint
<colname>
The midpoint value for each observation is
computed by adding half the level width to the
lower level bound.
Note: To maintain source information, value labels
are added to the new column, but they are turned off
by default.
Ranks
Ranked
<colname>
Provides a ranking for each of the corresponding
column’s values starting at 1. Duplicate response
values are assigned consecutive ranks in order of
their occurrence in the data table.
Ranks Averaged
RankAvgd
<colname>
If a value is unique, then the averaged rank is the
same as the rank. If a value occurs k times, the
average rank is computed as the sum of the value’s
ranks divided by k.
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Chapter 3
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Table 3.1 Descriptions of Save Commands (Continued)
Command
Column Added
to Data Table
Description
Prob Scores
Prob
<colname>
For N nonmissing scores, the probability score of a
value is computed as the averaged rank of that value
divided by N + 1. This column is similar to the
empirical cumulative distribution function.
Normal Quantiles
N-Quantile
<colname>
Saves the Normal quantiles to the data table. See
“Normal Quantile Plot” on page 74.
Standardized
Std <colname>
Saves standardized values to the data table. See
“Saving Standardized Data” on page 77.
Centered
Centered
<colname>
Saves values for centering on zero.
Spec Limits
(none)
Stores the specification limits applied in a capability
analysis as a column property of the corresponding
column in the current data table. Automatically
retrieves and displays the specification limits when
you repeat the capability analysis.
Script to Log
(none)
Prints the script to the log window. Run the script to
recreate the analysis.
Prediction Intervals
Prediction intervals concern a single observation, or the mean and standard deviation of the
next randomly selected sample. The calculations assume that the given sample is selected
randomly from a normal distribution. Select one-sided or two-sided prediction intervals.
When you select the Prediction Interval option for a variable, the Prediction Intervals window
appears. Use the window to specify the confidence level, the number of future samples, and
either a one-sided or two-sided limit.
Related Information
•
“Prediction Intervals” on page 78
•
“Example of Prediction Intervals” on page 69
Chapter 3
Basic Analysis
Distributions
Options for Continuous Variables
59
Tolerance Intervals
A tolerance interval contains at least a specified proportion of the population. It is a
confidence interval for a specified proportion of the population, not the mean, or standard
deviation. Complete discussions of tolerance intervals are found in Hahn and Meeker (1991)
and in Tamhane and Dunlop (2000).
When you select the Tolerance Interval option for a variable, the Tolerance Intervals window
appears. Use the window to specify the confidence level, the proportion to cover, and either a
one-sided or two-sided limit. The calculations are based on the assumption that the given
sample is selected randomly from a normal distribution.
Related Information
•
“Tolerance Intervals” on page 78
•
“Example of Tolerance Intervals” on page 70
Capability Analysis
The Capability Analysis option measures the conformance of a process to given specification
limits. By default, capability is calculated based on a normal distribution. However, if you fit a
distribution (continuous or discrete), then the capability report is produced based on that fit.
To use a fit distribution, select Spec Limits or set the Spec Limits column property. When you
select the Capability Analysis option for a variable, the Capability Analysis window appears.
Use the window to enter specification limits, distribution type, and information about sigma.
Note: To save the specification limits to the data table as a column property, select Save >
Spec Limits. When you repeat the capability analysis, the saved specification limits are
automatically retrieved.
The Capability Analysis report is organized into two sections: Capability Analysis and the
distribution type (Long Term Sigma, Specified Sigma, and so on).
Description of the Capability Analysis Window
<Distribution type> By default, the normal distribution is assumed when calculating the
capability statistics and the percent out of the specification limits. To perform a capability
analysis on non-normal distributions, see the description of Spec Limits under “Capability
Analysis” on page 79.
<Sigma type> Estimates sigma (σ) using the selected methods. See “Capability Analysis” on
page 79.
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Options for Continuous Variables
Chapter 3
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Description of the Capability Analysis Report
Specification Lists the specification limits.
Value Lists the values that you specified for each specification limit and the target.
Portion and % Actual
Portion labels describe the numbers in the % Actual column, as follows:
– Below LSL gives the percentage of the data that is below the lower specification limit.
– Above USL gives the percentage of the data that is above the upper specification limit.
– Total Outside gives the total percentage of the data that is either below LSL or above
USL.
Capability Type of process capability indices. See “Description of the Capability Analysis
Options” on page 60.
Note: There is a preference for Capability called Ppk Capability Labeling that labels the
long-term capability output with Ppk labels. Open the Preference window
(File > Preferences), then select Platforms > Distribution to see this preference.
Index
Process capability index values.
Upper CI Upper confidence interval.
Lower CI Lower confidence interval.
Portion and Percent Portion labels describe the numbers in the Percent column, as follows:
– Below LSL gives the percentage of the fitted distribution that is below the lower
specification limit.
– Above USL gives the percentage of the fitted distribution that is above the upper
specification limit.
– Total Outside gives the total percentage of the fitted distribution that is either below
LSL or above USL.
PPM (parts per million) The PPM value is the Percent column multiplied by 10,000.
Sigma Quality is frequently used in Six Sigma methods, and is also referred to
as the process sigma. See “Capability Analysis” on page 79.
Sigma Quality
Description of the Capability Analysis Options
Z Bench Shows the values (represented by Index) of the Benchmark Z statistics. According to
the AIAG Statistical Process Control manual, Z represents the number of standard
deviation units from the process average to a value of interest such as an engineering
specification. When used in capability assessment, Z USL is the distance to the upper
specification limit and Z LSL is the distance to the lower specification limit. See
“Capability Analysis” on page 79.
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61
Capability Animation Interactively change the specification limits and the process mean to see
the effects on the capability statistics. This option is available only for capability analyses
based on the Normal distribution.
Related Information
•
“Capability Analysis” on page 79
•
“Example of Capability Analysis” on page 71
Fit Distributions
Use the Continuous or Discrete Fit options to fit a distribution to a continuous or discrete
variable.
A curve is overlaid on the histogram, and a Parameter Estimates report is added to the report
window. A red triangle menu contains additional options. See “Fit Distribution Options” on
page 63.
Note: The Life Distribution platform also contains options for distribution fitting that might
use different parameterizations and allow for censoring. See the Life Distribution chapter in
the Reliability and Survival Methods book.
Continuous Fit
Use the Continuous Fit options to fit the following distributions to a continuous variable.
•
The Normal distribution is often used to model measures that are symmetric with most of
the values falling in the middle of the curve. JMP uses the unbiased estimate when
determining the parameters for the Normal distribution.
•
The LogNormal distribution is often used to model values that are constrained by zero but
have a few very large values. The LogNormal distribution can be obtained by
exponentiating the Normal distribution. JMP uses the maximum likelihood estimation
when determining the parameters for the LogNormal distribution.
•
The Weibull distribution, Weibull with threshold distribution, and Extreme Value
distribution often provide a good model for estimating the length of life, especially for
mechanical devices and in biology.
•
The Exponential distribution is especially useful for describing events that randomly occur
over time, such as survival data. The exponential distribution might also be useful for
modeling elapsed time between the occurrence of non-overlapping events, such as the
time between a user’s computer query and response of the server, the arrival of customers
at a service desk, or calls coming in at a switchboard.
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•
The Gamma distribution is bound by zero and has a flexible shape.
•
The Beta distribution is useful for modeling the behavior of random variables that are
constrained to fall in the interval 0,1. For example, proportions always fall between 0 and
1.
•
The Normal Mixtures distribution fits a mixture of normal distributions. This flexible
distribution is capable of fitting multi-modal data. You can also fit two or more
distributions by selecting the Normal 2 Mixture, Normal 3 Mixture, or Other options.
•
The Smooth Curve distribution fits a smooth curve using nonparametric density
estimation (kernel density estimation). The smooth curve is overlaid on the histogram and
a slider appears beneath the plot. Control the amount of smoothing by changing the kernel
standard deviation with the slider. The initial Kernel Std estimate is calculated from the
standard deviation of the data.
•
The Johnson Su, Johnson Sb, and Johnson Sl Distributions are useful for its data-fitting
capabilities because it supports every possible combination of skewness and kurtosis.
•
The Generalized Log (Glog) distribution is useful for fitting data that are rarely normally
distributed and often have non-constant variance, like biological assay data.
Comparing All Distributions
The All option fits all applicable continuous distributions to a variable. The Compare
Distributions report contains statistics about each fitted distribution. Use the check boxes to
show or hide a fit report and overlay curve for the selected distribution. By default, the best fit
distribution is selected.
The Show Distribution list is sorted by AICc in ascending order.
If your variable contains negative values, the Show Distribution list does not include those
distributions that require data with positive values. Only continuous distributions are fitted
by this command. Distributions with threshold parameters, like Beta and Johnson Sb, are not
included in the list of possible distributions.
Related Information
•
“Continuous Fit Distributions” on page 83
•
“Fitted Quantiles” on page 92
•
“Fit Distribution Options” on page 93
Discrete Fit
Use the Discrete Fit options to fit a distribution (such as Poisson or Binomial) to a discrete
variable. The available distributions are as follows:
•
Poisson
Chapter 3
Basic Analysis
•
Binomial
•
Gamma Poisson
•
Beta Binomial
Distributions
Fit Distributions
63
Related Information
•
“Discrete Fit Distributions” on page 90
•
“Fitted Quantiles” on page 92
•
“Fit Distribution Options” on page 93
Fit Distribution Options
Each fitted distribution report has a red triangle menu that contains additional options.
Diagnostic Plot Creates a quantile or a probability plot. See “Diagnostic Plot” on page 64.
Density Curve Uses the estimated parameters of the distribution to overlay a density curve
on the histogram.
Computes the goodness of fit test for the fitted distribution. See “Goodness
of Fit” on page 65.
Goodness of Fit
Enables you to fix parameters and re-estimate the non-fixed parameters. An
Adequacy LR (likelihood ratio) Test report also appears, which tests your new parameters
to determine whether they fit the data.
Fix Parameters
Quantiles Returns the un-scaled and un-centered quantiles for the specific lower probability
values that you specify.
Use this option when you do not know the specification limits
for a process and you want to use its distribution as a guideline for setting specification
limits.
Set Spec Limits for K Sigma
Usually specification limits are derived using engineering considerations. If there are no
engineering considerations, and if the data represents a trusted benchmark (well behaved
process), then quantiles from a fitted distribution are often used to help set specification
limits. See “Fit Distribution Options” on page 93.
Spec Limits Computes generalizations of the standard capability indices, based on the
specification limits and target you specify. See “Spec Limits” on page 65.
Save Fitted Quantiles Saves the fitted quantile values as a new column in the current data
table. See “Fitted Quantiles” on page 92.
Save Density Formula Creates a new column in the current data table that contains fitted
values that have been computed by the density formula. The density formula uses the
estimated parameter values.
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Saves the specification limits as a column property. See “Fit Distribution
Options” on page 93.
Save Spec Limits
Save Transformed Creates a new column and saves a formula. The formula can transform the
column to normality using the fitted distribution. This option is available only when one
of the Johnson distributions or the Glog distribution is fit.
Remove Fit Removes the distribution fit from the report window.
Diagnostic Plot
The Diagnostic Plot option creates a quantile or a probability plot. Depending on the fitted
distribution, the plot is one of four formats.
The fitted quantiles versus the data
•
Weibull with threshold
•
Gamma
•
Beta
•
Poisson
•
GammaPoisson
•
Binomial
•
BetaBinomial
The fitted probability versus the data
•
Normal
•
Normal Mixtures
•
Exponential
The fitted probability versus the data on log scale
•
Weibull
•
LogNormal
•
Extreme Value
The fitted probability versus the standard normal quantile
•
Johnson Sl
•
Johnson Sb
•
Johnson Su
•
Glog
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65
The following options are available in the Diagnostic Plot red triangle menu:
Rotate Reverses the x- and y-axes.
Draws Lilliefors 95% confidence limits for the Normal Quantile plot, and
95% equal precision bands with a = 0.001 and b = 0.99 for all other quantile plots (Meeker
and Escobar (1998)).
Confidence Limits
Line of Fit Draws the straight diagonal reference line. If a variable fits the selected
distribution, the values fall approximately on the reference line.
Median Reference Line
Draws a horizontal line at the median of the response.
Goodness of Fit
The Goodness of Fit option computes the goodness of fit test for the fitted distribution. The
goodness of fit tests are not Chi-square tests, but are EDF (Empirical Distribution Function)
tests. EDF tests offer advantages over the Chi-square tests, including improved power and
invariance with respect to histogram midpoints.
•
For Normal distributions, the Shapiro-Wilk test for normality is reported when the sample
size is less than or equal to 2000, and the KSL test is computed for samples that are greater
than 2000.
•
For discrete distributions (such as Poisson distributions) that have sample sizes less than
or equal to 30, the Goodness of Fit test is formed using two one-sided exact Kolmogorov
tests combined to form a near exact test. For details, see Conover 1972. For sample sizes
greater than 30, a Pearson Chi-squared goodness of fit test is performed.
Related Information
•
“Fit Distribution Options” on page 93
Spec Limits
The Spec Limits option launches a window requesting specification limits and target, and
then computes generalizations of the standard capability indices. This is done using the fact
that for the normal distribution, 3σ is both the distance from the lower 0.135 percentile to
median (or mean) and the distance from the median (or mean) to the upper 99.865 percentile.
These percentiles are estimated from the fitted distribution, and the appropriate
percentile-to-median distances are substituted for 3σ in the standard formulas.
Related Information
•
“Fit Distribution Options” on page 93
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Additional Examples of the Distribution Platform
Chapter 3
Basic Analysis
Additional Examples of the Distribution Platform
This section contains additional examples using the Distribution platform.
Example of Selecting Data in Multiple Histograms
1. Select Help > Sample Data Library and open Companies.jmp.
2. Select Analyze > Distribution.
3. Select Type and Size Co and click Y, Columns.
4. Click OK.
You want to see the type distribution of companies that are small.
5. Click on the bar next to small.
You can see that there are more small computer companies than there are pharmaceutical
companies. To broaden your selection, add medium companies.
6. Hold down the Shift key. In the Size Co histogram, click on the bar next to medium.
You can see the type distribution of small and medium sized companies. See Figure 3.12 at
left. To narrow your selection, you want to see the small and medium pharmaceutical
companies only.
7. Hold down the Control and Shift keys (on Windows) or the Command and Shift keys (on
Macintosh). In the Type histogram, click in the Computer bar to deselect it.
You can see how many of the small and medium companies are pharmaceutical
companies. See Figure 3.12 at right.
Figure 3.12 Selecting Data in Multiple Histograms
Broaden the selection using the
Shift key.
Narrow the selection using
Control + Shift (Windows) or
Command + Shift (Macintosh).
Chapter 3
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Additional Examples of the Distribution Platform
Examples of the Test Probabilities Option
Initiate a test probability report for a variable with more than two levels:
1. Select Help > Sample Data Library and open VA Lung Cancer.jmp.
2. Select Analyze > Distribution.
3. Select Cell Type and click Y, Columns.
4. Click OK.
5. From the red triangle menu next to Cell Type, select Test Probabilities.
See Figure 3.13 at left.
Initiate a test probability report for a variable with exactly two levels:
1. Select Help > Sample Data Library and open Penicillin.jmp.
2. Select Analyze > Distribution.
3. Select Response and click Y, Columns.
4. Click OK.
5. From the red triangle menu next to Response, select Test Probabilities.
See Figure 3.13 at right.
Figure 3.13 Examples of Test Probabilities Options
report options for a variable
with more than two levels
report options for a variable
with exactly two levels
67
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Chapter 3
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Example of Generating the Test Probabilities Report
To generate a test probabilities report for a variable with more than two levels:
1. Refer to Figure 3.13 at left. Type 0.25 in all four Hypoth Prob fields.
2. Click the Fix hypothesized values, rescale omitted button.
3. Click Done.
Likelihood Ratio and Pearson Chi-square tests are calculated. See Figure 3.14 at left.
To generate a test probabilities report for a variable with exactly two levels:
1. Refer to Figure 3.13 at right. Type 0.5 in both Hypoth Prob fields.
2. Click the probability less than hypothesized value button.
3. Click Done.
Exact probabilities are calculated for the binomial test. See Figure 3.14 at right.
Figure 3.14 Examples of Test Probabilities Reports
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Additional Examples of the Distribution Platform
69
Example of Prediction Intervals
Suppose you are interested in computing prediction intervals for the next 10 observations of
ozone level.
1. Select Help > Sample Data Library and open Cities.jmp.
2. Select Analyze > Distribution.
3. Select OZONE and click Y, Columns.
4. Click OK.
5. From the red triangle next to OZONE, select Prediction Interval.
Figure 3.15 The Prediction Intervals Window
6. In the Prediction Intervals window, type 10 next to Enter number of future samples.
7. Click OK.
Figure 3.16 Example of a Prediction Interval Report
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Chapter 3
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In this example, you can be 95% confident about the following:
•
Each of the next 10 observations will be between 0.013755 and 0.279995.
•
The mean of the next 10 observations will be between 0.115596 and 0.178154.
•
The standard deviation of the next 10 observations will be between 0.023975 and 0.069276.
Example of Tolerance Intervals
Suppose you want to estimate an interval that contains 90% of ozone level measurements.
1. Select Help > Sample Data Library and open Cities.jmp.
2. Select Analyze > Distribution.
3. Select OZONE and click Y, Columns.
4. Click OK.
5. From the red triangle menu next to OZONE, select Tolerance Interval.
Figure 3.17 The Tolerance Intervals Window
6. Keep the default selections, and click OK.
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Additional Examples of the Distribution Platform
71
Figure 3.18 Example of a Tolerance Interval Report
In this example, you can be 95% confident that at least 90% of the population lie between
0.057035 and 0.236715, based on the Lower TI (tolerance interval) and Upper TI values.
Example of Capability Analysis
Suppose you want to characterize the abrasion levels of the tires your company manufactures.
The lower and upper specification limits are 100 and 200, respectively.
1. Select Help > Sample Data Library and open Quality Control/Pickles.jmp.
2. Select Analyze > Distribution.
3. Select Acid and click Y, Columns.
4. Click OK.
5. From the red triangle menu next to Acid, select Capability Analysis.
6. Type 8 for the Lower Spec Limit.
7. Type 17 for the Upper Spec Limit.
8. Keep the rest of the default selections, and click OK.
9. From the red triangle menu next to Acid, select Histogram Options > Vertical.
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Chapter 3
Basic Analysis
Figure 3.19 Example of the Capability Analysis Report
The spec limits are added to the histogram so that the data can be visually compared to the
limits. As you can see, some of the acidity levels are below the lower spec limit, and some are
very close to the upper spec limit. The Capability Analysis results are added to the report. The
CPK value is 0.510, indicating a process that is not capable, relative to the given specification
limits.
Statistical Details for the Distribution Platform
This section contains statistical details for Distribution options and reports.
Standard Error Bars
Standard error bars are calculated using the standard error
np i ( 1 – p i ) where pi=ni/n.
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Statistical Details for the Distribution Platform
73
Quantiles
This section describes how quantiles are computed.
To compute the pth quantile of n non-missing values in a column, arrange the n values in
ascending order and call these column values y1, y2, ..., yn. Compute the rank number for the
pth quantile as p / 100(n + 1).
•
If the result is an integer, the pth quantile is that rank’s corresponding value.
•
If the result is not an integer, the pth quantile is found by interpolation. The pth quantile,
denoted qp, is computed as follows:
q p = ( 1 – f )y i + ( f )y i + 1
where:
– n is the number of non-missing values for a variable
– y1, y2, ..., yn represents the ordered values of the variable
– yn+1 is taken to be yn
– i is the integer part and f is the fractional part of (n+1)p.
– (n + 1)p = i + f
For example, suppose a data table has 15 rows and you want to find the 75th and 90th quantile
values of a continuous column. After the column is arranged in ascending order, the ranks
that contain these quantiles are computed as follows:
7590
-------( 15 + 1 ) = 12 and --------- ( 15 + 1 ) = 14.4
100
100
The value y12 is the 75th quantile. The 90th quantile is interpolated by computing a weighted
average of the 14th and 15th ranked values as y90 = 0.6y14 + 0.4y15.
Summary Statistics
This section contains statistical details for specific statistics in the Summary Statistics report.
Mean
The mean is the sum of the non-missing values divided by the number of non-missing values.
If you assigned a Weight or Freq variable, the mean is computed by JMP as follows:
1. Each column value is multiplied by its corresponding weight or frequency.
2. These values are added and divided by the sum of the weights or frequencies.
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Chapter 3
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Std Dev
The standard deviation measures the spread of a distribution around the mean. It is often
denoted as s and is the square root of the sample variance, denoted s2.
2
s =
s where
N
2
s =

i=1
2
wi ( yi – yw )
------------------------------N–1
y w = weighted mean
Std Err Mean
The standard error mean is computed by dividing the sample standard deviation, s, by the
square root of N. In the launch window, if you specified a column for Weight or Freq, then the
denominator is the square root of the sum of the weights or frequencies.
Skewness
Skewness is based on the third moment about the mean and is computed as follows:
3
--2 3
N
w
 i zi -----------------------------------( N – 1 )( N – 2 )
x –x
i where z = -----------i
s
and wi is a weight term (= 1 for equally weighted items).
Kurtosis
Kurtosis is based on the fourth moment about the mean and is computed as follows:
n(n + 1)
--------------------------------------------------(n – 1)(n – 2)(n – 3)
n

i=1
4
2
2 x i – x 
3(n – 1) w i ------------  – --------------------------------(n – 2)(n – 3)
 s 
where wi is a weight term (= 1 for equally weighted items). Using this formula, the Normal
distribution has a kurtosis of 0. This formula is often referred to as the excess kurtosis.
Normal Quantile Plot
The empirical cumulative probability for each value is computed as follows:
ri
------------N+1
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75
where ri is the rank of the ith observation, and N is the number of non-missing (and
nonexcluded) observations.
The normal quantile values are computed as follows:
Φ
ri
– 1  -------------
 N + 1
where Φ is the cumulative probability distribution function for the normal distribution.
These normal quantile values are Van Der Waerden approximations to the order statistics that
are expected for the normal distribution.
Wilcoxon Signed Rank Test
The Wilcoxon signed-rank test can be used to test for the median of a single population or to
test matched-pairs data for a common median. In the case of matched pairs, the test reduces to
testing the single population of paired differences for a median of 0. The test assumes that the
underlying population is symmetric.
The Wilcoxon test allows tied values. The test statistic is adjusted for differences of zero using
a method suggested by Pratt. See Lehman (2006), Pratt (1959), and Cureton (1967).
Testing for the Median of a Single Population
•
There are N observations:
X1, X2, ..., XN
•
The null hypothesis is:
H0: median = m
•
The differences between observations and the hypothesized value m are calculated as
follows:
Dj = Xj - m
Testing for the Equality of Two Population Medians with Matched Pairs Data
A special case of the Wilcoxon signed-rank test is applied to matched-pairs data.
•
There are N pairs of observations from two populations:
X1, X2, ..., XN and Y1, Y2, ..., YN
•
The null hypothesis is:
H0: medianX - Y = 0
•
The differences between pairs of observations are calculated as follows:
Dj = Xj -Yj
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Chapter 3
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Wilcoxon Signed-Rank Test Statistic
The test statistic is based on the sum of the signed ranks. Signed ranks are defined as follows:
•
•
•
The absolute values of the differences, D j , are ranked from smallest to largest.
The ranks start with the value 1, even if there are differences of zero.
When there are tied absolute differences, they are assigned the average, or midrank, of the
ranks of the observations.
Denote the rank or midrank for a difference D j by Rj. Define the signed rank for D j as
follows:
•
•
If the difference D j is positive, the signed rank is Rj.
If the difference D j is zero, the signed rank is 0.
•
If the difference D j is negative, the signed rank is -Rj.
The signed-rank statistic is computed as follows:
1
W = --2
N

signed ranks
j=1
Define the following:
d 0 is the number of signed ranks that equal zero
R+ is the sum of the positive signed ranks
Then the following holds:
+ 1
W = R – --- [ N ( N + 1 ) – d 0 ( d 0 + 1 ) ]
4
Wilcoxon Signed-Rank Test P-Values
For N ≤ 20 , exact p-values are calculated.
For N > 20, a Student’s t approximation to the statistic defined below is used. Note that a
correction for ties is applied. See Iman (1974) and Lehmann (1998).
Under the null hypothesis, the mean of W is zero. The variance of W is given by the following:
·
1
1
Var ( W ) = ------ N ( N + 1 ) ( 2N + 1 ) – d 0 ( d 0 + 1 ) ( 2d 0 + 1 ) – --24
2

i>0
·
di ( di + 1 ) ( di – 1 )
The last summation in the expression for Var(W) is a correction for ties. The notation di for i > 0
represents the number of values in the ith group of non-zero signed ranks. (If there are no ties
for a given signed rank, then di = 1 and the summand is 0.)
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77
The statistic t given by the following has an approximate t distribution with N - 1 degrees of
freedom:
t = W ⁄ Var ( W )
Standard Deviation Test
Here is the formula for calculating the Test Statistic:
2
(---------------------n – 1 )s
2
σ
The Test Statistic is distributed as a Chi-square variable with n - 1 degrees of freedom when
the population is normal.
The Min PValue is the p-value of the two-tailed test, and is calculated as follows:
2*min(p1,p2)
where p1 is the lower one-tail p-value and p2 is the upper one-tail p-value.
Normal Quantiles
The normal quantile values are computed as follows:
Φ
ri
– 1  -------------
 N + 1
where:
•
Φ is the cumulative probability distribution function for the normal distribution
•
ri is the rank of the ith observation
•
N is the number of non-missing observations
Saving Standardized Data
The standardized values are computed using the following formula:
X
– X- where:
------------SX
•
X is the original column
•
X is the mean of column X
•
S X is the standard deviation of column X
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Prediction Intervals
The formulas that JMP uses for computing prediction intervals are as follows:
•
For m future observations:
1
[ y , y˜m ] = X ± t ( 1 – α ⁄ 2m ;n – 1 ) × 1 + --- × s for m ≥ 1
m
n
˜
•
For the mean of m future observations:
1 1
[ Y l, Y u ] = X ± t ( 1 – α ⁄ 2, n – 1 ) × ---- + --- × s for m ≥ 1 .
m n
•
For the standard deviation of m future observations:
1
[ s l, s u ] = s × --------------------------------------------------------, s × F ( 1 – α ⁄ 2 ;( m – 1, n – 1 ) )
F ( 1 – α ⁄ 2 ;( n – 1 , m – 1 ) )
for m ≥ 2
where m = number of future observations, and n = number of points in current analysis
sample.
•
The one-sided intervals are formed by using 1-α in the quantile functions.
For references, see Hahn and Meeker (1991), pages 61-64.
Tolerance Intervals
This section contains statistical details for one-sided and two-sided tolerance intervals.
One-Sided Interval
The one-sided interval is computed as follows:
Upper Limit = x + g's
Lower Limit = x – g's
where
–1
g' = t ( 1 – α, n – 1, Φ ( p ) ⋅ n ) ⁄ n
t is the quantile from the non-central t-distribution
Φ
– 1 is the standard normal quantile.
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79
Two-Sided Interval
The two-sided interval is computed as follows:
[ T̃ p ,T̃ p ] = [ x – g ( 1 – α ⁄ 2 ;p ,n ) s , x + g ( 1 – α ⁄ 2 ;p ,n ) s ]
L
U
L
U
where
s = standard deviation
g ( 1 – α ⁄ 2 ;p,n ) is a constant.
To determine g, consider the fraction of the population captured by the tolerance interval.
Tamhane and Dunlop (2000) give this fraction as follows:
x – gs – μ
x + gs – μ
Φ  ------------------------ – Φ  ------------------------




σ
σ
where Φ denotes the standard normal c.d.f. (cumulative distribution function). Therefore,
g solves the following equation:
 X + gs – μ

X – gs – μ
P  Φ  ------------------------- – Φ  ------------------------- ≥ 1 – γ  = 1 – α




σ
σ


where 1-γ is the fraction of all future observations contained in the tolerance interval.
More information is given in Tables A.1a, A.1b, A.11a, and A.11b of Hahn and Meeker (1991).
Capability Analysis
All capability analyses use the same formulas. Options differ in how sigma (σ) is computed:
•
Long-term uses the overall sigma. This option is used for P pk statistics, and computes
sigma as follows:
n
σ =

i=1
2
( xi – x )
-------------------n–1
Note: There is a preference for Distribution called Ppk Capability Labeling that labels the
long-term capability output with Ppk labels. This option is found using File > Preferences,
then select Platforms > Distribution.
•
Specified Sigma enables you to type a specific, known sigma used for computing
capability analyses. Sigma is user-specified, and is therefore not computed.
•
Moving Range enables you to enter a range span, which computes sigma as follows:
80
Distributions
Statistical Details for the Distribution Platform
Chapter 3
Basic Analysis
R
σ = -------------- where
d2 ( n )
R is the average of the moving ranges
d2(n) is the expected value of the range of n independent normally distributed variables
with unit standard deviation.
•
Short Term Sigma, Group by Fixed Subgroup Size if r is the number of subgroups of size nj
and each ith subgroup is defined by the order of the data, sigma is computed as follows:
σ =
•
1
--r
nj
2
nj
( X ij – X i ) where
1
X i = -----  X
  --------------------------ij
n
nj – 1
j
j=1
i = 1j = 1
r
This formula is commonly referred to as the Root Mean Square Error, or RMSE.
Capability Index Definitions
Writing T for the target, LSL, and USL for the lower and upper specification limits, and Pα for
the α*100th percentile, the generalized capability indices are as follows:
P 0.5 – LSL
C pl = ------------------------------------P 0.5 – P 0.00135
USL – P 0.5
C pu = ------------------------------------P 0.99865 – P 0.5
USL – LSL
C p = -----------------------------------------------P 0.99865 – P 0.00135
USL – P 0.5 
 P 0.5 – LSL
C pk = min  ------------------------------------- ,-------------------------------------
 P 0.5 – P 0.00135 P 0.99865 – P 0.5
1--( USL + LSL ) – P 0.5
2
K = 2 × --------------------------------------------------------USL – LSL
T – LSL
USL – T
min  ------------------------------------- ,-------------------------------------
P – P

0.5
0.00135 P 0.99865 – P 0.5
C pm = ---------------------------------------------------------------------------------------------μ–T 2
1 +  -------------
σ
If the data are normally distributed, these formulas reduce to the formulas for standard
capability indices. See Table 3.2.
Chapter 3
Basic Analysis
Distributions
Statistical Details for the Distribution Platform
Note: The confidence intervals in the following table are computed using an alpha level of
0.05.
Table 3.2 Descriptions of Capability Indices and Computational Formulas
Index
Index Name
Formula
CP
process capability
ratio, Cp
(USL - LSL)/6s where:
CIs for CP
Lower CI on CP
Upper CI on CP
CPK (PPK for
AIAG)
process capability
index, Cpk
CIs for CPK
Lower CI
See Bissell (1990)
Upper CI
CPM
process capability
index, Cpm
•
USL is the upper spec limit
•
LSL is the lower spec limit
2
χ α ⁄ 2, n – 1
CP -------------------------n–1
2
χ 1 – α ⁄ 2, n – 1
CP ---------------------------------n–1
min(CPL, CPU)
1
1
Cˆ pk 1 – Φ – 1 ( 1 – α ⁄ 2 ) ----------------- + -------------------2
ˆ
2
(
n
– 1)
9n C pk
1
1
Cˆ pk 1 + Φ – 1 ( 1 – α ⁄ 2 ) ----------------- + -------------------2
ˆ
2
(
n
– 1)
9n C pk
min ( Target – LSL, USL – Target -)
-------------------------------------------------------------------------------------2
3 s + ( x – Target )
2
Note: CPM confidence intervals are not
reported when the target is not within the
Lower and Upper Spec Limits range. CPM
intervals are only reported when the target is
within this range. JMP writes a message to the
log to note why the CPM confidence intervals
are missing.
81
82
Distributions
Statistical Details for the Distribution Platform
Chapter 3
Basic Analysis
Table 3.2 Descriptions of Capability Indices and Computational Formulas (Continued)
Index
Index Name
CIs for CPM
Lower CI on CPM
Formula
χ
2
α ⁄ 2, γ , where γ
CPM -----------------γ
x – Target 2 2
n  1 +  -------------------------- 


 
s
=------------------------------------------------------x – Target 2

1 + 2 -------------------------

s
Upper CI on CPM
χ
2
1 – α ⁄ 2, γ
CPM -------------------------γ
where γ = same as above.
CPL
process capability
ratio of one-sided
lower spec
(mean - LSL)/3s
CPU
process capability
ratio of one-sided
upper spec
(USL - mean)/3s
•
A capability index of 1.33 is considered to be the minimum acceptable. For a normal
distribution, this gives an expected number of nonconforming units of about 6 per 100,000.
•
Exact 100(1 - α)% lower and upper confidence limits for CPL are computed using a
generalization of the method of Chou et al. (1990), who point out that the 100(1 - α) lower
confidence limit for CPL (denoted by CPLLCL) satisfies the following equation:
Pr { T n – 1 ( δ = 3 n )CPLLCL ≤ 3CPL n } = 1 – α
where Tn-1(δ) has a non-central t-distribution with n - 1 degrees of freedom and
noncentrality parameter δ.
•
Exact 100(1 - α)% lower and upper confidence limits for CPU are also computed using a
generalization of the method of Chou et al. (1990), who point out that the 100(1 - α) lower
confidence limit for CPU (denoted CPULCL) satisfies the following equation:
Pr { T n – 1 ( δ = 3 n )CPULCL ≥ 3CPU n } = 1 – α
where Tn-1(δ) has a non-central t-distribution with n - 1 degrees of freedom and
noncentrality parameter δ.
Chapter 3
Basic Analysis
Distributions
Statistical Details for the Distribution Platform
83
Note: Because of a lack of supporting research at the time of this writing, computing
confidence intervals for capability indices is not recommended, except for cases when the
capability indices are based on the standard deviation.
•
Sigma Quality is defined as the following
% outside
Sigma Quality = Normal Quantile  1 – -------------------------- + 1.5

100 
% above
Sigma Quality Above = Normal Quantile  1 – ---------------------- + 1.5

100 
% below
Sigma Quality Below = Normal Quantile  1 – ---------------------- + 1.5
100
For example, if there are 3 defects in n=1,000,000 observations, the formula yields 6.03, or a
6.03 sigma process. The results of the computations of the Sigma Quality Above USL and
Sigma Quality Below LSL column values do not sum to the Sigma Quality Total Outside
column value because calculating Sigma Quality involves finding normal distribution
quantiles, and is therefore not additive.
•
Here are the Benchmark Z formulas:
Z USL = (USL-Xbar)/sigma = 3 * CPU
Z LSL = (Xbar-LSL)/sigma = 3 * CPL
Z Bench = Inverse Cumulative Prob(1 - P(LSL) - P(USL))
where:
P(LSL) = Prob(X < LSL) = 1 - Cum Prob(Z LSL)
P(USL) = Prob(X > USL) = 1 - Cum Prob(Z USL).
Continuous Fit Distributions
This section contains statistical details for the options in the Continuous Fit menu.
Normal
The Normal fitting option estimates the parameters of the normal distribution. The normal
distribution is often used to model measures that are symmetric with most of the values
falling in the middle of the curve. Select the Normal fitting for any set of data and test how well
a normal distribution fits your data.
The parameters for the normal distribution are as follows:
•
μ (the mean) defines the location of the distribution on the x-axis
•
σ (standard deviation) defines the dispersion or spread of the distribution
84
Distributions
Statistical Details for the Distribution Platform
Chapter 3
Basic Analysis
The standard normal distribution occurs when μ = 0 and σ = 1 . The Parameter Estimates
table shows estimates of μ and σ, with upper and lower 95% confidence limits.
pdf:
1
( x – μ )2
----------------- exp – -------------------2σ 2
2πσ 2
for – ∞ < x < ∞ ; – ∞ < μ < ∞ ; 0 < σ
E(x) = μ
Var(x) = σ2
LogNormal
The LogNormal fitting option estimates the parameters μ (scale) and σ (shape) for the
two-parameter lognormal distribution. A variable Y is lognormal if and only if X = ln ( Y ) is
normal. The data must be greater than zero.
pdf:
– ( log ( x ) – μ ) 2
exp ------------------------------------1
2σ 2
-----------------------------------------------------------------x
σ 2π
for 0 ≤ x ; – ∞ < μ < ∞ ; 0 < σ
E(x) = exp ( μ + σ 2 ⁄ 2 )
Var(x) = exp ( 2 ( μ + σ 2 ) ) – exp ( 2μ + σ 2 )
Weibull, Weibull with Threshold, and Extreme Value
The Weibull distribution has different shapes depending on the values of α (scale) and β
(shape). It often provides a good model for estimating the length of life, especially for
mechanical devices and in biology. The Weibull option is the same as the Weibull with
threshold option, with a threshold (θ) parameter of zero. For the Weibull with threshold
option, JMP estimates the threshold as the minimum value. If you know what the threshold
should be, set it by using the Fix Parameters option. See “Fit Distribution Options” on page 63.
The pdf for the Weibull with threshold is as follows:
pdf:
βx–θ β
-----( x – θ ) β – 1 exp –  ------------
 α 
αβ
E(x) = θ + αΓ  1 + --1-

β
Var(x) =α 2  Γ  1 + --2- – Γ 2  1 + --1- 


β
β 

for α,β > 0; θ < x
Chapter 3
Basic Analysis
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Statistical Details for the Distribution Platform
85
where Γ ( . ) is the Gamma function.
The Extreme Value distribution is a two parameter Weibull (α, β) distribution with the
transformed parameters δ = 1 / β and λ = ln(α).
Exponential
The exponential distribution is especially useful for describing events that randomly occur
over time, such as survival data. The exponential distribution might also be useful for
modeling elapsed time between the occurrence of non-overlapping events, such as the time
between a user’s computer query and response of the server, the arrival of customers at a
service desk, or calls coming in at a switchboard.
The Exponential distribution is a special case of the two-parameter Weibull when β = 1 and α =
σ, and also a special case of the Gamma distribution when α = 1.
pdf:
1
--- exp ( – x ⁄ σ )
σ
for 0 < σ; 0 ≤ x
E(x) = σ
Var(x) = σ2
Devore (1995) notes that an exponential distribution is memoryless. Memoryless means that if
you check a component after t hours and it is still working, the distribution of additional
lifetime (the conditional probability of additional life given that the component has lived until
t) is the same as the original distribution.
Gamma
The Gamma fitting option estimates the gamma distribution parameters, α > 0 and σ > 0. The
parameter α, called alpha in the fitted gamma report, describes shape or curvature. The
parameter σ, called sigma, is the scale parameter of the distribution. A third parameter, θ,
called the Threshold, is the lower endpoint parameter. It is set to zero by default, unless there
are negative values. You can also set its value by using the Fix Parameters option. See “Fit
Distribution Options” on page 63.
pdf:
1
-------------------- ( x – θ ) α – 1 exp ( – ( x – θ ) ⁄ σ )
Γ ( α )σ α
for θ ≤ x ; 0 < α,σ
E(x) = ασ + θ
Var(x) = ασ2
•
The standard gamma distribution has σ = 1. Sigma is called the scale parameter because
values other than 1 stretch or compress the distribution along the x-axis.
•
The Chi-square χ (2ν ) distribution occurs when σ = 2, α = ν/2, and θ = 0.
86
Distributions
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•
Chapter 3
Basic Analysis
The exponential distribution is the family of gamma curves that occur when α = 1 and
θ = 0.
The standard gamma density function is strictly decreasing when α ≤ 1 . When α > 1 , the
density function begins at zero, increases to a maximum, and then decreases.
Beta
The standard beta distribution is useful for modeling the behavior of random variables that
are constrained to fall in the interval 0,1. For example, proportions always fall between 0 and
1. The Beta fitting option estimates two shape parameters, α > 0 and β > 0. There are also θ and
σ, which are used to define the lower threshold as θ, and the upper threshold as θ + σ. The
beta distribution has values only for the interval defined by θ ≤ x ≤ ( θ + σ ) . The θ is estimated
as the minimum value, and σ is estimated as the range. The standard beta distribution occurs
when θ = 0 and σ = 1.
Set parameters to fixed values by using the Fix Parameters option. The upper threshold must
be greater than or equal to the maximum data value, and the lower threshold must be less
than or equal to the minimum data value. For details about the Fix Parameters option, see “Fit
Distribution Options” on page 63.
pdf:
1
------------------------------------------- ( x – θ ) α – 1 ( θ + σ – x ) β – 1
B ( α, β )σ α + β – 1
for θ ≤ x ≤ θ + σ ; 0 < σ,α,β
α E(x) = θ + σ -----------α+β
σ 2 αβ
Var(x) = -----------------------------------------------( α + β )2( α + β + 1 )
where B ( . ) is the Beta function.
Normal Mixtures
The Normal Mixtures option fits a mixture of normal distributions. This flexible distribution is
capable of fitting multi-modal data.
Fit a mixture of two or three normal distributions by selecting the Normal 2 Mixture or Normal
3 Mixture options. Alternatively, you can fit a mixture of k normal distributions by selecting
the Other option. A separate mean, standard deviation, and proportion of the whole is
estimated for each group.
pdf:
x – μ 
----- φ  --------------i
σi  σi 
k π
i

i=1
Chapter 3
Basic Analysis
Distributions
Statistical Details for the Distribution Platform
E(x) = k

i =1
87
πi μi
 k
2

 πi ( μi2 + σi2 ) –   πi μ i
i=1
 i =1

Var(x) = k
where μi, σi, and πi are the respective mean, standard deviation, and proportion for the ith
group, and φ ( . ) is the standard normal pdf.
Smooth Curve
The Smooth Curve option fits a smooth curve using nonparametric density estimation (kernel
density estimation). The smooth curve is overlaid on the histogram and a slider appears
beneath the plot. Control the amount of smoothing by changing the kernel standard deviation
with the slider. The initial Kernel Std estimate is calculated from the standard deviation of the
data.
Johnson Su, Johnson Sb, Johnson Sl
The Johnson system of distributions contains three distributions that are all based on a
transformed normal distribution. These three distributions are the following:
•
Johnson Su, which is unbounded.
•
Johnson Sb, which has bounds on both tails defined by parameters that can be estimated.
•
Johnson Sl, which is bounded in one tail by a parameter that can be estimated. The
Johnson Sl family contains the family of lognormal distributions.
The S refers to system, the subscript of the range. Although we implement a different method,
information about selection criteria for a particular Johnson system can be found in Slifker and
Shapiro (1980).
Johnson distributions are popular because of their flexibility. In particular, the Johnson
distribution system is noted for its data-fitting capabilities because it supports every possible
combination of skewness and kurtosis.
If Z is a standard normal variate, then the system is defined as follows:
Z = γ + δf ( Y )
where, for the Johnson Su:
2
–1
f ( Y ) = ln  Y + 1 + Y  = sinh Y


88
Distributions
Statistical Details for the Distribution Platform
X–θ
Y = ------------σ
Chapter 3
Basic Analysis
–∞ < X < ∞
where, for the Johnson Sb:
Y
f ( Y ) = ln  -------------
1–Y
X–θ
Y = ------------σ
θ<X<θ+σ
and for the Johnson Sl, where σ = ± 1 .
f ( Y ) = ln ( Y )
X–θ
Y = ------------σ
θ<X<∞
–∞ < X < θ
if σ = 1
if σ = – 1
Johnson Su
pdf:
x–θ
– θ- 2 – 1 ⁄ 2
--δ- 1 +  x----------φ γ + δ sinh– 1  ------------
 σ 

σ
σ 
for – ∞ < x, θ, γ < ∞ ; 0 < θ,δ
Johnson Sb
pdf:
x–θ
δσ
φ γ + δ ln  --------------------------  ------------------------------------------------
 σ – ( x – θ )  ( x – θ ) ( σ – ( x – θ ) )
for θ < x < θ+σ; 0 < σ
Johnson Sl
pdf:
δ
x–θ
--------------- φ γ + δ ln  ------------
 σ 
x–θ
for θ < x if σ = 1; θ > x if σ = -1
where φ ( . ) is the standard normal pdf.
Note: The parameter confidence intervals are hidden in the default report. Parameter
confidence intervals are not very meaningful for Johnson distributions, because they are
transformations to normality. To show parameter confidence intervals, right-click in the report
and select Columns > Lower 95% and Upper 95%.
Generalized Log (Glog)
This distribution is useful for fitting data that are rarely normally distributed and often have
non-constant variance, like biological assay data. The Glog distribution is described with the
parameters μ (location), σ (scale), and λ (shape).
Chapter 3
Basic Analysis
pdf:
Distributions
Statistical Details for the Distribution Platform
89
1

x + x2 + λ2
x + x2 + λ2
φ  --- log  -------------------------------- – μ  -----------------------------------------------------------

2
σ
σ ( x 2 + λ 2 + x x 2 + λ 2 )
for 0 ≤ λ ; 0 < σ; – ∞ < μ < ∞
The Glog distribution is a transformation to normality, and comes from the following
relationship:
x + x 2 + λ 2
If z = --1- log  ------------------------------- – μ ~ N(0,1), then x ~ Glog(μ,σ,λ).


σ
2
When λ = 0, the Glog reduces to the LogNormal (μ,σ).
Note: The parameter confidence intervals are hidden in the default report. Parameter
confidence intervals are not very meaningful for the GLog distribution, because it is a
transformation to normality. To show parameter confidence intervals, right-click in the report
and select Columns > Lower 95% and Upper 95%.
All
In the Compare Distributions report, the ShowDistribution list is sorted by AICc in ascending
order.
The formula for AICc is as follows:
2ν ( ν + 1 )AICc = -2logL + 2ν + ------------------------n – (ν + 1)
where:
– logL is the log-likelihood
– n is the sample size
– ν is the number of parameters
If the column contains negative values, the Distribution list does not include those
distributions that require data with positive values. Only continuous distributions are listed.
Distributions with threshold parameters, such as Beta and Johnson Sb, are not included in the
list of possible distributions.
90
Distributions
Statistical Details for the Distribution Platform
Chapter 3
Basic Analysis
Discrete Fit Distributions
This section contains statistical details for the options in the Discrete Fit menu.
Poisson
The Poisson distribution has a single scale parameter λ > 0.
pmf:
e –λ λ x
--------------x!
for 0 ≤ λ < ∞ ; x = 0,1,2,...
E(x) = λ
Var(x) = λ
Since the Poisson distribution is a discrete distribution, the overlaid curve is a step function,
with jumps occurring at every integer.
Gamma Poisson
This distribution is useful when the data is a combination of several Poisson(μ) distributions,
each with a different μ. One example is the overall number of accidents combined from
multiple intersections, when the mean number of accidents (μ) varies between the
intersections.
The Gamma Poisson distribution results from assuming that x|μ follows a Poisson
distribution and μ follows a Gamma(α,τ). The Gamma Poisson has parameters λ = ατ and
σ = τ+1. The parameter σ is a dispersion parameter. If σ > 1, there is over dispersion, meaning
there is more variation in x than explained by the Poisson alone. If σ = 1, x reduces to
Poisson(λ).
pmf:
λ
λ Γ  x + ------------
----------
σ – 1  σ – 1 x – σ – 1
------------------------------------------- ------------ σ
λ  σ 
Γ ( x + 1 )Γ  ------------
 σ – 1
for 0 < λ ; 1 ≤ σ ; x = 0,1,2,...
E(x) = λ
Var(x) = λσ
where Γ ( . ) is the Gamma function.
Remember that x|μ ~ Poisson(μ), while μ~ Gamma(α,τ). The platform estimates λ = ατ and
σ = τ+1. To obtain estimates for α and τ, use the following formulas:
τ̂ = σ̂ – 1
Chapter 3
Basic Analysis
Distributions
Statistical Details for the Distribution Platform
91
λ̂
α̂ = --τ̂
If the estimate of σ is 1, the formulas do not work. In that case, the Gamma Poisson has
reduced to the Poisson(λ), and λ̂ is the estimate of λ.
If the estimate for α is an integer, the Gamma Poisson is equivalent to a Negative Binomial
with the following pmf:
y + r – 1 r
p ( 1 – p )y
p(y) = 

y 
for 0 ≤ y
with r = α and (1-p)/p = τ.
Run demoGammaPoisson.jsl in the JMP Samples/Scripts folder to compare a Gamma Poisson
distribution with parameters λ and σ to a Poisson distribution with parameter λ.
Binomial
The Binomial option accepts data in two formats: a constant sample size, or a column
containing sample sizes.
pmf:
 n p x ( 1 – p ) n – x
 x
for 0 ≤ p ≤ 1 ; x = 0,1,2,...,n
E(x) = np
Var(x) = np(1-p)
where n is the number of independent trials.
Note: The confidence interval for the binomial parameter is a Score interval. See Agresti
(1998).
Beta Binomial
This distribution is useful when the data is a combination of several Binomial(p) distributions,
each with a different p. One example is the overall number of defects combined from multiple
manufacturing lines, when the mean number of defects (p) varies between the lines.
The Beta Binomial distribution results from assuming that x|π follows a Binomial(n,π)
distribution and π follows a Beta(α,β). The Beta Binomial has parameters p = α/(α+β) and
δ = 1/(α+β+1). The parameter δ is a dispersion parameter. When δ > 0, there is over dispersion,
meaning there is more variation in x than explained by the Binomial alone. When δ < 0, there is
under dispersion. When δ = 0, x is distributed as Binomial(n,p). The Beta Binomial only exists
when n ≥ 2 .
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Distributions
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pmf:
Chapter 3
Basic Analysis
1
1
1
Γ  --- – 1 Γ x + p  --- – 1 Γ n – x + ( 1 – p )  --- – 1
δ

δ

δ

n
  ---------------------------------------------------------------------------------------------------------------------------- x
1
1
1
Γ p  --- – 1 Γ ( 1 – p )  --- – 1 Γ  n + --- – 1
δ

δ
 

δ
p
1–p
for 0 ≤ p ≤ 1 ; max (– --------------------, – ---------------------) ≤ δ ≤ 1 ; x = 0,1,2,...,n
n–p–1 n–2+p
E(x) = np
Var(x) = np(1-p)[1+(n-1)δ]
where Γ ( . ) is the Gamma function.
Remember that x|π ~ Binomial(n,π), while π ~ Beta(α,β). The parameters p = α/(α+β) and
δ = 1/(α+β+1) are estimated by the platform. To obtain estimates of α and β, use the following
formulas:
1 – δ̂
α̂ = p̂  -----------
 δ̂ 
1 – δ̂
β̂ = ( 1 – p̂ )  -----------
 δ̂ 
If the estimate of δ is 0, the formulas do not work. In that case, the Beta Binomial has reduced
to the Binomial(n,p), and p̂ is the estimate of p.
The confidence intervals for the Beta Binomial parameters are profile likelihood intervals.
Run demoBetaBinomial.jsl in the JMP Samples/Scripts folder to compare a Beta Binomial
distribution with dispersion parameter δ to a Binomial distribution with parameters p and
n = 20.
Fitted Quantiles
The fitted quantiles in the Diagnostic Plot and the fitted quantiles saved with the Save Fitted
Quantiles command are formed using the following method:
1. The data are sorted and ranked. Ties are assigned different ranks.
2. Compute the p[i] = rank[i]/(n+1).
3. Compute the quantile[i] = Quantiled(p[i]) where Quantiled is the quantile function for the
specific fitted distribution, and i = 1,2,...,n.
Chapter 3
Basic Analysis
Distributions
Statistical Details for the Distribution Platform
93
Fit Distribution Options
This section describes Goodness of Fit tests for fitting distributions and statistical details for
specification limits pertaining to fitted distributions.
Goodness of Fit
Table 3.3 Descriptions of JMP Goodness of Fit Tests
Distribution
Parameters
Goodness of Fit Test
Normala
μ and σ are unknown
Shapiro-Wilk (for n ≤ 2000)
Kolmogorov-Smirnov-Lillefors
(for n > 2000)
μ and σ are both known
Kolmogorov-Smirnov-Lillefors
either μ or σ is known
(none)
LogNormal
μ and σ are known or
unknown
Kolmogorovʹs D
Weibull
α and β known or
unknown
Cramér-von Mises W2
Weibull with threshold
α, β and θ known or
unknown
Cramér-von Mises W2
Extreme Value
α and β known or
unknown
Cramér-von Mises W2
Exponential
σ is known or unknown
Kolmogorovʹs D
Gamma
α and σ are known
Cramér-von Mises W2
either α or σ is unknown
(none)
α and β are known
Kolmogorovʹs D
either α or β is unknown
(none)
Binomial
ρ is known or unknown
and n is known
Kolmogorovʹs D (for n ≤ 30)
Pearson χ2 (for n > 30)
Beta Binomial
ρ and δ known or
unknown
Kolmogorovʹs D (for n ≤ 30)
Pearson χ2 (for n > 30)
Poisson
λ known or unknown
Kolmogorovʹs D (for n ≤ 30)
Pearson χ2 (for n > 30)
Beta
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Distributions
Statistical Details for the Distribution Platform
Chapter 3
Basic Analysis
Table 3.3 Descriptions of JMP Goodness of Fit Tests (Continued)
Distribution
Parameters
Goodness of Fit Test
Gamma Poisson
λ or σ known or unknown
Kolmogorovʹs D (for n ≤ 30)
Pearson χ2 (for n > 30)
a. For the three Johnson distributions and the Glog distribution, the data are transformed to
Normal, then the appropriate test of normality is performed.
Set Spec Limits for K Sigma
Type a K value and select one-sided or two-sided for your capability analysis. Tail
probabilities corresponding to K standard deviations are computed from the Normal
distribution. The probabilities are converted to quantiles for the specific distribution that you
have fitted. The resulting quantiles are used for specification limits in the capability analysis.
This option is similar to the Quantiles option, but you provide K instead of probabilities. K
corresponds to the number of standard deviations that the specification limits are away from
the mean.
For example, for a Normal distribution, where K=3, the 3 standard deviations below and
above the mean correspond to the 0.00135th quantile and 0.99865th quantile, respectively. The
lower specification limit is set at the 0.00135th quantile, and the upper specification limit is set
at the 0.99865th quantile of the fitted distribution. A capability analysis is returned based on
those specification limits.
Chapter 4
Introduction to Fit Y by X
Examine Relationships Between Two Variables
The Fit Y by X platform analyzes the pair of X and Y variables that you specify, by context,
based on modeling type.
Here are the four types of analyses:
•
Bivariate fitting
•
One-way analysis of variance
•
Logistic regression
•
Contingency table analysis
Figure 4.1 Examples of Four Types of Analyses
Oneway
Logistic
Contingency
Nominal or
Ordinal Y
Continuous Y
Bivariate
Continuous X
Nominal or Ordinal X
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Introduction to Fit Y by X
Overview of the Fit Y by X Platform
Chapter 4
Basic Analysis
Overview of the Fit Y by X Platform
The Fit Y by X platform is a collection of four specific platforms (or types of analyses).
Specific
Platform
Modeling Types
Description
Bivariate
Continuous Y by continuous X
Analyzes the relationship between two
continuous variables. See “Bivariate Analysis”.
Oneway
Continuous Y by nominal
or ordinal X
Analyzes how the distribution of a continuous
Y variable differs across groups defined by a
categorical X variable. See “Oneway Analysis”.
Logistic
Nominal or ordinal Y by
continuous X
Fits the probabilities for response categories to a
continuous X predictor. See “Logistic Analysis”.
Contingency
Nominal or ordinal Y by
nominal or ordinal X
Analyzes the distribution of a categorical
response variable Y as conditioned by the
values of a categorical X factor. See
“Contingency Analysis”.
Launch the Fit Y by X Platform
Launch the Fit Y by X platform by selecting Analyze > Fit Y by X.
Figure 4.2 The Fit Y by X Launch Window
Bivariate, Oneway, Logistic, Contingency This grid shows which analysis results from the
different combinations of data types. Once you have assigned your columns, the
applicable platform appears as a label above the grid.
Chapter 4
Basic Analysis
Block
Introduction to Fit Y by X
Launch the Fit Y by X Platform
97
(Optional, for Oneway and Contingency only):
– For the Oneway platform, identifies a second factor, which forms a two-way analysis
without interaction. The data should be balanced and have equal counts in each block
by group cell. If you specify a Block variable, the data should be balanced and have
equal counts in each block by group cell. In the plot, the values of the Y variable are
centered by the Block variable.
– For the Contingency platform, identifies a second factor and performs a
Cochran-Mantel-Haenszel test.
For more information about launch windows, see the Get Started chapter in the Using JMP
book.
Launch Specific Analyses from the JMP Starter Window
From the JMP Starter window, you can launch a specific analysis (Bivariate, Oneway, Logistic,
or Contingency). If you select this option, specify the correct modeling types (Y and X
variables) for the analysis. See Table 4.1.
To launch a specific analysis from the JMP Starter Window, click the Basic category, and select
a specific analysis.
Most of the platform launch options are the same. However, the naming for some of the Y and
X platform buttons is tailored for the specific analysis that you are performing.
Table 4.1 Platforms and Buttons
Platform or Analysis
Y Button
X Button
Fit Y by X
Y, Response
X, Factor
Bivariate
Y, Response
X, Regressor
Oneway
Y, Response
X, Grouping
Logistic
Y, Categorical Response
X, Continuous Regressor
Contingency
Y, Response Category
X, Grouping Category
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Launch the Fit Y by X Platform
Chapter 4
Basic Analysis
Chapter 5
Bivariate Analysis
Examine Relationships between Two Continuous Variables
The Bivariate platform shows the relationship between two continuous variables. It is the
continuous by continuous personality of the Fit Y by X platform. The word bivariate simply
means involving two variables instead of one (univariate) or many (multivariate).
The Bivariate analysis results appear in a scatterplot. Each point on the plot represents the X
and Y scores for a single subject; in other words, each point represents two variables. Using
the scatterplot, you can see at a glance the degree and pattern of the relationship between the
two variables. You can interactively add other types of fits, such as simple linear regression,
polynomial regression, and so on.
Figure 5.1 Example of Bivariate Analysis
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Example of Bivariate Analysis
Chapter 5
Basic Analysis
Example of Bivariate Analysis
This example uses the SAT.jmp sample data table. SAT test scores for students in the 50 U.S.
states, plus the District of Columbia, are divided into two areas: verbal and math. You want to
find out how the percentage of students taking the SAT tests is related to verbal test scores for
2004.
1. Select Help > Sample Data Library and open SAT.jmp.
2. Select Analyze > Fit Y by X.
3. Select 2004 Verbal and click Y, Response.
4. Select % Taking (2004) and click X, Factor.
5. Click OK.
Figure 5.2 Example of SAT Scores by Percent Taking
You can see that the verbal scores were higher when a smaller percentage of the population
took the test.
Launch the Bivariate Platform
You can perform a bivariate analysis using either the Fit Y by X platform or the Bivariate
platform. The two approaches give equivalent results.
•
To launch the Fit Y by X platform, select Analyze > Fit Y by X.
or
•
To launch the Bivariate platform, from the JMP Starter window, click on the Basic category
and click Bivariate.
Chapter 5
Basic Analysis
Bivariate Analysis
The Bivariate Plot
101
Figure 5.3 The Bivariate Launch Window
For information about this launch window, see “Introduction to Fit Y by X” chapter on
page 95.
After you click OK, the Bivariate plot appears. See “The Bivariate Plot” on page 101.
The Bivariate Plot
To produce the plot shown in Figure 5.4, follow the instructions in “Example of Bivariate
Analysis” on page 100.
Figure 5.4 The Bivariate Plot
The Bivariate report begins with a plot for each pair of X and Y variables. Replace variables in
the plot by dragging and dropping a variable, in one of two ways: swap existing variables by
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Fitting Commands and Options
Chapter 5
Basic Analysis
dragging and dropping a variable from one axis to the other axis; or, click on a variable in the
Columns panel of the associated data table and drag it onto an axis.
You can interact with this plot just as you can with other JMP plots (for example, resizing the
plot, highlighting points with the arrow or brush tool, and labeling points). For details about
these features, see the JMP Reports chapter in the Using JMP book.
You can fit curves on the plot and view statistical reports and additional menus using the
fitting commands that are located within the red triangle menu. See “Fitting Commands and
Options” on page 102.
Fitting Commands and Options
Note: The Fit Group menu appears if you have specified multiple Y or multiple X variables.
Menu options allow you to arrange reports or order them by RSquare. See the Standard Least
Squares chapter in the Fitting Linear Models book for more information.
The Bivariate Fit red triangle menu contains display options, fitting options, and control
options.
Show Points Hides or shows the points in the scatterplot. A check mark indicates that points
are shown.
Attaches histograms to the x- and y-axes of the scatterplot. A check mark
indicates that histogram borders are turned on. See “Histogram Borders” on page 105.
Histogram Borders
Note: When you apply only the Hidden row state to rows in the data table, the
corresponding points do not appear in the scatterplot. However, the histograms are
constructed using the hidden rows. If you want to exclude rows from the construction of
the histograms and from analysis results, apply the Exclude row state and select Redo >
Redo Analysis from the Bivariate red triangle menu.
Group By Lets you select a classification (or grouping) variable. A separate analysis is
computed for each level of the grouping variable, and regression curves or ellipses are
overlaid on the scatterplot. See “Group By” on page 121.
See the JMP Reports chapter in the Using JMP book for more information about the following
options:
Shows or hides the local data filter that enables you to filter the data used in
a specific report.
Local Data Filter
Contains options that enable you to repeat or relaunch the analysis. In platforms that
support the feature, the Automatic Recalc option immediately reflects the changes that
you make to the data table in the corresponding report window.
Redo
Chapter 5
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Bivariate Analysis
Fitting Commands and Options
103
Contains options that enable you to save a script that reproduces the report to
several destinations.
Save Script
Save By-Group Script Contains options that enable you to save a script that reproduces the
platform report for all levels of a By variable to several destinations. Available only when a
By variable is specified in the launch window.
Fit Mean Options
Each fitting command adds the following:
•
a line, curve, or distribution to the scatterplot
•
a red triangle menu to the report window
•
a specific report to the report window
Figure 5.5 Example of the Fit Mean Fitting Command
Fit Mean line
Fit Mean menu
Fit Mean report
The following Fit Mean options are available:
Fit Mean Adds a horizontal line to the scatterplot that represents the mean of the Y response
variable.See “Fit Mean” on page 105.
Fit Line Adds straight line fits to your scatterplot using least squares regression. See “Fit Line
and Fit Polynomial” on page 106.
Fit Polynomial Fits polynomial curves of a certain degree using least squares regression. See
“Fit Line and Fit Polynomial” on page 106.
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Fit Special Transforms Y and X. Transformations include: log, square root, square, reciprocal,
and exponential. You can also turn off center polynomials, constrain the intercept and the
slope, and fit polynomial models. See “Fit Special” on page 113.
Flexible Provides options that enable you to control the smoothness of the estimated
regression curve. See “Flexible” on page 114.
Provides options for orthogonal regression fits, which are useful when X is
assumed to vary. This option provides sub-options that reflect various assumptions about
the variances of X and Y. See “Fit Orthogonal” on page 117.
Fit Orthogonal
Provides options that reduce the influence of outliers in your data set on the fitted
model. See “Robust” on page 118.
Robust
Density Ellipse Plots density ellipsoids for the bivariate normal distribution fit to the X and Y
variables. See “Density Ellipse” on page 119.
Nonpar Density Plots density contours based on a smoothed surface. The contours describe
the density of data points. See “Nonpar Density” on page 120.
Note: You can remove a fit using the Remove Fit command. For details, see “Fitting Menu
Options” on page 122.
Fitting Command Categories
Fitting command categories include regression fits and density estimation.
Category
Description
Fitting Commands
Regression Fits
Regression methods fit a curve to the observed
data points. The fitting methods include least
squares fits as well as spline fits, kernel smoothing,
orthogonal fits, and robust fits.
Fit Mean
Fit Line
Fit Polynomial
Fit Special
Flexible
Fit Orthogonal
Robust
Density
Estimation
Density estimation fits a bivariate distribution to
the points. You can either select a bivariate normal
density, characterized by elliptical contours, or a
general nonparametric density.
Density Ellipse
Nonpar Density
Chapter 5
Basic Analysis
Bivariate Analysis
Histogram Borders
105
Fit the Same Command Multiple Times
You can select the same fitting command multiple times, and each new fit is overlaid on the
scatterplot. You can try fits, exclude points and refit, and you can compare them on the same
scatterplot.
To apply a fitting command to multiple analyses in your report window, hold down the Ctrl
key and select a fitting option.
Histogram Borders
The Histogram Borders option appends histograms to the x- and y-axes of the scatterplot. You
can use the histograms to visualize the marginal distributions of the X and Y variables.
Figure 5.6 Example of Histogram Borders
Fit Mean
Using the Fit Mean command, you can add a horizontal line to the scatterplot that represents
the mean of the Y response variable. You can start by fitting the mean and then use the mean
line as a reference for other fits (such as straight lines, confidence curves, polynomial curves,
and so on).
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Fit Line and Fit Polynomial
Chapter 5
Basic Analysis
Figure 5.7 Example of Fit Mean
Fit Mean line
Fit Mean menu
Fit Mean report
Fit Mean Report
The Fit Mean report shows summary statistics about the fit of the mean.
Mean Mean of the response variable. The predicted response when there are no specified
effects in the model.
Std Dev [RMSE] Standard deviation of the response variable. Square root of the mean square
error, also called the root mean square error (or RMSE).
Std Error Standard deviation of the response mean. Calculated by dividing the RMSE by the
square root of the number of values.
SSE Error sum of squares for the simple mean model. Appears as the sum of squares for
Error in the analysis of variance tables for each model fit.
Related Information
•
“Fitting Menus” on page 122
Fit Line and Fit Polynomial
Using the Fit Line command, you can add straight line fits to your scatterplot using least
squares regression. Using the Fit Polynomial command, you can fit polynomial curves of a
certain degree using least squares regression.
Chapter 5
Basic Analysis
Bivariate Analysis
Fit Line and Fit Polynomial
Figure 5.8 Example of Fit Line and Fit Polynomial
Figure 5.8 shows an example that compares a linear fit to the mean line and to a degree 2
polynomial fit.
Note the following information:
•
The Fit Line output is equivalent to a polynomial fit of degree 1.
•
The Fit Mean output is equivalent to a polynomial fit of degree 0.
Linear Fit and Polynomial Fit Reports
The Linear Fit and Polynomial Fit reports begin with the equation of fit.
Figure 5.9 Example of Equations of Fit
Note: You can edit the equation by clicking on it.
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Fit Line and Fit Polynomial
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Basic Analysis
Each Linear and Polynomial Fit Degree report contains at least three reports. A fourth report,
Lack of Fit, appears only if there are X replicates in your data.
Summary of Fit Report
The Summary of Fit reports show the numeric summaries of the response for the linear fit and
polynomial fit of degree 2 for the same data. You can compare multiple Summary of Fit
reports to see the improvement of one model over another, indicated by a larger RSquare
value and smaller Root Mean Square Error.
Figure 5.10 Summary of Fit Reports for Linear and Polynomial Fits
The Summary of Fit report contains the following columns:
RSquare Measures the proportion of the variation explained by the model. The remaining
variation is not explained by the model and attributed to random error. The RSquare is 1 if
the model fits perfectly.
Note: A low RSquare value suggests that there may be variables not in the model that
account for the unexplained variation. However, if your data are subject to a large amount
of inherent variation, even a useful regression model may have a low RSquare value. Read
the literature in your research area to learn about typical RSquare values.
The RSquare values in Figure 5.10 indicate that the polynomial fit of degree 2 gives a small
improvement over the linear fit. See “Summary of Fit Report” on page 134.
RSquare Adj Adjusts the RSquare value to make it more comparable over models with
different numbers of parameters by using the degrees of freedom in its computation. See
“Summary of Fit Report” on page 134.
Root Mean Square Error Estimates the standard deviation of the random error. It is the
square root of the mean square for Error in the Analysis of Variance report. See Figure 5.12.
Mean of Response Provides the sample mean (arithmetic average) of the response variable.
This is the predicted response when no model effects are specified.
Observations Provides the number of observations used to estimate the fit. If there is a
weight variable, this is the sum of the weights.
Chapter 5
Basic Analysis
Bivariate Analysis
Fit Line and Fit Polynomial
109
Lack of Fit Report
Note: The Lack of Fit report appears only if there are multiple rows that have the same x
value.
Using the Lack of Fit report, you can estimate the error, regardless of whether you have the
right form of the model. This occurs when multiple observations occur at the same x value.
The error that you measure for these exact replicates is called pure error. This is the portion of
the sample error that cannot be explained or predicted no matter what form of model is used.
However, a lack of fit test might not be of much use if it has only a few degrees of freedom for
it (few replicated x values).
Figure 5.11 Examples of Lack of Fit Reports for Linear and Polynomial Fits
The difference between the residual error from the model and the pure error is called the lack
of fit error. The lack of fit error can be significantly greater than the pure error if you have the
wrong functional form of the regressor. In that case, you should try a different type of model
fit. The Lack of Fit report tests whether the lack of fit error is zero.
The Lack of Fit report contains the following columns:
Source The three sources of variation: Lack of Fit, Pure Error, and Total Error.
DF
The degrees of freedom (DF) for each source of error.
– The Total Error DF is the degrees of freedom found on the Error line of the Analysis of
Variance table (shown under the “Analysis of Variance Report” on page 110). It is the
difference between the Total DF and the Model DF found in that table. The Error DF is
partitioned into degrees of freedom for lack of fit and for pure error.
– The Pure Error DF is pooled from each group where there are multiple rows with the
same values for each effect. See “Lack of Fit Report” on page 135.
– The Lack of Fit DF is the difference between the Total Error and Pure Error DF.
Sum of Squares The sum of squares (SS for short) for each source of error.
– The Total Error SS is the sum of squares found on the Error line of the corresponding
Analysis of Variance table, shown under “Analysis of Variance Report” on page 110.
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– The Pure Error SS is pooled from each group where there are multiple rows with the
same value for the x variable. This estimates the portion of the true random error that is
not explained by model x effect. See “Lack of Fit Report” on page 135.
– The Lack of Fit SS is the difference between the Total Error and Pure Error sum of
squares. If the lack of fit SS is large, the model might not be appropriate for the data.
The F-ratio described below tests whether the variation due to lack of fit is small
enough to be accepted as a negligible portion of the pure error.
Mean Square The sum of squares divided by its associated degrees of freedom. This
computation converts the sum of squares to an average (mean square). F-ratios for
statistical tests are the ratios of mean squares.
F Ratio The ratio of mean square for lack of fit to mean square for Pure Error. It tests the
hypothesis that the lack of fit error is zero.
Prob > F The probability of obtaining a greater F-value by chance alone if the variation due to
lack of fit variance and the pure error variance are the same. A high p value means that
there is not a significant lack of fit.
Max RSq The maximum R2 that can be achieved by a model using only the variables in the
model. See “Lack of Fit Report” on page 135.
Analysis of Variance Report
Analysis of variance (ANOVA) for a regression partitions the total variation of a sample into
components. These components are used to compute an F-ratio that evaluates the
effectiveness of the model. If the probability associated with the F-ratio is small, then the
model is considered a better statistical fit for the data than the response mean alone.
The Analysis of Variance reports in Figure 5.12 compare a linear fit (Fit Line) and a second
degree (Fit Polynomial). Both fits are statistically better from a horizontal line at the mean.
Figure 5.12 Examples of Analysis of Variance Reports for Linear and Polynomial Fits
The Analysis of Variance Report contains the following columns:
Source The three sources of variation: Model, Error, and C. Total.
Chapter 5
Basic Analysis
DF
Bivariate Analysis
Fit Line and Fit Polynomial
111
The degrees of freedom (DF) for each source of variation:
– A degree of freedom is subtracted from the total number of non missing values (N) for
each parameter estimate used in the computation. The computation of the total sample
variation uses an estimate of the mean. Therefore, one degree of freedom is subtracted
from the total, leaving 49. The total corrected degrees of freedom are partitioned into
the Model and Error terms.
– One degree of freedom from the total (shown on the Model line) is used to estimate a
single regression parameter (the slope) for the linear fit. Two degrees of freedom are
used to estimate the parameters ( β 1 and β 2 ) for a polynomial fit of degree 2.
– The Error degrees of freedom is the difference between C. Total df and Model df.
Sum of Squares The sum of squares (SS for short) for each source of variation:
– In this example, the total (C. Total) sum of squared distances of each response from the
sample mean is 57,278.157, as shown in Figure 5.12. That is the sum of squares for the
base model (or simple mean model) used for comparison with all other models.
– For the linear regression, the sum of squared distances from each point to the line of fit
reduces from 12,012.733. This is the residual or unexplained (Error) SS after fitting the
model. The residual SS for a second degree polynomial fit is 6,906.997, accounting for
slightly more variation than the linear fit. That is, the model accounts for more
variation because the model SS are higher for the second degree polynomial than the
linear fit. The C. total SS less the Error SS gives the sum of squares attributed to the
model.
Mean Square The sum of squares divided by its associated degrees of freedom. The F-ratio
for a statistical test is the ratio of the following mean squares:
– The Model mean square for the linear fit is 45,265.4. This value estimates the error
variance, but only under the hypothesis that the model parameters are zero.
– The Error mean square is 245.2. This value estimates the error variance.
F Ratio The model mean square divided by the error mean square. The underlying
hypothesis of the fit is that all the regression parameters (except the intercept) are zero. If
this hypothesis is true, then both the mean square for error and the mean square for model
estimate the error variance, and their ratio has an F-distribution. If a parameter is a
significant model effect, the F-ratio is usually higher than expected by chance alone.
Prob > F The observed significance probability (p-value) of obtaining a greater F-value by
chance alone if the specified model fits no better than the overall response mean. Observed
significance probabilities of 0.05 or less are often considered evidence of a regression
effect.
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Parameter Estimates Report
The terms in the Parameter Estimates report for a linear fit are the intercept and the single x
variable.
For a polynomial fit of order k, there is an estimate for the model intercept and a parameter
estimate for each of the k powers of the X variable.
Figure 5.13 Examples of Parameter Estimates Reports for Linear and Polynomial Fits
The Parameter Estimates report contains the following columns:
Term Lists the name of each parameter in the requested model. The intercept is a constant
term in all models.
Estimate Lists the parameter estimates of the linear model. The prediction formula is the
linear combination of these estimates with the values of their corresponding variables.
Std Error Lists the estimates of the standard errors of the parameter estimates. They are used
in constructing tests and confidence intervals.
t Ratio Lists the test statistics for the hypothesis that each parameter is zero. It is the ratio of
the parameter estimate to its standard error. If the hypothesis is true, then this statistic has
a Student’s t-distribution.
Prob>|t| Lists the observed significance probability calculated from each t-ratio. It is the
probability of getting, by chance alone, a t-ratio greater (in absolute value) than the
computed value, given a true null hypothesis. Often, a value below 0.05 (or sometimes
0.01) is interpreted as evidence that the parameter is significantly different from zero.
To reveal additional statistics, right-click in the report and select the Columns menu. Statistics
not shown by default are as follows:
Lower 95% The lower endpoint of the 95% confidence interval for the parameter estimate.
Upper 95%
The upper endpoint of the 95% confidence interval for the parameter estimate.
Std Beta The standardized parameter estimate. It is useful for comparing the effect of X
variables that are measured on different scales. See “Parameter Estimates Report” on
page 135.
Chapter 5
Basic Analysis
VIF
Bivariate Analysis
Fit Special
113
The variance inflation factor.
The design standard error for the parameter estimate. See “Parameter
Estimates Report” on page 135.
Design Std Error
Related Information
•
“Fit Line” on page 133
•
“Fitting Menus” on page 122
Fit Special
Using the Fit Special command, you can transform Y and X. Transformations include the
following: log, square root, square, reciprocal, and exponential. You can also constrain the
slope and intercept, fit a polynomial of specific degree, and center the polynomial.
The Specify Transformation or Constraint Window contains the following options:
Y Transformation Use these options to transform the Y variable.
X Transformation Use these options to transform the X variable.
Degree Use this option to fit a polynomial of the specified degree.
Centered Polynomial To turn off polynomial centering, deselect the Centered Polynomial
check box. See Figure 5.19. Note that for transformations of the X variable, polynomial
centering is not performed. Centering polynomials stabilizes the regression coefficients
and reduces multicollinearity.
Constrain Intercept to
Select this check box to constrain the model intercept to be the
specified value.
Constrain Slope to Select this check box to constrain the model slope to be the specified
value.
Fit Special Reports and Menus
Depending on your selections in the Fit Special window, you see different reports and menus.
The flowchart in Figure 5.14 shows you what reports and menus you see depending on your
choices.
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Figure 5.14 Example of Fit Special Flowchart
Transformation?
Yes
No
Transformed
Fit Report
and menu
Degree?
1
2-5
Linear Fit
Report and
menu
Polynomial
Fit Report
and menu
Transformed Fit Report
The Transformed Fit report contains the reports described in “Linear Fit and Polynomial Fit
Reports” on page 107. However, if you transformed Y, the Fit Measured on Original Scale
report appears. This shows the measures of fit based on the original Y variables, and the fitted
model transformed back to the original scale.
Related Information
•
“Example of the Fit Special Command” on page 126
•
“Linear Fit and Polynomial Fit Reports” on page 107
•
“Fitting Menus” on page 122
Flexible
Use the options in the Flexible menu to control the smoothness of the estimated regression
curve.
•
Fit Spline uses a penalized least squares approach. Adjust the degree of smoothness using
the parameter lambda.
•
Kernel Smoother is based on locally weighted fits. Control the influence of local behavior
using the parameter alpha.
•
Fit Each Value calculates the mean response at each X value.
Chapter 5
Basic Analysis
Bivariate Analysis
Flexible
115
Fit Spline
Using the Fit Spline command, you can fit a smoothing spline that varies in smoothness (or
flexibility) according to the lambda (λ) value. The lambda value is a tuning parameter in the
spline formula. As the value of λ decreases, the error term of the spline model has more
weight and the fit becomes more flexible and curved. As the value of λ increases, the fit
becomes stiff (less curved), approaching a straight line.
Note the following information:
•
The smoothing spline can help you see the expected value of the distribution of Y across X.
•
The points closest to each piece of the fitted curve have the most influence on it. The
influence increases as you lower the value of λ, producing a highly flexible curve.
•
If you want to use a lambda value that is not listed on the menu, select Fit Spline > Other. If
the scaling of the X variable changes, the fitted model also changes. To prevent this from
happening, select the Standardize X option. Note that the fitted model remains the same
for either the original X variable or the scaled X variable.
•
You might find it helpful to try several λ values. You can use the Lambda slider beneath
the Smoothing Spline report to experiment with different λ values. However, λ is not
invariant to the scaling of the data. For example, the λ value for an X measured in inches, is
not the same as the λ value for an X measured in centimeters.
Smoothing Spline Fit Report
The Smoothing Spline Fit report contains the R-Square for the spline fit and the Sum of
Squares Error. You can use these values to compare the spline fit to other fits, or to compare
different spline fits to each other.
R-Square Measures the proportion of variation accounted for by the smoothing spline
model. For more information, see “Smoothing Fit Reports” on page 136.
Sum of squared distances from each point to the fitted spline. It is the
unexplained error (residual) after fitting the spline model.
Sum of Squares Error
Enables you to change the λ value, either by entering a number, or by
moving the slider.
Change Lambda
Related Information
•
“Fitting Menus” on page 122
•
“Fit Spline” on page 133
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Kernel Smoother
The Kernel Smoother command produces a curve formed by repeatedly finding a locally
weighted fit of a simple curve (a line or a quadratic) at sampled points in the domain. The
many local fits (128 in total) are combined to produce the smooth curve over the entire
domain. This method is also called Loess or Lowess, which was originally an acronym for
Locally Weighted Scatterplot Smoother. See Cleveland (1979).
Use this method to quickly see the relationship between variables and to help you determine
the type of analysis or fit to perform.
Local Smoother Report
The Local Smoother report contains the R-Square for the smoother fit and the Sum of Squares
Error. You can use these values to compare the smoother fit to other fits, or to compare
different smoother fits to each other.
R-Square Measures the proportion of variation accounted for by the smoother model. For
more information, see “Smoothing Fit Reports” on page 136.
Sum of Squares Error Sum of squared distances from each point to the fitted smoother. It is
the unexplained error (residual) after fitting the smoother model.
Select the polynomial degree for each local fit. Quadratic polynomials can
track local bumpiness more smoothly. Lambda is the degree of certain polynomials that
are fitted by the method. Lambda can be 0, 1 or 2.
Local Fit (lambda)
Weight Function Specify how to weight the data in the neighborhood of each local fit. Loess
uses tri-cube. The weight function determines the influence that each xi and yi has on the
fitting of the line. The influence decreases as xi increases in distance from x and finally
becomes zero.
Smoothness (alpha) Controls how many points are part of each local fit. Use the slider or
type in a value directly. Alpha is a smoothing parameter. It can be any positive number,
but typical values are 1/4 to 1. As alpha increases, the curve becomes smoother.
Re-weights the points to de-emphasize points that are farther from the fitted
curve. Specify the number of times to repeat the process (number of passes). The goal is to
converge the curve and automatically filter out outliers by giving them small weights.
Robustness
Related Information
•
“Fitting Menus” on page 122
Chapter 5
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Bivariate Analysis
Fit Orthogonal
117
Fit Each Value
The Fit Each Value command fits a value to each unique X value. The fitted values are the
means of the response for each unique X value.
Fit Each Value Report
The Fit Each Value report shows summary statistics about the model fit.
Number of Observations Gives the total number of observations.
Number of Unique Values
Gives the number of unique X values.
Degrees of Freedom Gives the pure error degrees of freedom.
Sum of Squares Gives the pure error sum of squares.
Mean Square Gives the pure error mean square.
Related Information
•
“Fitting Menus” on page 122
Fit Orthogonal
The Fit Orthogonal command fits linear models that account for variability in X as well as Y.
Fit Orthogonal Options
Select one of the following options to specify a variance ratio.
Univariate Variances, Prin Comp Uses the univariate variance estimates computed from the
samples of X and Y. This turns out to be the standardized first principal component. This
option is not a good choice in a measurement systems application since the error variances
are not likely to be proportional to the population variances.
Equal Variances Uses 1 as the variance ratio, which assumes that the error variances are the
same. Using equal variances is equivalent to the non-standardized first principal
component line. Suppose that the scatterplot is scaled the same in the X and Y directions.
When you show a normal density ellipse, you see that this line is the longest axis of the
ellipse.
Fit X to Y Uses a variance ratio of zero, which indicates that Y effectively has no variance.
Specified Variance Ratio Lets you enter any ratio that you want, giving you the ability to
make use of known information about the measurement error in X and response error in
Y.
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Orthogonal Regression Report
The Orthogonal Regression report shows summary statistics about the orthogonal regression
model.
Variable Gives the names of the variables used to fit the line.
Mean Gives the mean of each variable.
Std Dev
Gives the standard deviation of each variable.
Variance Ratio Gives the variance ratio used to fit the line.
Correlation
Intercept
Gives the correlation between the two variables.
Gives the intercept of the fitted line.
Slope Gives the slope of the fitted line.
LowerCL
Gives the lower confidence limit for the slope.
UpperCL
Gives the upper confidence limit for the slope.
Alpha Enter the alpha level used in computing the confidence interval.
Related Information
•
“Fitting Menus” on page 122
•
“Fit Orthogonal” on page 133
•
“Example Using the Fit Orthogonal Command” on page 128
Robust
The Robust option provides two methods to reduce the influence of outliers in your data set.
Outliers can lead to incorrect estimates and decisions.
Fit Robust
The Fit Robust option reduces the influence of outliers in the response variable. The Huber
M-estimation method is used. Huber M-estimation finds parameter estimates that minimize
the Huber loss function, which penalizes outliers. The Huber loss function increases as a
quadratic for small errors and linearly for large errors. For more details about robust fitting,
see Huber (1973) and Huber and Ronchetti (2009).
Related Information
•
“Fitting Menus” on page 122
•
“Example Using the Fit Robust Command” on page 129
Chapter 5
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Bivariate Analysis
Density Ellipse
119
Fit Cauchy
Assumes that the errors have a Cauchy distribution. A Cauchy distribution has fatter tails
than the normal distribution, resulting in a reduced emphasis on outliers. This option can be
useful if you have a large proportion of outliers in your data. However, if your data are close
to normal with only a few outliers, this option can lead to incorrect inferences. The Cauchy
option estimates parameters using maximum likelihood and a Cauchy link function.
Density Ellipse
Using the Density Ellipse option, you can draw an ellipse (or ellipses) that contains the
specified mass of points. The number of points is determined by the probability that you select
from the Density Ellipse menu).
Figure 5.15 Example of Density Ellipses
The density ellipsoid is computed from the bivariate normal distribution fit to the X and Y
variables. The bivariate normal density is a function of the means and standard deviations of
the X and Y variables and the correlation between them. The Other selection lets you specify
any probability greater than zero and less than or equal to one.
These ellipses are both density contours and confidence curves. As confidence curves, they
show where a given percentage of the data is expected to lie, assuming the bivariate normal
distribution.
The density ellipsoid is a good graphical indicator of the correlation between two variables.
The ellipsoid collapses diagonally as the correlation between the two variables approaches
either 1 or –1. The ellipsoid is more circular (less diagonally oriented) if the two variables are
less correlated.
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Correlation Report
The Correlation report that accompanies each Density Ellipse fit shows the correlation
coefficient for the X and Y variables.
Note: To see a matrix of ellipses and correlations for many pairs of variables, use the
Multivariate command in the Analyze > Multivariate Methods menu.
Variable Gives the names of the variables used in creating the ellipse
Mean Gives the average of both the X and Y variable.
Std Dev
Gives the standard deviation of both the X and Y variable.
A discussion of the mean and standard deviation are in the section “The Summary
Statistics Report” on page 42 in the “Distributions” chapter.
The Pearson correlation coefficient. If there is an exact linear relationship
between two variables, the correlation is 1 or –1 depending on whether the variables are
positively or negatively related. If there is no relationship, the correlation tends toward
zero.
Correlation
For more information, see “Correlation Report” on page 136.
Signif. Prob Probability of obtaining, by chance alone, a correlation with greater absolute
value than the computed value if no linear relationship exists between the X and Y
variables.
Number Gives the number of observations used in the calculations.
Related Information
•
“Fitting Menus” on page 122
•
“Example of Group By Using Density Ellipses” on page 131
Nonpar Density
When a plot shows thousands of points, the mass of points can be too dark to show patterns in
density. Using the Nonpar Density (nonparametric density) option makes it easier to see the
patterns.
Nonpar Density estimates a smooth nonparametric bivariate surface that describes the density
of data points. The plot adds a set of contour lines showing the density (Figure 5.16). The
contour lines are quantile contours in 5% intervals. This means that about 5% of the points
generated from the estimated nonparametric distribution are below the lowest contour, 10%
are below the next contour, and so on. The highest contour has about 95% of the points below
it.
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Figure 5.16 Example of Nonpar Density
You can change the size of a nonparametric density contour grid to create smoother contours.
The default value is 51 points, which can create jagged contours around dense points.
Press Shift and select Nonpar Density from the Bivariate red triangle menu. Enter a larger
value than the default 51 points.
Nonparametric Bivariate Density Report
The nonparametric bivariate density report shows the standard deviations used in creating
the nonparametric density.
Related Information
•
“Fitting Menus” on page 122
Group By
Using the Group By option, you can select a classification (grouping) variable. When a
grouping variable is in effect, the Bivariate platform computes a separate analysis for each
level of the grouping variable. Regression curves or ellipses then appear on the scatterplot.
The fit for each level of the grouping variable is identified beneath the scatterplot, with
individual popup menus to save or remove fitting information.
The Group By option is checked in the Fitting menu when a grouping variable is in effect. You
can change the grouping variable by first selecting the Group By option to remove (uncheck)
the existing variable. Then, select the Group By option again and respond to its window as
before.
You might use the Group By option in these different ways:
•
An overlay of linear regression lines lets you compare slopes visually.
•
An overlay of density ellipses can show clusters of points by levels of a grouping variable.
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Related Information
•
“Example of Group By Using Density Ellipses” on page 131
•
“Example of Group By Using Regression Lines” on page 132
Fitting Menus
In addition to a report, each fitting command adds a fitting menu to the report window. The
following table shows the fitting menus that correspond to each fitting command.
Fitting Command
Fitting Menu
Fit Mean
Fit Mean
Fit Line
Linear Fit
Fit Polynomial
Polynomial Fit Degree=X*
Fit Special
Linear Fit
Polynomial Fit Degree=X*
Transformed Fit X*
Constrained Fits
Fit Spline
Smoothing Spline Fit, lambda=X*
Smoother
Local Smoother
Fit Each Value
Fit Each Value
Fit Orthogonal
Orthogonal Fit Ratio=X*
Fit Robust
Robust Fit
Fit Cauchy
Cauchy Fit
Density Ellipse
Bivariate Normal Ellipse P=X*
Nonpar Density
Quantile Density Colors
*X=variable character or number
Fitting Menu Options
The Fitting menu for the option that you have selected contains options that apply to that fit.
•
“Options That Apply to Most Fits” on page 123.
Chapter 5
Basic Analysis
Bivariate Analysis
Fitting Menus
•
“Options That Apply to Multiple Fits” on page 123.
•
“Options That Apply to Bivariate Normal Ellipse” on page 124.
•
“Options That Apply to Quantile Density Contours” on page 124.
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Options That Apply to Most Fits
Line of Fit Displays or hides the line or curve describing the model fit. For the Bivariate
Normal Ellipse report, this option shows or hides the ellipse representing the contour
border. Not applicable for Quantile Density Colors.
Lets you select from a palette of colors for assigning a color to each fit. Not
applicable for Quantile Density Colors.
Line Color
Line Style Lets you select from the palette of line styles for each fit. Not applicable for
Quantile Density Colors.
Line Width Gives three line widths for the line of fit. The default line width is the thinnest
line. Not applicable for Quantile Density Colors.
Report
Turns the fit’s report on and off. Does not modify the Bivariate plot.
Remove Fit Removes the fit from the graph and removes its report.
Options That Apply to Multiple Fits
Displays or hides the confidence limits for the expected value (mean). This
option is not available for the Fit Spline, Density Ellipse, Fit Each Value, and Fit
Orthogonal fits and is dimmed on those menus.
Confid Curves Fit
Confid Curves Indiv Displays or hides the confidence limits for an individual predicted value.
The confidence limits reflect variation in the error and variation in the parameter
estimates. This option is not available for the Fit Mean, Fit Spline, Density Ellipse, Fit Each
Value, and Fit Orthogonal fits and is dimmed on those menus.
Save Predicteds Creates a new column in the current data table called Predicted colname
where colname is the name of the Y variable. This column includes the prediction formula
and the computed sample predicted values. The prediction formula computes values
automatically for rows that you add to the table. This option is not available for the Fit
Each Value and Density Ellipse fits and is dimmed on those menus.
You can use the Save Predicteds and Save Residuals commands for each fit. If you use
these commands multiple times or with a grouping variable, it is best to rename the
resulting columns in the data table to reflect each fit.
Save Residuals Creates a new column in the current data table called Residuals colname
where colname is the name of the Y variable. Each value is the difference between the
actual (observed) value and its predicted value. Unlike the Save Predicteds command, this
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command does not create a formula in the new column. This option is not available for the
Fit Each Value and Density Ellipse fits and is dimmed on those menus.
You can use the Save Predicteds and Save Residuals commands for each fit. If you use
these commands multiple times or with a grouping variable, it is best to rename the
resulting columns in the data table to reflect each fit.
Creates a new column in the data table containing a formula
for the mean confidence intervals.
Mean Confidence Limit Formula
Indiv Confidence Limit Formula Creates a new column in the data table containing a formula
for the individual confidence intervals.
Produces four diagnostic plots:
residual by predicted, actual by predicted, residual by row, and a normal quantile plot of
the residuals. See “Diagnostics Plots” on page 125.
Plot Residuals (Linear, Polynomial, and Fit Special Only)
Set a Level Enables you to set the alpha level used in computing confidence bands for
various fits.
Draws the same curves as the Confid Curves Fit command and shades the
area between the curves.
Confid Shaded Fit
Draws the same curves as the Confid Curves Indiv command and shades
the area between the curves.
Confid Shaded Indiv
Save Coefficients Saves the spline coefficients as a new data table, with columns called X, A,
B, C, and D. The X column gives the knot points. A, B, C, and D are the intercept, linear,
quadratic, and cubic coefficients of the third-degree polynomial. These coefficients span
from the corresponding value in the X column to the next highest value.
Options That Apply to Bivariate Normal Ellipse
Shaded Contour Shades the area inside the density ellipse.
Select Points Inside Selects the points inside the ellipse.
Select Points Outside Selects the points outside the ellipse.
Options That Apply to Quantile Density Contours
Displays a slider for each variable, where you can change the standard
deviation that defines the range of X and Y values for determining the density of contour
lines.
Kernel Control
5% Contours
Shows or hides the 5% contour lines.
Contour Lines Shows or hides the 10% contour lines.
Contour Fill
Fills the areas between the contour lines.
Select Points by Density Selects points that fall in a user-specified quantile range.
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Color by Density Quantile Colors the points according to density.
Save Density Quantile
Creates a new column containing the density quantile each point is in.
Mesh Plot Is a three-dimensional plot of the density over a grid of the two analysis variables.
See Figure 5.17.
Modal Clustering
Creates a new column in the current data table and fills it with cluster
values.
Note: If you save the modal clustering values first and then save the density grid, the grid
table also contains the cluster values. The cluster values are useful for coloring and
marking points in plots.
Saves the density estimates and the quantiles associated with them in a
new data table. The grid data can be used to visualize the density in other ways, such as
with the Scatterplot 3D or the Contour Plot platforms.
Save Density Grid
Figure 5.17 Example of a Mesh Plot
Diagnostics Plots
The Plot Residuals option creates residual plots and other plots to diagnose the model fit. The
following plots are available:
Residual by Predicted Plot is a plot of the residuals vs. the predicted values. A histogram of
the residuals is also created.
Actual by Predicted Plot
is a plot of the actual values vs. the predicted values.
Residual by Row Plot is a plot of the residual values vs. the row number.
Residual by X Plot
is a plot of the residual values vs. the X variable.
Residual Normal Quantile Plot
is a Normal quantile plot of the residuals.
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Additional Examples of the Bivariate Platform
Chapter 5
Basic Analysis
Additional Examples of the Bivariate Platform
This section contains additional examples using the fitting commands in the Bivariate
platform.
Example of the Fit Special Command
To transform Y as log and X as square root, proceed as follows:
1. Select Help > Sample Data Library and open SAT.jmp.
2. Select Analyze > Fit Y by X.
3. Select 2004 Verbal and click Y, Response.
4. Select % Taking (2004) and click X, Factor.
5. Click OK.
Figure 5.18 Example of SAT Scores by Percent Taking
6. From the red triangle menu for Bivariate Fit, select Fit Special. The Specify Transformation
or Constraint window appears. For a description of this window, see “Fit Special” on
page 113.
Figure 5.19 The Specify Transformation or Constraint Window
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Bivariate Analysis
Additional Examples of the Bivariate Platform
127
7. Within Y Transformation, select Natural Logarithm: log(y).
8. Within X Transformation, select Square Root: sqrt(x).
9. Click OK.
Figure 5.20 Example of Fit Special Report
Figure 5.20 shows the fitted line plotted on the original scale. The model appears to fit the data
well, as the plotted line goes through the cloud of points.
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Example Using the Fit Orthogonal Command
This example involves two parts. First, standardize the variables using the Distribution
platform. Then, use the standardized variables to fit the orthogonal model.
Standardize the Variables
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Distribution.
3. Select height and weight and click Y, Columns.
4. Click OK.
5. Hold down the Ctrl key. On the red triangle menu next to height, select Save >
Standardized.
Holding down the Ctrl key broadcasts the operation to all variables in the report window.
Notice that in the Big Class.jmp sample data table, two new columns have been added.
6. Close the Distribution report window.
Use the Standardized Variables to Fit the Orthogonal Model
1. From the Big Class.jmp sample data table, select Analyze > Fit Y by X.
2. Select Std weight and click Y, Response.
3. Select Std height and click X, Factor.
4. Click OK.
5. From the red triangle menu, select Fit Line.
6. From the red triangle menu, select Fit Orthogonal. Then select each of the following:
– Equal Variances
– Fit X to Y
– Specified Variance Ratio and type 0.2.
– Specified Variance Ratio and type 5.
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Additional Examples of the Bivariate Platform
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Figure 5.21 Example of Orthogonal Fitting Options
Fit X to Y
Fit Line
The scatterplot in Figure 5.21 shows the standardized height and weight values with various
line fits that illustrate the behavior of the orthogonal variance ratio selections. The standard
linear regression (Fit Line) occurs when the variance of the X variable is considered to be very
small. Fit X to Y is the opposite extreme, when the variation of the Y variable is ignored. All
other lines fall between these two extremes and shift as the variance ratio changes. As the
variance ratio increases, the variation in the Y response dominates and the slope of the fitted
line shifts closer to the Y by X fit. Likewise, when you decrease the ratio, the slope of the line
shifts closer to the X by Y fit.
Example Using the Fit Robust Command
The data in the Weight Measurements.jmp sample data table shows the height and weight
measurements taken by 40 students.
1. Select Help > Sample Data Library and open Weight Measurements.jmp.
2. Select Analyze > Fit Y by X.
3. Select weight and click Y, Response.
4. Select height and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Fit Line.
7. From the red triangle menu, select Robust > Fit Robust.
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Figure 5.22 Example of Robust Fit
If you look at the standard Analysis of Variance report, you might wrongly conclude that
height and weight do not have a linear relationship, since the p-value is 0.1203. However,
when you look at the Robust Fit report, you would probably conclude that they do have a
linear relationship, because the p-value there is 0.0489. It appears that some of the
measurements are unusually low, perhaps due to incorrect user input. These measurements
were unduly influencing the analysis.
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Additional Examples of the Bivariate Platform
131
Example of Group By Using Density Ellipses
This example uses the Hot Dogs.jmp sample data table. The Type column identifies three
different types of hot dogs: beef, meat, or poultry. You want to group the three types of hot
dogs according to their cost variables.
1. Select Help > Sample Data Library and open Hot Dogs.jmp.
2. Select Analyze > Fit Y by X.
3. Select $/oz and click Y, Response.
4. Select $/lb Protein and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Group By.
7. From the list, select Type.
8. Click OK. If you look at the Group By option again, you see it has a check mark next to it.
9. From the red triangle menu, select Density Ellipse > 0.90.
To color the points according to Type, proceed as follows:
10. Right-click on the scatterplot and select Row Legend.
11. Select Type in the column list and click OK.
Figure 5.23 Example of Group By
The ellipses in Figure 5.23 show clearly how the different types of hot dogs cluster with
respect to the cost variables.
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Example of Group By Using Regression Lines
Another use for grouped regression is overlaying lines to compare slopes of different groups.
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Fit Y by X.
3. Select weight and click Y, Response.
4. Select height and click X, Factor.
5. Click OK.
To create the example on the left in Figure 5.24:
6. Select Fit Line from the red triangle menu.
To create the example on the right in Figure 5.24:
7. From the Linear Fit menu, select Remove Fit.
8. From the red triangle menu, select Group By.
9. From the list, select sex.
10. Click OK.
11. Select Fit Line from the red triangle menu.
Figure 5.24 Example of Regression Analysis for Whole Sample and Grouped Sample
The scatterplot to the left in Figure 5.24 has a single regression line that relates weight to
height. The scatterplot to the right shows separate regression lines for males and females.
Chapter 5
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Bivariate Analysis
Statistical Details for the Bivariate Platform
133
Statistical Details for the Bivariate Platform
This section contains statistical details for selected commands and reports.
Fit Line
The Fit Line command finds the parameters β 0 and β 1 for the straight line that fits the points
to minimize the residual sum of squares. The model for the ith row is written
yi = β0 + β1 xi + εi .
A polynomial of degree 2 is a parabola; a polynomial of degree 3 is a cubic curve. For degree k,
the model for the ith observation is as follows:
k
yi =

j
βj xi + εi
j=0
Fit Spline
The cubic spline method uses a set of third-degree polynomials spliced together such that the
resulting curve is continuous and smooth at the splices (knot points). The estimation is done
by minimizing an objective function that is a combination of the sum of squares error and a
penalty for curvature integrated over the curve extent. See the paper by Reinsch (1967) or the
text by Eubank (1988) for a description of this method.
Fit Orthogonal
Standard least square fitting assumes that the X variable is fixed and the Y variable is a
function of X plus error. If there is random variation in the measurement of X, you should fit a
line that minimizes the sum of the squared perpendicular differences. See Figure 5.25.
However, the perpendicular distance depends on how X and Y are scaled, and the scaling for
the perpendicular is reserved as a statistical issue, not a graphical one.
Figure 5.25 Line Perpendicular to the Line of Fit
y distance
orthogonal
distance
x distance
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The fit requires that you specify the ratio of the variance of the error in Y to the error in X. This
is the variance of the error, not the variance of the sample points, so you must choose carefully.
The ratio ( σ 2y ) ⁄ ( σ 2x ) is infinite in standard least squares because σ 2x is zero. If you do an
orthogonal fit with a large error ratio, the fitted line approaches the standard least squares line
of fit. If you specify a ratio of zero, the fit is equivalent to the regression of X on Y, instead of Y
on X.
The most common use of this technique is in comparing two measurement systems that both
have errors in measuring the same value. Thus, the Y response error and the X measurement
error are both the same type of measurement error. Where do you get the measurement error
variances? You cannot get them from bivariate data because you cannot tell which
measurement system produces what proportion of the error. So, you either must blindly
assume some ratio like 1, or you must rely on separate repeated measurements of the same
unit by the two measurement systems.
An advantage to this approach is that the computations give you predicted values for both Y
and X; the predicted values are the point on the line that is closest to the data point, where
closeness is relative to the variance ratio.
Confidence limits are calculated as described in Tan and Iglewicz (1999).
Summary of Fit Report
RSquare
Using quantities from the corresponding analysis of variance table, the RSquare for any
continuous response fit is calculated as follows:
Sum of Squares for Model
-----------------------------------------------------------------------Sum of Squares for C. Total
RSquare Adj
The RSquare Adj is a ratio of mean squares instead of sums of squares and is calculated as
follows:
Mean Square for Error
1 – -----------------------------------------------------------------Mean Square for C. Total
The mean square for Error is in the Analysis of Variance report. See Figure 5.12. You can
compute the mean square for C. Total as the Sum of Squares for C. Total divided by its
respective degrees of freedom.
Chapter 5
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Bivariate Analysis
Statistical Details for the Bivariate Platform
135
Lack of Fit Report
Pure Error DF
For the Pure Error DF, consider the multiple instances in the Big Class.jmp sample data table
where more than one subject has the same value of height. In general, if there are g groups
having multiple rows with identical values for each effect, the pooled DF, denoted DFp, is as
follows:
g
DF p =

( ni – 1 )
i=1
ni is the number of subjects in the ith group.
Pure Error SS
For the Pure Error SS, in general, if there are g groups having multiple rows with the same x
value, the pooled SS, denoted SSp, is written as follows:
g
SS p =

SS i
i=1
where SSi is the sum of squares for the ith group corrected for its mean.
Max RSq
Because Pure Error is invariant to the form of the model and is the minimum possible variance,
Max RSq is calculated as follows:
SS ( Pure error )
1 – --------------------------------------------------------------------SS ( Total for whole model )
Parameter Estimates Report
Std Beta
Std Beta is calculated as follows:
β̂ ( s x ⁄ s y )
where β̂ is the estimated parameter, sx and sy are the standard deviations of the X and Y
variables.
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Design Std Error
Design Std Error is calculated as the standard error of the parameter estimate divided by the
RMSE.
Smoothing Fit Reports
R-Square is equal to 1-(SSE/C.Total SS), where C.Total SS is available in the Fit Line ANOVA
report.
Correlation Report
The Pearson correlation coefficient is denoted r, and is computed as follows:
2
s xy
 wi ( xi – xi ) ( yi – yi )
2
r xy = --------------- where s xy = ----------------------------------------------------df
2 2
sx sy
Where w i is either the weight of the ith observation if a weight column is specified, or 1 if no
weight column is assigned.
Chapter 6
Oneway Analysis
Examine Relationships between a Continuous Y and a Categorical X
Variable
Using the Oneway or Fit Y by X platform, you can explore how the distribution of a
continuous Y variable differs across groups defined by a single categorical X variable. For
example, you might want to find out how different categories of the same type of drug (X)
affect patient pain levels on a numbered scale (Y).
The Oneway platform is the continuous by nominal or ordinal personality of the Fit Y by X
platform. The analysis results appear in a plot, and you can interactively add additional
analyses, such as the following:
•
a one-way analysis of variance to fit means and to test that they are equal
•
nonparametric tests
•
a test for homogeneity of variance
•
multiple-comparison tests on means, with means comparison circles
•
outlier box plots overlaid on each group
•
power details for the one-way layout
Figure 6.1 Oneway Analysis
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Overview of Oneway Analysis
Chapter 6
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Overview of Oneway Analysis
A one-way analysis of variance tests for differences between group means. The total
variability in the response is partitioned into two parts: within-group variability and
between-group variability. If the between-group variability is large relative to the
within-group variability, then the differences between the group means are considered to be
significant.
Example of Oneway Analysis
This example uses the Analgesics.jmp sample data table. Thirty-three subjects were
administered three different types of analgesics (A, B, and C). The subjects were asked to rate
their pain levels on a sliding scale. You want to find out if the means for A, B, and C are
significantly different.
1. Select Help > Sample Data Library and open Analgesics.jmp.
2. Select Analyze > Fit Y by X.
3. Select pain and click Y, Response.
4. Select drug and click X, Factor.
5. Click OK.
Figure 6.2 Example of Oneway Analysis
You notice that one drug (A) has consistently lower scores than the other drugs. You also
notice that the x-axis ticks are unequally spaced. The length between the ticks is proportional
to the number of scores (observations) for each drug.
Perform an analysis of variance on the data.
6. From the red triangle menu for Oneway Analysis, select Means/Anova.
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Note: If the X factor has only two levels, the Means/Anova option appears as
Means/Anova/Pooled t, and adds a pooled t-test report to the report window.
Figure 6.3 Example of the Means/Anova Option
Note the following observations:
•
Mean diamonds representing confidence intervals appear.
– The line near the center of each diamond represents the group mean. At a glance, you
can see that the mean for each drug looks significantly different.
– The vertical span of each diamond represents the 95% confidence interval for the mean
of each group.
See “Mean Diamonds and X-Axis Proportional” on page 150.
•
The Summary of Fit table provides overall summary information about the analysis.
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•
The Analysis of Variance report shows the standard ANOVA information. You notice that
the Prob > F (the p-value) is 0.0053, which supports your visual conclusion that there are
significant differences between the drugs.
•
The Means for Oneway Anova report shows the mean, sample size, and standard error for
each level of the categorical factor.
Launch the Oneway Platform
You can perform a Oneway analysis using either the Fit Y by X platform or the Oneway
platform. The two approaches are equivalent.
•
To launch the Fit Y by X platform, select Analyze > Fit Y by X.
or
•
To launch the Oneway platform, from the JMP Starter window, click on the Basic category
and click Oneway.
Figure 6.4 The Oneway Launch Window
For more information about this launch window, see “Introduction to Fit Y by X” chapter on
page 95.
After you click OK, the Oneway report window appears. See “The Oneway Plot” on page 140.
The Oneway Plot
The Oneway plot shows the response points for each X factor value. You can compare the
distribution of the response across the levels of the X factor. The distinct values of X are
sometimes called levels.
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Replace variables in the plot in one of two ways: swap existing variables by dragging and
dropping a variable from one axis to the other axis; or, click on a variable in the Columns
panel of the associated data table and drag it onto an axis.
You can add reports, additional plots, and tests to the report window using the options in the
red triangle menu for Oneway Analysis. See “Oneway Platform Options” on page 141.
To produce the plot shown in Figure 6.5, follow the instructions in “Example of Oneway
Analysis” on page 138.
Figure 6.5 The Oneway Plot
Note: Any rows that are excluded in the data table are also hidden in the Oneway plot.
Oneway Platform Options
Note: The Fit Group menu appears if you have specified multiple Y or X variables. Menu
options allow you to arrange reports or order them by RSquare. See the Standard Least
Squares chapter in the Fitting Linear Models book for more information.
When you select a platform option, objects might be added to the plot, and a report is added
to the report window.
Table 6.1 Examples of Options and Elements
Platform Option
Object Added to Plot
Report Added to Report Window
Quantiles
Box plots
Quantiles report
Means/Anova
Mean diamonds
Oneway ANOVA reports
Means and Std Dev
Mean lines, error bars, and
standard deviation lines
Means and Std Deviations
report
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Table 6.1 Examples of Options and Elements (Continued)
Platform Option
Object Added to Plot
Report Added to Report Window
Compare Means
Comparison circles
Means Comparison reports
(except Nonparametric
Multiple Comparisons option)
The red triangle menu for Oneway Analysis provides the following options. Some options
might not appear unless specific conditions are met.
Quantiles Lists the following quantiles for each group:
– 0% (Minimum)
– 10%
– 25%
– 50% (Median)
– 75%
– 90%
– 100% (Maximum)
Activates Box Plots from the Display Options menu. See “Quantiles” on page 146.
Means/Anova Fits means for each group and performs a one-way analysis of variance to test
if there are differences among the means. See “Means/Anova and Means/Anova/Pooled t”
on page 147.
If the X factor has two levels, the menu option changes to Means/Anova/Pooled t.
Means and Std Dev Gives summary statistics for each group. The standard errors for the
means use individual group standard deviations rather than the pooled estimate of the
standard deviation.
The plot now contains mean lines, error bars, and standard deviation lines. For a brief
description of these elements, see “Display Options” on page 144. For more details about
these elements, see “Mean Lines, Error Bars, and Standard Deviation Lines” on page 151.
Produces a t-test report assuming that the variances are not equal. See “The t-test
Report” on page 148.
t test
This option appears only if the X factor has two levels.
Analysis of Means Methods Provides five commands for performing Analysis of Means
(ANOM) procedures. There are commands for comparing means, variances, and ranges.
See “Analysis of Means Methods” on page 152.
Compare Means Provides multiple-comparison methods for comparing sets of group means.
See “Compare Means” on page 155.
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Nonparametric Provides nonparametric comparisons of group locations. See
“Nonparametric Tests” on page 161.
Performs four tests for equality of group variances. Also gives the Welch
test, which is an ANOVA test for comparing means when the variances within groups are
not equal. See “Unequal Variances” on page 167.
Unequal Variances
Equivalence Test Tests that a difference is less than a threshold value. See “Equivalence Test”
on page 170.
Provides two methods for reducing the influence of outliers on your data. See
“Robust” on page 170.
Robust
Power Provides calculations of statistical power and other details about a given hypothesis
test. See “Power” on page 171.
The Power Details window and reports also appear within the Fit Model platform. For
further discussion and examples of power calculations, see the Statistical Details appendix
in the Fitting Linear Models book.
Set α Level
You can select an option from the most common alpha levels or specify any level
with the Other selection. Changing the alpha level results in the following actions:
– recalculates confidence limits
– adjusts the mean diamonds on the plot (if they are showing)
– modifies the upper and lower confidence level values in reports
– changes the critical number and comparison circles for all Compare Means reports
– changes the critical number for all Nonparametric Multiple Comparison reports
Normal Quantile Plot Provides the following options for plotting the quantiles of the data in
each group:
– Plot Actual by Quantile generates a quantile plot with the response variable on the
y-axis and quantiles on the x-axis. The plot shows quantiles computed within each
level of the categorical X factor.
– Plot Quantile by Actual reverses the x- and y-axes.
– Line of Fit draws straight diagonal reference lines on the plot for each level of the X
variable. This option is available only once you have created a plot (Actual by Quantile
or Quantile by Actual).
Plots the cumulative distribution function for all of the groups in the Oneway
report. See “CDF Plot” on page 173.
CDF Plot
Densities Compares densities across groups. See “Densities” on page 173.
Matching Column Specify a matching variable to perform a matching model analysis. Use
this option when the data in your Oneway analysis comes from matched (paired) data,
such as when observations in different groups come from the same subject.
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The plot now contains matching lines that connect the matching points. See “Matching
Column” on page 173.
Save Saves the following quantities as new columns in the current data table:
– Save Residuals saves values computed as the response variable minus the mean of the
response variable within each level of the factor variable.
– Save Standardized saves standardized values of the response variable computed
within each level of the factor variable. This is the centered response divided by the
standard deviation within each level.
– Save Normal Quantiles saves normal quantile values computed within each level of the
categorical factor variable.
– Save Predicted saves the predicted mean of the response variable for each level of the
factor variable.
Display Options
Adds or removes elements from the plot. See “Display Options” on
page 144.
See the JMP Reports chapter in the Using JMP book for more information about the following
options:
Contains options that enable you to repeat or relaunch the analysis. In platforms that
support the feature, the Automatic Recalc option immediately reflects the changes that
you make to the data table in the corresponding report window.
Redo
Contains options that enable you to save a script that reproduces the report to
several destinations.
Save Script
Save By-Group Script Contains options that enable you to save a script that reproduces the
platform report for all levels of a By variable to several destinations. Available only when a
By variable is specified in the launch window.
Display Options
Using Display Options, you can add or remove elements from a plot. Some options might not
appear unless they are relevant.
All Graphs Shows or hides all graphs.
Points
Shows or hides data points on the plot.
Box Plots Shows or hides outlier box plots for each group. For an example, see “Conduct the
Oneway Analysis” on page 200.
Draws a horizontal line through the mean of each group proportional to its
x-axis. The top and bottom points of the mean diamond show the upper and lower 95%
confidence points for each group. See “Mean Diamonds and X-Axis Proportional” on
page 150.
Mean Diamonds
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Draws a line at the mean of each group. See “Mean Lines, Error Bars, and
Standard Deviation Lines” on page 151.
Mean Lines
Mean CI Lines Draws lines at the upper and lower 95% confidence levels for each group.
Mean Error Bars Identifies the mean of each group and shows error bars one standard error
above and below the mean. See “Mean Lines, Error Bars, and Standard Deviation Lines”
on page 151.
Grand Mean Draws the overall mean of the Y variable on the plot.
Std Dev Lines Shows lines one standard deviation above and below the mean of each group.
See “Mean Lines, Error Bars, and Standard Deviation Lines” on page 151.
Shows or hides comparison circles. This option is available only when
one of the Compare Means options is selected. See “Comparison Circles” on page 202. For
an example, see “Conduct the Oneway Analysis” on page 200.
Comparison Circles
Connect Means
Connects the group means with a straight line.
Mean of Means Draws a line at the mean of the group means.
X-Axis proportional Makes the spacing on the x-axis proportional to the sample size of each
level. See “Mean Diamonds and X-Axis Proportional” on page 150.
Points Spread Spreads points over the width of the interval
Adds small spaces between points that overlay on the same y value. The
horizontal adjustment of points varies from 0.375 to 0.625 with a 4*(Uniform-0.5)5
distribution.
Points Jittered
Matching Lines (Only appears when the Matching Column option is selected.) Connects
matching points.
(Only appears when the Matching Column option is selected.) Draws
dotted lines to connect cell means from missing cells in the table. The values used as the
endpoints of the lines are obtained using a two-way ANOVA model.
Matching Dotted Lines
Histograms Draws side-by-side histograms to the right of the original plot.
Robust Mean Lines (Appears only when a Robust option is selected.) Draws a line at the
robust mean of each group.
Legend Displays a legend for the Normal Quantile Plot, CDF Plot, and Densities options.
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Quantiles
The Quantiles report lists selected percentiles for each level of the X factor variable. The
median is the 50th percentile, and the 25th and 75th percentiles are called the quartiles.
The Quantiles option adds the following elements to the plot:
•
the grand mean representing the overall mean of the Y variable
•
outlier box plots summarizing the distribution of points at each factor level
Figure 6.6 Outlier Box Plot and Grand Mean
outlier box plot
grand mean
Note: To hide these elements, click the red triangle next to Oneway Analysis and select
Display Options > Box Plots or Grand Mean.
Outlier Box Plots
The outlier box plot is a graphical summary of the distribution of data. Note the following
aspects about outlier box plots (see Figure 6.7):
•
The horizontal line within the box represents the median sample value.
•
The ends of the box represent the 75th and 25th quantiles, also expressed as the 3rd and 1st
quartile, respectively.
•
The difference between the 1st and 3rd quartiles is called the interquartile range.
•
Each box has lines, sometimes called whiskers, that extend from each end. The whiskers
extend from the ends of the box to the outermost data point that falls within the distances
computed as follows:
3rd quartile + 1.5*(interquartile range)
1st quartile - 1.5*(interquartile range)
If the data points do not reach the computed ranges, then the whiskers are determined by
the upper and lower data point values (not including outliers).
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Figure 6.7 Examples of Outlier Box Plots
whisker
median
sample
value
75% quantile
or 3rd quartile
25% quantile
or 1st quartile
Means/Anova and Means/Anova/Pooled t
The Means/Anova option performs an analysis of variance. If the X factor contains exactly two
levels, this option appears as Means/Anova/Pooled t. In addition to the other reports, a t-test
report assuming pooled (or equal) variances appears.
Mean diamonds are added to the Oneway plot See “Display Options” on page 144 and “Mean
Diamonds and X-Axis Proportional” on page 150.
Reports See “The Summary of Fit Report” on page 147, “The Analysis of Variance Report”
on page 149, “The Means for Oneway Anova Report” on page 150, “The t-test Report” on
page 148, and “The Block Means Report” on page 150.
– The t-test report appears only if the Means/Anova/Pooled t option is selected.
– The Block Means report appears only if you have specified a Block variable in the
launch window.
The Summary of Fit Report
The Summary of Fit report shows a summary for a one-way analysis of variance.
Rsquare Measures the proportion of the variation accounted for by fitting means to each
factor level. The remaining variation is attributed to random error. The R2 value is 1 if
fitting the group means account for all the variation with no error. An R2 of 0 indicates that
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the fit serves no better as a prediction model than the overall response mean. For more
information, see “Summary of Fit Report” on page 204.
R2 is also called the coefficient of determination.
Note: A low RSquare value suggests that there may be variables not in the model that
account for the unexplained variation. However, if your data are subject to a large amount
of inherent variation, even a useful ANOVA model may have a low RSquare value. Read
the literature in your research area to learn about typical RSquare values.
Adjusts R2 to make it more comparable over models with different numbers of
parameters by using the degrees of freedom in its computation. For more information, see
“Summary of Fit Report” on page 204.
Adj Rsquare
Root Mean Square Error Estimates the standard deviation of the random error. It is the
square root of the mean square for Error found in the Analysis of Variance report.
Mean of Response Overall mean (arithmetic average) of the response variable.
Observations (or Sum Wgts) Number of observations used in estimating the fit. If weights are
used, this is the sum of the weights. See “Summary of Fit Report” on page 204.
The t-test Report
Note: This option is applicable only for the Means/Anova/Pooled t option.
There are two types of t-Tests:
•
Equal variances. If you select the Means/Anova/Pooled t option, a t-Test report appears.
This t-Test assumes equal variances.
•
Unequal variances. If you select the t-Test option from the red triangle menu, a t-Test
report appears. This t-Test assumes unequal variances.
The t-test report contains the following columns:
Shows the sampling distribution of the difference in the means, assuming the null
hypothesis is true. The vertical red line is the actual difference in the means. The shaded
areas correspond to the p-values.
t Test plot
Difference Shows the estimated difference between the two X levels. In the plots, the
Difference value appears as a red line that compares the two levels.
Std Err Dif
Shows the standard error of the difference.
Upper CL Dif
Shows the upper confidence limit for the difference.
Lower CL Dif
Shows the lower confidence limit for the difference.
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Confidence Shows the level of confidence (1-alpha). To change the level of confidence, select
a new alpha level from the Set α Level command from the platform red triangle menu.
t Ratio Value of the t-statistic.
DF
The degrees of freedom used in the t-test.
Prob > |t| The p-value associated with a two-tailed test.
Prob > t The p-value associated with a lower-tailed test.
Prob < t The p-value associated with an upper-tailed test.
The Analysis of Variance Report
The Analysis of Variance report partitions the total variation of a sample into two components.
The ratio of the two mean squares forms the F ratio. If the probability associated with the F
ratio is small, then the model is a better fit statistically than the overall response mean.
Note: If you specified a Block column, then the Analysis of Variance report includes the Block
variable.
Source Lists the three sources of variation, which are the model source, Error, and C. Total
(corrected total).
DF
Records an associated degrees of freedom (DF for short) for each source of variation:
– The degrees of freedom for C. Total are N - 1, where N is the total number of
observations used in the analysis.
– If the X factor has k levels, then the model has k - 1 degrees of freedom.
The Error degrees of freedom is the difference between the C. Total degrees of freedom and
the Model degrees of freedom (in other words, N - k).
Sum of Squares Records a sum of squares (SS for short) for each source of variation:
– The total (C. Total) sum of squares of each response from the overall response mean.
The C. Total sum of squares is the base model used for comparison with all other
models.
– The sum of squared distances from each point to its respective group mean. This is the
remaining unexplained Error (residual) SS after fitting the analysis of variance model.
The total SS minus the error SS gives the sum of squares attributed to the model. This tells
you how much of the total variation is explained by the model.
Mean Square Is a sum of squares divided by its associated degrees of freedom:
– The Model mean square estimates the variance of the error, but only under the
hypothesis that the group means are equal.
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– The Error mean square estimates the variance of the error term independently of the
model mean square and is unconditioned by any model hypothesis.
F Ratio Model mean square divided by the error mean square. If the hypothesis that the
group means are equal (there is no real difference between them) is true, then both the
mean square for error and the mean square for model estimate the error variance. Their
ratio has an F distribution. If the analysis of variance model results in a significant
reduction of variation from the total, the F ratio is higher than expected.
Prob>F Probability of obtaining (by chance alone) an F value greater than the one calculated
if, in reality, there is no difference in the population group means. Observed significance
probabilities of 0.05 or less are often considered evidence that there are differences in the
group means.
The Means for Oneway Anova Report
The Means for Oneway Anova report summarizes response information for each level of the
nominal or ordinal factor.
Level
Lists the levels of the X variable.
Number Lists the number of observations in each group.
Mean Lists the mean of each group.
Std Error Lists the estimates of the standard deviations for the group means. This standard
error is estimated assuming that the variance of the response is the same in each level. It is
the root mean square error found in the Summary of Fit report divided by the square root
of the number of values used to compute the group mean.
Lower 95% and Upper 95% Lists the lower and upper 95% confidence interval for the group
means.
The Block Means Report
If you have specified a Block variable on the launch window, the Means/Anova and
Means/Anova/Pooled t commands produce a Block Means report. This report shows the
means for each block and the number of observations in each block.
Mean Diamonds and X-Axis Proportional
A mean diamond illustrates a sample mean and confidence interval.
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Figure 6.8 Examples of Mean Diamonds and X-Axis Proportional Options
overlap marks
95%
confidence
interval
x-axis proportional
x-axis not proportional
group mean
Note the following observations:
•
The top and bottom of each diamond represent the (1-alpha)x100 confidence interval for
each group. The confidence interval computation assumes that the variances are equal
across observations. Therefore, the height of the diamond is proportional to the reciprocal
of the square root of the number of observations in the group.
•
If the X-Axis proportional option is selected, the horizontal extent of each group along the
x-axis (the horizontal size of the diamond) is proportional to the sample size for each level
of the X variable. Therefore, the narrower diamonds are usually taller, because fewer data
points results in a wider confidence interval.
•
The mean line across the middle of each diamond represents the group mean.
•
Overlap marks appear as lines above and below the group mean. For groups with equal
sample sizes, overlapping marks indicate that the two group means are not significantly
different at the given confidence level. Overlap marks are computed as group
mean ± ( 2 ) ⁄ 2 × CI ⁄ 2 . Overlap marks in one diamond that are closer to the mean of
another diamond than that diamond’s overlap marks indicate that those two groups are
not different at the given confidence level.
•
The mean diamonds automatically appear when you select the Means/Anova/Pooled t or
Means/Anova option from the platform menu. However, you can show or hide them at
any time by selecting Display Options > Mean Diamonds from the red triangle menu.
Mean Lines, Error Bars, and Standard Deviation Lines
Show mean lines by selecting Display Options > Mean Lines. Mean lines indicate the mean of
the response for each level of the X variable.
Mean error bars and standard deviation lines appear when you select the Means and Std Dev
option from the red triangle menu. See Figure 6.9. To turn each option on or off singly, select
Display Options > Mean Error Bars or Std Dev Lines.
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Figure 6.9 Mean Lines, Mean Error Bars, and Std Dev Lines
std dev lines
mean error bar
mean line
Analysis of Means Methods
Analysis of means (ANOM) methods compare means and variances and other measures of
location and scale across several groups. You might want to use these methods under these
circumstances:
•
to test whether any of the group means are statistically different from the overall (sample)
mean
•
to test whether any of the group standard deviations are statistically different from the
root mean square error (RMSE)
•
to test whether any of the group ranges are statistically different from the overall mean of
the ranges
Note: Within the Contingency platform, you can use the Analysis of Means for Proportions
when the response has two categories. For details, see the “Contingency Analysis” chapter on
page 209.
For a description of ANOM methods and to see how JMP implements ANOM, see the book by
Nelson et al. (2005).
Analysis of Means for Location
You can test whether groups have a common mean or center value using the following
options:
•
ANOM
•
ANOM with Transformed Ranks
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ANOM
Use ANOM to compare group means to the overall mean. This method assumes that your
data are approximately normally distributed. See “Example of an Analysis of Means Chart”
on page 174.
ANOM with Transformed Ranks
This is the nonparametric version of the ANOM analysis. Use this method if your data is clearly
non-normal and cannot be transformed to normality. ANOM with Transformed Ranks
compares each group’s mean transformed rank to the overall mean transformed rank. The
ANOM test involves applying the usual ANOM procedure and critical values to the
transformed observations.
Transformed Ranks
Suppose that there are n observations. The transformed observations are computed as follows:
•
Rank all observations from smallest to largest, accounting for ties. For tied observations,
assign each one the average of the block of ranks that they share.
•
Denote the ranks by R1, R2, ..., Rn.
•
The transformed rank corresponding to the ith observations is:
Ri
Transformed R i = Normal Quantile  ---------------- + 0.5
 2n + 1
The ANOM procedure is applied to the values Transformed Ri. Since the ranks have a uniform
distribution, the transformed ranks have a folded normal distribution. For details, see Nelson
et al. (2005).
Analysis of Means for Scale
You can test for homogeneity of variation within groups using the following options:
•
ANOM for Variances
•
ANOM for Variances with Levene (ADM)
•
ANOM for Ranges
ANOM for Variances
Use this method to compare group standard deviations (or variances) to the root mean square
error (or mean square error). This method assumes that your data is approximately normally
distributed. To use this method, each group must have at least four observations. For details
about the ANOM for Variances test, see Wludyka and Nelson (1997) and Nelson et al. (2005).
For an example, see “Example of an Analysis of Means for Variances Chart” on page 175.
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ANOM for Variances with Levene (ADM)
This method provides a robust test that compares the group means of the absolute deviations
from the median (ADM) to the overall mean ADM. Use ANOM for Variances with Levene
(ADM) if you suspect that your data is non-normal and cannot be transformed to normality.
ANOM for Variances with Levene (ADM) is a nonparametric analog of the ANOM for
Variances analysis. For details about the ANOM for Variances with Levene (ADM) test, see
Levene (1960) or Brown and Forsythe (1974).
ANOM for Ranges
Use this test to compare group ranges to the mean of the group ranges. This is a test for scale
differences based on the range as the measure of spread. For details, see Wheeler (2003).
Note: ANOM for Ranges is available only for balanced designs and specific group sizes. See
“Restrictions for ANOM for Ranges Test” on page 154.
Restrictions for ANOM for Ranges Test
Unlike the other ANOM decision limits, the decision limits for the ANOM for Ranges chart
uses only tabled critical values. For this reason, ANOM for Ranges is available only for the
following:
•
groups of equal sizes
•
groups specifically of the following sizes: 2–10, 12, 15, and 20
•
number of groups between 2 and 30
•
alpha levels of 0.10, 0.05, and 0.01
Analysis of Means Charts
Each Analysis of Means Methods option adds a chart to the report window that shows the
following:
•
an upper decision limit (UDL)
•
a lower decision limit (LDL)
•
a horizontal (center) line that falls between the decision limits and is positioned as follows:
– ANOM: the overall mean
– ANOM with Transformed Ranks: the overall mean of the transformed ranks
– ANOM for Variances: the root mean square error (or MSE when in variance scale)
– ANOM for Variances with Levene (ADM): the overall absolute deviation from the
mean
– ANOM for Ranges: the mean of the group ranges
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If a group’s plotted statistic falls outside of the decision limits, then the test indicates that there
is a statistical difference between that group’s statistic and the overall average of the statistic
for all the groups.
Analysis of Means Options
Each Analysis of Means Methods option adds an Analysis of Means red triangle menu to the
report window.
Select an option from the most common alpha levels or specify any level with
the Other selection. Changing the alpha level modifies the upper and lower decision limits.
Set Alpha Level
Note: For ANOM for Ranges, only the selections 0.10, 0.05, and 0.01 are available.
Show Summary Report The reports are based on the Analysis of Means method:
– For ANOM, creates a report showing group means and decision limits.
– For ANOM with Transformed Ranks, creates a report showing group mean
transformed ranks and decision limits.
– For ANOM for Variances, creates a report showing group standard deviations (or
variances) and decision limits.
– For ANOM for Variances with Levene (ADM), creates a report showing group mean
ADMs and decision limits.
– For ANOM for Ranges, creates a report showing group ranges and decision limits.
Graph in Variance Scale (Only for ANOM for Variances) Changes the scale of the y-axis from
standard deviations to variances.
Display Options
Display options include the following:
– Show Decision Limits shows or hides decision limit lines.
– Show Decision Limit Shading shows or hides decision limit shading.
– Show Center Line shows or hides the center line statistic.
– Point Options: Show Needles shows the needles. This is the default option. Show
Connected Points shows a line connecting the means for each group. Show Only Points
shows only the points representing the means for each group.
Compare Means
Note: Another method for comparing means is ANOM. See “Analysis of Means Methods” on
page 152.
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Use the Compare Means options to perform multiple comparisons of group means. All of
these methods use pooled variance estimates for the means. Each Compare Means option
adds comparison circles next to the plot and specific reports to the report window. For details
about comparison circles, see “Using Comparison Circles” on page 157.
Option
Description
Nonparametric Menu Option
Each Pair,
Student’s t
Computes individual pairwise
comparisons using Student’s t-tests.
If you make many pairwise tests,
there is no protection across the
inferences. Therefore, the
alpha-size (Type I error rate) across
the hypothesis tests is higher than
that for individual tests. See “Each
Pair, Student’s t” on page 158.
Nonparametric > Nonparametric
Multiple Comparisons > Wilcoxon
Each Pair
All Pairs, Tukey
HSD
Shows a test that is sized for all
differences among the means. This
is the Tukey or Tukey-Kramer HSD
(honestly significant difference)
test. (Tukey 1953, Kramer 1956).
This test is an exact alpha-level test
if the sample sizes are the same,
and conservative if the sample sizes
are different (Hayter 1984). See “All
Pairs, Tukey HSD” on page 158.
Nonparametric > Nonparametric
Multiple Comparisons >
Steel-Dwass All Pairs
With Best, Hsu
MCB
Tests whether the means are less
than the unknown maximum or
greater than the unknown
minimum. This is the Hsu MCB test
(Hsu, 1996 and Hsu, 1981). See
“With Best, Hsu MCB” on page 159.
none
With Control,
Dunnett’s
Tests whether the means are
different from the mean of a control
group. This is Dunnett’s test
(Dunnett 1955). See “With Control,
Dunnett’s” on page 160.
Nonparametric > Nonparametric
Multiple Comparisons > Steel With
Control
Note: If you have specified a Block column, then the multiple comparison methods are
performed on data that has been adjusted for the Block means.
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Related Information
•
“Example Contrasting All of the Compare Means Tests” on page 182
Using Comparison Circles
Each multiple comparison test begins with a comparison circles plot, which is a visual
representation of group mean comparisons. Figure 6.10 shows the comparison circles for the
All Pairs, Tukey HSD method. Other comparison tests lengthen or shorten the radii of the
circles.
Figure 6.10 Visual Comparison of Group Means
Compare each pair of group means visually by examining the intersection of the comparison
circles. The outside angle of intersection tells you whether the group means are significantly
different. See Figure 6.11.
•
Circles for means that are significantly different either do not intersect, or intersect
slightly, so that the outside angle of intersection is less than 90 degrees.
•
If the circles intersect by an angle of more than 90 degrees, or if they are nested, the means
are not significantly different.
Figure 6.11 Angles of Intersection and Significance
angle greater
than 90 degrees
angle equal to
90 degrees
angle less than
90 degrees
not significantly
different
borderline
significantly
different
significantly
different
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If the intersection angle is close to 90 degrees, you can verify whether the means are
significantly different by clicking on the comparison circle to select it. See Figure 6.12. To
deselect circles, click in the white space outside the circles.
Figure 6.12 Highlighting Comparison Circles
Groups that are different
from the selected group
appear as thick gray
circles.
Groups that are not
different from the
selected group appear
as thin red circles.
The selected group
appears as a thick red
circle.
Related Information
•
“Comparison Circles” on page 202
Each Pair, Student’s t
The Each Pair, Student’s t test shows the Student’s t-test for each pair of group levels and tests
only individual comparisons.
Related Information
•
“Example of the Each Pair, Student’s t Test” on page 176
All Pairs, Tukey HSD
The All Pairs, Tukey HSD test (also called Tukey-Kramer) protects the significance tests of all
combinations of pairs, and the HSD intervals become greater than the Student’s t pairwise
LSDs. Graphically, the comparison circles become larger and differences are less significant.
The q statistic is calculated as follows: q* = (1/sqrt(2)) * q where q is the required percentile of
the studentized range distribution. For more details, see the description of the T statistic by
Neter, Wasserman, and Kutner (1990).
Related Information
•
“Example of the All Pairs, Tukey HSD Test” on page 178
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159
With Best, Hsu MCB
The With Best, Hsu MCB test determines whether the mean for a given level exceeds the
maximum mean of the remaining levels, or is smaller than the minimum mean of the
remaining levels. See Hsu, 1996.
The quantiles for the Hsu MCB test vary by the level of the categorical variable. Unless the
sample sizes are equal across levels, the comparison circle technique is not exact. The radius of
a comparison circle is given by the standard error of the level multiplied by the largest
quantile value. Use the p-values of the tests to obtain precise assessments of significant
differences. See “Comparison with Max and Min” on page 159.
Note: Means that are not regarded as the maximum or the minimum by MCB are also the
means that are not contained in the selected subset of Gupta (1965) of potential maximums or
minimum means.
Confidence Quantile
This report gives the quantiles for each level of the categorical variable. These correspond to
the specified value of Alpha.
Comparison with Max and Min
The report shows p-values for one-sided Dunnett tests. For each level other than the best, the
p-value given is for a test that compares the mean of the sample best level to the mean of each
remaining level treated as a control (potentially best) level. The p-value for the sample best
level is obtained by comparing the mean of the second sample best level to the mean of the
sample best level treated as a control.
The report shows three columns.
Level The level of the categorical variable.
with Max p-Value For each level of the categorical variable, this column gives a p-value for a
test that the mean of that level exceeds the maximum mean of the remaining levels. Use
the tests in this column to screen out levels whose means are significantly smaller than the
(unknown) largest true mean.
with Min p-Value For each level of the categorical variable, this column gives a p-value for a
test that the mean of that level is smaller than the minimum mean of the remaining levels.
Use the tests in this column to screen out levels whose means are significantly greater than
the (unknown) smallest true mean.
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LSD Threshold Matrix
The first report shown is for the maximum and the second is for the minimum.
For the maximum report, a column shows the row mean minus the column mean minus the
LSD. If a value is positive, the row mean is significantly higher than the mean for the column,
and the mean for the column is not the maximum.
For the minimum report, a column shows the row mean minus the column mean plus the LSD.
If a value is negative, the row mean is significantly less than the mean for the column, and the
mean for the column is not the minimum.
Related Information
•
“Example of the With Best, Hsu MCB Test” on page 180
With Control, Dunnett’s
The With Control, Dunnett’s test compares a set of means against the mean of a control group.
The LSDs that it produces are between the Student’s t and Tukey-Kramer LSDs, because they
are sized to refrain from an intermediate number of comparisons.
In the Dunnett’s report, the d quantile appears, and can be used in a manner similar to a
Student’s t-statistic. The LSD threshold matrix shows the absolute value of the difference
minus the LSD. If a value is positive, its mean is more than the LSD apart from the control
group mean and is therefore significantly different.
Related Information
•
“Example of the With Control, Dunnett’s Test” on page 181
Compare Means Options
The Means Comparisons reports for all four tests contain a red triangle menu with
customization options.
Difference Matrix
Shows a table of all differences of means.
Confidence Quantile Shows the t-value or other corresponding quantiles used for confidence
intervals.
Shows a matrix showing if a difference exceeds the least significant
difference for all comparisons.
LSD Threshold Matrix
Connecting Letters Report Shows the traditional letter-coded report where means that are
not sharing a letter are significantly different.
Note: Not available for With Best, Hsu MCB and With Control, Dunnett’s.
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Ordered Differences Report Shows all the positive-side differences with their confidence
interval band overlaid on the plot. Confidence intervals that do not fully contain their
corresponding bar are significantly different from each other.
Note: Not available for With Best, Hsu MCB and With Control, Dunnett’s.
Detailed Comparisons Report Shows a detailed report for each comparison. Each section
shows the difference between the levels, standard error and confidence intervals, t-ratios,
p-values, and degrees of freedom. A plot illustrating the comparison appears on the right
of each report.
Note: Not available for All Pairs, Tukey HSD, With Best, Hsu MCB, and With Control,
Dunnett’s.
Nonparametric Tests
Nonparametric tests are useful when the usual analysis of variance assumption of normality is
not viable. The Nonparametric option provides several methods for testing the hypothesis of
equal means or medians across groups. Nonparametric multiple comparison procedures are
also available to control the overall error rate for pairwise comparisons. Nonparametric tests
use functions of the response ranks, called rank scores. See Hajek (1969) and SAS Institute
(2008).
Note: If you specify a Block column, the nonparametric tests are conducted on data values
that are centered using the block means.
Performs a test based on Wilcoxon rank scores. The Wilcoxon rank scores are
the simple ranks of the data. The Wilcoxon test is the most powerful rank test for errors
with logistic distributions. If the factor has more than two levels, the Kruskal-Wallis test is
performed. For information about the report, see “The Wilcoxon, Median, and Van der
Waerden Test Reports” on page 162. For an example, see “Example of the Nonparametric
Wilcoxon Test” on page 183.
Wilcoxon Test
The Wilcoxon test is also called the Mann-Whitney test.
Performs a test based on Median rank scores. The Median rank scores are either
1 or 0, depending on whether a rank is above or below the median rank. The Median test is
the most powerful rank test for errors with double-exponential distributions. For
information about the report, see “The Wilcoxon, Median, and Van der Waerden Test
Reports” on page 162.
Median Test
Performs a test based on Van der Waerden rank scores. The Van der
Waerden rank scores are the ranks of the data divided by one plus the number of
observations transformed to a normal score by applying the inverse of the normal
van der Waerden Test
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distribution function. The Van der Waerden test is the most powerful rank test for errors
with normal distributions. For information about the report, see “The Wilcoxon, Median,
and Van der Waerden Test Reports” on page 162.
Performs a test based on the empirical distribution function, which
tests whether the distribution of the response is the same across the groups. Both an
approximate and an exact test are given. This test is available only when the X factor has
two levels. For information about the report, see “Kolmogorov-Smirnov Two-Sample Test
Report” on page 163.
Kolmogorov Smirnov Test
Exact Test Provides options for performing exact versions of the Wilcoxon, Median,
van der Waerden, and Kolmogorov-Smirnov tests. These options are available only when
the X factor has two levels. Results for both the approximate and the exact test are given.
For information about the report, see “ 2-Sample, Exact Test” on page 163. For an example
involving the Wilcoxon Exact Test, see “Example of the Nonparametric Wilcoxon Test” on
page 183.
The Wilcoxon, Median, and Van der Waerden Test Reports
For each test, the report shows the descriptive statistics followed by the test results. Test
results appear in the 1-Way Test, ChiSquare Approximation report and, if the X variable has
exactly two levels, a 2-Sample Test, Normal Approximation report also appears. The
descriptive statistics are the following:
Level
Count
The levels of X.
The frequencies of each level.
Score Sum
The sum of the rank score for each level.
Expected Score The expected score under the null hypothesis that there is no difference
among class levels.
Score Mean
The mean rank score for each level.
(Mean-Mean0)/Std0 The standardized score. Mean0 is the mean score expected under the
null hypothesis. Std0 is the standard deviation of the score sum expected under the null
hypothesis. The null hypothesis is that the group means or medians are in the same
location across groups.
2-Sample Test, Normal Approximation
When you have exactly two levels of X, a 2-Sample Test, Normal Approximation report
appears. This report gives the following:
S
Gives the sum of the rank scores for the level with the smaller number of observations.
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Z
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163
Gives the test statistic for the normal approximation test. For details, see “Two-Sample
Normal Approximations” on page 207.
Gives the p-value, based on a standard normal distribution, for the normal
approximation test.
Prob>|Z|
1-Way Test, ChiSquare Approximation
This report gives results for a chi-square test for location. For details, see Conover (1999).
Gives the values of the chi-square test statistic. For details, see “One-Way
ChiSquare Approximations” on page 207.
ChiSquare
DF
Gives the degrees of freedom for the test.
Prob>ChiSq Gives the p-value for the test. The p-value is based on a ChiSquare distribution
with degrees of freedom equal to the number of levels of X minus 1.
2-Sample, Exact Test
If your data are sparse, skewed, or heavily tied, exact tests might be more suitable than
approximations based on asymptotic behavior. When you have exactly two levels of X, JMP
computes test statistics for exact tests. Select Nonparametric > Exact Test and select the test of
your choice. A 2-Sample: Exact Test report appears. This report gives the following:
S
Gives the sum of the rank scores for the observations in the smaller group. If the two levels
of X have the same numbers of observations, then the value of S corresponds to the last
level of X in the value ordering.
Prob≤ S Gives a one-sided p-value for the test.
Prob ≥ |S-Mean| Gives a two-sided p-value for the test.
Kolmogorov-Smirnov Two-Sample Test Report
The Kolmogorov-Smirnov test is available only when X has exactly two levels. The report
shows descriptive statistics followed by test results. The descriptive statistics are the
following:
Level The two levels of X.
Count
The frequencies of each level.
EDF at Maximum For a level of X, gives the value of the empirical cumulative distribution
function (EDF) for that level at the value of X for which the difference between the two
EDFs is a maximum. For the row named Total, gives the value of the pooled EDF (the EDF
for the entire data set) at the value of X for which the difference between the two EDFs is a
maximum.
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Deviation from Mean at Maximum For each level, gives the value obtained as follows:
– Compute the difference between the EDF at Maximum for the given level and the EDF
at maximum for the pooled data set (Total).
– Multiply this difference by the square root of the number of observations in that level,
given as Count.
Kolmogorov-Smirnov Asymptotic Test
This report gives the details for the test.
KS A Kolmogorov-Smirnov statistic computed as follows:
2
1
KS = max ---  n i ( F i ( x j ) – F ( x j ) )
j
n
i
The formula uses the following notation:
– xj, j = 1, ..., n are the observations
– ni is the number of observations in the ith level of X
– F is the pooled cumulative empirical distribution function
– Fi is the cumulative empirical distribution function for the ith level of X
This version of the Kolmogorov-Smirnov statistic applies even when there are more than
two levels of X. Note, however, that JMP only performs the Kolmogorov-Smirnov analysis
when X has only two levels of X.
KSa An asymptotic Kolmogorov-Smirnov statistic computed as KS n , where n is the total
number of observations.
The maximum absolute deviation between the EDFs for the two levels. This is
the version of the Kolmogorov-Smirnov statistic typically used to compare two samples.
D=max|F1-F2|
Prob > D The p-value for the test. This is the probability that D exceeds the computed value
under the null hypothesis of no difference between the levels.
D+ = max(F1-F2) A one-sided test statistic for the alternative hypothesis that the level of the
first group exceeds the level of the second group.
Prob > D+ The p-value for the test of D+.
D- = max(F2-F1) A one-sided test statistic for the alternative hypothesis that the level of the
second group exceeds the level of the first group
Prob > D- The p-value for the test of D-.
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Kolmogorov-Smirnov Exact Test
For the Kolmogorov-Smirnov exact test, the report gives the same statistics as does the
asymptotic test, but the p-values are computed to be exact.
Nonparametric Multiple Comparisons
This option provides several methods for performing nonparametric multiple comparisons.
These tests are based on ranks and, except for the Wilcoxon Each Pair test, control for the
overall experimentwise error rate. For details about these tests, see See Dunn (1964) and Hsu
(1996). For information about the reports, see “Nonparametric Multiple Comparisons
Procedures” on page 165.
For the Wilcoxon, Median, and Van der Waerden tests, if the X factor has more than two levels,
a chi-square approximation to the one-way test is performed. If the X factor has two levels, a
normal approximation to the two-sample test is performed, in addition to the chi-square
approximation to the one-way test.
Tip: While Friedman’s test for nonparametric repeated measures ANOVA is not directly
supported in JMP, it can be performed as follows: Calculate the ranks within each block.
Define this new column to have an ordinal modeling type. Enter the ranks as Y in the Fit Y by
X platform. Enter one of the effects as a blocking variable. Obtain Cochran-Mantel-Haenszel
statistics.
Nonparametric Multiple Comparisons Procedures
Wilcoxon Each Pair Performs the Wilcoxon test on each pair. This procedure does not control
for the overall alpha level. This is the nonparametric version of the Each Pair, Student’s t
option found on the Compare Means menu. See “Wilcoxon Each Pair, Steel-Dwass All
Pairs, and Steel with Control” on page 166.
Steel-Dwass All Pairs Performs the Steel-Dwass test on each pair. This is the nonparametric
version of the All Pairs, Tukey HSD option found on the Compare Means menu. See
“Wilcoxon Each Pair, Steel-Dwass All Pairs, and Steel with Control” on page 166.
Compares each level to a control level. This is the nonparametric version
of the With Control, Dunnett’s option found on the Compare Means menu. See “Wilcoxon
Each Pair, Steel-Dwass All Pairs, and Steel with Control” on page 166.
Steel With Control
Compares each level to a control level, similar to the Steel
With Control option. The Dunn method computes ranks for all the data, not just the pair
being compared. The reported p-Value reflects a Bonferroni adjustment. It is the
unadjusted p-value multiplied by the number of comparisons. If the adjusted p-value
exceeds 1, it is reported as 1. See “Dunn All Pairs for Joint Ranks and Dunn with Control
for Joint Ranks” on page 167.
Dunn With Control for Joint Ranks
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Dunn All Pairs for Joint Ranks Performs a comparison of each pair, similar to the Steel-Dwass
All Pairs option. The Dunn method computes ranks for all the data, not just the pair being
compared. The reported p-value reflects a Bonferroni adjustment. It is the unadjusted
p-value multiplied by the number of comparisons. If the adjusted p-value exceeds 1, it is
reported as 1. See “Dunn All Pairs for Joint Ranks and Dunn with Control for Joint Ranks”
on page 167.
Wilcoxon Each Pair, Steel-Dwass All Pairs, and Steel with Control
The reports for these multiple comparison procedures give test results and confidence
intervals. For these tests, observations are ranked within the sample obtained by combining
only the two levels used in a given comparison.
q* The quantile used in computing the confidence intervals.
Alpha The alpha level used in computing the confidence interval. You can change the
confidence level by selecting the Set α Level option from the Oneway menu.
Level
- Level
The first level of the X variable used in the pairwise comparison.
The second level of the X variable used in the pairwise comparison.
Score Mean Difference The mean of the rank score of the observations in the first level
(Level) minus the mean of the rank scores of the observations in the second level (-Level),
where a continuity correction is applied.
Denote the number of observations in the first level by n1 and the number in the second
level by n2. The observations are ranked within the sample consisting of these two levels.
Tied ranks are averaged. Denote the sum of the ranks for the first level by ScoreSum1 and
for the second level by ScoreSum2.
If the difference in mean scores is positive, then the Score Mean Difference is given as
follows:
Score Mean Difference = (ScoreSum1 - 0.5)/n1 - (ScoreSum2 + 0.5)/n2
If the difference in mean scores is negative, then the Score Mean Difference is given as
follows:
Score Mean Difference = (ScoreSum1 + 0.5)/n1 - (ScoreSum2 -0.5)/n2
Std Error Dif The standard error of the Score Mean Difference.
Z
The standardized test statistic, which has an asymptotic standard normal distribution
under the null hypothesis of no difference in means.
p-Value The p-value for the asymptotic test based on Z.
Hodges-Lehmann The Hodges-Lehmann estimator of the location shift. All paired
differences consisting of observations in the first level minus observations in the second
level are constructed. The Hodges-Lehmann estimator is the median of these differences.
The bar graph to the right of Upper CL shows the size of the Hodges-Lehmann estimate.
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Lower CL The lower confidence limit for the Hodges-Lehmann statistic.
Note: Not computed if group sample sizes are large enough to cause memory issues.
Upper CL The upper confidence limit for the Hodges-Lehmann statistic.
Note: Not computed if group sample sizes are large enough to cause memory issues.
~Difference A bar graph showing the size of the Hodges-Lehmann estimate for each
comparison.
Dunn All Pairs for Joint Ranks and Dunn with Control for Joint Ranks
These comp are based on the rank of an observation in the entire data set. For the Dunn with
Control for Joint Ranks tests, you must select a control level.
Level The first level of the X variable used in the pairwise comparison.
- Level
he second level of the X variable used in the pairwise comparison.
Score Mean Difference The mean of the rank score of the observations in the first level
(Level) minus the mean of the rank scores of the observations in the second level (-Level),
where a continuity correction is applied. The ranks are obtained by ranking the
observations within the entire sample. Tied ranks are averaged. The continuity correction
is described in “Score Mean Difference” on page 166.
Std Error Dif The standard error of the Score Mean Difference.
Z
The standardized test statistic, which has an asymptotic standard normal distribution
under the null hypothesis of no difference in means.
p-Value The p-value for the asymptotic test based on Z.
Unequal Variances
When the variances across groups are not equal, the usual analysis of variance assumptions
are not satisfied and the ANOVA F test is not valid. JMP provides four tests for equality of
group variances and an ANOVA that is valid when the group sample variances are unequal.
The concept behind the first three tests of equal variances is to perform an analysis of variance
on a new response variable constructed to measure the spread in each group. The fourth test is
Bartlett’s test, which is similar to the likelihood ratio test under normal distributions.
Note: Another method to test for unequal variances is ANOMV. See “Analysis of Means
Methods” on page 152.
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The following Tests for Equal Variances are available:
Constructs a dependent variable so that the group means of the new variable equal
the group sample variances of the original response. An ANOVA on the O’Brien variable
is actually an ANOVA on the group sample variances (O’Brien 1979, Olejnik, and Algina
1987).
O’Brien
Brown-Forsythe Shows the F test from an ANOVA where the response is the absolute value
of the difference of each observation and the group median (Brown and Forsythe 1974).
Levene Shows the F test from an ANOVA where the response is the absolute value of the
difference of each observation and the group mean (Levene 1960). The spread is measured
2
as z ij = y ij – y i (as opposed to the SAS default zij2 = ( y ij – y i ) ).
Bartlett Compares the weighted arithmetic average of the sample variances to the weighted
geometric average of the sample variances. The geometric average is always less than or
equal to the arithmetic average with equality holding only when all sample variances are
equal. The more variation there is among the group variances, the more these two
averages differ. A function of these two averages is created, which approximates a
χ2-distribution (or, in fact, an F distribution under a certain formulation). Large values
correspond to large values of the arithmetic or geometric ratio, and therefore to widely
varying group variances. Dividing the Bartlett Chi-square test statistic by the degrees of
freedom gives the F value shown in the table. Bartlett’s test is not very robust to violations
of the normality assumption (Bartlett and Kendall 1946).
If there are only two groups tested, then a standard F test for unequal variances is also
performed. The F test is the ratio of the larger to the smaller variance estimate. The p-value
from the F distribution is doubled to make it a two-sided test.
Note: If you have specified a Block column, then the variance tests are performed on data after
it has been adjusted for the Block means.
Tests That the Variances Are Equal Report
The Tests That the Variances Are Equal report shows the differences between group means to
the grand mean and to the median, and gives a summary of testing procedures.
If the equal variances test reveals that the group variances are significantly different, use
Welch’s test instead of the regular ANOVA test. The Welch statistic is based on the usual
ANOVA F test. However, the means are weighted by the reciprocal of the group mean
variances (Welch 1951; Brown and Forsythe 1974b; Asiribo, Osebekwin, and Gurland 1990). If
there are only two levels, the Welch ANOVA is equivalent to an unequal variance t-test.
Description of the Variances Are Equal Report
Level
Lists the factor levels occurring in the data.
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Records the frequencies of each level.
Records the standard deviations of the response for each factor level. The standard
deviations are equal to the means of the O’Brien variable. If a level occurs only once in the
data, no standard deviation is calculated.
Std Dev
Records the mean absolute difference of the response and group mean.
The mean absolute differences are equal to the group means of the Levene variable.
MeanAbsDif to Mean
Records the absolute difference of the response and group median.
The mean absolute differences are equal to the group means of the Brown-Forsythe
variable.
MeanAbsDif to Median
Test Lists the names of the tests performed.
F Ratio Records a calculated F statistic for each test. See “Tests That the Variances Are Equal”
on page 204.
Records the degrees of freedom in the numerator for each test. If a factor has k levels,
the numerator has k - 1 degrees of freedom. Levels occurring only once in the data are not
used in calculating test statistics for O’Brien, Brown-Forsythe, or Levene. The numerator
degrees of freedom in this situation is the number of levels used in calculations minus one.
DFNum
DFDen Records the degrees of freedom used in the denominator for each test. For O’Brien,
Brown-Forsythe, and Levene, a degree of freedom is subtracted for each factor level used
in calculating the test statistic. If a factor has k levels, the denominator degrees of freedom
is n - k.
p-Value Probability of obtaining, by chance alone, an F value larger than the one calculated if
in reality the variances are equal across all levels.
Description of the Welch’s Test Report
F Ratio Shows the F test statistic for the equal variance test. See “Tests That the Variances Are
Equal” on page 204.
Records the degrees of freedom in the numerator of the test. If a factor has k levels,
the numerator has k - 1 degrees of freedom. Levels occurring only once in the data are not
used in calculating the Welch ANOVA. The numerator degrees of freedom in this situation
is the number of levels used in calculations minus one.
DFNum
DFDen Records the degrees of freedom in the denominator of the test. See “Tests That the
Variances Are Equal” on page 204.
Prob>F Probability of obtaining, by chance alone, an F value larger than the one calculated if
in reality the means are equal across all levels. Observed significance probabilities of 0.05
or less are considered evidence of unequal means across the levels.
Shows the relationship between the F ratio and the t Test. Calculated as the square root
of the F ratio. Appears only if the X factor has two levels.
t Test
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Related Information
•
“Example of the Unequal Variances Option” on page 186
•
“Tests That the Variances Are Equal” on page 204
Equivalence Test
Equivalence tests assess whether there is a practical difference in means. You must pick a
threshold difference for which smaller differences are considered practically equivalent. The
most straightforward test to construct uses two one-sided t-tests from both sides of the
difference interval. If both tests reject (or conclude that the difference in the means differs
significantly from the threshold), then the groups are practically equivalent. The Equivalence
Test option uses the Two One-Sided Tests (TOST) approach.
Related Information
•
“Example of an Equivalence Test” on page 187
Robust
Outliers can lead to incorrect estimates and decisions. The Robust option provides two
methods to reduce the influence of outliers in your data set: Robust Fit and Cauchy Fit.
Robust Fit
The Robust Fit option reduces the influence of outliers in the response variable. The Huber
M-estimation method is used. Huber M-estimation finds parameter estimates that minimize
the Huber loss function, which penalizes outliers. The Huber loss function increases as a
quadratic for small errors and linearly for large errors. For more details about robust fitting,
see Huber (1973) and Huber and Ronchetti (2009).
Related Information
•
“Example of the Robust Fit Option” on page 188
Cauchy Fit
The Cauchy fit option assumes that the errors have a Cauchy distribution. A Cauchy
distribution has fatter tails than the normal distribution, resulting in a reduced emphasis on
outliers. This option can be useful if you have a large proportion of outliers in your data.
However, if your data are close to normal with only a few outliers, this option can lead to
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incorrect inferences. The Cauchy option estimates parameters using maximum likelihood and
a Cauchy link function.
Power
The Power option calculates statistical power and other details about a given hypothesis test.
•
LSV (the Least Significant Value) is the value of some parameter or function of parameters
that would produce a certain p-value alpha. Said another way, you want to know how
small an effect would be declared significant at some p-value alpha. The LSV provides a
measuring stick for significance on the scale of the parameter, rather than on a probability
scale. It shows how sensitive the design and data are.
•
LSN (the Least Significant Number) is the total number of observations that would
produce a specified p-value alpha given that the data has the same form. The LSN is
defined as the number of observations needed to reduce the variance of the estimates
enough to achieve a significant result with the given values of alpha, sigma, and delta (the
significance level, the standard deviation of the error, and the effect size). If you need more
data to achieve significance, the LSN helps tell you how many more. The LSN is the total
number of observations that yields approximately 50% power.
•
Power is the probability of getting significance (p-value < alpha) when a real difference
exists between groups. It is a function of the sample size, the effect size, the standard
deviation of the error, and the significance level. The power tells you how likely your
experiment is to detect a difference (effect size), at a given alpha level.
Note: When there are only two groups in a one-way layout, the LSV computed by the power
facility is the same as the least significant difference (LSD) shown in the multiple-comparison
tables.
Power Details Window and Reports
The Power Details window and reports are the same as those in the general fitting platform
launched by the Fit Model platform. For more details about power calculation, see the
Statistical Details appendix in the Fitting Linear Models book.
For each of four columns Alpha, Sigma, Delta, and Number, fill in a single value, two values,
or the start, stop, and increment for a sequence of values. See Figure 6.31. Power calculations
are performed on all possible combinations of the values that you specify.
Alpha (α) Significance level, between 0 and 1 (usually 0.05, 0.01, or 0.10). Initially, a value of
0.05 shows.
Standard error of the residual error in the model. Initially, RMSE, the estimate from
the square root of the mean square error is supplied here.
Sigma (σ)
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Delta (δ) Raw effect size. For details about effect size computations, see the Standard Least
Squares chapter in the Fitting Linear Models book. The first position is initially set to the
square root of the sums of squares for the hypothesis divided by the square root of n; that
is, δ = SS ⁄ n .
Number (n) Total sample size across all groups. Initially, the actual sample size is put in the
first position.
Solves for the power (the probability of a significant result) as a function of
all four values: α, σ, δ, and n.
Solve for Power
Solve for Least Significant Number Solves for the number of observations needed to achieve
approximately 50% power given α, σ, and δ.
Solve for Least Significant Value Solves for the value of the parameter or linear test that
produces a p-value of α. This is a function of α, σ, n, and the standard error of the estimate.
This feature is available only when the X factor has two levels and is usually used for
individual parameters.
Adjusted Power and Confidence Interval When you look at power retrospectively, you use
estimates of the standard error and the test parameters.
– Adjusted power is the power calculated from a more unbiased estimate of the
non-centrality parameter.
– The confidence interval for the adjusted power is based on the confidence interval for
the non-centrality estimate.
Adjusted power and confidence limits are computed only for the original Delta, because
that is where the random variation is.
Related Information
•
“Example of the Power Option” on page 190
•
“Power” on page 203
Normal Quantile Plot
You can create two types of normal quantile plots:
•
Plot Actual by Quantile creates a plot of the response values versus the normal quantile
values. The quantiles are computed and plotted separately for each level of the X variable.
•
Plot Quantile by Actual creates a plot of the normal quantile values versus the response
values. The quantiles are computed and plotted separately for each level of the X variable.
The Line of Fit option shows or hides the lines of fit on the quantile plots.
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Related Information
•
“Example of a Normal Quantile Plot” on page 191
CDF Plot
A CDF plot shows the cumulative distribution function for all of the groups in the Oneway
report. CDF plots are useful if you want to compare the distributions of the response across
levels of the X factor.
Related Information
•
“Example of a CDF Plot” on page 192
Densities
The Densities options provide several ways to compare the distribution and composition of
the response across the levels of the X factor. There are three density options:
•
Compare Densities shows a smooth curve estimating the density of each group. The
smooth curve is the density estimate for each group.
•
Composition of Densities shows the summed densities, weighted by each group’s counts.
At each X value, the Composition of Densities plot shows how each group contributes to
the total.
•
Proportion of Densities shows the contribution of the group as a proportion of the total at
each X level.
Related Information
•
“Example of the Densities Options” on page 193
Matching Column
Use the Matching Column option to specify a matching (ID) variable for a matching model
analysis. The Matching Column option addresses the case when the data in a one-way analysis
come from matched (paired) data, such as when observations in different groups come from
the same subject.
Note: A special case of matching leads to the paired t-test. The Matched Pairs platform
handles this type of data, but the data must be organized with the pairs in different columns,
not in different rows.
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The Matching Column option performs two primary actions:
•
It fits an additive model (using an iterative proportional fitting algorithm) that includes
both the grouping variable (the X variable in the Fit Y by X analysis) and the matching
variable that you select. The iterative proportional fitting algorithm makes a difference if
there are hundreds of subjects, because the equivalent linear model would be very slow
and would require huge memory resources.
•
It draws lines between the points that match across the groups. If there are multiple
observations with the same matching ID value, lines are drawn from the mean of the
group of observations.
The Matching Column option automatically activates the Matching Lines option connecting the
matching points. To turn the lines off, select Display Options > Matching Lines.
The Matching Fit report shows the effects with F tests. These are equivalent to the tests that
you get with the Fit Model platform if you run two models, one with the interaction term and
one without. If there are only two levels, then the F test is equivalent to the paired t-test.
Note: For details about the Fit Model platform, see the Model Specification chapter in the
Fitting Linear Models book.
Related Information
•
“Example of the Matching Column Option” on page 194
Additional Examples of the Oneway Platform
This section contains additional examples of selected options and reports in the Oneway
platform.
Example of an Analysis of Means Chart
1. Select Help > Sample Data Library and open Analgesics.jmp.
2. Select Analyze > Fit Y by X.
3. Select pain and click Y, Response.
4. Select drug and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Analysis of Means Methods > ANOM.
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Figure 6.13 Example of Analysis of Means Chart
For the example in Figure 6.13, the means for drug A and C are statistically different from the
overall mean. The drug A mean is lower and the drug C mean is higher. Note the decision
limits for the drug types are not the same, due to different sample sizes.
Example of an Analysis of Means for Variances Chart
This example uses the Spring Data.jmp sample data table. Four different brands of springs
were tested to see what weight is required to extend a spring 0.10 inches. Six springs of each
brand were tested. The data was checked for normality, since the ANOMV test is not robust to
non-normality. Examine the brands to determine whether the variability is significantly
different between brands.
1. Select Help > Sample Data Library and open Spring Data.jmp.
2. Select Analyze > Fit Y by X.
3. Select Weight and click Y, Response.
4. Select Brand and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Analysis of Means Methods > ANOM for Variances.
7. From the red triangle menu next to Analysis of Means for Variances, select Show Summary
Report.
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Figure 6.14 Example of Analysis of Means for Variances Chart
From Figure 6.14, notice that the standard deviation for Brand 2 exceeds the lower decision
limit. Therefore, Brand 2 has significantly lower variance than the other brands.
Example of the Each Pair, Student’s t Test
This example uses the Big Class.jmp sample data table. It shows a one-way layout of weight by
age, and shows the group comparison using comparison circles that illustrate all possible
t-tests.
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Fit Y by X.
3. Select weight and click Y, Response.
4. Select age and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Compare Means > Each Pair, Student’s t.
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Figure 6.15 Example of Each Pair, Student’s t Comparison Circles
The means comparison method can be thought of as seeing if the actual difference in the
means is greater than the difference that would be significant. This difference is called the LSD
(least significant difference). The LSD term is used for Student’s t intervals and in context with
intervals for other tests. In the comparison circles graph, the distance between the circles’
centers represent the actual difference. The LSD is what the distance would be if the circles
intersected at right angles.
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Figure 6.16 Example of Means Comparisons Report for Each Pair, Student’s t
In Figure 6.16, the LSD threshold table shows the difference between the absolute difference in
the means and the LSD (least significant difference). If the values are positive, the difference in
the two means is larger than the LSD, and the two groups are significantly different.
Example of the All Pairs, Tukey HSD Test
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Fit Y by X.
3. Select weight and click Y, Response.
4. Select age and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Compare Means > All Pairs, Tukey HSD.
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Figure 6.17 Example of All Pairs, Tukey HSD Comparison Circles
Figure 6.18 Example of Means Comparisons Report for All Pairs, Tukey HSD
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In Figure 6.18, the Tukey-Kramer HSD Threshold matrix shows the actual absolute difference
in the means minus the HSD, which is the difference that would be significant. Pairs with a
positive value are significantly different. The q* (appearing above the HSD Threshold Matrix
table) is the quantile that is used to scale the HSDs. It has a computational role comparable to
a Student’s t.
Example of the With Best, Hsu MCB Test
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Fit Y by X.
3. Select weight and click Y, Response.
4. Select age and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Compare Means > With Best, Hsu MCB.
Figure 6.19 Examples of With Best, Hsu MCB Comparison Circles
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Figure 6.20 Example of Means Comparisons Report for With Best, Hsu MCB
The Comparison with Max and Min report compares the mean of each level to the maximum
and the minimum of the means of the remaining levels. For example, the mean for age 15
differs significantly from the maximum of the means of the remaining levels. The mean for
age 17 differs significantly from the minimum of the means of the remaining levels. The
maximum mean could occur for age 16 or age 17, because neither mean differs significantly
from the maximum mean. By the same reasoning, the minimum mean could correspond to
any of the ages other than age 17.
Example of the With Control, Dunnett’s Test
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Fit Y by X.
3. Select weight and click Y, Response.
4. Select age and click X, Factor.
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5. Click OK.
6. From the red triangle menu, select Compare Means > With Control, Dunnett’s.
7. Select the group to use as the control group. In this example, select age 12.
Alternatively, click on a row to highlight it in the scatterplot before selecting the Compare
Means > With Control, Dunnett’s option. The test uses the selected row as the control
group.
8. Click OK.
Figure 6.21 Example of With Control, Dunnett’s Comparison Circles
Using the comparison circles in Figure 6.21, you can conclude that level 17 is the only level
that is significantly different from the control level of 12.
Example Contrasting All of the Compare Means Tests
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Fit Y by X.
3. Select weight and click Y, Response.
4. Select age and click X, Factor.
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5. Click OK.
6. From the red triangle menu, select each one of the Compare Means options.
Although the four methods all test differences between group means, different results can
occur. Figure 6.22 shows the comparison circles for all four tests, with the age 17 group as the
control group.
Figure 6.22 Comparison Circles for Four Multiple Comparison Tests
From Figure 6.22, notice that for the Student’s t and Hsu methods, age group 15 (the third
circle from the top) is significantly different from the control group and appears gray. But, for
the Tukey and Dunnett method, age group 15 is not significantly different, and appears red.
Example of the Nonparametric Wilcoxon Test
Suppose you want to test whether the mean profit earned by companies differs by type of
company. In Companies.jmp, the data consist of various metrics on two types of companies,
Pharmaceutical (12 companies) and Computer (20 companies).
1. Select Help > Sample Data Library and open Companies.jmp.
2. Select Analyze > Fit Y by X.
3. Select Profits ($M) and click Y, Response.
4. Select Type and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Display Options > Box Plots.
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Figure 6.23 Computer Company Profit Distribution
The box plots suggest that the distributions are not normal or even symmetric. There is a very
large value for the company in row 32 that might affect parametric tests.
7. From the red triangle menu, select Means/ANOVA/Pooled t.
Figure 6.24 Company Analysis of Variance
The F test shows no significance because the p-value is large (p = 0.1163). This might be due
to the large value in row 32 and the possible violation of the normality assumption.
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8. From the red triangle menu, select t Test.
Figure 6.25 t-Test Results
The Prob > |t| for a two-sided test is 0.0671. The t Test does not assume equal variances,
but the unequal variances t-test is also a parametric test.
9. From the red triangle menu, select Nonparametric > Wilcoxon Test.
Figure 6.26 Wilcoxon Test Results
The Wilcoxon test is a nonparametric test. It is based on ranks and so is resistant to
outliers. Also, it does not require normality.
Both the normal and the chi-square approximations for the Wilcoxon test statistic indicate
significance at a p-value of 0.0010. You conclude that there is a significant difference in the
location of the distributions, and conclude that mean profit differs based on company
type.
The normal and chi-square tests are based on the asymptotic distributions of the test
statistics. If you have JMP Pro, you can conduct an exact test.
10.
From the red triangle menu, select Nonparametric > Exact Test > Wilcoxon Exact Test.
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Figure 6.27 Wilcoxon Exact Test Results
The observed value of the test statistic is S = 283. This is the sum of the midranks for the
level of Type with the smaller sample size (pharmaceuticals). The probability of observing
an absolute difference from the mean midrank that exceeds the absolute value of S minus
the mean of the midranks is 0.0005. This is a two-sided test for a difference in location and
supports rejecting the hypothesis that profits do not differ by type of company.
In this example, the nonparametric tests are more appropriate than the normality-based
ANOVA test and the unequal variances t-test. The nonparametric tests are resistant to the
large value in row 32 and do not require the underlying normality of Profits ($M) for each
group.
Example of the Unequal Variances Option
Suppose you want to test whether two variances (males and females) are equal, instead of two
means.
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Fit Y by X.
3. Select height and click Y, Response.
4. Select sex and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Unequal Variances.
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Figure 6.28 Example of the Unequal Variances Report
Since the p-value from the 2-sided F-Test is large, you cannot conclude that the variances are
unequal.
Example of an Equivalence Test
This example uses the Big Class.jmp sample data table. Examine if the difference in height
between males and females is less than 6 inches.
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Fit Y by X.
3. Select height and click Y, Response.
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4. Select sex and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Equivalence Test.
7. Type 6 as the difference considered practically zero.
8. Click OK.
Figure 6.29 Example of an Equivalence Test
From Figure 6.29, notice the following:
•
The test at the Lower Threshold is the upper one-tailed test for the null hypothesis that the
mean difference is -6.
•
The test at the Upper Threshold is the lower one-tailed test for the null hypothesis that the
mean difference is 6.
•
For both tests, the p-value is small. Therefore, you can conclude that the difference in
population means is significantly located somewhere from 6 to -6. For your purposes, you
can declare the means to be practically equivalent.
Example of the Robust Fit Option
The data in the Drug Toxicity.jmp sample data table shows the toxicity levels for three different
formulations of a drug.
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1. Select Help > Sample Data Library and open Drug Toxicity.jmp.
2. Select Analyze > Fit Y by X.
3. Select Toxicity and click Y, Response.
4. Select Formulation and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Means/Anova.
7. From the red triangle menu, select Robust > Robust Fit.
Figure 6.30 Example of Robust Fit
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If you look at the standard Analysis of Variance report, you might wrongly conclude that
there is a difference between the three formulations, since the p-value is 0.0319. However,
when you look at the Robust Fit report, you would not conclude that the three formulations
are significantly different, because the p-value there is 0.21755. It appears that the toxicity for a
few of the observations is unusually high, creating the undue influence on the data.
Example of the Power Option
1. Select Help > Sample Data Library and open Typing Data.jmp.
2. Select Analyze > Fit Y by X.
3. Select speed and click Y, Response.
4. Select brand and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Power.
7. Within the From row, type 2 for Delta (the third box) and type 11 for Number.
8. Within the To row, type 6 for Delta, and type 17 in the Number box.
9. Within the By row, type 2 for both Delta and Number.
10. Select the Solve for Power check box.
Figure 6.31 Example of the Power Details Window
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11. Click Done.
Note: The Done button remains dimmed until all of the necessary options are applied.
Power is computed for each combination of Delta and Number, and appears in the Power
report.
To plot the Power values:
12. From the red triangle menu at the bottom of the report, select Power Plot.
Figure 6.32 Example of the Power Report
13. You might need to click and drag vertically on the Power axis to see all of the data in the
plot.
Power is plotted for each combination of Delta and Number. As you might expect, the power
rises for larger Number (sample sizes) values and for larger Delta values (difference in
means).
Example of a Normal Quantile Plot
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Fit Y by X.
3. Select height and click Y, Response.
4. Select sex and click X, Factor.
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5. Click OK.
6. From the red triangle menu, select Normal Quantile Plot > Plot Actual by Quantile.
Figure 6.33 Example of a Normal Quantile Plot
From Figure 6.33, notice the following:
•
The Line of Fit appears by default.
•
The data points track very closely to the line of fit, indicating a normal distribution.
Example of a CDF Plot
1. Select Help > Sample Data Library and open Analgesics.jmp.
2. Select Analyze > Fit Y by X.
3. Select pain and click Y, Response.
4. Select drug and click X, Factor.
5. Click OK.
6. From the red triangle menu, select CDF Plot.
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Figure 6.34 Example of a CDF Plot
The levels of the X variables in the initial Oneway analysis appear in the CDF plot as different
curves. The horizontal axis of the CDF plot uses the y value in the initial Oneway analysis.
Example of the Densities Options
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Fit Y by X.
3. Select height and click Y, Response.
4. Select sex and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Densities > Compare Densities, Densities >
Composition of Densities, and Densities > Proportion of Densities.
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Figure 6.35 Example of the Densities Options
contribution of
males to height
density
contribution of
females to
height density
females contributed
about 61% to height
density at this X level
Example of the Matching Column Option
This example uses the Matching.jmp sample data table, which contains data on six animals and
the miles that they travel during different seasons.
1. Select Help > Sample Data Library and open Matching.jmp.
2. Select Analyze > Fit Y by X.
3. Select miles and click Y, Response.
4. Select season and click X, Factor.
5. Click OK.
6. From the red triangle menu, select Matching Column.
7. Select subject as the matching column.
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8. Click OK.
Figure 6.36 Example of the Matching Column Report
The plot graphs the miles traveled by season, with subject as the matching variable. The labels
next to the first measurement for each subject on the graph are determined by the species and
subject variables.
The Matching Fit report shows the season and subject effects with F tests. These are
equivalent to the tests that you get with the Fit Model platform if you run two models, one
with the interaction term and one without. If there are only two levels, then the F test is
equivalent to the paired t-test.
Note: For details about the Fit Model platform, see the Model Specification chapter in the
Fitting Linear Models book.
Example of Stacking Data for a Oneway Analysis
When your data are in a format other than a JMP data table, sometimes they are arranged so
that a row contains information for multiple observations. To analyze the data in JMP, you
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must import the data and restructure it so that each row of the JMP data table contains
information for a single observation. For example, suppose that your data are in a
spreadsheet. The data for parts produced on three production lines are arranged in three sets
of columns. In your JMP data table, you need to stack the data from the three production lines
into a single set of columns so that each row represents the data for a single part.
Description and Goals
This example uses the file Fill Weights.xlsx, which contains the weights of cereal boxes
randomly sampled from three different production lines. Figure 6.37 shows the format of the
data.
•
The ID columns contain an identifier for each cereal box that was measured.
•
The Line columns contain the weights (in ounces) for boxes sampled from the
corresponding production line.
Figure 6.37 Data Format
The target fill weight for the boxes is 12.5 ounces. Although you are interested in whether the
three production lines are meeting the target, initially you want to see whether the three lines
are achieving the same mean fill rate. You can use Oneway to test for differences among the
mean fill weights.
To use the Oneway platform, you need to do the following:
1. Import the data into JMP. See “Import the Data” on page 196.
2. Reshape the data so that each row in the JMP data table reflects only a single observation.
Reshaping the data requires that you stack the cereal box IDs, the line identifiers, and the
weights into columns. See “Stack the Data” on page 198.
Import the Data
This example illustrates two ways to import data from Microsoft Excel into JMP. Select one
method or explore both:
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•
Use the File > Open option to import data from a Microsoft Excel file using the Excel
Import Wizard. See “Import the Data Using the Excel Import Wizard” on page 197. This
method is convenient for any Excel file.
•
Copy and paste data from Microsoft Excel into a new JMP data table. See “Copy and Paste
the Data from Excel” on page 197. You can use this method with small data files.
For more information on how to import data from Microsoft Excel, see the Import Your Data
chapter in the Using JMP book.
Import the Data Using the Excel Import Wizard
1. Select Help > Sample Data Library and open Fill Weights.xlsx located in the Samples/Import
Data folder.
The file opens in the Excel Import Wizard.
2. Type 3 next to Column headers start on on row.
In the Excel file, row 1 contains information about the table and row 2 is blank. The
column header information starts on row 3.
3. Type 2 for Number of rows with column headers.
In the Excel file, rows 3 and 4 both contain column header information.
4. Click Import.
Figure 6.38 JMP Table Created Using Excel Import Wizard
The data are placed in seven rows and multiple IDs appear in each row. For each of the
three lines, there are an ID and Weight column, giving a total of six columns.
Notice that the “Weights” part of the ID column name is unnecessary and misleading. You
could rename the columns now, but it will be more efficient to rename the columns after
you stack the data.
5. Proceed to “Stack the Data” on page 198.
Copy and Paste the Data from Excel
1. Open Fill Weights.xlsx in Microsoft Excel.
2. Select the data inside the table but exclude the unnecessary “Weights” heading.
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3. Right-click and select Copy.
4. In JMP, select File > New > Data Table.
5. Select Edit > Paste with Column Names.
The Edit > Paste with Column Names option is used when you have column names
included in the selection on the clipboard.
Figure 6.39 JMP Table Created Using Paste with Column Names
6. Proceed to “Stack the Data” on page 198.
Stack the Data
Use the Stack option to place one observation in each row of a new data table. For more
information on the Stack option, see the Reshape Data chapter in the Using JMP book.
1. In the JMP data table, select Tables > Stack.
2. Select all six columns and click Stack Columns.
3. Select Multiple Series Stack.
You are stacking two series, ID and Line, so you do not change the Number of Series,
which is set to 2 by default. The columns that contain the series are not contiguous. They
alternate (ID, Line A, ID, Line B, ID, Line C). For this reason, you do not check Contiguous.
4. Deselect Stack By Row.
5. Select Eliminate Missing Rows.
6. Enter Stacked next to Output table name.
7. Click OK.
In the new data table, Data and Data 2 are columns containing the ID and Weight data.
8. Right-click the Label column heading and select Delete Columns.
The entries in the Label column were the column headings for the box IDs in the imported
data table. These entries are not needed.
9. Rename each column by double-clicking on the column header. Change the column names
as follows:
– Data to ID
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– Label 2 to Line
– Data 2 to Weight
10. In the Columns panel, click on the icon to the left of ID and select Nominal.
Although ID is given as a number, it is an identifier and should be treated as nominal when
modeling. This is not an issue in this example, but it is good practice to assign the
appropriate modeling type to a column.
11. (Only applies if you imported the data from Excel using File > Open.) Do the following:
1. Click the Line column header to select the column and select Cols > Recode.
2. Change the values in the New Values column to match those in Figure 6.40 below.
Figure 6.40 Recode Column Values
3. Click Done > In place.
Your new data table is now properly structured for JMP analysis. Each row contains data for a
single cereal box. The first column gives the box ID, the second gives the production line, and
the third gives the weight of the box (Figure 6.41).
Figure 6.41 Recoded Data Table
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Conduct the Oneway Analysis
For this part of the example, you will:
•
Conduct a Oneway Analysis of Variance to test for differences in the mean fill weights
among the three production lines.
•
Obtain Comparison Circles to explore which lines might differ.
•
Label points by ID in case you want to reweigh or further examine their boxes.
Before beginning, verify that you are using the Stacked data table.
1. Select Analyze > Fit Y by X.
2. Select Weight and click Y, Response.
3. Select Line and click X, Factor.
4. Click OK.
5. From the red triangle menu, select Means/Anova.
The mean diamonds in the plot show 95% confidence intervals for the production line
means. The points that fall outside the mean diamonds might seem like outliers, however
they are not. To see this, add box plots to the plot.
6. From the red triangle menu, select Display Options > Box Plots.
All points fall within the box plots boundaries, therefore they are not outliers.
7. From the data table, in the Columns panel, right-click ID and select Label/Unlabel.
8. In the plot, hover over the points with your mouse to see their ID values, as well as their
Line and Weight data. See Figure 6.42.
9. From the red triangle menu, select Compare Means > All Pairs, Tukey HSD.
Comparison circles appear in a panel to the right of the plot.
10. Click the bottom comparison circle.
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Additional Examples of the Oneway Platform
201
Figure 6.42 Oneway Analysis of Weight by Line
In the Analysis of Variance report, the p-value of 0.0102 provides evidence that the means are
not all equal. In the plot, the comparison circle for Line C is selected and appears red (in JMP
default colors). Since the circle for Line B appears as thick gray, the mean for Line C differs
from the mean for Line B at the 0.05 significance level. The means for Lines A and B do not
show a statistically significant difference.
The mean diamonds shown in the plot span 95% confidence intervals for the means. The
numeric bounds for the 95% confidence intervals are given in the Means for Oneway ANOVA
report. Both of these indicate that the confidence intervals for Lines B and C do not contain the
target fill weight of 12.5: Line B appears to overfill and Line C appears to underfill. For these
two production lines, the underlying causes that result in off-target fill weights must be
addressed.
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Statistical Details for the Oneway Platform
The following sections provide statistical details for selected options and reports.
Comparison Circles
One approach to comparing two means is to determine whether their actual difference is
greater than their least significant difference (LSD). This least significant difference is a Student’s
t-statistic multiplied by the standard error of the difference of the two means and is written as
follows:
LSD = t α ⁄ 2 std ( μ̂ 1 – μ̂ 2 )
The standard error of the difference of two independent means is calculated from the
following relationship:
2
2
[ std ( μ̂ 1 – μ̂ 2 ) ] = [ std ( μ̂ 1 ) ] + [ std ( μ̂ 2 ) ]
2
When the means are un correlated, these quantities have the following relationship:
2
2
2
2
LSD = [ t α ⁄ 2 std ( ( μ̂ 1 – μ̂ 2 ) ) ] = [ t α ⁄ 2 std ( μ̂ 1 ) ] + [ t α ⁄ 2 stdμˆ 2 ]
These squared values form a Pythagorean relationship, illustrated graphically by the right
triangle shown in Figure 6.43.
Figure 6.43 Relationship of the Difference between Two Means
t α ⋅ std ( μ̂ 1 )
--2
t α ⋅ std ( μ̂ 1 – μ̂ 2 )
--2
t α ⋅ std ( μ̂ 2 )
--2
The hypotenuse of this triangle is a measuring stick for comparing means. The means are
significantly different if and only if the actual difference is greater than the hypotenuse (LSD).
Suppose that you have two means that are exactly on the borderline, where the actual
difference is the same as the least significant difference. Draw the triangle with vertices at the
means measured on a vertical scale. Also, draw circles around each mean so that the diameter
of each is equal to the confidence interval for that mean.
Chapter 6
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203
Figure 6.44 Geometric Relationship of t-test Statistics
μ̂ 1
t α ⋅ std ( μ̂ 1 )
--2
t α ⋅ std ( μ̂ 1 – μ̂ 2 )
--2
μ̂ 2
t α ⋅ std ( μ̂ 2 )
--2
The radius of each circle is the length of the corresponding leg of the triangle, which is
t
std ( μ̂ i ) .
α⁄2
The circles must intersect at the same right angle as the triangle legs, giving the following
relationship:
•
If the means differ exactly by their least significant difference, then the confidence interval
circles around each mean intersect at a right angle. That is, the angle of the tangents is a
right angle.
Now, consider the way that these circles must intersect if the means are different by greater
than or less than the least significant difference:
•
If the circles intersect so that the outside angle is greater than a right angle, then the means
are not significantly different. If the circles intersect so that the outside angle is less than a
right angle, then the means are significantly different. An outside angle of less than 90
degrees indicates that the means are farther apart than the least significant difference.
•
If the circles do not intersect, then they are significantly different. If they nest, they are not
significantly different. See Figure 6.11.
The same graphical technique works for many multiple-comparison tests, substituting a
different probability quantile value for the Student’s t.
Power
To compute power, you make use of the noncentral F distribution. The formula (O’Brien and
Lohr 1984) is given as follows:
Power = Prob(F > Fcrit, ν1, ν2, nc)
where:
•
F is distributed as the noncentral F ( nc, v 1, v ) and F crit = F ( 1 – α, v1, v2 ) is the 1 - α
2
quantile of the F distribution with ν1 and ν2 degrees of freedom.
•
ν1 = r -1 is the numerator df.
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•
ν2 = r(n -1) is the denominator df.
•
n is the number per group.
•
r is the number of groups.
•
nc = n(CSS)/σ2 is the non-centrality parameter.
r
CSS =

( μg – μ )
2
Chapter 6
Basic Analysis
is the corrected sum of squares.
g=1
•
μg is the mean of the gth group.
•
μ is the overall mean.
•
σ2 is estimated by the mean squared error (MSE).
Summary of Fit Report
Rsquare
Using quantities from the Analysis of Variance report for the model, the R2 for any continuous
response fit is always calculated as follows:
Sum of Squares (Model)----------------------------------------------------------------Sum of Squares (C Total)
Adj Rsquare
Adj Rsquare is a ratio of mean squares instead of sums of squares and is calculated as follows:
Mean Square (Error)
1 – ----------------------------------------------------------Mean Square (C Total)
The mean square for Error is found in the Analysis of Variance report and the mean square for
C. Total can be computed as the C. Total Sum of Squares divided by its respective degrees of
freedom. See “The Analysis of Variance Report” on page 149.
Tests That the Variances Are Equal
F Ratio
O’Brien’s test constructs a dependent variable so that the group means of the new variable
equal the group sample variances of the original response. The O’Brien variable is computed
as follows:
Chapter 6
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Statistical Details for the Oneway Platform
2
205
2
( n ij – 1.5 )n ij ( y ijk – y·· ) – 0.5s ij ( n ij – 1 )
ij
r ijk = ------------------------------------------------------------------------------------------------------( n ij – 1 ) ( n ij – 2 )
where n represents the number of yijk observations.
Brown-Forsythe is the model F statistic from an ANOVA on z ij = y ij – ỹ i where ỹ i is the
median response for the ith level.
The Levene F is the model F statistic from an ANOVA on z ij = y ij – y i. where y i. is the mean
response for the ith level.
Bartlett’s test is calculated as follows:
 vi 
v log   ---- s 2i  –  v i log ( s i2 )
 i v  i
T = ----------------------------------------------------------------------- where v i = n i – 1 and v =
 vi
1 1
  ---- – --v-
i
v
 i i

-
1 +  ------------------ 3( k – 1 )


and ni is the count on the ith level and si2 is the response sample variance on the ith level. The
Bartlett statistic has a χ2-distribution. Dividing the Chi-square test statistic by the degrees of
freedom results in the reported F value.
Welch’s Test F Ratio
The Welch’s Test F Ratio is computed as follows:
 wi ( y i – ỹ.. )
2
i
-----------------------------------k–1
n
where w = -----i , u =
i
2
----------------------------------------------------------------------F =
s
w 2

 1 – ------i


2( k – 2 )
u
 1 + -------------------  ----------------------2
n
–

i 1
k –1 i






i
 wi
i
, ỹ =
..
w i y i.
,
 -----------u
i
and ni is the count on the ith level, y i. is the mean response for the ith level, and si2 is the
response sample variance for the ith level.
Welch’s Test DF Den
The Welch approximation for the denominator degrees of freedom is as follows:
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Chapter 6
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1
df = -------------------------------------------------------w 2
 1 – ------i

3
u --------------
 2   ---------------------k – 1 i ni – 1
where wi, ni, and u are defined as in the F ratio formula.
Nonparametric Test Statistics
This section provides formulas for the test statistics used in the Wilcoxon, Median, and van
der Waerden tests.
Notation
The tests are based on scores and use the following notation.
j = 1,...,n The observations in the entire sample.
n1, n2, ..., nk
The numbers of observations in each of the k levels of X.
Rj The midrank of the jth observation. The midrank is the observation’s rank if it is not tied
and its average rank if it is tied.
α A function of the midranks used to define scores for the various tests.
α(Rj) The function α applied to the midrank of the jth observation.
The function α defines scores as follows:
Wilcoxon scores
α ( Rj ) = Rj
Median scores
 1 if R > ( n + 1 ) ⁄ 2
j


α ( R j ) = 0 if R j < ( n + 1 ) ⁄ 2

 t if R = median

j
Let nt denote the number of observations tied at the median. Then nt is given by the
following:
 ( n ⁄ 2 ) if n is even
t
t
nt = 
 ( n + 1 ) ⁄ ( 2n ) if n is odd
t
t
t

van der Waerden scores
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207
α ( R j ) = Standard Normal Quantile ( R j ⁄ ( n + 1 ) )
Two-Sample Normal Approximations
Tests based on the normal approximation are given only when X has exactly two levels. The
notation used in this section is defined in “Notation” on page 206. The statistics that appear in
the Two-Sample Normal Approximation report are defined below.
S
The statistic S is the sum of the values α(Rj) for the observations in the smaller group. If the
two levels of X have the same numbers of observations, then the value of S corresponds to
the last level of X in the value ordering.
Z
The value of Z is given as follows:
Z = ( S – E ( S ) ) ⁄ Var ( S )
Note: The Wilcoxon test adds a continuity correction. If (S - E(S)) is greater than zero, then
0.5 is subtracted from the numerator. If (S - E(S)) is less than zero, then 0.5 is added to the
numerator.
E(S) The expected value of S under the null hypothesis. Denote the number of
observations in the smaller level, or in the last level in the value ordering if the two
groups have the same number of observations, by nl:
nl
E ( S ) = ----n
n

α ( Rj )
j=1
Var(S) Define ave to be the average score across all observations. Then the variance of S is
given as follows:
n1 n2
Var ( S ) = -------------------n(n – 1)
n

( α ( R j ) – ave )
2
j=1
One-Way ChiSquare Approximations
Note: The ChiSquare test based on the Wilcoxon scores is knows as the Kruskal-Wallis test.
The notation used in this section is defined in “Notation” on page 206. The following
quantities are used in calculating the ChiSquare statistic:
Ti The total of the scores for the ith level of X.
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E(Ti) The expected value of the total score for level i under the null hypothesis of no
difference in levels, given as follows:
ni
E ( T i ) = ----n
n

α ( Rj )
j=1
Var(T) Define ave to be the average score across all observations. Then the variance of T is
given as follows:
1
Var ( T ) = ----------------(n – 1)
n

( α ( R j ) – ave )
2
j=1
The value of the test statistic is given below. This statistic is asymptotically ChiSquare on k - 1
degrees of freedom.
 k

2
C =   ( T i – E ( T i ) ) ⁄ n i ⁄ Var ( T )


i = 1

Chapter 7
Contingency Analysis
Examine Relationships between Two Categorical Variables
The Contingency or Fit Y by X platform lets you explore the distribution of a categorical
(nominal or ordinal) variable Y across the levels of a second categorical variable X. The
Contingency platform is the categorical by categorical personality of the Fit Y by X platform. The
analysis results include a mosaic plot, frequency counts, and proportions. You can
interactively perform additional analyses and tests on your data, such as an Analysis of Means
for Proportions, a correspondence analysis plot, and so on.
Figure 7.1 Example of Contingency Analysis
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Example of Contingency Analysis
Chapter 7
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Example of Contingency Analysis
This example uses the Car Poll.jmp sample data table, which contains data collected from car
polls. The data include respondent attributes: sex, marital status, and age. The data also
include attributes of the respondent’s car: country of origin, the size, and the type. Examine
the relationship between car sizes (small, medium, and large) and the cars’ country of origin.
1. Select Help > Sample Data Library and open Car Poll.jmp.
2. Select Analyze > Fit Y by X.
3. Select size and click Y, Response.
4. Select country and click X, Factor.
5. Click OK.
Figure 7.2 Example of Contingency Analysis
Chapter 7
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Contingency Analysis
Launch the Contingency Platform
211
From the mosaic plot and legend in Figure 7.2, notice the following:
•
Very few Japanese cars fall into the Large size category.
•
The majority of the European cars fall into the Small and Medium size categories.
•
The majority of the American cars fall into the Large and Medium size categories.
Launch the Contingency Platform
You can perform a contingency analysis using either the Fit Y by X platform or the
Contingency platform. The two approaches are equivalent.
•
To launch the Fit Y by X platform, select Analyze > Fit Y by X.
or
•
To launch the Contingency platform, from the JMP Starter window, click on the Basic
category and click Contingency.
Figure 7.3 The Contingency Launch Window
For more information about this launch window, see “Introduction to Fit Y by X” chapter on
page 95.
After you click OK, the Contingency report window appears. See “The Contingency Report”
on page 212.
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The Contingency Report
Chapter 7
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The Contingency Report
To produce the plot shown in Figure 7.4, follow the instructions in “Example of Contingency
Analysis” on page 210.
Figure 7.4 Example of a Contingency Report
Note: Any rows that are excluded in the data table are also hidden in the Mosaic Plot.
The Contingency report initially shows a Mosaic Plot, a Contingency Table, and a Tests report.
You can add other analyses and tests using the options that are located within the red triangle
menu. For details about all of these reports and options, see “Contingency Platform Options”
on page 213.
Chapter 7
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Contingency Analysis
Contingency Platform Options
213
Contingency Platform Options
Note: The Fit Group menu appears if you have specified multiple Y variables. Menu options
allow you to arrange reports or order them by RSquare. See the Standard Least Squares
chapter in the Fitting Linear Models book for more information.
Use the platform options within the red triangle menu next to Contingency Analysis to
perform additional analyses and tests on your data.
Mosaic Plot A graphical representation of the data in the Contingency Table. See “Mosaic
Plot” on page 214.
Contingency Table A two-way frequency table. There is a row for each factor level and a
column for each response level. See “Contingency Table” on page 216.
Analogous to the Analysis of Variance table for continuous data. The tests show that
the response level rates are the same across X levels. See “Tests” on page 218.
Tests
Set α level Changes the alpha level used in confidence intervals. Select one of the common
values (0.10, 0.05, 0.01) or select a specific value using the Other option.
Analysis of Means for Proportions Only appears if the response has exactly two levels.
Compares response proportions for the X levels to the overall response proportion. See
“Analysis of Means for Proportions” on page 219.
Correspondence Analysis Shows which rows or columns of a frequency table have similar
patterns of counts. In the correspondence analysis plot, there is a point for each row and
for each column of the contingency table. See “Correspondence Analysis” on page 220.
Cochran Mantel Haenszel Tests if there is a relationship between two categorical variables
after blocking across a third classification. See “Cochran-Mantel-Haenszel Test” on
page 221.
Only appears when both the X and Y variables have the same levels.
Displays the Kappa statistic (Agresti 1990), its standard error, confidence interval,
hypothesis test, and Bowker’s test of symmetry, also know as McNemarʹs test. See
“Agreement Statistic” on page 222.
Agreement Statistic
Calculates risk ratios. Appears only when both the X and Y variables have only
two levels. See “Relative Risk” on page 222.
Relative Risk
Odds Ratio Appears only when there are exactly two levels for each variable. Produces a
report of the odds ratio. For more information, see “Odds Ratio Option” on page 236.
The report also gives a confidence interval for this ratio. You can change the alpha level
using the Set α Level option.
Two Sample Test for Proportions Performs a two-sample test for proportions. This test
compares the proportions of the Y variable between the two levels of the X variable.
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Mosaic Plot
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Appears only when both the X and Y variables have only two levels. See “Two Sample Test
for Proportions” on page 223.
Describes the association between the variables in the contingency
table. See “Measures of Association” on page 223.
Measures of Association
Tests for trends in binomial proportions across levels of a
single variable. This test is appropriate only when one variable has two levels and the
other variable is ordinal. See “Cochran Armitage Trend Test” on page 225.
Cochran Armitage Trend Test
Exact Test
Provides exact versions of the following tests:
– Fisher’s Test
– Cochran Armitage Trend Test
– Agreement Test
See “Exact Test” on page 225.
Mosaic Plot
The mosaic plot is a graphical representation of the two-way frequency table or Contingency
Table. A mosaic plot is divided into rectangles, so that the vertical length of each rectangle is
proportional to the proportions of the Y variable in each level of the X variable. The mosaic
plot was introduced by Hartigan and Kleiner in 1981 and refined by Friendly (1994).
To produce the plot shown in Figure 7.5, follow the instructions in “Example of Contingency
Analysis” on page 210.
Figure 7.5 Example of a Mosaic Plot
Chapter 7
Basic Analysis
Contingency Analysis
Mosaic Plot
215
Note the following about the mosaic plot in Figure 7.5:
•
The proportions on the x-axis represent the number of observations for each level of the X
variable, which is country.
•
The proportions on the y-axis at right represent the overall proportions of Small, Medium,
and Large cars for the combined levels (American, European, and Japanese).
•
The scale of the y-axis at left shows the response probability, with the whole axis being a
probability of one (representing the total sample).
Clicking on a rectangle in the mosaic plot highlights the selection and highlights the
corresponding data in the associated data table.
Replace variables in the mosaic plot by dragging and dropping a variable, in one of two ways:
swap existing variables by dragging and dropping a variable from one axis to the other axis;
or, click on a variable in the Columns panel of the associated data table and drag it onto an
axis.
Pop-Up Menu
Right-click on the mosaic plot to change colors and label the cells.
Set Colors
Shows the current assignment of colors to levels. See “Set Colors” on page 215.
Cell Labeling Specify a label to be drawn in the mosaic plot. Select one of the following
options:
Unlabeled
Shows no labels, and removes any of the other options.
Show Counts Shows the number of observations in each cell.
Show Percents Shows the percent of observations in each cell.
Show Labels Shows the levels of the Y variable corresponding to each cell.
Show Row Labels
Shows the row labels for all of the rows represented by the cell.
Note: For descriptions of the remainder of the right-click options, see the JMP Reports chapter
in the Using JMP book.
Set Colors
When you select the Set Colors option, the Select Colors for Values window appears.
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Chapter 7
Basic Analysis
Figure 7.6 Select Colors for Values Window
The default mosaic colors depend on whether the response column is ordinal or nominal, and
whether there is an existing Value Colors column property. To change the color for any level,
click on the oval in the second column of colors and pick a new color.
Description of the Select Colors for Values Window
Macros
Computes a color gradient between any two levels, as follows:
– If you select a range of levels (by dragging the mouse over the levels that you want to
select, or pressing the Shift key and clicking the first and last level), the Gradient
Between Selected Points option applies a color gradient to the levels that you have
selected.
– The Gradient Between Ends option applies a gradient to all levels of the variable.
– Undo any of your changes by selecting Revert to Old Colors.
Color Theme Changes the colors for each value based on a color theme.
Save Colors to Column If you select this check box, a new column property (Value Colors) is
added to the column in the associated data table. To edit this property from the data table,
select Cols > Column Info.
Contingency Table
The Contingency Table is a two-way frequency table. There is a row for each factor level and a
column for each response level.
To produce the plot shown in Figure 7.7, follow the instructions in “Example of Contingency
Analysis” on page 210.
Chapter 7
Basic Analysis
Contingency Analysis
Contingency Table
217
Figure 7.7 Example of a Contingency Table
Note the following about Contingency tables:
•
The Count, Total%, Col%, and Row% correspond to the data within each cell that has row
and column headings (such as the cell under American and Large).
•
The last column contains the total counts for each row and percentages for each row.
•
The bottom row contains total counts for each column and percentages for each column.
For example, in Figure 7.7, focus on the cars that are large and come from America. The
following table explains the conclusions that you can make about these cars using the
Contingency Table.
Table 7.1 Conclusions Based on Example of a Contingency Table
Number
Description
Label in Table
36
Number of cars that are both large and come from
America
Count
11.88%
Percentage of all cars that are both large and come
from America (36/303)a.
Total%
85.71%
Percentage of large cars that come from America
(36/42)b
Col%
31.30%
Percentage of American cars that are large (36/115)c.
Row%
37.95%
Percentage of all cars that come from America
(115/303).
(none)
13.86%
Percentage of all cars that are large (42/303).
(none)
a. 303 is the total number of cars in the poll.
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b. 42 is the total number of large cars in the poll.
c. 115 is the total number of American cars in the poll.
Tip: To show or hide data in the Contingency Table, from the red triangle menu next to
Contingency Table, select the option that you want to show or hide.
Description of the Contingency Table
Count
Total%
Cell frequency, margin total frequencies, and grand total (total sample size).
Percent of cell counts and margin totals to the grand total.
Row% Percent of each cell count to its row total.
Col% Percent of each cell count to its column total.
Expected Expected frequency (E) of each cell under the assumption of independence.
Computed as the product of the corresponding row total and column total divided by the
grand total.
Deviation
Observed cell frequency (O) minus the expected cell frequency (E).
Cell Chi Square Chi-square values computed for each cell as (O - E)2 / E.
Col Cum Cumulative column total.
Col Cum% Cumulative column percentage.
Row Cum Cumulative row total.
Row Cum% Cumulative row percentage.
Tests
The Tests report shows the results for two tests to determine whether the response level rates
are the same across X levels.
To produce the report shown in Figure 7.8, follow the instructions in “Example of
Contingency Analysis” on page 210.
Figure 7.8 Example of a Tests Report
Chapter 7
Basic Analysis
Contingency Analysis
Analysis of Means for Proportions
219
Note the following about the Chi-square statistics:
•
When both categorical variables are responses (Y variables), the Chi-square statistics test
that they are independent.
•
You might have a Y variable with a fixed X variable. In this case, the Chi-square statistics
test that the distribution of the Y variable is the same across each X level.
Description of the Tests Report
N
Total number of observations.
DF
Records the degrees of freedom associated with the test. The degrees of freedom are
equal to (c - 1)(r - 1), where c is the number of columns and r is the number of rows.
-LogLike Negative log-likelihood, which measures fit and uncertainty (much like sums of
squares in continuous response situations).
Rsquare (U) Portion of the total uncertainty attributed to the model fit.
– An R2 of 1 means that the factors completely predict the categorical response.
– An R2 of 0 means that there is no gain from using the model instead of fixed
background response rates.
For more information, see “Tests Report” on page 237.
Test Lists two Chi-square statistical tests of the hypothesis that the response rates are the
same in each sample category. For more information, see “Tests Report” on page 237.
Prob>ChiSq Lists the probability of obtaining, by chance alone, a Chi-square value greater
than the one computed if no relationship exists between the response and factor. If both
variables have only two levels, Fisher’s exact probabilities for the one-tailed tests and the
two-tailed test also appear.
Fisher’s Exact Test
This report gives the results of Fisher’s exact test for a 2x2 table. The results appear
automatically for 2x2 tables. For more details about Fisher’s exact test, and for details about
the test for r x c tables, see “Exact Test” on page 225.
Analysis of Means for Proportions
If the response has two levels, you can use this option to compare response proportions for the
X levels to the overall response proportion. This method uses the normal approximation to the
binomial. Therefore, if the sample sizes are too small, a warning appears in the results.
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Note: For a description of Analysis of Means methods, see the document by Nelson, Wludyka,
and Copeland (2005).
Set Alpha Level
Selects the alpha level used in the analysis.
Show Summary Report Produces a report that shows the response proportions with decision
limits for each level of the X variable. The report indicates whether a limit has been
exceeded.
Switch Response Level for Proportion
Changes the response category used in the analysis.
Shows or hides the decision limits, decision limit shading, center line, and
point options.
Display Options
Related Information
•
“Example of Analysis of Means for Proportions” on page 226
Correspondence Analysis
Correspondence analysis is a graphical technique to show which rows or columns of a
frequency table have similar patterns of counts. In the correspondence analysis plot, there is a
point for each row and for each column. Use Correspondence Analysis when you have many
levels, making it difficult to derive useful information from the mosaic plot.
Understanding Correspondence Analysis Plots
The row profile can be defined as the set of rowwise rates, or in other words, the counts in a
row divided by the total count for that row. If two rows have very similar row profiles, their
points in the correspondence analysis plot are close together. Squared distances between row
points are approximately proportional to Chi-square distances that test the homogeneity
between the pair of rows.
Column and row profiles are alike because the problem is defined symmetrically. The distance
between a row point and a column point has no meaning. However, the directions of columns
and rows from the origin are meaningful, and the relationships help interpret the plot.
Correspondence Analysis Options
Use the options in the red triangle menu next to Correspondence Analysis to produce a 3-D
scatterplot and add column properties to the data table.
3D Correspondence Analysis Produces a 3-D scatterplot.
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Save Value Ordering Takes the order of the levels sorted by the first correspondence score
coefficient and makes a column property for both the X and Y columns.
The Details Report
The Details report contains statistical information about the correspondence analysis and
shows the values used in the plot.
Provides the singular value decomposition of the contingency table. For the
formula, see “Details Report in Correspondence Analysis” on page 238.
Singular Value
Inertia Lists the square of the singular values, reflecting the relative variation accounted for
in the canonical dimensions.
Portion
Portion of inertia with respect to the total inertia.
Cumulative Shows the cumulative portion of inertia. If the first two singular values capture
the bulk of the inertia, then the 2-D correspondence analysis plot is sufficient to show the
relationships in the table.
X variable c1, c2, c3 The values plotted on the Correspondence Analysis plot (Figure 7.11).
Y variable c1, c2, c3 The values plotted on the Correspondence Analysis plot (Figure 7.11).
Related Information
•
“Example of Correspondence Analysis” on page 227
Cochran-Mantel-Haenszel Test
The Cochran-Mantel-Haenszel test discovers if there is a relationship between two categorical
variables after blocking across a third classification.
Correlation of Scores Applicable when both Y or X are ordinal or interval. The alternative
hypothesis is that there is a linear association between Y and X in at least one level of the
blocking variable.
Row Score by Col Categories Applicable when Y is ordinal or interval. The alternative
hypothesis is that, for at least one level of the blocking variable, the mean scores of the r
rows are unequal.
Col Score by Row Categories Applicable when X is ordinal or interval. The alternative
hypothesis is that, for at least one level of the blocking variable, the mean scores of the c
columns are unequal.
General Assoc. of Categories Tests that for at least one level of the blocking variable, there is
some type of association between X and Y.
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Related Information
•
“Example of a Cochran Mantel Haenszel Test” on page 229
Agreement Statistic
When the two variables have the same levels, the Agreement Statistic option is available. This
option shows the Kappa statistic (Agresti 1990), its standard error, confidence interval,
hypothesis test, and Bowker’s test of symmetry.
The Kappa statistic and associated p-value given in this section are approximate. An exact
version of the agreement test is available. See “Exact Test” on page 225.
Kappa Shows the Kappa statistic.
Std Err Shows the standard error of the Kappa statistic.
Lower 95% Shows the lower endpoint of the confidence interval for Kappa.
Upper 95%
Shows the upper endpoint of the confidence interval for Kappa.
Prob>Z Shows the p-value for a one-sided test for Kappa. The null hypothesis tests if Kappa
equals zero.
Prob>|Z|
Shows the p-value for a two-sided test for Kappa.
ChiSquare Shows the test statistic for Bowker’s test. For Bowker’s test of symmetry, the null
hypothesis is that the probabilities in the square table satisfy symmetry, or that pij=pji for all
pairs of table cells. When both X and Y have two levels, this test is equal to McNemar’s
test.
Prob>ChiSq Shows the p-value for the Bowker’s test.
Related Information
•
“Example of the Agreement Statistic Option” on page 231
•
“Agreement Statistic Option” on page 236
Relative Risk
Calculate risk ratios for 2x2 contingency tables using the Relative Risk option. Confidence
intervals also appear in the report. You can find more information about this method in
Agresti (1990) section 3.4.2.
The Choose Relative Risk Categories window appears when you select the Relative Risk
option. You can select a single response and factor combination, or you can calculate the risk
ratios for all combinations of response and factor levels.
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Related Information
•
“Example of the Relative Risk Option” on page 232
Two Sample Test for Proportions
When both the X and Y variables have two levels, you can request a confidence interval for a
difference between two proportions. It also outputs the test corresponding to the confidence
interval.
Description Shows the test being performed.
Proportion Difference Shows the difference in the proportions between the levels of the X
variable.
Lower 95% Shows the lower endpoint of the confidence interval for the difference. Based on
the adjusted Wald confidence interval.
Shows the upper endpoint of the confidence interval for the difference. Based on
the adjusted Wald confidence interval.
Upper 95%
Adjusted Wald Test Shows two-tailed and one-tailed tests.
Prob Shows the p-values for the tests.
Response <variable> category of interest
Select which response level to use in the test.
Related Information
•
“Example of a Two Sample Test for Proportions” on page 233
Measures of Association
You can request several statistics that describe the association between the variables in the
contingency table by selecting the Measures of Association option.
Gamma Based on the number of concordant and discordant pairs and ignores tied pairs.
Takes values in the range -1 to 1.
Kendall’s Tau-b Similar to Gamma and uses a correction for ties. Takes values in the range -1
to 1.
Stuart’s Tau-c Similar to Gamma and uses an adjustment for table size and a correction for
ties. Takes values in the range -1 to 1.
Somers’ D An asymmetric modification of Tau-b.
– The C|R denotes that the row variable X is regarded as an independent variable and
the column variable Y is regarded as dependent.
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– Similarly, the R|C denotes that the column variable Y is regarded as an independent
variable and the row variable X is dependent.
Somers’ D differs from Tau-b in that it uses a correction for ties only when the pair is tied
on the independent variable. It takes values in the range -1 to 1.
Lambda Asymmetric Differs for C|R and R|C.
– For C|R, is interpreted as the probable improvement in predicting the column variable
Y given knowledge of the row variable X.
– For R|C, is interpreted as the probable improvement in predicting the row variable X
given knowledge about the column variable Y.
Takes values in the range 0 to 1.
Lambda Symmetric Loosely interpreted as the average of the two Lambda Asymmetric
measures. Takes values in the range 0 to 1.
Uncertainty Coef –For C|R, is the proportion of uncertainty in the column variable Y that is
explained by the row variable X.
– For R|C, is interpreted as the proportion of uncertainty in the row variable X that is
explained by the column variable Y.
Takes values in the range 0 to 1.
Uncertainty Coef Symmetric
Symmetric version of the two Uncertainty Coef measures. Takes
values in the range 0 to 1.
Notes:
Each statistic appears with its standard error and confidence interval.
•
Gamma, Kendall’s Tau-b, Stuart’s Tau-c, and Somers’ D are measures of ordinal association
that consider whether the variable Y tends to increase as X increases. They classify pairs of
observations as concordant or discordant. A pair is concordant if an observation with a
larger value of X also has a larger value of Y. A pair is discordant if an observation with a
larger value of X has a smaller value of Y. These measures are appropriate only when both
variables are ordinal.
•
The Lambda and Uncertainty measures are appropriate for ordinal and nominal variables.
For details about measures of association, see the following references:
•
Brown and Benedetti (1977)
•
Goodman and Kruskal (1979)
•
Kendall and Stuart (1979)
•
Snedecor and Cochran (1980)
•
Somers (1962)
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Cochran Armitage Trend Test
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Related Information
•
“Example of the Measures of Association Option” on page 234
Cochran Armitage Trend Test
This Cochran Armitage Trend tests for trends in binomial proportions across the levels of a
single variable. This test is appropriate only when one variable has two levels and the other
variable is ordinal. The two-level variable represents the response, and the other represents an
explanatory variable with ordered levels. The null hypothesis is the hypothesis of no trend,
which means that the binomial proportion is the same for all levels of the explanatory
variable.
The test statistic and p-values given in this section are approximate. An exact version of the
trend test is available. See “Exact Test” on page 225.
Related Information
•
“Example of the Cochran Armitage Trend Test” on page 235
Exact Test
The following Exact tests are available in the Contingency platform:
Performs Fisher’s Exact test for an r x c table. This is a test for association
between two variables. Fisher’s exact test assumes that the row and column totals are
fixed, and uses the hypergeometric distribution to compute probabilities.
Fisher’s Exact Test
This test does not depend on any large-sample distribution assumptions. This means it is
appropriate for situations where the Likelihood Ratio and Pearson tests become less
reliable, like for small sample sizes or sparse tables.
The report includes the following information:
Table Probability (P) Gives the probability for the observed table. This is not the p-value
for the test.
Two-sided Prob ≤ P Gives the p-value for the two-sided test.
For 2x2 tables, the Fisher’s Exact test is automatically performed, unless one row or
column contains all zeros (in this case, the test can not be calculated). See “Tests” on
page 218.
Exact Cochran Armitage Trend Test Performs the exact version of the Cochran Armitage
Trend Test. This test is available only when one of the variables has two levels. For more
details about the trend test, see “Cochran Armitage Trend Test” on page 225.
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Exact Agreement Test Performs an exact test for testing agreement between variables. This is
an exact test for the Kappa statistic. This is available only when the two variables have the
same levels. For more details about agreement testing, see “Agreement Statistic” on
page 222.
Additional Examples of the Contingency Platform
This section contains additional examples using the options in the Contingency platform.
Example of Analysis of Means for Proportions
This example uses the Office Visits.jmp sample data table, which records late and on-time
appointments for six clinics in a geographic region. 60 random appointments were selected
from 1 week of records for each of the six clinics. To be considered on-time, the patient must
be taken to an exam room within five minutes of their scheduled appointment time. Examine
the proportion of patients that arrived on-time to their appointment.
1. Select Help > Sample Data Library and open Office Visits.jmp.
2. Select Analyze > Fit Y by X.
3. Select On Time and click Y, Response.
4. Select Clinic and click X, Factor.
5. Select Frequency and click Freq.
6. Click OK.
7. From the red triangle menu next to Contingency Analysis, select Analysis of Means for
Proportions.
8. From the red triangle menu next to Analysis of Means for Proportions, select Show
Summary Report and Switch Response Level for Proportion.
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Figure 7.9 Example of Analysis of Means for Proportions
Figure 7.9 shows the proportion of patients who were on-time from each clinic. From
Figure 7.9, notice the following:
•
The proportion of on-time arrivals is the highest for clinic F, followed by clinic B.
•
Clinic D has the lowest proportion of on-time arrivals, followed by clinic A.
•
Clinic E and clinic C are close to the average, and do not exceed the decision limits.
Example of Correspondence Analysis
This example uses the Cheese.jmp sample data table, which is taken from the Newell cheese
tasting experiment, reported in McCullagh and Nelder (1989). The experiment records counts
more than nine different response levels across four different cheese additives.
1. Select Help > Sample Data Library and open Cheese.jmp.
2. Select Analyze > Fit Y by X.
3. Select Response and click Y, Response.
The Response values range from one to nine, where one is the least liked, and nine is the
best liked.
4. Select Cheese and click X, Factor.
A, B, C, and D represent four different cheese additives.
5. Select Count and click Freq.
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6. Click OK.
Figure 7.10 Mosaic Plot for the Cheese Data
From the mosaic plot in Figure 7.10, you notice that the distributions do not appear alike.
However, it is challenging to make sense of the mosaic plot across nine levels. A
correspondence analysis can help define relationships in this type of situation.
7. To see the correspondence analysis plot, from the red triangle menu next to Contingency
Analysis, select Correspondence Analysis.
Figure 7.11 Example of a Correspondence Analysis Plot
least liked
neutral
most liked
Figure 7.11 shows the correspondence analysis graphically, with the plot axes labeled c1 and
c2. Notice the following:
•
c1 seems to correspond to a general satisfaction level. The cheeses on the c1 axis go from
least liked at the top to most liked at the bottom.
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Additional Examples of the Contingency Platform
•
Cheese D is the most liked cheese, with responses of 8 and 9.
•
Cheese B is the least liked cheese, with responses of 1,2, and 3.
•
Cheeses C and A are in the middle, with responses of 4,5,6, and 7.
229
8. From the red triangle menu next to Correspondence Analysis, select 3D Correspondence
Analysis.
Figure 7.12 Example of a 3-D Scatterplot
From Figure 7.12, notice the following:
•
Looking at the c1 axis, responses 1 through 5 appear to the right of 0 (positive). Responses
6 through 9 appear to the left of 0 (negative).
•
Looking at the c2 axis, A and C appear to the right of 0 (positive). B and D appear to the
left of 0 (negative).
•
You can conclude that c1 corresponds to the general satisfaction (from least to most liked).
Example of a Cochran Mantel Haenszel Test
This example uses the Hot Dogs.jmp sample data table. Examine the relationship between hot
dog type and taste.
1. Select Help > Sample Data Library and open Hot Dogs.jmp.
2. Select Analyze > Fit Y by X.
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3. Select Type and click Y, Response.
4. Select Taste and click X, Factor.
5. Click OK.
6. From the red triangle menu next to Contingency Analysis, select
Cochran Mantel Haenszel.
7. Select Protein/Fat as the grouping variable and click OK.
Figure 7.13 Example of a Cochran-Mantel-Haenszel Test
From Figure 7.13, you notice the following:
•
The Tests report shows a marginally significant Chi-square probability of about 0.0799,
indicating some significance in the relationship between hot dog taste and type.
•
The Cochran Mantel Haenszel report shows that the p-value for the general association of
categories is 0.2816, which is much larger than 5%.
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Example of the Agreement Statistic Option
This example uses the Attribute Gauge.jmp sample data table. The data gives results from three
people (raters) rating fifty parts three times each. Examine the relationship between raters A
and B.
1. Select Help > Sample Data Library and open Attribute Gauge.jmp.
2. Select Analyze > Fit Y by X.
3. Select A and click Y, Response.
4. Select B and click X, Factor.
5. Click OK.
6. From the red triangle menu next to Contingency Analysis, select Agreement Statistic.
Figure 7.14 Example of the Agreement Statistic Report
From Figure 7.14, you notice that the agreement statistic of 0.86 is high (close to 1) and the
p-value of <.0001 is small. This reinforces the high agreement seen by looking at the diagonal
of the contingency table. Agreement between the raters occurs when both raters give a rating
of 0 or both give a rating of 1.
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Example of the Relative Risk Option
This example uses the Car Poll.jmp sample data table. Examine the relative probabilities of
being married and single for the participants in the poll.
1. Select Help > Sample Data Library and open Car Poll.jmp.
2. Select Analyze > Fit Y by X.
3. Select marital status and click Y, Response.
4. Select sex and click X, Factor.
5. Click OK.
6. From the red triangle menu next to Contingency Analysis, select Relative Risk.
The Choose Relative Risk Categories window appears.
Figure 7.15 The Choose Relative Risk Categories Window
Note the following about the Choose Relative Risk Categories window:
•
If you are interested in only a single response and factor combination, you can select that
here. For example, if you clicked OK in the window in Figure 7.15, the calculation would
be as follows:
P ( Y = Married X = Female )
--------------------------------------------------------------------------P ( Y = Married X = Male )
•
If you would like to calculate the risk ratios for all ( 2 × 2 =4) combinations of response and
factor levels, select the Calculate All Combinations check box. See Figure 7.16.
7. Ask for all combinations by selecting the Calculate All Combinations check box. Leave all
other default selections as is.
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Figure 7.16 Example of the Risk Ratio Report
To see how the relative risk is calculated, proceed as follows:
1. Examine the first entry in the Relative Risk report, which is
P(Married|Female)/P(Married|Male).
2. You can find these probabilities in the Contingency Table. Since the probabilities are
computed based on two levels of sex, which differs across the rows of the table, use the
Row% to read the probabilities, as follows:
P(Married|Female)=0.6884
P(Married|Male) = 0.6121
Therefore, the calculations are as follows:
0.6884
0.6121
P(Married|Female)/P(Married|Male) = ---------------- = 1.1247
Example of a Two Sample Test for Proportions
This example uses the Car Poll.jmp sample data table. Examine the probability of being
married for males and females.
1. Select Help > Sample Data Library and open Car Poll.jmp.
2. Select Analyze > Fit Y by X.
3. Select marital status and click Y, Response.
4. Select sex and click X, Factor.
5. Click OK.
6. From the red triangle menu next to Contingency Analysis, select Two Sample Test for
Proportions.
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Chapter 7
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Figure 7.17 Example of the Two Sample Test for Proportions Report
In this example, you are comparing the probability of being married between females and
males. See the Row% in the Contingency Table to obtain the following:
P(Married|Female)=0.6884
P(Married|Male) = 0.6121
The difference between these two numbers, 0.0763, is the Proportion Difference shown in the
report. The two-sided confidence interval is [-0.03175, 0.181621]. The p-value by the adjusted
Wald method corresponding to the confidence interval is 0.1686, which is close to the p-value
(0.1665) by Pearson’s Chi-square test. Generally, Pearson’s Chi-square test is more popular
than the modified Wald’s test for testing the difference of two proportions.
Example of the Measures of Association Option
This example uses the Car Poll.jmp sample data table. Examine the probability of being
married for males and females.
1. Select Help > Sample Data Library and open Car Poll.jmp.
2. Select Analyze > Fit Y by X.
3. Select marital status and click Y, Response.
4. Select sex and click X, Factor.
5. Click OK.
6. From the red triangle menu next to Contingency Analysis, select Measures of Association.
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Additional Examples of the Contingency Platform
235
Figure 7.18 Example of the Measures of Association Report
Since the variables that you want to examine (sex and marital status) are nominal, use the
Lambda and Uncertainty measures. All of them are small, so it seems that there is a weak
association between sex and marital status.
Example of the Cochran Armitage Trend Test
1. Select Help > Sample Data Library and open Car Poll.jmp.
For the purposes of this test, change size to an ordinal variable:
2. In the Columns panel, right-click on the icon next to size and select Ordinal.
3. Select Analyze > Fit Y by X.
4. Select sex and click Y, Response.
5. Select size and click X, Factor.
6. Click OK.
7. From the red triangle menu next to Contingency Analysis, select Cochran Armitage Trend
Test.
Figure 7.19 Example of the Cochran Armitage Trend Test Report
The two-sided p-value (0.7094) is large. From this, you cannot conclude that there is a
relationship in the proportion of male and females that purchase different sizes of cars.
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Statistical Details for the Contingency Platform
Chapter 7
Basic Analysis
Statistical Details for the Contingency Platform
This section contains statistical details for selected options and reports in the Contingency
platform.
Agreement Statistic Option
Viewing the two response variables as two independent ratings of the n subjects, the Kappa
coefficient equals +1 when there is complete agreement of the raters. When the observed
agreement exceeds chance agreement, the Kappa coefficient is positive, with its magnitude
reflecting the strength of agreement. Although unusual in practice, Kappa is negative when
the observed agreement is less than chance agreement. The minimum value of Kappa is
between -1 and 0, depending on the marginal proportions.
Quantities associated with the Kappa statistic are computed as follows:
P0 – Pe
κ̂ = ------------------ where P 0 =
1 – Pe
 p ii and Pe
i
=
 p i. p .i
i
The asymptotic variance of the simple kappa coefficient is estimated by the following:
2
2
A+B–C
var = -------------------------- where A =  p [ 1 – ( p + p ) ( 1 – κ̂ ) ] , B = ( 1 – κ̂ ) 2   p ij ( p .i + p j. ) and
ii
i.
.i
2
i≠j
( 1 – Pe ) n
i
C = [ κ̂ – P e ( 1 – κ̂ ) ]
2
See Fleiss, Cohen, and Everitt (1969).
For Bowker’s test of symmetry, the null hypothesis is that the probabilities in the two-by-two
table satisfy symmetry (pij=pji).
Odds Ratio Option
The Odds Ratio is calculated as follows:
p 11 × p 22
----------------------p 12 × p 21
where pij is the count in the ith row and jth column of the 2x2 table.
Chapter 7
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Statistical Details for the Contingency Platform
237
Tests Report
Rsquare (U)
Rsquare (U) is computed as follows:
– log likelihood for Model
------------------------------------------------------------------------------------------– log likelihood for Corrected Total
The total negative log-likelihood is found by fitting fixed response rates across the total
sample.
Test
The two Chi-square tests are as follows:
The Likelihood Ratio Chi-square test is computed as twice the negative log-likelihood for
Model in the Tests table. Some books use the notation G2 for this statistic. The difference of
two negative log-likelihoods, one with whole-population response probabilities and one with
each-population response rates, is written as follows:
2
G = 2
 ( –nij ) ln ( p j ) –  –nij ln ( p ij ) where pij
ij
ij
n ij
Nj
= ------ and p j = -----N
N
This formula can be more compactly written as follows:
 n ij
2
n ln -----G = 2   ij  e 
ij
i j
The Pearson Chi-square is calculated by summing the squares of the differences between the
observed and expected cell counts. The Pearson Chi-square exploits the property that
frequency counts tend to a normal distribution in very large samples. The familiar form of this
Chi-square statistic is as follows:
2
χ =
(O – E)
2
 --------------------E
where O is the observed cell counts and E is the expected cell counts. The summation is over
all cells. There is no continuity correction done here, as is sometimes done in 2-by-2 tables.
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Details Report in Correspondence Analysis
Lists the singular values of the following equation:
– 0.5
Dr
– 0.5
( P – rc' ) D c
where:
•
P is the matrix of counts divided by the total frequency
•
r and c are row and column sums of P
•
the Ds are diagonal matrices of the values of r and c
Chapter 7
Basic Analysis
Chapter 8
Logistic Analysis
Examine Relationships between a Categorical Y and a Continuous X
Variable
The Logistic platform fits the probabilities for response categories to a continuous x predictor.
The fitted model estimates probabilities for each x value. The Logistic platform is the nominal
or ordinal by continuous personality of the Fit Y by X platform. There is a distinction between
nominal and ordinal responses on this platform:
•
Nominal logistic regression estimates a set of curves to partition the probability among the
responses.
•
Ordinal logistic regression models the probability of being less than or equal to a given
response. This has the effect of estimating a single logistic curve, which is shifted
horizontally to produce probabilities for the ordered categories. This model is less
complex and is recommended for ordered responses.
Figure 8.1 Examples of Logistic Regression
Ordinal Logistic Regression
Nominal Logistic Regression
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Overview of Logistic Regression
Chapter 8
Basic Analysis
Overview of Logistic Regression
Logistic regression has a long tradition with widely varying applications such as modeling
dose-response data and purchase-choice data. Unfortunately, many introductory statistics
courses do not cover this fairly simple method. Many texts in categorical statistics cover it
(Agresti 1998), in addition to texts on logistic regression (Hosmer and Lemeshow 1989). Some
analysts use the method with a different distribution function, the normal. In that case, it is
called probit analysis. Some analysts use discriminant analysis instead of logistic regression
because they prefer to think of the continuous variables as Ys and the categories as Xs and
work backwards. However, discriminant analysis assumes that the continuous data are
normally distributed random responses, rather than fixed regressors.
Simple logistic regression is a more graphical and simplified version of the general facility for
categorical responses in the Fit Model platform. For examples of more complex logistic
regression models, see the Logistic Regression chapter in the Fitting Linear Models book.
Nominal Logistic Regression
Nominal logistic regression estimates the probability of choosing one of the response levels as
a smooth function of the x factor. The fitted probabilities must be between 0 and 1, and must
sum to 1 across the response levels for a given factor value.
In a logistic probability plot, the y-axis represents probability. For k response levels, k - 1
smooth curves partition the total probability (which equals 1) among the response levels. The
fitting principle for a logistic regression minimizes the sum of the negative natural logarithms
of the probabilities fitted to the response events that occur (that is, maximum likelihood).
Ordinal Logistic Regression
When Y is ordinal, a modified version of logistic regression is used for fitting. The cumulative
probability of being at or below each response level is modeled by a curve. The curves are the
same for each level except that they are shifted to the right or left.
The ordinal logistic model fits a different intercept, but the same slope, for each of r - 1
cumulative logistic comparisons, where r is the number of response levels. Each parameter
estimate can be examined and tested individually, although this is seldom of much interest.
The ordinal model is preferred to the nominal model when it is appropriate because it has
fewer parameters to estimate. In fact, it is practical to fit ordinal responses with hundreds of
response levels.
Chapter 8
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Logistic Analysis
Example of Nominal Logistic Regression
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Example of Nominal Logistic Regression
This example uses the Penicillin.jmp sample data table. The data in this example comes from an
experiment where 5 groups, each containing 12 rabbits, were injected with streptococcus
bacteria. Once the rabbits were confirmed to have the bacteria in their system, they were given
different doses of penicillin. You want to find out whether the natural log (In(dose)) of dosage
amounts has any effect on whether the rabbits are cured.
1. Select Help > Sample Data Library and open Penicillin.jmp.
2. Select Analyze > Fit Y by X.
3. Select Response and click Y, Response.
4. Select In(Dose) and click X, Factor.
Notice that JMP automatically fills in Count for Freq. Count was previously assigned the
role of Freq.
5. Click OK.
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Figure 8.2 Example of Nominal Logistic Report
The plot shows the fitted model, which is the predicted probability of being cured, as a
function of ln(dose). The p-value is significant, indicating that the dosage amounts have a
significant effect on whether the rabbits are cured.
Tip: To change the response level that is analyzed, use the Value Ordering column property.
Launch the Logistic Platform
To perform a Logistic analysis, do the following:
1. Select Analyze > Fit Y by X.
2. Enter a nominal or ordinal column for Y, Response.
3. Enter a continuous column for X, factor.
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The schematic indicates that you will be performing a logistic analysis.
Note: You can also launch a logistic analysis from the JMP Starter window. See “Launch
Specific Analyses from the JMP Starter Window” on page 97.
Figure 8.3 Fit Y by X Logistic Launch Window
When the response is binary and has a nominal modeling type, a Target Level menu appears
in the launch window. Use this menu to specify the level of the response whose probability
you want to model.
For more information about the Fit Y by X launch window, see “Introduction to Fit Y by X”
chapter on page 95.
After you click OK, the Logistic report appears. See “The Logistic Report” on page 243.
Data Structure
Your data can consist of unsummarized or summarized data:
Unsummarized data There is one row for each observation containing its X value and its Y
value.
Summarized data Each row represents a set of observations with common X and Y values.
The data table must contain a column that gives the counts for each row. Enter this column
as Freq in the launch window.
The Logistic Report
To produce the plot shown in Figure 8.4, follow the instructions in “Example of Nominal
Logistic Regression” on page 241.
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Figure 8.4 Example of a Logistic Report
The Logistic report window contains the Logistic plot, the Iterations report, the Whole Model
Test report, and the Parameter Estimates report.
Note: The red triangle menu provides more options that can add to the initial report window.
See “Logistic Platform Options” on page 247.
Logistic Plot
The logistic probability plot gives a complete picture of what the logistic model is fitting. At
each x value, the probability scale in the y direction is divided up (partitioned) into
probabilities for each response category. The probabilities are measured as the vertical
distance between the curves, with the total across all Y category probabilities summing to 1.
Replace variables in the plot in one of two ways: swap existing variables by dragging and
dropping a variable from one axis to the other axis; or, click on a variable in the Columns
panel of the associated data table and drag it onto an axis.
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Related Information
•
“Additional Example of a Logistic Plot” on page 252
Iterations
The Iterations report shows each iteration and the evaluated criteria that determine whether
the model has converged. Iterations appear only for nominal logistic regression.
Whole Model Test
The Whole Model Test report shows if the model fits better than constant response
probabilities. This report is analogous to the Analysis of Variance report for a continuous
response model. It is a specific likelihood ratio Chi-square test that evaluates how well the
categorical model fits the data.
The negative sum of natural logs of the observed probabilities is called the negative
log-likelihood (–LogLikelihood). The negative log-likelihood for categorical data plays the
same role as sums of squares in continuous data: twice the difference in the negative
log-likelihood from the model fitted by the data and the model with equal probabilities is a
Chi-square statistic. This test statistic examines the hypothesis that the x variable has no effect
on the responses.
Values of the RSquare (U) (sometimes denoted as R2) range from 0 to 1. High R2 values are
indicative of a good model fit, and are rare in categorical models.
The Whole Model Test report contains the following columns:
Model Sometimes called Source.
– The Reduced model only contains an intercept.
– The Full model contains all of the effects as well as the intercept.
– The Difference is the difference of the log-likelihoods of the full and reduced models.
DF
Records the degrees of freedom associated with the model.
–LogLikelihood
Measures variation, sometimes called uncertainty, in the sample.
Full (the full model) is the negative log-likelihood (or uncertainty) calculated after fitting
the model. The fitting process involves predicting response rates with a linear model and a
logistic response function. This value is minimized by the fitting process.
Reduced (the reduced model) is the negative log-likelihood (or uncertainty) for the case
when the probabilities are estimated by fixed background rates. This is the background
uncertainty when the model has no effects.
The difference of these two negative log-likelihoods is the reduction due to fitting the
model. Two times this value is the likelihood ratio Chi-square test statistic.
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For more information, see the Statistical Details appendix in the Fitting Linear Models book.
The likelihood ratio Chi-square test of the hypothesis that the model fits no better
than fixed response rates across the whole sample. It is twice the –LogLikelihood for the
Difference Model. It is two times the difference of two negative log-likelihoods, one with
whole-population response probabilities and one with each-population response rates.
For more information, see “Whole Model Test Report” on page 258.
Chi-Square
Prob>ChiSq The observed significance probability, often called the p value, for the
Chi-square test. It is the probability of getting, by chance alone, a Chi-square value greater
than the one computed. Models are often judged significant if this probability is below
0.05.
RSquare (U) The proportion of the total uncertainty that is attributed to the model fit, defined
as the Difference negative log-likelihood value divided by the Reduced negative
log-likelihood value. An RSquare (U) value of 1 indicates that the predicted probabilities
for events that occur are equal to one: There is no uncertainty in predicted probabilities.
Because certainty in the predicted probabilities is rare for logistic models, RSquare (U)
tends to be small. For more information, see “Whole Model Test Report” on page 258.
Note: RSquare (U) is also know as McFadden’s pseudo R-square.
AICc The corrected Akaike Information Criterion. See the Statistical Details appendix in the
Fitting Linear Models book.
BIC The Bayesian Information Criterion. See the Statistical Details appendix in the Fitting
Linear Models book.
Observations Sometimes called Sum Wgts. The total sample size used in computations. If
you specified a Weight variable, this is the sum of the weights.
Measure The available measures of fit are as follows:
compares the log-likelihoods from the fitted model and the constant
probability model.
Entropy RSquare
Generalized RSquare is a measure that can be applied to general regression models. It is
based on the likelihood function L and is scaled to have a maximum value of 1. The
Generalized RSquare measure simplifies to the traditional RSquare for continuous
normal responses in the standard least squares setting. Generalized RSquare is also
known as the Nagelkerke or Craig and Uhler R2, which is a normalized version of Cox
and Snell’s pseudo R2. See Nagelkerke (1991).
Mean -Log p is the average of -log(p), where p is the fitted probability associated with the
event that occurred.
RMSE is the root mean square error, where the differences are between the response and
p (the fitted probability for the event that actually occurred).
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Mean Abs Dev is the average of the absolute values of the differences between the
response and p (the fitted probability for the event that actually occurred).
Misclassification Rate is the rate for which the response category with the highest fitted
probability is not the observed category.
For Entropy RSquare and Generalized RSquare, values closer to 1 indicate a better fit. For
Mean -Log p, RMSE, Mean Abs Dev, and Misclassification Rate, smaller values indicate a
better fit.
Training The value of the measure of fit.
Definition The algebraic definition of the measure of fit.
Parameter Estimates
The nominal logistic model fits a parameter for the intercept and slope for each of k – 1 logistic
comparisons, where k is the number of response levels. The Parameter Estimates report lists
these estimates. Each parameter estimate can be examined and tested individually, although
this is seldom of much interest.
Term Lists each parameter in the logistic model. There is an intercept and a slope term for the
factor at each level of the response variable, except the last level.
Estimate Lists the parameter estimates given by the logistic model.
Std Error Lists the standard error of each parameter estimate. They are used to compute the
statistical tests that compare each term to zero.
Lists the Wald tests for the hypotheses that each of the parameters is zero. The
Wald Chi-square is computed as (Estimate/Std Error)2.
Chi-Square
Prob>ChiSq Lists the observed significance probabilities for the Chi-square tests.
Covariance of Estimates
Reports the estimated variances of the parameter estimates, and the estimated covariances
between the parameter estimates. The square root of the variance estimates is the same as
those given in the Std Error section.
Logistic Platform Options
Note: The Fit Group menu appears if you have specified multiple Y variables. Menu options
allow you to arrange reports or order them by RSquare. See the Standard Least Squares
chapter in the Fitting Linear Models book for more information.
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Odds Ratios Adds odds ratios to the Parameter Estimates report. For more details, see the
Logistic Regression chapter in the Fitting Linear Models book.
This option is available only for a response with two levels.
Prediction of x values from given y values. For more information, see
“Inverse Prediction” on page 250.
Inverse Prediction
This option is available only for a response with two levels.
Logistic Plot Shows or hides the logistic plot.
Plot Options
The Plot Options menu includes the following options:
Show Points Toggles the points on or off.
Show Rate Curve Is useful only if you have several points for each x-value. In these cases,
you get reasonable estimates of the rate at each value, and compare this rate with the
fitted logistic curve. To prevent too many degenerate points, usually at zero or one,
JMP only shows the rate value if there are at least three points at the x-value.
Line Color
Enables you to pick the color of the plot curves.
ROC Curve A Receiver Operating Characteristic curve is a plot of sensitivity by (1 –
specificity) for each value of x. See “ROC Curves” on page 249.
Lift Curve Produces a lift curve for the model. A lift curve shows the same information as a
ROC curve, but in a way to dramatize the richness of the ordering at the beginning. The
Y-axis shows the ratio of how rich that portion of the population is in the chosen response
level compared to the rate of that response level as a whole. See the Logistic Regression
chapter in the Fitting Linear Models book for details about lift curves.
Save Probability Formula Creates new data table columns that contain formulas. See “Save
Probability Formula” on page 250.
See the JMP Reports chapter in the Using JMP book for more information about the following
options:
Contains options that enable you to repeat or relaunch the analysis. In platforms that
support the feature, the Automatic Recalc option immediately reflects the changes that
you make to the data table in the corresponding report window.
Redo
Contains options that enable you to save a script that reproduces the report to
several destinations.
Save Script
Save By-Group Script Contains options that enable you to save a script that reproduces the
platform report for all levels of a By variable to several destinations. Available only when a
By variable is specified in the launch window.
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ROC Curves
Suppose you have an x value that is a diagnostic measurement and you want to determine a
threshold value of x that indicates the following:
•
A condition exists if the x value is greater than the threshold.
•
A condition does not exist if the x value is less than the threshold.
For example, you could measure a blood component level as a diagnostic test to predict a type
of cancer. Now consider the diagnostic test as you vary the threshold and, thus, cause more or
fewer false positives and false negatives. You then plot those rates. The ideal is to have a very
narrow range of x criterion values that best divides true negatives and true positives. The
Receiver Operating Characteristic (ROC) curve shows how rapidly this transition happens,
with the goal being to have diagnostics that maximize the area under the curve.
Two standard definitions used in medicine are as follows:
•
Sensitivity, the probability that a given x value (a test or measure) correctly predicts an
existing condition. For a given x, the probability of incorrectly predicting the existence of a
condition is 1 – sensitivity.
•
Specificity, the probability that a test correctly predicts that a condition does not exist.
A ROC curve is a plot of sensitivity by (1 – specificity) for each value of x. The area under the
ROC curve is a common index used to summarize the information contained in the curve.
When you do a simple logistic regression with a binary outcome, there is a platform option to
request a ROC curve for that analysis. After selecting the ROC Curve option, a window asks
you to specify which level to use as positive.
If a test predicted perfectly, it would have a value above which the entire abnormal population
would fall and below which all normal values would fall. It would be perfectly sensitive and
then pass through the point (0,1) on the grid. The closer the ROC curve comes to this ideal
point, the better its discriminating ability. A test with no predictive ability produces a curve
that follows the diagonal of the grid (DeLong, et al. 1988).
The ROC curve is a graphical representation of the relationship between false-positive and
true-positive rates. A standard way to evaluate the relationship is with the area under the
curve, shown below the plot in the report. In the plot, a yellow line is drawn at a 45 degree
angle tangent to the ROC Curve. This marks a good cutoff point under the assumption that
false negatives and false positives have similar costs.
Related Information
•
“Example of ROC Curves” on page 254
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Save Probability Formula
The Save Probability Formula option creates new data table columns. These data table
columns save the following:
•
formulas for linear combinations (typically called logits) of the x factor
•
prediction formulas for the response level probabilities
•
a prediction formula that gives the most likely response
Inverse Prediction
Inverse prediction is the opposite of prediction. It is the prediction of x values from given y
values. But in logistic regression, instead of a y value, you have the probability attributed to
one of the Y levels. This feature only works when there are two response categories (a binary
response).
The Fit Model platform also has an option that gives an inverse prediction with confidence
limits. The Standard Least Squares chapter in the Fitting Linear Models book gives more
information about inverse prediction.
Related Information
•
“Example of Inverse Prediction Using the Crosshair Tool” on page 255
•
“Example of Inverse Prediction Using the Inverse Prediction Option” on page 256
Additional Examples of Logistic Regression
This section contains additional examples using logistic regression.
Example of Ordinal Logistic Regression
This example uses the AdverseR.jmp sample data table to illustrate an ordinal logistic
regression. Suppose you want to model the severity of an adverse event as a function of
treatment duration value.
1. Select Help > Sample Data Library and open AdverseR.jmp.
2. Right-click on the icon to the left of ADR SEVERITY and change the modeling type to
ordinal.
3. Select Analyze > Fit Y by X.
4. Select ADR SEVERITY and click Y, Response.
5. Select ADR DURATION and click X, Factor.
6. Click OK.
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Figure 8.5 Example of Ordinal Logistic Report
You interpret this report the same way as the nominal report. See “The Logistic Report” on
page 243.
In the plot, markers for the data are drawn at their x-coordinate. When several data points
appear at the same y position, the points are jittered. That is, small spaces appear between the
data points so you can see each point more clearly.
Where there are many points, the curves are pushed apart. Where there are few to no points,
the curves are close together. The data pushes the curves in that way because the criterion that
is maximized is the product of the probabilities fitted by the model. The fit tries to avoid
points attributed to have a small probability, which are points crowded by the curves of fit.
See the Fitting Linear Models book for more information about computational details.
For details about the Whole Model Test report and the Parameter Estimates report, see “The
Logistic Report” on page 243. In the Parameter Estimates report, an intercept parameter is
estimated for every response level except the last, but there is only one slope parameter. The
intercept parameters show the spacing of the response levels. They always increase
monotonically.
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Additional Example of a Logistic Plot
This example uses the Car Physical Data.jmp sample data table to show an additional example
of a logistic plot. Suppose you want to use weight to predict car size (Type) for 116 cars. Car
size can be one of the following, from smallest to largest: Sporty, Small, Compact, Medium, or
Large.
1. Select Help > Sample Data Library and open Car Physical Data.jmp.
2. In the Columns panel, right-click on the icon to the left of Type, and select Ordinal.
3. Right-click on Type, and select Column Info.
4. From the Column Properties menu, select Value Ordering.
5. Move the data in the following top-down order: Sporty, Small, Compact, Medium, Large.
6. Click OK.
7. Select Analyze > Fit Y by X.
8. Select Type and click Y, Response.
9. Select Weight and click X, Factor.
10. Click OK.
The report window appears.
Figure 8.6 Example of Type by Weight Logistic Plot
In Figure 8.6, note the following observations:
•
The first (bottom) curve represents the probability that a car at a given weight is Sporty.
•
The second curve represents the probability that a car is Small or Sporty. Looking only at
the distance between the first and second curves corresponds to the probability of being
Small.
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•
As you might expect, heavier cars are more likely to be Large.
•
Markers for the data are drawn at their x-coordinate, with the y position jittered randomly
within the range corresponding to the response category for that row.
If the x -variable has no effect on the response, then the fitted lines are horizontal and the
probabilities are constant for each response across the continuous factor range. Figure 8.7
shows a logistic plot where Weight is not useful for predicting Type.
Figure 8.7 Examples of Sample Data Table and Logistic Plot Showing No y by x Relationship
Note: To re-create the plots in Figure 8.7 and Figure 8.8, you must first create the data tables
shown here, and then perform steps 7-10 at the beginning of this section.
If the response is completely predicted by the value of the factor, then the logistic curves are
effectively vertical. The prediction of a response is near certain (the probability is almost 1) at
each of the factor levels. Figure 8.8 shows a logistic plot where Weight almost perfectly
predicts Type.
Note: In this case, the parameter estimates become very large and are marked unstable in the
regression report.
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Figure 8.8 Examples of Sample Data Table and Logistic Plot Showing an Almost Perfect y by x
Relationship
Example of ROC Curves
To demonstrate ROC curves, proceed as follows:
1. Select Help > Sample Data Library and open Penicillin.jmp.
2. Select Analyze > Fit Y by X.
3. Select Response and click Y, Response.
4. Select In(Dose) and click X, Factor.
Notice that JMP automatically fills in Count for Freq. Count was previously assigned the
role of Freq.
5. Click OK.
6. From the red triangle menu, select ROC Curve.
7. Select Cured as the positive.
8. Click OK.
Note: This example shows a ROC Curve for a nominal response. For details about ordinal
ROC curves, see the Partition chapter in the Predictive and Specialized Modeling book.
The results for the response by In(Dose) example are shown here. The ROC curve plots the
probabilities described above, for predicting response. Note that in the ROC Table, the row
with the highest Sens-(1-Spec) is marked with an asterisk.
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Figure 8.9 Examples of ROC Curve and Table
Since the ROC curve is well above a diagonal line, you conclude that the model has good
predictive ability.
Example of Inverse Prediction Using the Crosshair Tool
In a study of rabbits who were given penicillin, you want to know what dose of penicillin
results in a 0.5 probability of curing a rabbit. In this case, the inverse prediction for 0.5 is called
the ED50, the effective dose corresponding to a 50% survival rate. Use the crosshair tool to
visually approximate an inverse prediction.
To see which value of In(dose) is equally likely either to cure or to be lethal, proceed as follows:
1. Select Help > Sample Data Library and open Penicillin.jmp.
2. Select Analyze > Fit Y by X.
3. Select Response and click Y, Response.
4. Select In(Dose) and click X, Factor.
Notice that JMP automatically fills in Count for Freq. Count was previously assigned the
role of Freq.
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5. Click OK.
6. Click on the crosshairs tool.
7. Place the horizontal crosshair line at about 0.5 on the vertical (Response) probability axis.
8. Move the cross-hair intersection to the prediction line, and read the In(dose) value that
shows on the horizontal axis.
In this example, a rabbit with a In(dose) of approximately -0.9 is equally likely to be cured as it
is to die.
Figure 8.10 Example of Crosshair Tool on Logistic Plot
Example of Inverse Prediction Using the Inverse Prediction Option
If your response has exactly two levels, the Inverse Prediction option enables you to request an
exact inverse prediction. You are given the x value corresponding to a given probability of the
lower response category, as well as a confidence interval for that x value.
To use the Inverse Prediction option, proceed as follows:
1. Select Help > Sample Data Library and open Penicillin.jmp.
2. Select Analyze > Fit Y by X.
3. Select Response and click Y, Response.
4. Select In(Dose) and click X, Factor.
Notice that JMP automatically fills in Count for Freq. Count was previously assigned the
role of Freq.
5. Click OK.
6. From the red triangle menu, select Inverse Prediction. See Figure 8.11.
7. Type 0.95 for the Confidence Level.
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8. Select Two sided for the confidence interval.
9. Request the response probability of interest. Type 0.5 and 0.9 for this example, which
indicates you are requesting the values for ln(Dose) that correspond to a 0.5 and 0.9
probability of being cured.
10. Click OK.
The Inverse Prediction plot appears.
Figure 8.11 Inverse Prediction Window
Figure 8.12 Example of Inverse Prediction Plot
The estimates of the x values and the confidence intervals are shown in the report as well as in
the probability plot. For example, the value of ln(Dose) that results in a 90% probability of
being cured is estimated to be between -0.526 and 0.783.
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Chi-Square
The Chi-Square statistic is sometimes denoted G2 and is written as follows:
2
G = 2 (  – ln p ( background ) –  – ln p ( model ) )
where the summations are over all observations instead of all cells.
RSquare (U)
The ratio of this test statistic to the background log-likelihood is subtracted from 1 to calculate
R2. More simply, RSquare (U) is computed as follows:
negative
log-likelihood for Difference-------------------------------------------------------------------------------------------------negative log-likelihood for Reduced
using quantities from the Whole Model Test report.
Note: RSquare (U) is also known as McFadden’s pseudo R-square.
Chapter 9
Tabulate
Create Summary Tables Interactively
Use the Tabulate platform to interactively construct tables of descriptive statistics. The
Tabulate platform is an easy and flexible way to present summary data in tabular form. Tables
are built from grouping columns, analysis columns, and statistics keywords.
Figure 9.1 Tabulate Examples
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Example of the Tabulate Platform
You have data containing height measurements for male and female students. You want to
create a table that shows the mean height for males and females and the aggregate mean for
both sexes. You want the table to look like Figure 9.2.
Figure 9.2 Table Showing Mean Height
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Tabulate.
Since height is the variable you are examining, you want it to appear at the top of the table.
3. Click height and drag it into the Drop zone for columns.
Figure 9.3 Height Variable Added
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Example of the Tabulate Platform
You want the statistics by sex, and you want sex to appear on the side.
4. Click sex and drag it into the blank cell next to the number 2502.
Figure 9.4 Sex Variable Added
Instead of the sum, you want it to show the mean.
5. Click Mean and drag it on top of Sum.
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Figure 9.5 Mean Statistic Added
You also want to see the combined mean for males and females.
6. Click All and drag it on top of sex. Or, you can simply select the Add Aggregate Statistics
check box.
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Launch the Tabulate Platform
Figure 9.6 All Statistic Added
7. (Optional) Click Done.
The completed table shows the mean height for females, males, and the combined mean
height for both.
Launch the Tabulate Platform
To launch the Tabulate platform, select Analyze > Tabulate.
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Figure 9.7 The Tabulate Interactive Table
Note: For details about red triangle options, see “Tabulate Platform Options” on page 273.
The Tabulate window contains the following options:
Switch between the two modes. Use the interactive table mode to
drag and drop items, creating a custom table. Use the dialog mode to create a simple table
using a fixed format. See “Use the Dialog” on page 265.
Interactive table/dialog
Statistics options Lists standard statistics. Drag any statistic from the list to the table to
incorporate it. See “Add Statistics” on page 266.
Drop zone for columns
Drag and drop columns or statistics here to create columns.
Note: If the data table contains columns with names equal to those in the Statistics options, be
sure to drag and drop the column name from the column list; otherwise, JMP may substitute
the statistic of the same name in the table.
Drop zone for rows Drag and drop columns or statistics here to create rows.
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Resulting cells Shows the resulting cells based on the columns or statistics that you drag and
drop.
Freq Identifies the data table column whose values assign a frequency to each row. This
option is useful when a frequency is assigned to each row in summarized data.
Weight Identifies the data table column whose variables assign weight (such as importance
or influence) to the data.
Page Column Generates separate tables for each category of a nominal or ordinal column.
See “Example Using a Page Column” on page 281.
Include missing for grouping columns Creates a separate group for missing values in
grouping columns. When unchecked, missing values are not included in the table. Note
that any missing value codes that you have defined as column properties are taken into
account.
Changes the order of the table to be in ascending order
of the values of the grouping columns.
Order by count of grouping columns
Add Aggregate Statistics Adds aggregate statistics for all rows and columns.
Default Statistics Enables you to change the default statistics that appear when you drag and
drop analysis or non-analysis (for example, grouping) columns.
Enables you to change the numeric format for displaying specific statistics.
See “Change Numeric Formats” on page 268.
Change Format
Change Plot Scale (Only appears if Show Chart is selected from the red triangle menu.)
Enables you to specify a uniform custom scale.
Uniform plot scale (Only appears if Show Chart is selected from the red triangle menu.)
Deselect this box for each column of bars to use the scale determined separately from the
data in each displayed column.
Use the Dialog
If you prefer not to drag and drop and build the table interactively, you can create a simple
table using the Dialog interface. After selecting Analyze > Tabulate, select Dialog from the
menu, as shown in Figure 9.8. You can make changes to the table by selecting Show Control
Panel from the red triangle menu, and then drag and drop new items into the table.
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Figure 9.8 Using the Dialog
The dialog contains the following options:
Include marginal statistics Aggregates summary information for categories of a grouping
column.
For quantile statistics, enter value (%) Type the value at which the specific percentage of the
argument is less than or equal to. For example, 75% of the data is less than the 75th
quantile. This applies to all grouping columns.
Statistics Once you’ve selected a column, select a standard statistic to apply to that column.
See “Add Statistics” on page 266.
Grouping (row labels) Select the column to use as the row label.
Grouping (column labels)
Select the column to use as the column label.
Add Statistics
Tip: You can select both a column and a statistic at the same time and drag them into the table.
Tabulate supports a list of standard statistics. The list is displayed in the control panel. You can
drag any keyword from that list to the table, just like you do with the columns. Note the
following:
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•
The statistics associated with each cell are calculated on values of the analysis columns
from all observations in that category, as defined by the grouping columns.
•
All of the requested statistics have to reside in the same dimension, either in the row table
or in the column table.
•
If you drag a continuous column into a data area, it is treated as an analysis column.
Tabulate uses the following keywords:
N
Provides the number of nonmissing values in the column. This is the default statistic when
there is no analysis column.
Mean Provides the arithmetic mean of a column’s values. It is the sum of nonmissing values
(and if defined, multiplied by the weight variable) divided by the Sum Wgt.
Provides the sample standard deviation, computed for the nonmissing values. It is
the square root of the sample variance.
Std Dev
Min
Provides the smallest nonmissing value in a column.
Max Provides the largest nonmissing value in a column.
Range Provides the difference between Max and Min.
Computes the percentage of total of the whole population. The denominator used
in the computation is the total of all the included observations, and the numerator is the
total for the category. If there is no analysis column, the % of Total is the percentage of total
of counts. If there is an analysis column, the % of Total is the percentage of the total of the
sum of the analysis column. Thus, the denominator is the sum of the analysis column over
all the included observations, and the numerator is the sum of the analysis column for that
category. You can request different percentages by dragging the keyword into the table.
% of Total
– Dropping one or more grouping columns from the table to the % of Total heading
changes the denominator definition. For this, Tabulate uses the sum of these grouping
columns for the denominator.
– To get the percentage of the column total, drag all the grouping columns on the row
table and drop them onto the % of Total heading (same as Column %). Similarly, to get
the percentage of the row total, drag all grouping columns on the column table and
drop them onto the % of Total heading (same as Row %).
N Missing
Provides the number of missing values.
N Categories Provides the number of distinct categories.
Sum Provides the sum of all values in the column. This is the default statistic for analysis
columns when there are no other statistics for the table.
Provides the sum of all weight values in a column. Or, if no column is assigned the
weight role, Sum Wgt is the total number of nonmissing values.
Sum Wgt
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Variance Provides the sample variance, computed for the nonmissing values. It is the sum of
squared deviations from the mean, divided by the number of nonmissing values minus
one.
Std Err Provides the standard error of the mean. It is the standard deviation divided by the
square root of N. If a column is assigned the role of weight, then the denominator is the
square root of the sum of the weights.
CV (Coefficient of Variation) Provides the measure of dispersion, which is the standard
deviation divided by the mean multiplied by one hundred.
Median Provides the 50th percentile, which is the value where half the data are below and
half are above or equal to the 50th quantile (median).
Geometric Mean Provides the nth root of the product of n numbers. Zero and negative
numbers are treated like missing. For example, you might want to compare two
companies based on varying metrics that come from different ranges. The statistic is also
helpful when the data contains a large value in a skewed distribution.
Interquartile Range Provides the difference between the 3rd quartile and 1st quartile.
Quantiles Provides the value at which the specific percentage of the argument is less than or
equal to. For example, 75% of the data is less than the 75th quantile. You can request
different quantiles by clicking and dragging the Quantiles keyword into the table, and then
entering the quantile into the box that appears.
Column % Provides the percent of each cell count to its column total if there is no analysis
column. If there is an analysis column, the Column % is the percent of the column total of
the sum of the analysis column. For tables with statistics on the top, you can add Column
% to tables with multiple row tables (stacked vertically).
Row % Provides the percent of each cell count to its row total if there is no analysis column.
If there is an analysis column, the Row % is the percent of the row total of the sum of the
analysis column. For tables with statistics on the side, you can add Row % to tables with
multiple column tables (side by side tables).
All
Aggregates summary information for categories of a grouping column.
Change Numeric Formats
The formats of each cell depend on the analysis column and the statistics. For counts, the
default format has no decimal digits. For each cell defined by some statistics, JMP tries to
determine a reasonable format using the format of the analysis column and the statistics
requested. To override the default format:
1. Click the Change Format button at the bottom of the Tabulate window.
2. In the panel that appears, enter the field width, a comma, and then the number of decimal
places that you want displayed in the table. See Figure 9.9.
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3. To exhibit the cell value in Percent format, add a comma after the number of decimal
places and type the word Percent.
4. (Optional) If you would like JMP to determine the best format for you to use, type the
word Best in the text box.
JMP now considers the precision of each cell value and selects the best way to show it.
5. Click OK to implement the changes and close the Format section, or click Set Format to see
the changes implemented without closing the Format section.
Figure 9.9 Changing Numeric Formats
The Tabulate Output
The Tabulate output consists of one or more column tables concatenated side by side, and one
or more row tables concatenated top to bottom. The output might have only a column table or
a row table.
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Figure 9.10 Tabulate Output
Creating a table interactively is an iterative process:
•
Click the items (columns or statistics) from the appropriate list, and drag them into the
drop zone (for rows or columns). See “Edit Tables” on page 273, and “Column and Row
Tables” on page 272.
•
Add to the table by repeating the drag and drop process. The table updates to reflect the
latest addition. If there are already column headings or row labels, you can decide where
the addition goes relative to the existing items.
Note the following about clicking and dragging:
•
JMP uses the modeling type to determine a column’s role. Continuous columns are
assumed to be analysis columns. See “Analysis Columns” on page 271. Ordinal or nominal
columns are assumed to be grouping columns. See “Grouping Columns” on page 271.
•
When you drag and drop multiple columns into the initial table:
– If the columns share a set of common values, they are combined into a single table. A
crosstabulation of the column names and the categories gathered from these columns is
generated. Each cell is defined by one of the columns and one of the categories.
– If the columns do not share common values, they are put into separate tables.
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– You can always change the default action by right-clicking on a column and selecting
Combine Tables or Separate Tables. For more details, see “Right-Click Menu for
Columns” on page 275.
•
To nest columns, create a table with the first column, and then drag the additional columns
into the first column.
•
In a properly created table, all grouping columns are together, all analysis columns are
together, and all statistics are together. Therefore, JMP does not intersperse a statistics
keyword within a list of analysis columns. JMP also does not insert an analysis column
within a list of grouping columns.
•
You can drag columns from the Table panel in the data table onto a Tabulate table instead
of using the Tabulate Control Panel.
Note: The Tabulate table is updated when you add data to the open data table, delete rows,
and recode the data.
Analysis Columns
Analysis columns are any numeric columns for which you want to compute statistics. They
are continuous columns. Tabulate computes statistics on the analysis columns for each
category formed from the grouping columns.
Note that all the analysis columns have to reside in the same dimension, either in the row
table or in the column table.
Grouping Columns
Grouping columns are columns that you want to use to classify your data into categories of
information. They can have character, integer, or even decimal values, but the number of
unique values should be limited. Grouping columns are either nominal or ordinal.
Note the following:
•
If grouping columns are nested, Tabulate constructs distinct categories from the
hierarchical nesting of the values of the columns. For example, from the grouping columns
Sex with values F and M, and the grouping column Marital Status with values Married
and Single, Tabulate constructs four distinct categories: F and Married, F and Single, M
and Married, M and Single.
•
You can specify grouping columns for column tables as well as row tables. Together they
generate the categories that define each table cell.
•
Tabulate does not include observations with a missing value for one or more grouping
columns by default. You can include them by checking the Include missing for grouping
columns option.
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To specify codes or values that should be treated as missing, use the Missing Value Codes
column property. You can include these by checking the Include missing for grouping
columns option. For more details about Missing Value Codes, see The Column Info
Window chapter in the Using JMP book.
Column and Row Tables
In Tabulate, a table is defined by its column headings and row labels. These sub-tables are
referred to as row tables and column tables. See Figure 9.11.
Example of Row and Column Tables
1. Select Help > Sample Data Library and open Car Poll.jmp.
2. Select Analyze > Tabulate.
3. Drag size into the Drop zone for rows.
4. Drag country to the left of the size heading.
5. Drag Mean over the N heading.
6. Drag Std Dev below the Mean heading.
7. Drag age above the Mean heading.
8. Drag type to the far right of the table.
9. Drag sex under the table.
Figure 9.11 Row and Column Tables
two column tables
two row tables
For multiple column tables, the labels on the side are shared across the column tables. In this
instance, country and sex are shared across the tables. Similarly, for multiple row tables, the
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headings on the top are shared among the row tables. In this instance, both age and type are
shared among the tables.
Edit Tables
There are several ways to edit the items that you add to a table.
Delete Items
After you add items to the table, you can remove them in any one of the following ways:
•
Drag the item away from the table.
•
To remove the last item, click Undo.
•
Right-click on an item and select Delete.
Remove Column Labels
Grouping columns display the column name on top of the categories associated with that
column. For some columns, the column name might seem redundant. Remove the column
name from the column table by right-clicking on the column name and selecting Remove
Column Label. To re-insert the column label, right-click on one of its associated categories and
select Restore Column Label.
Edit Statistical Key Words and Labels
You can edit a statistical key word or a statistical label. For example, instead of Mean, you
might want to use the word Average. Right-click on the word that you want to edit and select
Change Item Label. In the box that appears, type the new label. Alternatively, you can type
directly into the edit box.
If you change one statistics keyword to another statistics keyword, JMP assumes that you
actually want to change the statistics, not just the label. It would be as if you have deleted the
statistics from the table and added the latter.
Tabulate Platform Options
The following options are available from the red triangle menu next to Tabulate:
Show Table Displays the summarized data in tabular form.
Show Chart Displays the summarized data in bar charts that mirrors the table of summary
statistics. The simple bar chart enables visual comparison of the relative magnitude of the
summary statistics. By default, all columns of bars share the same scale. You can have each
column of bars use the scale determined separately from the data in each displayed
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column, by clearing the Uniform plot scale check box. You can specify a uniform custom
scale using the Change Plot Scale button. The charts are either 0-based or centered on 0. If
the data are all nonnegative, or all non-positive, the charts baseline is at 0. Otherwise, the
charts are centered on 0.
Show Control Panel Displays the control panel for further interaction.
Show Shading Displays gray shading boxes in the table when there are multiple rows.
Show Tooltip Displays tips that appear when you move the mouse over areas of the table.
Displays the control area that lets you create a test build using a
random sample from the original table. This is particularly useful when you have large
amounts of data. See “Show Test Build Panel” on page 274.
Show Test Build Panel
Makes a data table from the report. There is one data table for each row
table, because labels of different row tables might not be mapped to the same structure.
Make Into Data Table
Full Path Column Name Uses the fully qualified column names of grouping columns for the
column name in the created data table.
See the JMP Reports chapter in the Using JMP book for more information about the following
options:
Shows or hides the local data filter that enables you to filter the data used in
a specific report.
Local Data Filter
Contains options that enable you to repeat or relaunch the analysis. In platforms that
support the feature, the Automatic Recalc option immediately reflects the changes that
you make to the data table in the corresponding report window.
Redo
Contains options that enable you to save a script that reproduces the report to
several destinations.
Save Script
For a description of the options in the Select Columns red triangle menu, see the Get Started
chapter in the Using JMP book.
Show Test Build Panel
If you have a very large data table, you might want to use a small subset of the data table to try
out different table layouts to find one that best shows the summary information. In this case,
JMP generates a random subset of the size as specified and uses that subset when it builds the
table. To use the test build feature:
1. From the red triangle menu next to Tabulate, select Show Test Build Panel.
2. Enter the size of the sample that you want in the box under Sample Size (>1) or Sampling
Rate (<1), as shown in Figure 9.12. The size of the sample can be either the proportion of
the active table that you enter or the number of rows from the active table.
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Figure 9.12 The Test Build Panel
3. Click Resample.
4. To see the sampled data in a JMP data table, click the Test Data View button. When you
dismiss the test build panel, Tabulate uses the full data table to regenerate the tables as
designed.
Right-Click Menu for Columns
Right-click on a column in Tabulate to see the following options:
Delete Deletes the selected column.
Use as Grouping column Changes the analysis column to a grouping column.
Use as Analysis column Changes the grouping column to an analysis column.
Change Item Label (Appears only for separate or nested columns) Type a new label.
(Appears only for separate or nested columns)
Combines separate or nested columns. See “Example of Combining Columns into a Single
Table” on page 279.
Combine Tables (Columns by Categories)
Separate Tables (Appears only for combined tables) Creates a separate table for each
column.
Nest Grouping Columns
Nests grouping columns vertically or horizontally.
Additional Examples of the Tabulate Platform
This example contains the following steps:
1. “Create a Table of Counts” on page 276
2. “Create a Table Showing Statistics” on page 277
3. “Rearrange the Table Contents” on page 278
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Create a Table of Counts
Suppose that you would like to view a table that contains counts for how many people in the
survey own Japanese, European, and American cars, broken down by the size of the car. You
want the table to look Figure 9.3.
Figure 9.13 Table Showing Counts of Car Ownership
1. Select Help > Sample Data Library and open Car Poll.jmp.
2. Select Analyze > Tabulate.
3. Click country and drag it into the Drop zone for rows.
4. Click size and drag it to the right of the country heading.
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Figure 9.14 Country and Size Added to the Table
Create a Table Showing Statistics
Suppose that you would like to see the mean (average) and the standard deviation of the age
of people who own each size car. You want the table to look like Figure 9.15.
Figure 9.15 Table Showing Mean and Standard Deviation by Age
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1. Start from Figure 9.14. Click age and drag it to the right of the size heading.
2. Click Mean and drag it over Sum.
3. Click Std Dev and drag it below Mean.
Std Dev is placed below Mean in the table. Dropping Std Dev above Mean places Std Dev
above Mean in the table.
Figure 9.16 Age, Mean, and Std Dev Added to the Table
Rearrange the Table Contents
Suppose that you would prefer size to be on top, showing a crosstab layout. You want the
table to look like Figure 9.17.
Figure 9.17 Size on Top
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To rearrange the table contents, proceed as follows:
1. Start from Figure 9.16. Click on the size heading and drag it to the right of the table
headings. See Figure 9.18.
Figure 9.18 Moving size
2. Click on age and drag it under the Large Medium Small heading.
3. Select both Mean and Std Dev, and then drag them under the Large heading.
Now your table clearly presents the data. It is easier to see the mean and standard deviation of
the car owner age broken down by car size and country.
Example of Combining Columns into a Single Table
You have data from students indicating the importance of self-reported factors in children’s
popularity (grades, sports, looks, money). Suppose that you want to see all of these factors in a
single, combined table with additional statistics and factors. You want the table to look like
Figure 9.19.
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Figure 9.19 Adding Demographic Data
1. Select Help > Sample Data Library and open Children’s Popularity.jmp.
2. Select Analyze > Tabulate.
3. Select Grades, Sports, Looks, and Money and drag them into the Drop zone for rows.
Figure 9.20 Columns by Categories
Notice that a single, combined table appears.
Tabulate the percentage of the one to four ratings of each category.
4. Drag Gender into the empty heading at left.
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5. Drag % of Total above the numbered headings.
6. Drag All beside the number 4.
Figure 9.21 Gender, % of Total, and All Added to the Table
Break down the tabulation further by adding demographic data.
7. Drag Urban/Rural below the % of Total heading.
Figure 9.22 Urban/Rural Added to the Table
You can see that for boys in rural, suburban, and urban areas, sports are the most important
factor for popularity. For girls in rural, suburban, and urban areas, looks are the most
important factor for popularity.
Example Using a Page Column
You have data containing height measurements for male and female students. You want to
create a table that shows the mean height by the age of the students. Then you want to stratify
your data by sex in different tables. To do so, add the stratification column as a page column,
which will build the pages for each group. You want the table to look like Figure 9.23.
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Figure 9.23 Mean Height of Students by Sex
Females
Males
1. Select Help > Sample Data Library and open Big Class.jmp.
2. Select Analyze > Tabulate.
Since height is the variable you are examining, you want it to appear at the top of the table.
3. Click height and drag it into the Drop zone for columns.
You want the statistics by age, and you want age to appear on the side.
4. Click age and drag it into the blank cell next to the number 2502.
5. Click sex and drag it into Page Column.
6. Select F from the Page Column list to show the mean heights for only females.
7. Select M from the Page Column list to show the mean heights for only males. You can also
select None Selected to show all values.
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Figure 9.24 Using a Page Column
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Chapter 10
Simulate
Answer Challenging Questions with Parametric Resampling
The Simulate platform is available only in JMP Pro.
The Simulate feature provides powerful parametric and nonparametric simulation capability.
Use Simulate to do the following:
•
Expand on the bootstrap to provide parametric bootstrapping.
•
Obtain power calculations in nonstandard situations.
•
Approximate the distribution of statistics, such as predicted values, and confidence
intervals, in nonstandard situations.
•
Conduct permutation tests.
•
Explore the effect of assumptions about predictors on models.
•
Explore various “what if” scenarios relative to your models.
•
Evaluate new or existing statistical methods.
The Simulate option is available in many reports, including all of those that support Bootstrap.
To access the Simulate option, right-click in a report.
Figure 10.1 Power Analysis Using Simulate
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Overview of Simulate
Simulate provides simulated results for a column of statistics in a report. Right-click on a
column of statistics in a report and select Simulate. In the Simulate window, specify a column
in your data table that forms the basis for your simulation. This is the column that you switch
out. This column can have any role in the analysis. In particular, it can be a response or a
predictor in a model. You then specify a column in your data table that contains a formula that
you want to use for the simulation. This is the column that you switch in. It functions as a
surrogate for the column that you switched out.
Note: Your data table must contain a column that has a random component.
The method works as follows. A column of simulated values is generated based on the
formula in the formula column that you switch in. The entire analysis that generated the
report containing the statistics of interest is rerun using this new column of simulated values
to replace the column that you switched out. This process is repeated N times, where N is the
total number of samples that you specify.
The Simulate analysis produces an output data table showing a summary of the analysis.
•
Each row of the data table represents the results of the analysis for one column of
simulated values.
•
There is a column for each row of the report table involved in the simulation.
•
There are scripts to facilitate your analysis.
Tip: Simulate reruns the entire analysis that appears in the platform report from which
Simulate is invoked. As a result, Simulate may run slowly for your selected column because of
extraneous analyses in the report. If Simulate is taking a long time, remove extraneous options
from the platform report prior to running Simulate.
Examples That Use Simulate
This section provides several examples of the use of Simulate. Additional examples, also listed
below, can be found in other books:
•
“Construct an Accurate Confidence Interval for Variance Components” on page 287
•
“Conduct a Permutation Test” on page 292
•
“Explore Retaining a Factor in Generalized Regression” on page 295
•
“Conduct Prospective Power Analysis for a Nonlinear Model” on page 300
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For an example that shows how to simulate a confidence interval for Ppk and the percent
nonconforming for a non-normal variable, see the Process Capability chapter in the
Quality and Process Methods book.
Construct an Accurate Confidence Interval for Variance Components
In this example, you are interested in the effects of temperature, time, and the amount of
catalyst on a reaction. Temperature is a very-hard-to-change variable (whole plot factor), time
is hard-to-change (subplot factor), and the amount of catalyst is easy-to-change. For
information on whole plot and subplot factors, see the Custom Designs chapter in the Design
of Experiments Guide.
Your goal is to obtain accurate confidence intervals for the whole-plot and sub-plot variance
components. Previous studies have suggested that the whole-plot standard deviation is about
twice the error standard deviation, while the sub-plot error is about 1.5 times the error
standard deviation. The Wald intervals given in the REML report, which assume that the
variance components are asymptotically normal, have poor coverage properties. You will
obtain confidence intervals by simulating the distributions of the variance components.
In this example, you will do the following:
•
Construct a custom design for your split-split-plot experiment. See “Construct the Design”
on page 287.
•
Fit a model using the REML method. See “Fit the Model” on page 290.
•
Simulate variance component estimates in order to obtain percentile confidence intervals
for the variance components. See “Explore Power” on page 290.
Construct the Design
If you prefer to skip the steps in this section, select Help > Sample Data Library and open
Design Experiment/Catalyst Design.jmp. In the Catalyst Design.jmp data table, click the green
triangle next to the DOE Simulate script. Then go to “Fit the Model” on page 290.
1. Select DOE > Custom Design.
2. In the Factors outline, type 3 next to Add N Factors.
3. Click Add Factor > Continuous.
4. Double-click to rename these factors Temperature, Time, and Catalyst.
Keep the default Values of –1 and 1 for these factors.
5. For Temperature, click Easy and select Very Hard.
This defines Temperature to be a whole plot factor.
6. For Time, click Easy and select Hard for Time.
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This defines Time to be a subplot factor.
7. Click Continue.
8. In the Model outline, select Interactions > 2nd.
This adds all two-way interactions to the model.
9. Click the Custom Design red triangle and select Simulate Responses.
This opens the Simulate Responses window after you select Make Table to construct the
design table.
Note: Setting the Random Seed in step 10 and Number of Starts in step 11 reproduces the
exact results shown in this example. In constructing a design on your own, these steps are
not necessary.
10. (Optional) Click the Custom Design red triangle and select Set Random Seed. Type 12345
and click OK.
11. (Optional) Click the Custom Design red triangle and select Number of Starts. Type 1000
and click OK.
12. Click Make Design.
13. Click Make Table.
Note: The entries in your Y and Y Simulated columns will differ from those that appear in
Figure 10.2.
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Figure 10.2 Design Table
Figure 10.3 Simulate Responses Window
The design table and a Simulate Responses window appear. Notice that the design table
contains a DOE Simulate script. At any time, you can run this script to specify different
parameter values.
Continue to the next section, where you specify standard deviations for the whole plot and
subplot errors, and fit a REML model to the first set of simulated values.
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Fit the Model
Assume that the whole plot and subplot errors are normal. Based on your estimates of their
standard deviations, if the error standard deviation is about 1 unit, the whole plot standard
deviation is about 2 units and the subplot standard deviation is about 1.5 units. Since you are
only interested in the whole- and sub-plot variation, you do not need to change the values
assigned to Effects in the Simulate Responses outline.
1. In the Distribution panel (Figure 10.3), next to Whole Plots σ, type 2.
Notice that the Normal distribution is selected by default, indicating that normal error will
be added to the formula.
2. Next to Subplots σ, type 1.5.
3. Click Apply.
In the data table, the formula for Y Simulated updates to reflect your specifications. To
view the formula, click on the plus sign to the right of the column name in the Columns
panel.
4. In the data table, click the green triangle next to the Model script.
5. Click the Y variable next to the Y button and click Remove.
6. Click Y Simulated and click the Y button.
This action replaces Y with a column that contains a simulation formula.
7. Click Run.
The model that is fit is based on a single set of simulated responses.
Note: Because the values in Y Simulated are randomly generated, the entries in your report
will differ from those that appear in Figure 10.4.
Figure 10.4 REML Report Showing Wald Confidence Intervals
Explore Power
Next, simulate values for the variance components and use these to construct simulated
percentile confidence intervals.
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1. In the REML Variance Components Estimates outline, right-click in the Var Component
column and select Simulate.
Figure 10.5 Simulate Window
In your simulations, you replace the column Y Simulated, which you used to run your
model, with a new instance of the column Y Simulated, which generates a new column of
simulated values for each simulation. The column on which you right-clicked and that
appears as selected, Var Component, will be simulated for each effect listed in the
Parameter Estimates table.
2. Next to Number of Samples, enter 200.
3. (Optional) Next to Random Seed, enter 456.
This reproduces the values shown in Figure 10.6, except for the values in row 1.
4. Click OK.
The entries in your row 1 will differ from those that appear in Figure 10.6.
Figure 10.6 Table of Simulated Results for Var Component (Partial View)
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The first row of the Fit Least Squares Simulate Results (Var Component) data table
contains the initial values of Var Component and is excluded. The remaining rows contain
simulated values.
5. Run the Distribution script.
Figure 10.7 Distribution Plots for Variance Components (Partial View)
For each variance component, a 95% percent confidence interval is constructed using the
2.5% and 97.5% quantiles (see the Simulation Results report in Figure 10.7). Compare these
to the intervals given in the REML report (Figure 10.4):
– The simulated 95% percentile interval for the whole-plot variance component is -3.296
to 22.863. The Wald interval given in the REML report is -1.973 to 5.086.
– The simulated 95% percentile interval for the sub-plot variance component is -0.332 to
10.459. The Wald interval given in the REML report is -0.605 to 1.223.
The intervals that you obtain using simulation are considerably wider than the REML
interval calculated from your single set of values. For more accurate intervals, consider
running a larger number of simulations.
Conduct a Permutation Test
In this example, you are studying the effects of three drugs on pain. You are interested in
whether they differ in their effects. Because you have a very small sample size and are
somewhat concerned about violations of the usual ANOVA assumptions, you will use
Simulate to conduct a permutation test.
First, you construct a formula that randomly shuffles the pain measurements among the three
drugs. Under the null hypothesis of no effect, any of these allocations is as likely as any other.
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It follows that the F ratios obtained in this manner approximate the distribution of F ratios
under the null hypothesis. Finally, you compare the observed value of the F ratio to the null
distribution obtained by simulation.
Define the Simulation Formula
1. Select Help > Sample Data Library and open Analgesics.jmp.
2. Select Cols > New Columns.
3. Type Pain Shuffled for Column Name.
4. From the Column Properties list, select Formula.
5. In the function list, select Row > Col Stored Value.
6. In the Columns list, double-click pain.
7. Click the insert key (^) in the list of symbols above the editor panel.
8. From the list of functions, select Random > Col Shuffle.
Figure 10.8 Completed Formula
This formula randomly shuffles the entries in the pain column.
9. Click OK in the Formula Editor window.
10. Click OK in the Column Info window.
Perform the Permutation Test
1. Select Analyze > Fit Y by X.
2. Select pain and click Y, Response.
3. Select drug and click X, Factor.
4. Click OK.
5. Click the Oneway Analysis red triangle and select Means/Anova.
Figure 10.9 Analysis of Variance Report
Notice that the F ratio is 6.2780.
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6. In the Analysis of Variance outline, right-click on the F Ratio column and select Simulate.
7. In the Column to Switch Out list, click pain.
8. In the Column to Switch In list, click Pain Shuffled.
9. Next to Number of Samples, enter 1000.
10. (Optional) Next to Random Seed, enter 456.
This reproduces the values in this example.
Figure 10.10 Completed Simulate Window
11. Click OK.
In the table of simulated results, the columns for C. Total and Error are empty, since the
F Ratio value in the Analysis of Variance table only applies to drug.
12. In the table of simulated values, run the Distribution script.
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Figure 10.11 Simulated Distribution of F Ratios under the Null Distribution
The observed F ratio value of 6.2780 is represented with a red line in the histogram. This
value falls in the upper 0.5% of the simulated null distribution of F ratios. This presents
strong evidence that the three drugs differ in their effects on pain.
Explore Retaining a Factor in Generalized Regression
In this example, a pharmaceutical manufacturer has historical information on the dissolution
of a tablet inside the body and various factors that may affect the dissolution rate. A tablet
with a dissolution rate below 70 is considered defective. You want to understand which
factors affect dissolution rate.
In this example, you will do the following:
•
Construct a generalized regression model.
•
Fit a reduced model using the non-zeroed terms.
•
Based on the reduced model, use simulation to explore the likelihood that one of the
factors is included in the model.
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Fit the Model
In this section, you fit a model using generalized regression. If you prefer not to work through
the steps in this section, click the green triangle next to the Generalized Regression script in
the Tablet Production.jmp data table to obtain the model.
1. Select Help > Sample Data Library and open Tablet Production.jmp.
2. Select Analyze > Fit Model.
3. Click Dissolution and click Y.
4. Select Mill Time through Atomizer Pressure and click Add.
5. From the Personality menu, select Generalized Regression.
6. Click Run.
7. In the Model Launch panel, click Go.
Figure 10.12 Model Based on Adaptive Lasso
You are interested in the parameter estimates shown in the Adaptive Lasso with AICc
Validation report. Based on the non-zero parameter estimates, the model suggests that Mill
Time, Screen Size, Blend Time, Blend Speed, Compressor, Coating Viscosity, and Spray Rate
are related to Dissolution.
Reduce the Model
Before reducing the model, ensure that no columns are selected in the Tablet Production.jmp
data table. Selected columns are not deselected in the first step below. Ensuring that no
columns are selected prevents the inadvertent inclusion of columns with zeroed terms.
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If you prefer not to work through the steps in this section, click the green triangle next to the
Generalized Regression Reduced Model script in the Tablet Production.jmp data table to obtain
the reduced model.
1. Click the red triangle next to Adaptive Lasso with AICc Validation and select Relaunch
with Active Effects.
This opens a Fit Model window that places the terms with nonzero coefficient estimates in
the Parameter Estimates reports into the Construct Model Effects list. The response is
entered as Y. The Generalized Regression personality is selected.
2. Click Run.
3. In the Model Launch panel, click Go.
Figure 10.13 Reduced Model Using Adaptive Lasso
Notice that the estimate for Blend Speed has a confidence interval (Lower 95%) that comes
very close to including zero. Next, perform a simulation study to see how often Blend Speed
would be included in the model had other data values from the dissolution distribution been
observed.
Explore the Inclusion of Blend Speed in the Model
Use the report for the reduced model (Figure 10.13) in the steps below.
1. Click the red triangle next to Adaptive Lasso with AICc Validation and select Save
Columns > Save Simulation Formula.
This adds a new column called Dissolution Simulation Formula to the Tablet Production.jmp
data table.
2. (Optional) In the data table Columns panel, click the plus sign to the right of Dissolution
Simulation Formula.
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Figure 10.14 Simulation Formula
For each row, this formula simulates a value that could be obtained given the model and
the distribution of Dissolution, which is estimated to be Normal with standard deviation
about 1.998.
3. Click Cancel.
4. Go back to the reduced model report window. In the Parameter Estimates for Original
Predictors report, right-click in the Estimate column and select Simulate.
5. Next to Number of Samples, enter 300.
For the simulation, you ask JMP to replace the Dissolution column in each of 300 analyses
with values simulated using the Dissolution Simulation Formula column.
6. (Optional) Set the Random Seed to 123.
This reproduces the values in this example.
Figure 10.15 Completed Simulation Window
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7. Click OK.
The first row of the table contains the initial values of the Estimates and is excluded. The
remaining rows contain simulated values.
8. Run the Distribution script.
9. Press the Ctrl key, click the Intercept red triangle menu, and select Display Options >
Customize Summary Statistics.
10. Select N Zero.
11. Click OK.
12. Scroll to the Distribution report for Blend Speed.
Figure 10.16 Histogram of Simulated Blend Speed Coefficient Estimates
•
The Summary Statistics report shows that for 97/300 = 32.3% of the simulations, the Blend
Speed estimates are zero. This information suggests that it might make sense to retain
Blend Speed in the model.
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Conduct Prospective Power Analysis for a Nonlinear Model
In this example, you are interested in the main effects of six continuous factors on whether a
part passes or fails inspection. The response is binomial and you can afford a total of 60 runs.
You will do the following:
1. Construct a custom design for your experiment. See “Construct the Design” on page 301.
Note: Although a custom design is not optimal for a non-linear situation, in this example,
for simplicity, you will use the Custom Design platform rather than the Nonlinear Design
platform. For an example illustrating why a design constructed using the Nonlinear
Design platform is better than an orthogonal design, see the Nonlinear Designs chapter in
the Design of Experiments Guide.
2. Fit a logistic model using the Generalized Linear Model personality. See “Fit the
Generalized Linear Model” on page 304.
3. Simulate likelihood ratio test p-values to explore the power of detecting a difference over a
range of probability values that is determined by the linear predictor. See “Explore Power”
on page 305.
Plan for the Example
You will model the probability of a success using a generalized linear model with the logit as a
link function. The logit link function fits a logistic model:
1
π ( X ) = ---------------------------------------------------------------------–( β0 + β1 X1 + … + β6 X6 )
1+e
where π(X) denotes the probability that a part passes at the given design settings
X = (X1, X2, ..., X6).
Denote the linear predictor by L(X):
L ( X ) = β0 + β1 X1 + … + β6 X6
You will explore power for the following values of the coefficients of the linear predictor:
Coefficient
Value
β0
0
β1
1
β2
.9
β3
.8
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Coefficient
Value
β4
.7
β5
.6
β6
.5
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Because the intercept in the linear predictor is 0, when all factors are set to 0, the probability of
a passing part equals 50%. The probabilities associated with the levels of the ith factor, when all
other factors are set to 0, are given below.
Factor
Percent Passing at
Difference
Percent Passing at
Xi = 1
Xi = -1
X1
73.11%
26.89%
46.2%
X2
71.09%
28.91%
42.2%
X3
69.00%
31.00%
38.0%
X4
66.82%
33.18%
33.6%
X5
64.56%
35.43%
29.1%
X6
62.25%
37.75%
24.5%
For example, when all factors other than X1 are set to 0, the difference in pass rates that you
want to detect is 46.2%. The smallest difference in pass rates that you want to detect occurs
when all factors other than X6 are set to zero and that difference is 24.5%.
Construct the Design
Note: If you prefer to skip the steps in this section, select Help > Sample Data Library and open
Design Experiment/Binomial Experiment.jmp. Click the green triangle next to the DOE Simulate
script and then go to “Define Simulated Responses” on page 303.
1. Select DOE > Custom Design.
2. In the Factors outline, type 6 next to Add N Factors.
3. Click Add Factor > Continuous.
4. Click Continue.
You are constructing a main effects design, so do not make any changes to the Model
outline.
5. Under Number of Runs, type 60 next to User Specified.
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6. Click the Custom Design red triangle and select Simulate Responses.
This opens the Simulate Responses window after you select Make Table to construct the
design table.
Note: Setting the Random Seed in step 7 and Number of Starts in step 8 reproduces the
exact results shown in this example. In constructing a design on your own, these steps are
not necessary.
7. (Optional) Click the Custom Design red triangle and select Set Random Seed. Type 12345
and click OK.
8. (Optional) Click the Custom Design red triangle and select Number of Starts. Type 1 and
click OK.
9. Click Make Design.
10. Click Make Table.
Note: The entries in your Y and Y Simulated columns will differ from those that appear in
Figure 10.17.
Figure 10.17 Partial View of Design Table
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Figure 10.18 Simulate Responses Window
The design table and a Simulate Responses window appear. Two columns are added to the
design table:
– Y contains a set of values simulated according to the specifications in the Simulate
Responses window.
– Y Simulated contains a formula that calculates its values using the formula for the
model that is specified in the Simulate Responses window. To view the formula, click
on the plus sign to the right of the column name in the Columns panel.
Continue to the next section, where you simulate binomial responses and fit a generalized
linear model to these simulated responses.
Define Simulated Responses
Your plan is to simulate binomial response data where the probability of success is given by a
logistic model. For more information on Simulate Response, see the Custom Designs chapter
in the Design of Experiments Guide.
Note: If you prefer to skip the steps in this section, click the green triangle next to the Simulate
Model Responses script. Then go to “Fit the Generalized Linear Model” on page 304.
1. In the Simulate Responses window (Figure 10.18), type the following values under Y:
– Next to Intercept, type 0.
– Next to X1, 1 is entered by default. Keep that value.
– Next to X2, type 0.9.
– Next to X3, type 0.8.
– Next to X4, type 0.7.
– Next to X5, type 0.6.
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– Next to X6, type 0.5.
2. In the Distribution outline, select Binomial.
Leave the value for N set to 1, indicating that there is only one unit per trial.
Figure 10.19 Completed Simulate Responses Window
3. Click Apply.
In the design data table, the Y Simulated column is replaced with a formula column that
generates binomial values. A column called Y N Trials indicates the number of trials for
each run.
4. (Optional) Click on the plus sign to the right of Y Simulated in the Columns panel.
Figure 10.20 Random Binomial Formula for Y Simulated
5. Click Cancel.
Fit the Generalized Linear Model
1. In the data table, click the green triangle next to the Model script.
2. Click the Y variable next to the Y button and click Remove.
3. Click Y Simulated and click the Y button.
You are replacing Y with a column that contains randomly generated binomial values.
4. From the Personality menu, select Generalized Linear Model.
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5. From the Distribution menu, select Binomial.
Notice that the Logit function appears in the Link Function menu.
6. Click Run.
The model that is fit is based on a single set of simulated binomial responses.
Explore Power
Next, explore the power of tests to detect a difference over the range of probability values
determined by the linear predictor with the coefficient values given in “Plan for the Example”
on page 300.
1. In the Effect Tests outline, right-click in the Prob>ChiSq column and select Simulate.
Figure 10.21 Simulate Window
The column Y Simulated under the Column to Switch Out contains the values that were
used to fit the model. When you select the column Y Simulated under Column to Switch In,
for each simulation, you are telling JMP to replace the values in Y Simulated with a new
column of values that are simulated using the formula in the column Y Simulated.
The column you have selected in the report, Prob>ChiSq, is the p-value for a likelihood
ratio test of whether the associated main effect is 0. The Prob>ChiSq value will be
simulated for each effect listed in the Effect Tests table.
2. Next to Number of Samples, enter 500.
3. (Optional) Next to Random Seed, enter 123 and then click outside the text box.
4. Click OK.
A Generalized Linear Model Simulate Results data table appears.
Note: Because response values are simulated, your simulated p-values will differ from
those shown in Figure 10.22.
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Figure 10.22 Table of Simulated Results, Partial View
The first row of the table contains the initial values of Prob>ChiSq and is excluded. The
remaining 500 rows contain simulated values.
5. Run the Power Analysis script.
Note: Because response values are simulated, your simulated power results will differ
from those shown in Figure 10.23.
Figure 10.23 Distribution Plots for the First Three Effects
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The histograms plot the 500 simulated Prob>ChiSq values for each main effect. The
Simulated Power outline shows the simulated Rejection Rate in the 500 simulations.
For easier viewing, stack the reports and de-select the plots, as follows.
6. Click the Distributions red triangle and select Stack.
7. Press Ctrl and click the X1 red triangle, and de-select Outlier Box Plot.
8. Press Ctrl and click the X1 red triangle, then select Histogram Options and de-select
Histogram.
Note: Because response values are simulated, your simulated power results will differ
from those shown in Figure 10.24.
Figure 10.24 Power Results for the First Three Effects
In the Simulated Power outlines, the Rejection Rate for each row gives the proportion of
p-values that are smaller that the corresponding Alpha. For example, for X3, which
corresponds to a coefficient value of 0.8 and a probability difference of 38%, the simulated
power for a 0.05 significance level is 379/500 = 0.758. Table 10.1 summarizes the estimated
power at the 0.05 significance level for all effects. Notice how power decreases as the
Difference to Detect decreases. Also notice that the power to detect an effect as large as
24.5% (X6) is only approximately 0.37.
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Note: Because response values are simulated, your simulated power results will differ
from those shown in Table 10.1.
Table 10.1 Simulated Power at Significance Level 0.05
Factor
Percent Passing
at Xi = 1
Percent Passing
at Xi = -1
Difference to
Detect
Simulated Power (Rejection
Rate) at Alpha=0.05
X1
73.11%
26.89%
46.2%
0.852
X2
71.09%
28.91%
42.2%
0.828
X3
69.00%
31.00%
38.0%
0.758
X4
66.82%
33.18%
33.6%
0.654
X5
64.56%
35.43%
29.1%
0.488
X6
62.25%
37.75%
24.5%
0.372
Launch the Simulate Window
To launch the Simulate window, right-click on a column of calculated values in a report
window and select Simulate. Simulate is available in many reports, including all reports that
support bootstrapping. To use Simulate, the data table must contain a formula with a random
component that simulates data.
The Simulate Window
Figure 10.25 Simulate Window for Tablet Production.jmp
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The Simulate window contains these panels and options:
Column to Switch Out The column that is replaced by the Column to Switch In.
Column to Switch In The column that replaces replace the Column to Switch Out. The
analysis will be repeated with values simulated according to the formula in the Column to
Switch In. Only columns with formulas are listed in the Column to Switch In panel.
Number of Samples Number of times that the report is re-run for a set of simulated data. The
default value is 2500.
A value that controls the simulated results. The random seed makes the
results reproducible.
Random Seed
When you click OK in the Simulation window, a window that shows a progress bar and a Stop
Early button appears. The number of the sample being simulated is shown above the progress
bar. If you click Stop Early, the simulated values that have been computed up to that point are
presented in a Simulate Results table. The window also shows you which analyses are being
run at any given time.
The Simulate Results Table
Simulate results appear in a table. Note the following:
•
The first row of the table contains the values for the table items that appear in the report.
For this reason, the first row is always excluded.
•
The remaining rows give the simulation results. The number of remaining rows is equal to
the Number of Samples you specified in the Simulate launch window.
•
The rows in the report are identified by the first column in the report table that contains
the selected column of calculated values. A column appears in the simulated results table
for each item in this first column.
•
The table contains a Distribution script that constructs a Distribution report. This report
contains histograms, quantiles, summary statistics, and simulation results for each column
in the simulated results data table. In addition to the standard Distribution report, the
report contains the following items:
– A red line that denotes the original estimate appears on the histogram.
– A Simulation Results report containing the original estimate, as well as confidence
intervals and empirical p-values for the simulation. See “Simulation Results Report” on
page 310.
– If the values in the simulated results data table have a PValue format, a Simulated
Power report is also provided. See “Simulated Power Report” on page 310.
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The table contains a Power Analysis script only if you have simulated a column of p-values.
This script constructs a Distribution report showing histograms of p-values and provides a
Simulated Power report. See “Simulated Power Report” on page 310.
Simulation Results Report
The value of the original estimate, also shown in the first row of the
Simulate Results data table. This estimate is labeled Y0.
Original Estimate
Confidence Intervals Lower and upper limits for quantile-based confidence intervals at the
following significance levels: 0.05, 0.10, 0.20, and 0.50.
Empirical p-Values The empirical p-values for a two-sided test and both one-sided tests that
compare the simulated values to the original estimate. These p-values are computed as the
proportions of the simulated values that fall in the ranges that are specified in the Test
column of the report.
Simulated Power Report
Alpha The significance level: 0.01, 0.05, 0.10, and 0.20.
Rejection Count The number of simulations where the test rejects at the corresponding
significance level.
Rejection Rate The proportion of simulations where the test rejects at the corresponding
significance level.
Lower 95% and Upper 95% Lower and upper limits for a 95% confidence interval for the
simulated rejection rate. The interval is computed using the Wilson score method. See
Wilson (1927).
Tip: Increase the Number of Samples for a narrower confidence interval.
Chapter 11
Bootstrapping
Approximate the Distribution of a Statistic through Resampling
Bootstrapping is available only in JMP Pro.
Bootstrapping is a resampling method for approximating the sampling distribution of a
statistic. You can use bootstrapping to estimate the distribution of a statistic and its properties,
such as its mean, bias, standard error, and confidence intervals. Bootstrapping is especially
useful in the following situations:
•
The theoretical distribution of the statistic is complicated or unknown.
•
Inference using parametric methods is not possible because of violations of assumptions.
Note: Bootstrap is available only from a right-click in a report. It is not a platform command.
Figure 11.1 Bootstrapping Results for a Slope Parameter
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Overview of Bootstrapping
Bootstrapping repeatedly resamples the observations that are used in your report to construct
an estimate of the distribution of a statistic or statistics. The observations are assumed to be
independent.
In the simple bootstrap, the n observations are resampled with replacement to produce a
bootstrap sample of size n. Note that some observations might not appear in the bootstrap
sample, and others might appear multiple times. The number of times that an observation
occurs in the bootstrap sample is called its bootstrap weight. For each bootstrap iteration, the
entire analysis that produced the statistic of interest is rerun with these changes:
•
the bootstrap sample of n observations is the data set
•
the bootstrap weight is a frequency variable in the analysis platform
This process is repeated to produce a distribution of values for the statistic or statistics of
interest.
However, the simple bootstrap can sometimes be inadequate. For example, suppose your data
set is small or you have a logistic regression setting where you can encounter separation
issues. In such cases, JMP enables you to conduct Bayesian bootstrapping using fractional
weights. When fractional weights are used, a fractional weight is associated with each
observation. The fractional weights sum to n. The statistic of interest is computed by treating
the fractional weights as a frequency variable in the analysis platform. For information about
fractional weights, see “Fractional Weights” on page 315 and “Calculation of Fractional
Weights” on page 324.
To run a bootstrap analysis in a report, right-click in a table column that contains the statistic
that you want to bootstrap and select Bootstrap.
Note: Bootstrap is available only from a right-click in a report. It is not a platform command.
JMP provides bootstrapping in most statistical platforms. The observations that comprise the
sample are all observations that are used in the calculations for the statistics of interest. If the
report uses a frequency column, the observations in that column are treated as if they were
repeated the number of times indicated by the Freq variable. If the report uses a Weight
variable, Bootstrap treats it as it was treated in the calculations for the report.
Tip: Bootstrap reruns the entire analysis that appears in the platform report from which
Bootstrap is invoked. As a result, Bootstrap might run slowly for your selected column
because of extraneous analyses in the report. If Bootstrap is running slowly, remove
extraneous options from the platform report before running Bootstrap.
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Example of Bootstrapping
This example uses the Car Physical Data.jmp sample data table. A tire manufacturer wants to
predict an engine’s horsepower from the engine’s displacement (in3). The company is most
interested in estimating the slope of the relationship between the variables. The slope values
help the company predict the corresponding change in horsepower when the displacement
changes.
In this example, the regression assumption of homogeneity of variance is violated, so the
confidence limits from the regression analysis for the slope might be misleading. For this
reason, the company uses a bootstrap estimate of the confidence interval for the slope.
1. Select Help > Sample Data Library and open Car Physical Data.jmp.
2. Select Analyze > Fit Y by X.
3. Select Horsepower and click Y, Response.
4. Select Displacement and click X, Factor.
5. Click OK.
6. Select Fit Line from the Bivariate Fit red triangle menu.
The slope estimate is 0.503787, approximately 0.504.
7. (Optional) Right-click in the Parameter Estimates report and select Columns > Lower 95%.
8. (Optional) Right-click in the Parameter Estimates report and select Columns > Upper 95%.
The confidence limits from the regression analysis for the slope are 0.4249038 and
0.5826711.
9. Right-click the Estimate column in the Parameter Estimates report and select Bootstrap
(Figure 11.2).
Figure 11.2 The Bootstrap Option
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The column that you right-click is relevant when the Split Selected Column option is
selected. For more information, see “Bootstrapping Window Options” on page 315.
10. Type 1000 for the Number of Bootstrap Samples.
11. (Optional) To match the results in Figure 11.3, type 12345 for the Random Seed.
12. Click OK.
The bootstrap process runs and produces a Bootstrap Results data table with unstacked
results for the slope and intercept.
Next, analyze the bootstrapped slope.
13. In the Bootstrap Results table, run the Distribution script.
The Distribution report includes the Bootstrap Confidence Limits report (Figure 11.3).
Figure 11.3 Bootstrap Report
The estimate of the slope (step 6) is 0.504. Based on the bootstrap results for 95% coverage, the
company can estimate the slope to be between 0.40028 and 0.61892. When the displacement is
changed by one unit, with 95% confidence, the horsepower changes by some amount between
0.40028 and 0.61892. The bootstrap confidence interval for the slope (0.400 to 0.619) is slightly
wider than the confidence interval (0.425 to 0.583) obtained using the usual regression
assumptions in step 7 and step 8.
Note: The BC Lower and BC Upper columns in the Bootstrap Confidence Limits report refer
to bias-corrected intervals. See “Bias-Corrected Percentile Intervals” on page 324.
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To perform a bootstrap analysis, right-click on a numeric column of sample statistics in a table
in a report window and select Bootstrap. The selected column is highlighted, and the
Bootstrapping window appears. After you select options and click OK in the Bootstrapping
window, bootstrap results for every statistic in the column appear in the default results table.
The Bootstrapping window contains the following options:
Sets the number of times that you want to resample the data
and compute the statistics. A larger number results in more precise estimates of the
statistics’ properties. By default, the number of bootstrap samples is set to 2500.
Number of Bootstrap Samples
Sets a random seed that you can re-enter in subsequent runs of the bootstrap
analysis to duplicate your current results. By default, no seed is set.
Random Seed
Performs a Bayesian bootstrap analysis. In each bootstrap iteration, each
observation is assigned a weight that is calculated as described in “Calculation of
Fractional Weights” on page 324. The weighted observations are used in computing the
statistics of interest. By default, the Fractional Weights option is not selected and a simple
bootstrap analysis is conducted.
Fractional Weights
Tip: Use the Fractional Weights option if the number of observations that are used in your
analysis is small or if you are concerned about separation in a logistic regression setting.
Suppose that Fractional Weights is selected. For each bootstrap iteration, each observation
that is used in the report is assigned a nonzero weight. These weights sum to n, the
number of observations used in the calculations of the statistics of interest. For more
information about how the weights are calculated and used, see “Calculation of Fractional
Weights” on page 324.
Split Selected Column Places bootstrap results for each statistic in the column that you
selected for bootstrapping into a separate column in the Bootstrap Results table. Each row
of the Bootstrap Results table (other than the first) corresponds to a single bootstrap
sample.
If you deselect this option, a Stacked Bootstrap Results table appears. For each bootstrap
iteration, this table contains results for the entire report table that contains the column that
you selected for bootstrapping. Results for each row of the report table appear as rows in
the Stacked Bootstrap Results table. Each column in the report table defines a column in
the Stacked Bootstrap Results table. For an example, see “Stacked Results Table” on
page 316.
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(Applicable only if the Split Selected Column option is
selected.) Determines the number of results tables produced by Bootstrap.
Discard Stacked Table if Split Works
If the Discard Stacked Table if Split Works option is not selected, then two Bootstrap tables
are shown:
– The Stacked Bootstrap Results table, which contains bootstrap results for each row of
the table containing the column that you selected for bootstrapping. This table gives
bootstrap results for every statistic in the report, where each column is defined by a
statistic.
– The unstacked Bootstrap Results table, which is obtained by splitting the stacked table
and providing results only for the column that is selected in the original report.
If the Discard Stacked Table if Split Works option is selected and if the Split Selected
Column operation is successful, the Stacked Bootstrap Results table is not shown.
Stacked Results Table
The initial results of a bootstrap analysis appear in a stacked results table (Figure 11.4). This
table might not appear if you have selected the Discard Stacked Table if Split Works option.
Figure 11.4 shows a bootstrap table that is based on the Parameter Estimates report obtained
by fitting a Bivariate model in Fit Y by X to Car Physical Data.jmp. See “Overview of
Bootstrapping” on page 312.
Figure 11.4 Stacked Bootstrap Results Table
Note the following about the stacked results table:
•
For each bootstrap sample, there is a row for each value given in the first column of the
report table. These values are shown in a column whose name is the name of the first
column in the report table. In this example, for each bootstrap sample there is a row
containing results for each Term: Intercept and Displacement, which appear in the Term
column.
•
The data table columns that are used in the analysis appear in the table. In this example, X
is Displacement, and Y is Horsepower.
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•
There is a column for every column in the report table that you are bootstrapping. In this
example, the columns are ~Bias, Estimate, Std Error, t Ratio, and Prob>|t|. Note that ~Bias is
a column in the Fit Y by X report that is hidden unless one of the parameter estimates is
biased.
•
The BootID• column identifies the bootstrap sample. The rows where BootID• = 0
correspond to the original estimates. Those rows are marked with an X and have the
excluded row state. In this example, each bootstrap sample is used to calculate results for
two rows: the results for Intercept and the results for Displacement.
•
The data table name begins with “Stacked Bootstrap Results”.
If you selected the Split Selected Column option, an unstacked results table might also appear.
See “Unstacked Bootstrap Results Table” on page 317.
Unstacked Bootstrap Results Table
Select Split Selected Column to create a bootstrap table that contains separate columns for the
report column that you selected. Each column corresponds to a Term in the report table. For
example, in Figure 11.5, the Estimate column from Figure 11.4 is split into two columns
(Displacement and Intercept), one for each level of Term.
Figure 11.5 Unstacked Bootstrap Results Table
Note the following about the unstacked results table:
•
There is a single row for each bootstrap sample.
•
The data table columns used in the analysis appear in the table. In this example, X is
Displacement, and Y is Horsepower.
•
There is a column for each row of the report that was bootstrapped.
•
If you specified a Random Seed in the Bootstrapping window, the bootstrap results table
contains a table variable called Random Seed that gives its value.
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•
The unstacked bootstrap results table contains a Source table script and a Distribution
table script. The Distribution table script enables you to quickly obtain statistics based on
the bootstrap samples, including bootstrap confidence intervals.
•
The BootID• column identifies the bootstrap sample. The row where BootID• = 0
corresponds to the original estimates. That row is marked with an X and has the excluded
row state. In the unstacked bootstrap table, each row is calculated from a single bootstrap
sample.
•
The data table name ends with “Bootstrap Results (<colname>)”, where <colname>
identifies the column in the report that was bootstrapped.
Analysis of Bootstrap Results
Analyze your bootstrap results using the Distribution platform:
•
If your analysis produced an unstacked bootstrap results table, run the Distribution script
in the table.
•
If your analysis produced a stacked bootstrap results table, select Analyze > Distribution
and assign the columns of interest to the appropriate roles. In most cases, it is appropriate
to assign the column that corresponds to the first column in the report table to the By role.
The Distribution platform provides summary statistics for your bootstrap results. It also
produces a Bootstrap Confidence Limits report for any table that contains a BootID• column
(Figure 11.6).
You can use the Distribution report to obtain two types of bootstrap confidence intervals:
•
The Quantiles report provides percentile intervals. For example, to construct a 95%
confidence interval using the percentile method, use the 2.5% and 97.5% quantiles as the
interval bounds.
•
The Bootstrap Confidence Limits report provides bias-corrected percentile intervals. The
report shows intervals with 95%, 90%, 80%, and 50% coverage levels. The BC Lower and
BC Upper columns show the lower and upper endpoints, respectively. For more
information about the computation of the bias-corrected percentile intervals, see
“Bias-Corrected Percentile Intervals” on page 324.
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Figure 11.6 Bootstrap Confidence Limits Report
The Original Estimate at the bottom of the Bootstrap Confidence Limits report is the estimate
of the statistic using the original data.
For more information about interpreting the Bootstrap Confidence Limits report, see
“Overview of Bootstrapping” on page 312. Efron (1981) describes the methods for both the
percentile interval and the bias-corrected percentile interval.
Additional Example of Bootstrapping
This example illustrates the benefits of the Fractional Weights (Bayesian Bootstrap) option for
a small data table. The data consist of a response, Y, measured on three samples of each of
seven different soil types. A scientist is interested in finding a confidence interval for the mean
response for the wabash soil type.
Because each soil type has only three observations, the simple bootstrap has the potential to
exclude all three of the observations for wabash from a bootstrap sample. The Fractional
Weights option ensures that all observations for every soil type are represented in all
bootstrap samples.
The scientist examines the distribution of wabash sample means from both bootstrap
methods:
•
“Simple Bootstrap Analysis” on page 320
•
“Bayesian Bootstrap Analysis” on page 322
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1. Select Help > Sample Data Library and open Snapdragon.jmp.
2. Select Analyze > Fit Y by X.
3. Select Y and click Y, Response.
4. Select Soil and click X, Factor.
5. Click OK.
6. Select Means/Anova from the Oneway Analysis red triangle menu.
7. In the Means for Oneway Anova report, right-click the Mean column and select Bootstrap.
8. Type 1000 for the Number of Bootstrap Samples.
9. (Optional) To match the results in Figure 11.7, type 12345 for the Random Seed.
10. Click OK.
Figure 11.7 Bootstrap Results for a Simple Bootstrap
The missing values in Figure 11.7 represent bootstrap iterations in which none of the
observations for a given soil type were selected for the bootstrap sample.
11. Select Analyze > Distribution.
12. Select wabash and click Y, Columns.
13. Click OK.
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Figure 11.8 Distribution of wabash Means from a Simple Bootstrap
Figure 11.8 shows the distribution of wabash means from the simple bootstrap analysis.
Notice the following:
– The Summary Statistics report indicates that the number of rows containing bootstrap
means for wabash is N = 961. Although you conducted 1,000 iterations, 39 bootstrap
samples did not contain any of the three observations for wabash.
– The histogram of sample means is not smooth, with peaks at the two extremes. The
three values for wabash are 38.2, 37.8, and 31.9. The peak at the low end of the
distribution results from bootstrap samples that contain only the value 31.9. The peak
at the high end results from bootstrap samples that contain one or both of the values
38.2 and 37.8.
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Next, use the Fractional Weights (Bayesian Bootstrap) option to avoid obtaining missing
values from the bootstrap samples and to smooth the distribution of bootstrapped means.
Bayesian Bootstrap Analysis
1. In the Oneway Analysis report, right-click the Mean column in the Means for Oneway
Anova report and select Bootstrap.
2. Type 1000 for the Number of Bootstrap Samples.
3. (Optional) To match the results in Figure 11.9, type 12345 for the Random Seed.
4. Select the Fractional Weights option.
5. Click OK.
Figure 11.9 Bootstrap Results for a Bayesian Bootstrap
There are no missing values in the Bayesian Bootstrap results table. All 21 rows in the
Snapdragon.jmp data table are included, with varying bootstrap weights, in each bootstrap
sample.
6. Select Analyze > Distribution.
7. Select wabash and click Y, Columns.
8. Click OK.
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Figure 11.10 Distribution of wabash Means from a Bayesian Bootstrap
The Bayesian Bootstrap produces a much smoother distribution for the wabash sample
means. All 1,000 bootstrap samples include the three observations for wabash. For each
iteration, the wabash sample mean is calculated using different fractional weights.
The Bootstrap Confidence Limits report shows that a 95% confidence interval for the mean
is 32.6396 to 37.8168.
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This section provides details about the Bootstrapping methods.
Calculation of Fractional Weights
The Fractional Weights option is based on the Bayesian bootstrap (Rubin, 1981). The number
of times that an observation occurs in a given bootstrap sample is called its bootstrap weight. In
the simple bootstrap, the bootstrap weights for each bootstrap sample are determined using
simple random sampling with replacement.
In the Bayesian approach, sampling probabilities are treated as unknown parameters and
their posterior distribution is obtained using a non-informative prior. Estimates of the
probabilities are obtained by sampling from this posterior distribution. These estimates are
used to construct the bootstrap weights, as follows:
•
Randomly generate a vector of n values from a gamma distribution with shape parameter
equal to (n - 1)/n and scale parameter equal to 1.
Note: Rubin (1981) uses 1 as the gamma shape parameter. The shape parameter that is
used in JMP Pro ensures that the mean and variance of the fractional weights are equal to
the mean and variance of the simple bootstrap weights.
•
Compute S = sum of the n values.
•
Compute the fractional weights by multiplying the vector of n values by n/S.
•
If a Freq or Weight variable is specified for the analysis, multiply the fractional weights by
the Freq or Weight values on a row-by-row basis. The sum of the values of the Freq or
Weight variable must be greater than 1.
This procedure scales the fractional weights to have mean and variance equal to those of the
simple bootstrap weights. The fractional bootstrap weights are positive, sum to n, and have a
mean of 1.
Bias-Corrected Percentile Intervals
This section describes the calculation of the bias-corrected (BC) confidence intervals that
appear in the Bootstrap Confidence Limits report when you run the Distribution script in the
Bootstrap Results table. Bias-corrected percentile intervals improve on the ability of percentile
intervals in accounting for asymmetry in the bootstrap distribution. See Efron (1981).
Notation
•
p* is the proportion of bootstrap samples with an estimate of the statistic of interest that is
less than or equal to the original estimate.
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z0 is the p* quantile of a standard normal distribution.
•
zα is the α quantile of a standard normal distribution.
Bias-Corrected Confidence Interval Endpoints
The endpoints of a (1 - α) bias-corrected confidence intervals are given by quantiles of the
bootstrap distribution:
•
The lower endpoint is the Φ ( 2z 0 + z α ) quantile.
•
The upper endpoint is the Φ ( 2z 0 + z 1 – α ) quantile.
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Chapter 12
Text Explorer
Explore Unstructured Text in Your Data
Many features in this platform are available only in JMP Pro and noted with this icon.
The Text Explorer platform enables you to analyze unstructured text, such as comment fields
in surveys or incident reports. Interact with the text data by using tools to combine similar
terms, recode misspecified terms, and understand the underlying patterns in your textual
data.
The JMP Pro version of the platform also contains analysis tools that use singular value
decomposition (SVD) to group similar documents into topics. You can cluster text documents
or cluster terms that are in a collection of documents. You can also cluster documents using
latent class analysis.
Figure 12.1 SVD Plots in Text Explorer
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Text Explorer Platform Overview
Unstructured text data are common. For example, unstructured text data could result from a
free response field in a survey, product review comments, or incident reports. The Text
Explorer platform enables you to explore unstructured text in order to better understand its
meaning. Text analysis is often an iterative process, so you might alternate between curating
and analyzing the list of terms.
Curating the List of Terms
Text analysis uses some unique terminology. A term or token is the smallest piece of text,
similar to a word in a sentence. However, you can define terms in many ways, including
through the use of regular expressions; the process of breaking the text into terms is called
tokenization.
•
A phrase is a short collection of terms; the platform has options to manage phrases that are
specified as terms in and of themselves.
•
A document refers to a collection of words; in a JMP data table, the unstructured text in
each row of the text column corresponds to a document.
•
A corpus refers to a collection of documents.
It is often desirable to exclude some common words from the analysis. These excluded words
are called stop words. The platform has a default list of stop words, but you can also add
specific words as stop words. Although stop words are not eligible to be terms, they can be
used in phrases.
You can also recode terms; this is useful for combining synonyms into one common term.
Stemming is the process of combining words with identical beginnings (stems) by removing the
endings that differ. This results in “jump”, “jumped”, and “jumping” all being treated as the
term “jump·”. The stemming procedure is similar to the procedure used in the Snowball string
processing language. When a phrase is stemmed, each word in the phrase is stemmed as it
would be stemmed as a stand-alone term.
Analyzing the List of Terms
Text analysis in the Text Explorer platform uses a bag of words approach. Other than in the
formation of phrases, the order of terms is ignored. The analysis is based on the term counts.
After you curate the list of terms through the use of regular expressions, stop words, recoding,
and stemming, you can perform analyses on the curated list of terms. The analysis options in
the platform are based on the document term matrix (DTM). Each row in the DTM corresponds
to a document (a cell in a text column of a JMP data table). Each column in the DTM
corresponds to a term from the curated term list. This approach implements the bag of words
approach since it ignores word ordering. In its simplest form, each cell of the DTM contains
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the frequency (number of occurrences) of the column’s term in the row’s document. There are
various other weighting schemes; these are described in “Save Options” on page 351.
The analysis options that are available in the platform first perform a singular value
decomposition (SVD) on the document term matrix. This can greatly reduce the number of
columns needed to represent the term information in the data. For more details about singular
value decomposition, see the Statistical Details appendix in the Multivariate Methods book.
Hierarchical clustering options are available for clustering the terms and for clustering the
documents. These options enable you to group similar terms or documents together.
Platform Workflow
The expected steps for using the Text Explorer platform are as follows:
1. Specify the method for tokenizing (either built-in or customized regular expression).
2. Use the report to specify additional stop words, add phrases to the term list, perform
recodes of terms, and specify exceptions to stemming rules.
3. Specify the preference for stemming.
4. Use word and phrase counts, SVD, and clustering approaches to identify important terms
and phrases.
Note: The SVD and clustering options are available only in JMP Pro.
5. Save results for use in further analysis: the term table, the DTM, the singular values, or
other results.
Note: The option to save the singular values is available only in JMP Pro.
6. Save Phrase, Recode, and Stop Words properties for use in future analyses of similar text
data.
Text Processing Steps
The text is processed in three stages: tokenizing, phrasing, and terming.
Tokenizing Stage
The Tokenizing stage performs the following operations:
1. Convert text to lowercase.
2. Apply Tokenizing method (either Basic Words or Regex) to group characters into tokens.
3. Recode tokens based on specified recode definitions. Note that recoding occurs before
stemming.
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Phrasing Stage
The Phrasing stage collects phrases that occur in the corpus (collection of documents) and
enables you to specify that individual phrases be treated as terms. Phrases cannot start or end
with a stop word, but they can contain a stop word.
Terming Stage
The Terming stage creates the Term List from the tokens and phrases that result from the
previous stages.
For each token, the Terming stage performs the following operations:
1. Check that the minimum and maximum length requirements specified in the launch
window are met. Tokens that contain only numbers are excluded from this operation.
2. Check that the token is qualified to become a term; tokens parsed by the Basic Words
tokenization method must contain at least one alphabetical or Unicode character. Tokens
that contain only numbers are excluded from this operation. The Regex tokenization
method uses regular expressions to determine what characters are part of a token.
3. Check that the token is not a stop word.
4. Apply stemming and stem exceptions.
For each phrase that you add, the Terming stage performs the following operations:
1. Add the phrase to the Term List. Phrases should apply stemming to each word in the
phrase that is stemmed in the Term List. Phrases that have different raw tokens but the
same stems are combined in the Term List.
2. Remove token term occurrences that appear in the phrase.
Example of the Text Explorer Platform
In this example, you want to explore the text responses from a survey about pets.
1. Select Help > Sample Data Library and open Pet Survey.jmp.
2. Select Analyze > Text Explorer.
3. Select Survey Response and click Text Columns.
4. Click OK.
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Figure 12.2 Example of Initial Text Explorer Report
At a glance, you can see that there are 372 unique terms in 194 documents. In all, there are
1921 tokenized terms. The most common term is “cat”, and it occurs 55 times.
5. From the red triangle menu next to Text Explorer for Survey Response, select Term Options
> Stemming > Stem All Terms.
6. In the Phrase List table, select cat food and dog food, right-click on the selection, and select
Add Phrase.
The terms cat food and dog food are included in the Term List.
7. Scroll down in the Term List and find the cat and dog food entries.
You can see that there are four occurrences of each phrase.
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Figure 12.3 Term List after Modifications and Scrolling
In the Phrase List, cat food and dog food are gray, since they are now locally being treated
as terms in this Text Explorer report.
8.
9.
From the red triangle menu next to Text Explorer for Survey Response, select Latent
Semantic Analysis, SVD.
Click OK to accept the default values.
Two SVD Plots appear in the report, as shown in Figure 12.4. The one on the left shows the
first two singular vectors in the document space. The one on the right shows the first two
singular vectors in the term space.
Figure 12.4 SVD Plots
10.
Select the three right-most points in the left SVD Plot.
These three points represent survey responses that are clustered away from the rest of the
points. To further investigate this cluster, you read the text of these responses.
11.
Click the Show Text button that is below the SVD Plots outline title.
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Figure 12.5 Text of Selected Documents
A window appears that contains the text of the three documents represented by the
selected points. These survey responses are similar in that they all refer to some
combination of “funny”, “cat”, and “video”. These documents have larger positive values
for the first singular vector than the rest of the documents, which indicates that they are
different from the rest of the documents in that dimension.
Further investigation of the singular vector dimensions could lead to interpretations of
what the dimensions represent. For example, many of the documents on the far right of
the plot are responses that are about cats. On the far left, many of the responses are about
dogs. Therefore, the first singular vector is picking up differences based on whether the
response was about a cat or a dog.
Launch the Text Explorer Platform
Launch the Text Explorer platform by selecting Analyze > Text Explorer.
Figure 12.6 The Text Explorer Launch Window
The Text Explorer launch window contains the following options:
Text Columns Assigns the columns that contain text data. If you specify multiple columns, a
separate analysis is created for each column.
ID Assigns a column used to identify separate respondents in the Save Stacked DTM for
Association output data table. This output data table is suitable for association analysis.
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Identifies a column that creates a report consisting of separate analyses for each level of
the variable. If more than one By variable is assigned, a separate report is produced for
each possible combination of the levels of the By variables.
Note: If you specify a By variable, the Customize Regex option and settings apply to all levels
of the By variables.
Specifies the language used for text processing. This affects stemming and the
built-in lists of stop words, recodes, and phrases. This option is independent of the
language in which JMP is running. Unless the Language platform preference is set, the
Language option is set according to the JMP Display Language preference. However, the
Language option in Text Explorer does not support Chinese, Japanese, or Korean. If the
JMP Display Language is Chinese, Japanese, or Korean, this option defaults to English.
Language
Specifies a maximum number of words that a phrase can contain
to be included as a phrase in the analysis.
Maximum Words per Phrase
Maximum Number of Phrases Specifies the maximum number of phrases that appear in the
Phrase List.
Minimum Characters per Word Specifies the number of characters that a word must contain
to be included as a term in the analysis.
Specifies the largest number of characters (up to 2000) that a
word can contain to be included as a term in the analysis.
Maximum Characters per Word
Stemming Specifies a method for combining terms with similar beginning characters but
different endings. The following options are available:
No Stemming No terms are combined.
Stem for Combining
Stems only the terms where two or more terms stem to the same
term.
Stem All Terms Stems all terms.
Note: The use of the Stemming option also affects phrases that have been added to the
Term List. Phrase identification occurs after terms within a phrase have been stemmed.
For example, “dogs bark” and “dog barks” would both match the specified phrase “dog·
bark·”.
Tokenizing Specifies a method for parsing the text into terms or tokens. The following
tokenization options are available:
Regex Parses text using a default set of built-in regular expressions. If you want to add
to, remove, or edit the set of regular expressions used to parse the text, select the
Customize Regex option. See “Customize Regex: Regular Expression Editor” on
page 335.
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Basic Words Text is parsed into words based on a set of characters that typically separate
words. These characters include spaces, tabs, new lines, and most punctuation marks.
If you want numbers to be parsed into terms for the analysis, select the Treat Numbers
as Words option. If you do not select this option, pieces of text between delimiters that
contain only numbers are ignored in the tokenizing step.
Tip: You can view the default set of delimiters using the Display Options > Show Delimiters
option in a Text Explorer report that uses the Basic Words Tokenizing method.
Customize Regex (Available only with the Regex Tokenizing method.) Enables you to use the
Text Explorer Regular Expression Editor window to modify the Regex settings. Use this
option to accommodate non-traditional words. Examples include phone numbers or
words formed by a combination of characters and numbers. Using the Customize Regex
option is not recommended unless the default Regex method is not giving you the results
that you need. See “Customize Regex: Regular Expression Editor” on page 335.
Treat Numbers as Words (Available only with the Basic Words Tokenizing method.) Allows
numbers to be tokenized as terms in the analysis. When this option is selected, the
Minimum Characters per Word setting is ignored for terms that contain numeric digits.
After you click OK on the launch window, the Text Explorer Regular Expression Editor
window appears if you selected Customize Regex in the launch window. Otherwise, the Text
Explorer report appears.
Note: The processing of text input is not case-sensitive. All text is converted to lowercase
internally prior to tokenization. This conversion affects the processing of regular expressions
and the aggregation of terms in the Text Explorer output.
Customize Regex: Regular Expression Editor
When you select the Customize Regex option, the Text Explorer Regular Expression Editor
appears. Use this window to parse text documents using a wide variety of built-in regular
expressions, such as phone numbers, times, or monetary values. You can also create your own
regular expression definitions.
Note: Using the Customize Regex option is recommended only if you are not getting desired
results from the default Regex method.
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Figure 12.7 Text Explorer Regular Expression Editor
Parsing with the Script Editor Box
The script editor box at the top of the window shows you how the parsing would proceed for
sample text. The results of parsing the regular expressions in the Regex Editor list are
highlighted in colors that correspond to the colors in the Regex Editor list.
•
Click the Previous and Next row buttons to populate the script editor box with text from
your own data. This enables you to see how a given row of text data is parsed.
•
Click the Save to Column button to save a new column to the data table that contains the
result of the regular expression tokenization. For more information about specifying the
result of the regular expression, see “Editing the Regular Expressions” on page 337.
Note: The Save to Column button uses only the regular expression to match text. The
following settings are not used: stop words, recodes, stemming, phrases, or minimum and
maximum characters per word to modify the output of the regular expression.
Adding Regular Expressions
To add a regular expression to be used in tokenization, click Copy From Library. The Regex
Library Selections window appears. This window contains all the built-in regular expressions
as well as any recently modified regular expressions that you created in previous instances of
the Regular Expression Editor. Built-in regular expressions are labeled. Custom regular
expressions that are saved in your library are labeled with the name that you specified. Only
the most recent expression for a given name is stored in the Regex Library. Use the Delete
Selected Item button to remove a custom regular expression from the Regex Library. The
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Regex Library for each user is stored as a JSL file in a directory called TextExplorer. The
location of this directory is based on your computer’s operating system, as follows:
•
Windows: "C:/Users/<username>/AppData/Roaming/SAS/JMP/TextExplorer/"
•
Macintosh: "/Users/<username>/Library/Application Support/JMP/TextExplorer/"
These files can be shared with other users, but you should not edit the file directly. Use the
Regular Expression Editor instead.
Editing the Regular Expressions
Terms are tokenized by processing the regular expressions in the order specified in the Regex
Editor panel. To change the order of the regular expressions, select a regular expression in the
list and click the up or down arrow buttons below the list. You can also drag and drop items in
the regular expression list to change the order of execution. A blue triangle represents the
currently selected regular expression. To remove a regular expression and exclude it from the
tokenization, select it in the list and click the minus sign below the list. The “Leftover” regular
expression cannot be removed and must appear last in the sequence of regular expressions.
When you select a regular expression in the list, the editable fields in the Regex Editor panel
refer to the selected regular expression. Click and type in any of these fields to edit them.
Each regular expression has the following attributes:
Title Specifies a name used to identify the regular expression in the current window (as well
as in the Regex Library later).
Specifies the regular expression definition. The regular expression must have at least
one set of parentheses to designate the regular expression capture.
Regex
Result Specifies what replaces the text matched by the regular expression. This value can be
static text, blank, or the value of the regular expression capture. The regular expression
capture is defined as the result of the Regex definition:
– To replace the matched text with static text, specify the static text in the Result field.
– To ignore the matched text, leave the Result field blank.
– To keep the text that results from the outer-most parentheses in the regular expression,
use “\1” (without quotation marks) in the Result field.
– To keep the entire result of the regular expression, use “\0” (without quotation marks)
in the Result field.
Example (Optional) Specifies an example text string with colors indicating the behavior of
the regular expression.
Comment (Optional) Specifies a comment to explain the regular expression and its behavior.
Color Specifies the color used to identify matches of the regular expression in the text in the
Script Editor box and in the Example field.
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Note: If the regular expression definition in the Regex field is invalid, a red X appears next to
the name of the regular expression in the list of regular expressions.
Creating a Custom Regular Expression
Follow these steps to create your own custom regular expression:
1. Click Copy From Library.
2. In the Regex Library Selections window, select a regular expression that seems close to
what you need.
3. Click OK.
4. Edit the Regex definition in the Regex Editor panel.
5. Give your custom regular expression a different name in the Title field.
Tip: When editing the Regex definition field, it is helpful to have the Log window open and
visible. Some error messages appear only in the Log window. To open the Log window, select
View > Log. There are many Internet resources available for troubleshooting regular
expressions, such as www.regexr.com.
The Word Separator List
The Word Separator List button enables you to specify a list of characters that occur between
words in the tokenization process. The between-word characters cannot begin a word, but they
can appear inside a word if one of the regular expressions allows it. You can add or remove
characters from the list in the window that appears when you click the button. By default, the
only character in the list is a whitespace character. In the Separator Characters window, click
the Reset button to undo any modifications to the list of separator characters. Modifications to
the list of separator characters are applied only to the current regular expression tokenization.
The processing of the specified regular expressions and the required “Leftover” regular
expression proceeds as follows:
1. Compare the current character in the text stream to the list of separator characters.
– If the character is in the list of separator characters, ignore the character, process any
accumulated characters in the “Leftover” temporary string, move to the next character,
and repeat step 1.
– If the character is not in the list of separator characters, go to step 2.
2. Compare the string starting at the current character to each regular expression (one at a
time, up to, but not including, the “Leftover” regular expression).
– If the string starting at the current character matches one of the regular expressions, the
following events occur. Any accumulated characters in the “Leftover” temporary
string are processed. The value of the Result field is saved as a term. The current
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character in the text stream becomes the character following the matched string. The
processing returns to step 1.
– If the string starting at the current character does not match any of the regular
expressions up to the “Leftover” regular expression, go to step 3.
3. Collect characters into the “Leftover” temporary string by appending the current
character and setting the current character to the next character in the text stream. Return
to step 1.
– The “Leftover” temporary string is accumulated one character at a time, until one of
the other regular expressions produces a match.
– The default Result of the “Leftover” regular expression is to discard the accumulated
“Leftover” temporary string.
Tips:
•
If you set the Result of the “Leftover” regular expression to \1, you might want to add
more separator characters, such as punctuation marks. This ensures that your results do
not include the specified punctuation marks.
•
You might want to add more regular expressions from the Regex Library or create custom
ones to capture terms of interest rather than changing the Result of the “Leftover” regular
expression to \1.
The processing follows the above steps until reaching the end of the text string for each row in
the data table.
Saving the Results to a Column in the Data Table
Click the Save to Column button to save to the data table a new column that contains the
results of the regular expression tokenization. The new column is a character column with the
same name as the text column specified in the Text Explorer launch window; a number is
appended to the name so that the column names are unique.
Note: When you save the results of the custom regular expression tokenization to a column in
the data table, the regular expression process is run on the original text in each row of the data
table. It is not run on the version of the text string that was converted to lowercase.
Closing the Text Explorer Regular Expression Editor
After you click OK in the Text Explorer Regular Expression Editor window, the following
events occur:
1. The custom regular expressions defined in the Text Explorer Regular Expression Editor
window are saved to the Regex Library.
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Caution: The custom Regex Library is saved only when you click OK and there are
customized regular expressions. The most recently saved regular expressions will be
available next time. Use unique names to keep additional regular expressions in the Regex
Library. To ensure that a regular expression is available later, you can save a script from
the Text Explorer report window.
2. The Text Explorer report appears. The report shows the result of using the specified
regular expression settings to tokenize the text.
The Text Explorer Report
The Text Explorer report window contains the Summary Counts report and the Term and
Phrase Lists report.
Figure 12.8 Example of a Text Explorer Report
Summary Counts Report
The first table in the Text Explorer report window contains the following summary statistics:
Number of Terms
the number of terms in the Term List.
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Number of Cases the number of documents in the corpus.
Total Tokens
the total number of terms in the corpus.
Tokens per Case the number of tokens divided by the number of cases.
Number of Non-empty Cases the number of documents in the corpus that contain at least one
term.
Portion Non-empty per Case the proportion of documents in the corpus that contain at least
one term.
Term and Phrase Lists
The Term and Phrase Lists report contains tables of terms and phrases found in the text after
tokenization has occurred. See Figure 12.8 for an example of the Term and Phrase Lists report.
The Count column in the Term List indicates the number of occurrences of the term in the
corpus. The Count column in the Phrase List indicates the number of occurrences of the
phrase in the corpus; the N column indicates the number of words in the phrase.
By default, the Terms List is sorted in descending count order; terms that are tied in count are
sorted alphabetically. The Phrases List is sorted in descending count order; phrases that are
tied in count are then sorted in descending length (N) order. Further ties in the Phrases List
are sorted alphabetically. The sort order of each list can be changed to alphabetical sorting
using the options in each list.
The phrases that appear in the Phrase List are determined by the settings of the Maximum
Words per Phrase and Maximum Number of Phrases options in the launch window. Phrases
that occur only one time in the data table do not appear in the Phrase List.
Phrases can be specified as terms at various scopes. Phrases in the Phrase List that have been
specified as terms are colored based on the scope of the phrase specification. See Table 12.1.
For more information about specifying phrases in different scopes, see “Term Options
Management Windows” on page 346.
Table 12.1 Colors for Specified Phrases
Scope
Color
Built-in
Red
User Library
Green
Column Property
Orange
Local
Gray
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Actions for Terms and Phrases
You can access options in the Term List and Phrase List tables by selecting items and then
right-clicking in the left-most column of each table. You can save each table as a data table by
right-clicking in the Count column of each table and selecting Make into Data Table.
Term List Pop-up Menu Options
When you right-click in the Term column of the Term List table, a pop-up menu appears with
the following options:
Select Rows Selects rows in the data table that contain the selected terms.
Show Text Shows the documents that contain the selected terms.
Note: By default, only the first 10,000 documents are shown. If there are more than 10,000
documents that contain a selected term, a window appears that enables you to increase
this limit.
Alphabetical Order Toggles the sort order of the Term List between alphabetical order and
descending Count order.
Find Enables you to search for a string in the Term List. When results of a Find operation are
visible, you can return to the full list of terms by selecting Find again.
Copy Places the selected terms onto the clipboard.
Color
Enables you to assign a color to the selected terms.
Label
Places labels on the corresponding points in the Term SVD Plot for the selected terms.
Containing Phrases
Selects the phrases in the Phrase List table that contain the selected
terms.
Saves an indicator column to the data table for each term selected in the Term
List. The value of the indicator column for each row is 1 if the document in that row
contains the term and 0 otherwise.
Save Indicators
Save Formula Saves a column formula to the data table for each term selected in the Term
List. The column formula for each row evaluates to 1 if the document in that row contains
the term and 0 otherwise. This is useful for new documents.
Recode Enables you to change the values for one or more terms. Select the terms in the list
before selecting this option. After you select this option, the Recode window appears. See
the Enter and Edit Data chapter in the Using JMP book.
Add Stop Word Adds the selected terms to the list of stop words and removes those terms
from the Term List. This action also updates the Phrase List.
Add Stem Exception
stemming.
Adds the selected terms to the list of terms that are excluded from
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Make into Data Table Creates a JMP data table from the report table.
Make Combined Data Table Searches the report for other tables like the one you selected and
combines them into a single JMP data table.
Phrase List Pop-up Menu Options
When you right-click in the Phrase column of the Phrase List table, a pop-up menu appears
with the following options:
Select Rows Selects rows in the data table that contain the selected phrases.
Show Text
Shows the documents that contain the selected phrases.
Saves an indicator column to the data table for each phrase selected in the
Phrase List. The value of the indicator column for each row is 1 if the document in that row
contains the phrase and 0 otherwise.
Save Indicators
Alphabetical Order Toggles the sort order of the Phrase List between alphabetical order and
descending Count order.
Enables you to search for a string in the Phrase List. When results of a Find operation
are visible, you can return to the full list of phrases by selecting Find again.
Find
Copy Places the selected phrases onto the clipboard.
Select Contains Selects larger phrases in the Phrase List that contain the selected phrase.
Select Contained Selects smaller phrases in the Phrase List that are contained by the selected
phrase.
Add Phrase Adds the selected phrases to the Term List and updates the Term Counts
accordingly.
Add Stop Word Adds the selected phrases to the list of stop words and removes those
phrases from the Phrase List. This action also updates the Term List.
Make into Data Table Creates a JMP data table from the report table.
Make Combined Data Table Searches the report for other tables like the one you selected and
combines them into a single JMP data table.
Text Explorer Platform Options
This section describes the options available in the Text Explorer platform.
Text Preparation Options
The Text Explorer red triangle menu contains the following options for text preparation:
Display Options
Shows a submenu of options to control the report display.
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Shows or hides the Word Cloud report. The Word Cloud red triangle
menu enables you to change the layout and font for the word cloud. See “Word Cloud
Options” on page 345.
Show Word Cloud
The word cloud can be interactively resized by changing the width. The height is then
determined automatically. The rows in the Term List are linked to the terms in the
Word Cloud.
Show Term List
Shows or hides the Term List.
Show Phrase List Shows or hides the Phrase List.
Shows buttons in the Term and Phrase Lists report
corresponding to the options available in the pop-up menus for each list. See “Term
and Phrase Lists” on page 341.
Show Term and Phrase Options
Show Summary Counts Shows or hides the Summary Counts table. See “Summary
Counts Report” on page 340.
Show Stop Words Shows or hides a list of the stop words used in the analysis. A built-in
list of stop words is used initially. To add a stop word, right-click on it in the Term List
and select Add Stop Word from the pop-up menu. See “Term Options Management
Windows” on page 346.
Show Recodes Shows or hides a list of the recoded terms. See “Term Options
Management Windows” on page 346.
Show Specified Phrases Shows or hides a list of the phrases that have been specified by
the user to be treated as terms. See “Term Options Management Windows” on
page 346.
Show Stem Exceptions Shows or hides the terms that are excluded from stemming. See
“Term Options Management Windows” on page 346.
Show Delimiters Shows or hides the delimiters used by the Basic Words Tokenizing
method. To modify the set of delimiters used, you must use the Add Delimiters() or
Set Delimiters() messages in JSL. This option is available only when the selected
Tokenizing method is Basic Words.
Show Stem Report Shows or hides the Stemming report that contains two tables of
stemming results. The table on the left maps each stem to the corresponding terms. The
table on the right maps each term to its corresponding stem. This option is available
only when the selected Stemming method is not No Stemming.
Show Selected Rows Opens a window that contains the text of the documents that are in
the currently selected rows.
Term Options Shows a submenu of options that apply to the Term List.
Stemming See the description of stemming options in “Launch the Text Explorer
Platform” on page 333.
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Include Builtin Stop Words Specifies if the stop words used in the tokenizing process
include built-in stop words or not.
Specifies if the phrases used in the tokenizing process include
built-in phrases or not.
Include Builtin Phrases
Manage Stop Words Shows a window that enables you to add or remove stop words. The
changes made can be applied at the User, Column, and Local levels. You can also
specify Local Exceptions that exclude stop words that are specified in any of the other
levels. See “Term Options Management Windows” on page 346.
Manage Recodes Shows a window that enables you to add or remove recodes. The
changes made can be applied at the User, Column, and Local levels. You can also
specify Local Exceptions that exclude recodes that are specified in any of the other
levels. See “Term Options Management Windows” on page 346.
Manage Phrases Shows a window that enables you to add or remove the phrases that are
treated as terms. The changes made can be applied at the User, Column, and Local
levels. You can also specify Local Exceptions that exclude phrases that are specified in
any of the other levels. See “Term Options Management Windows” on page 346.
Manage Stem Exceptions Shows a window that enables you to add or remove exceptions
to stemming. The changes made can be applied at the User, Column, and Local levels.
You can also specify Local Exceptions that exclude stem exceptions that are specified in
any of the other levels. See “Term Options Management Windows” on page 346.
Parsing Options Shows a submenu of options that apply to parsing and tokenization.
Tokenizing See the description of tokenizing options in “Launch the Text Explorer
Platform” on page 333.
Customize Regex Shows the Customize Regex window. This option enables you to
modify the Regex settings for the current Text Explorer report. This option is available
only when the selected Tokenizing method is Regex.
Note: If you specified a By variable in the platform launch window, the Customize Regex
option automatically broadcasts to all level of the By variables.
Treat Numbers as Words Allows numbers to be tokenized as terms in the analysis. Note
that this option is affected by the setting for Minimum characters per word. This option
is available only when the selected Tokenizing method is Basic Words.
Word Cloud Options
The Word Cloud red triangle menu contains the following options:
Layout (Set to Ordered by default.) Specifies the arrangement of the terms in the Word
Cloud.
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Presents the terms in horizontal lines ordered from most to least frequent.
Alphabetical
Presents the terms in horizontal lines sorted in ascending alphabetical order.
Centered Presents the terms in a cloud and sized by frequency
Coloring
(Set to None by default.) Specifies the coloring of the terms in the Word Cloud.
None Colors each term black. You can change this color in the Legend.
Arbitrary Grays
Arbitrary Colors
Colors each term in varying shades of gray.
Colors each term in various colors. You can adjust the colors in the
Legend.
By column values Colors each term on a gradient color scale. The scale is based on the
difference between the overall mean of another column and the value for a term in the
Score Terms by Column command. You can adjust the colors and gradient in the
Legend.
Font
Specifies the font, style, and size of the terms in the Word Cloud.
Show Legend (On by default.) Shows or hides the legend for the Word Cloud.
Term Options Management Windows
Phrase, stop word, recode, and stem exception information can be specified for many different
scopes. They can be stored in the following locations: the Text Explorer user library (User
scope), a column property for the analysis column (Column scope), or in a platform script
(Local scope). You can save the local specifications and local exceptions for a specific instance
of Text Explorer by saving the script for the Text Explorer report.
The Term Options management windows are four similar windows that enable you to
manage the collections of stop words, recodes, phrases, and stem exceptions. Figure 12.9
shows the Manage Stop Words window. The Manage Phrases and Manage Stem Exceptions
are identical to the Manage Stop Words window. The Manage Recodes window differs
slightly. See “Manage Recodes” on page 348.
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Figure 12.9 Manage Stop Words Window
Manage Stop Words
The Manage Stop Words window contains multiple lists of stop words that represent the
different scopes (or locations) of specified stop words. Below each list is a text edit box and an
add button. These controls enable you to add custom stop words to each scope. You can move
stop words from one scope to another by dragging them. You can copy and paste items from
one list to another list. Two buttons at the bottom of the window move the selected items from
one scope to the next, either left or right. The X button removes the selected items from their
current scope. You can edit existing items in the lists by double-clicking on an item and
changing the text.
Specifies the list of Built-in stop words and to which language the user library
selections are saved. If you select Apply Items for Language, the changes are saved to the
master user library. The Language setting applies only to the Built-in and User scopes.
Language
Lists the built-in list of stop words for the specified language. You can
exclude a built-in stop word by placing it in the Local Exceptions list.
Built-in (Locked)
User Lists the stop words in the user library for the specified language.
Column Lists the stop words in the “Stop Words” column property for the text column.
Lists the stop words in the local scope. They can be specified when Text Explorer is
launched via JSL. These stop words are used only in the current Text Explorer platform
report.
Local
Lists words that are not treated as stop words in the current Text Explorer
platform. They can be specified when Text Explorer is launched via JSL. The words listed
in Local Exceptions override words listed in all of the other scopes.
Local Exceptions
Enables you to import stop words from a text file. The stop words are copied to the
clipboard. You can paste them into any of the lists other than Built-in.
Import
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Export Enables you to export stop words to the clipboard or to a text file. An Export window
appears that enables you to select the scopes for which you would like to export stop
words and the location of the export.
The user library files are located in a TextExplorer directory. The location of this directory is
based on your computer’s operating system:
•
Windows: "C:/Users/<username>/AppData/Roaming/SAS/JMP/TextExplorer/<lang>/"
•
Macintosh: "/Users/<username>/Library/Application Support/JMP/TextExplorer/<lang>/"
The master user library files are located in the TextExplorer directory itself. These files are not
language-specific.
When you click OK, changes to the User and Column lists are saved to the user library and the
column properties, respectively. Anything specified in the Local and Local Exceptions lists is
saved only when you save the script of the Text Explorer report.
If saving Stop Words to the user library, the file is named stopwords.txt. If saving to a column
property, the property is called “Stop Words”.
Manage Recodes
The Manage Recodes window differs slightly from the Manage Stop Words window. Instead
of one text edit box below each list, there are two text edit boxes. The old value (specified in
the top box) is recoded to the new value (specified in the bottom box).
If saving Recodes to the user library, the file is named recodes.txt. If saving to a column
property, the property is called “Recodes”.
Manage Phrases
If saving Phrases to the user library, the file is named phrases.txt. If saving to a column
property, the property is called “Phrases”.
Manage Stem Exceptions
If saving Stem Exceptions to the user library, the file is named stemExceptions.txt. If saving to
a column property, the property is called “Stem Exceptions”.
Note: The Local Exceptions list in the Manage Stem Exceptions window lists stem exceptions
that are excluded from the stem exception list. The words in this list are involved in the
stemming operation.
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Text Analysis Options
The Text Explorer red triangle menu contains the following analysis options:
Latent Class Analysis Performs a latent class analysis on the binary weighted document term
matrix using sparse matrix routines.
When you select Latent Class Analysis from the Text Explorer red triangle menu, a
Specifications window appears with the following options:
Maximum Number of Terms The maximum number of terms included in the latent class
analysis.
Minimum Term Frequency The minimum number of occurrences a term must have to be
included in the latent class analysis.
Number of Clusters The number of clusters in the latent class analysis.
See“Latent Class Analysis” on page 354.
Performs a sparse singular value decomposition of the
document term matrix. See “Latent Semantic Analysis (SVD)” on page 355.
Latent Semantic Analysis, SVD
Performs a varimax rotated singular value decomposition of the
document term matrix to produce groups of terms called topics. See “Topic Analysis” on
page 356.
Topic Analysis, Rotated SVD
Cluster Terms Shows or hides a hierarchical clustering analysis of the terms in the data. To
the right of the dendrogram, there are options to set the number of clusters and save the
clusters to a data table. For each term, this data table contains its frequency, the number of
documents that contain it, and its assigned cluster.
Cluster Documents Shows or hides a hierarchical clustering analysis of the documents in the
data. To the right of the dendrogram, there are options to set the number of clusters, save
the clusters to a column in the data table, and show the documents in a selected branch of
the dendrogram plot.
SVD Scatterplot Matrix (Available after selecting Latent Semantic Analysis, SVD.) Shows or
hides a scatterplot matrix of the term and document singular value decomposition vectors.
You are prompted to select the size of the scatterplot matrix when you select this option.
This scatterplot matrix enables you to visualize more than the first two dimensions of the
singular value decomposition. The Show Text button opens a window that contains the
text of the selected documents.
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Figure 12.10 SVD Scatterplots of Document and Term Spaces
Topic Scatterplot Matrix (Available after selecting Topic Analysis, Rotated SVD.) Shows or
hides a scatterplot matrix of the rotated singular value decomposition vectors. The Show
Text button opens a window that contains the text of the selected documents.
Singular Value Decomposition Specifications Windows
The analysis options in the Text Explorer platform are based on the Document Term Matrix
(DTM). The DTM is formed by creating a column for each term in the Term List (up to a
specified Maximum Number of Terms). Each text document (equivalent to a row in the data
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table) corresponds to a row of the DTM. The values in the cells of the DTM depend on the type
of weighting specified by the user in the Specifications window.
Figure 12.11 shows the Singular Value Decomposition Specifications window. When you
select options from the Text Explorer red triangle menu that perform a singular value
decomposition on the document term matrix, the Specifications window appears with the
following options:
Maximum Number of Terms The maximum number of terms included in the singular value
decomposition.
Minimum Term Frequency The minimum number of occurrences a term must have to be
included in the singular value decomposition.
The weighting scheme that determines the values that go into the cells of the
document term matrix. The weighting scheme options are described in “Document Term
Matrix Specifications Window” on page 353.
Weighting
The number of singular vectors in the singular value
decomposition. The default value is the minimum of the number of documents, the
number of terms, or 100.
Number of Singular Vectors
Centering and Scaling Options for centering and scaling of the document term matrix. You
can choose between Centered, Centered and Scaled, and Uncentered. By default, the
document term matrix is both centered and scaled.
Figure 12.11 SVD Specification Window
Save Options
The Text Explorer red triangle menu contains the following options that are available to save
information to data tables, table columns, and column properties:
Save Document Term Matrix Saves columns to the data table for each column of the
document term matrix (up to a specified Maximum Number of Terms).
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Save Document Singular Vectors Saves a user-specified number of singular vectors from
the document singular value decomposition as columns to the data table.
Save Document Topic Vectors Saves a user-specified number of singular vectors from
the rotated singular value decomposition as columns to the data table.
Save Stacked DTM for Association Saves a stacked version of the document-term matrix
to a JMP data table. The stacked format is appropriate for analysis in the Association
Analysis platform. For more information, see the Association Analysis chapter in the
Predictive and Specialized Modeling book. If you specify an ID variable in the Text Explorer
launch window, the ID variable is used to identify the rows that each term came from in
the original text data table. The stacked table also contains a table script to launch
Association Analysis.
Save DTM Formula Saves a vector-valued formula column to the data table. The length of the
vector depends on user-specified options for the maximum number of terms, the
minimum term frequency, and the weighting. The resulting column uses the Text Score()
JSL function. For more information about this function, see Help > Scripting Index.
Save Singular Vector Formula (Available only after the Latent Semantic Analysis, SVD
option has been selected.) Saves a vector-valued formula column containing the document
singular value decomposition to the data table. The resulting column uses the Text
Score() JSL function. For more information about this function, see Help > Scripting
Index.
Save Topic Vector Formula (Available only after the Topic Analysis, Rotated SVD option
has been selected.) Saves a vector-valued formula column containing the rotated singular
value decomposition to the data table. The resulting column uses the Text Score() JSL
function. For more information about this function, see Help > Scripting Index.
Save Term Table Creates a JMP data table that contains each term from the Term List, the
number of occurrences, and the number of documents that contain each term. This data
table also contains a column containing scores for each term if the Score Terms by Column
option has been selected.
Save Term Singular Vectors Saves a user-specified number of singular vectors from the
terms singular value decomposition as columns to a new data table where each row
corresponds to a term. If a Term Table data table is already open, this option saves the
columns to that data table.
Save Term Topic Vectors (Available only after the Topic Analysis, Rotated SVD option
has been selected and the Term Table has been created.) Saves the topic vectors as columns
to the data table created by the Save Term Table command.
Score Terms by Column Saves the Term List table with scores based on values in a specified
column to a JMP data table. The scores for each term are the mean value of the specified
column weighted by the number of occurrences of the term in each row.
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Document Term Matrix Specifications Window
When you select the Save Document Term Matrix and Save DTM Formula options from the
Text Explorer red triangle menu, the Document Term Matrix Specifications window appears
with the following options:
Maximum Number of Terms The maximum number of terms included in the document term
matrix.
Minimum Term Frequency The minimum number of occurrences a term must have to be
included in the document term matrix.
The weighting scheme that determines the values that go into the cells of the
document term matrix.
Weighting
The following options are available for Weighting:
Binary
Assigns 1 if a term occurs in each document and 0 otherwise.
Ternary Assigns 2 if a term occurs more than once in each document, 1 if it occurs only
once and 0 otherwise.
Frequency
Assigns the count of a term’s occurrence in each document.
Assigns log10( 1 + x ), where x is the count of a term’s occurrence in each
document.
Log Freq
Assigns TF * log( nDoc / nDocTerm ). Abbreviation for term frequency - inverse
document frequency. This is the default weighting. The terms in the formula are defined
as follows:
TF IDF
– TF = frequency of the term in the document
– nDoc = number of documents in the corpus
– nDocTerm = number of documents that contain the term
Report Options
See the JMP Reports chapter in the Using JMP book for more information about the following
options:
Shows or hides the local data filter that enables you to filter the data used in
a specific report.
Local Data Filter
Contains options that enable you to repeat or relaunch the analysis. In platforms that
support the feature, the Automatic Recalc option immediately reflects the changes that
you make to the data table in the corresponding report window.
Redo
Contains options that enable you to save a script that reproduces the report to
several destinations.
Save Script
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Save By-Group Script Contains options that enable you to save a script that reproduces the
platform report for all levels of a By variable to several destinations. Available only when a
By variable is specified in the launch window.
Latent Class Analysis
Latent class analysis enables you to group the documents from the corpus into clusters of
similar documents. The Latent Class Analysis report contains the Bayesian Information
Criterion (BIC) value for the model and a Show Text button. If one or more clusters in the
Cluster Mixture Probabilities table is selected, the Show Text button opens a window that
contains the text of the documents that are deemed most likely to belong to the selected
cluster.
The Latent Class Analysis red triangle menu contains the following options:
Specifies the contents of the Latent Class Analysis report. By default, all of
the report options are shown.
Display Options
Mixture Probabilities by Cluster Shows or hides a table of the probability of an
observation belonging to each cluster.
Tip: You can select one or more rows in the Mixture Probabilities by Cluster table to select
the observations assigned to the corresponding clusters.
Term Probabilities by Cluster Shows or hides a table of terms with an estimate for each
cluster of the conditional probability that a document contains the term, given that the
document belongs to a particular cluster. By default, the terms in this table are sorted
by descending frequency in the corpus.
The Cluster Most Characteristic column shows the cluster that the term occurs in at the
highest rate.
The Cluster Most Probable column shows the cluster in which a randomly chosen
document that contains the term is most likely to be found.
Top Terms per Cluster
Show or hides a table of the ten most frequent terms in each
cluster.
MDS Plot Shows or hides a multidimensional scaling plot, which is a two-dimensional
representation of the proximity of the clusters. For more information about MDS plots,
see the Multidimensional Scaling chapter in the Consumer Research book. The Show
Text button opens a window that contains the text of the selected documents.
Cluster Probabilities by Row Shows or hides the Mixture Probabilities table, which
displays probabilities of cluster membership for each row. The Most Likely Cluster
column indicates the cluster with the highest probability of membership for each row.
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Save Probabilities Saves the values in the Mixture Probabilities table to the corresponding
rows in the data table.
Save Probability Formulas Saves a formula column to the data table for each cluster as well as
a formula column for the most likely cluster.
The score formula that is saved uses the Text Score() JSL function with the weighting
argument set to “LCA”.
Color by Cluster
Colors each row in the data table according to its most likely cluster.
Remove Removes the Latent Class Analysis report from the Text Explorer report.
For more information about latent class analysis, see the Latent Class Analysis chapter in the
Multivariate Methods book.
Note: The LCA algorithm that is used in the Text Explorer platform takes advantage of the
specific structure of the document term matrix. For this reason, the LCA results in the Text
Explorer platform do not exactly match the results in the Latent Class Analysis platform.
Latent Semantic Analysis (SVD)
Latent semantic analysis is a family of analysis techniques centered around computing a
partial singular value decomposition (SVD) of the document term matrix (DTM). This
decomposition reduces the text data into a manageable number of dimensions for analysis.
Latent semantic analysis is similar to principal components analysis.
The singular value decomposition approximates the DTM using three matrices: U, S, and V‘.
The relationship between these matrices is defined as follows:
DTM ≈ U * S * V‘
Define nDoc as the number of documents (rows) in the DTM, nTerm as the number of terms
(columns) in the DTM, and nVec as the specified number of singular vectors. Note that nVec
must be less than or equal to min(nDoc, nTerm). It follows that U is an nDoc by nVec matrix. S is
a diagonal matrix of dimension nVec. The diagonal entries in S are the singular values in the
SVD. V‘ is an nVec by nTerm matrix. The rows in V‘ are the singular vectors
The singular vectors capture connections among different words with similar meanings or
topic areas. If three words tend to appear in the same documents, the SVD is likely to produce
a singular vector in V‘ with large values for those three words. The U singular vectors
represent the documents projected into this new term space.
Latent semantic analysis also captures indirect connections. If two words never appear
together in the same document, but they generally appear in documents with another third
word, the SVD is able to capture some of that connection. If two documents have no words in
356
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Topic Analysis
Chapter 12
Basic Analysis
common but contain words that are connected in the dimension-reduced space, they map to
similar vectors in the SVD output.
The SVD transforms text data into a fixed-dimensional vector space, making it amenable to all
types of clustering, classification, and regression techniques. The Save options enable you to
export this vector space to be analyzed in other JMP platforms.
The DTM, by default, is centered and scaled before the singular value decomposition.
Centering and scaling can be turned off in the Specifications window. However, the SVD
implementation takes full advantage of the sparsity of the DTM even when the DTM is
centered.
SVD Plots Report
The Latent Semantic Analysis option produces two SVD Plots. The first plot shows the first
two singular vectors for the documents, that is, the first two columns of the U matrix. The
second plot shows the first two singular vectors for the terms, that is the first two rows of the
V‘ matrix. The points in both of these plots can be selected. For the document SVD plot, the
points correspond to rows in the data table. For the term SVD plot, the points correspond to
rows in the Term List table.
Below the document and term SVD plots, a table of the singular values appears. These are the
diagonal entries of the S matrix in the singular value decomposition of the document term
matrix.
Topic Analysis
The Topic Analysis option performs a varimax rotation on the singular value decomposition
(SVD) of the document term matrix (DTM). You must specify a number of rotated singular
vectors, which corresponds to the number of topics. After you specify a number of topics, the
Topic Words and Topic Scores reports appear. Topic analysis is similar to factor analysis.
The varimax rotation takes a set of singular vectors and rotates them to make them point more
directly in the coordinate directions (toward the terms). This rotation makes the vectors help
explain the text as each rotated vector orients toward a set of words. Negative values indicate
a repulsion force. The terms with negative values occur in a topic less frequently compared to
the terms with positive values.
Topic Words Report
The Topic Words report shows the terms that have the highest topic scores in each topic. There
are additional reports that show the components of the rotated singular value decomposition
Chapter 12
Basic Analysis
Text Explorer
Additional Example of the Text Explorer Platform
357
The report shows a table of terms in each topic that have the largest scores in absolute value.
Each table is sorted in descending order by the absolute value of the score. These tables can be
used to determine conceptual themes that correspond to each topic.
The Topic Words report also contains the following matrix reports:
Transform Contains the rotation matrix for the varimax rotation.
Rotated V Matrix Contains a matrix of term scores for each topic. The rotated V matrix results
from a varimax rotation of the V matrix in the SVD analysis. See “Latent Semantic
Analysis (SVD)” on page 355.
Contains a matrix of document scores for each topic. Documents with
higher scores in a topic are more likely to be associated with that topic.
Rotated U Matrix
Topic Portion
Contains the topic portion values for each topic.
Topic Scores Report
The Topic Scores report contains a Show Text button and a topic scores plot. The Show Text
button opens a window that contains the text of the selected documents.
The topic scores plot is a visual representation of the rotated U matrix. Each panel in the plot
corresponds to one of the topics, or one of the rows of the rotated U matrix. Within each panel,
each point corresponds to one of the document in the corpus, or one of the columns in the
rotated U matrix.
Additional Example of the Text Explorer Platform
This example looks at aircraft incident reports from the National Transportation Safety Board
for events occurring in 2001 in the United States. You want to explore the text that contains a
description of the results of the investigation into the cause of each incident. You also want to
find themes in the collection of incident reports.
1. Select Help > Sample Data Library and open Aircraft Incidents.jmp.
2. Select Rows > Color or Mark by Column.
3. Select Fatal from the columns list and click OK.
The rows that contain accidents involving fatalities are colored red.
4. Select Analyze > Text Explorer.
5. Select Narrative Cause from the Select Columns list and click Text Columns.
6. From the Stemming list, select Stem All Terms.
7. From the Tokenizing list, select Basic Words.
358
Text Explorer
Additional Example of the Text Explorer Platform
Chapter 12
Basic Analysis
8. Click OK.
Figure 12.12 Text Explorer Report for Narrative Cause
From the report, you see that there are almost 51,000 tokens and about 1,900 unique terms.
9. Select the top three terms in the Term List.
Because there are approximately 51,000 tokens, the counts for these three terms represent
almost 2% or more of all the terms.
10. Right-click on the selected terms and select Add Stop Word.
Because these terms occur frequently compared to other terms, they do not provide much
information to differentiate among documents.
11.
From the red triangle menu next to Text Explorer for Narrative Cause, select Latent
Semantic Analysis, SVD.
This is the first analysis step toward topic analysis, which performs a rotation of the SVD.
12.
In the Specifications window, type 50 for Minimum Term Frequency.
Because there are approximately 51,000 tokens, this frequency is equivalent to a term that
represents at least 0.1% of all the terms.
13.
Click OK.
Chapter 12
Basic Analysis
Text Explorer
Additional Example of the Text Explorer Platform
359
Figure 12.13 SVD Plots for Narrative Cause
There is not a lot of difference in the document SVD plot between fatal and non-fatal
incidents.
14.
From the Text Explorer red triangle menu, select Topic Analysis, Rotated SVD.
You want to look for groups of terms that form topics.
15.
Type 5 for Number of Topics.
16.
Click OK.
Figure 12.14 Topic Words for Narrative Cause
The highest scoring words for each topic enable you to interpret whether the topic is
capturing a theme in the incident reports.
For example, Topic 1 has high scores for power, loss, and engine, indicating a theme of
losing power to the engine as a cause of the incident. This corresponds to the phrase “loss
of engine power” occurring 273 times in the set of incident reports.
360
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Additional Example of the Text Explorer Platform
Chapter 12
Basic Analysis
Based on the words with high scores in Topic 3, it can be described as being related to
incidents that involved darkness or low altitude.
At this stage of the text analysis, you have many choices in how to proceed. Text analysis is an
iterative process, so you might use topic information to further curate your term list by adding
stop words or specifying phrases. You might save the weighted document-term matrix, the
vectors from the SVD or rotated SVD as numeric columns in your data table and use them in
other JMP analysis platforms. When you use these columns in other platforms, you can also
include other columns from your data table in further analyses.
Appendix A
Statistical Details
Basic Analysis
This appendix contains statistical details that apply to multiple platforms in JMP Pro.
362
Statistical Details
Platforms That Support Validation
Appendix A
Basic Analysis
Platforms That Support Validation
The table below lists the types of crossvalidation available in each platform.
Parent
Platform
Platform
Use Excluded
Rows as
Validation
Holdback
Random
Validation
Holdback
K-Fold
Cross-Validat
ion
Validation Role
Column
Fit Model
Fit Least
Squares
No
No
No
Yes (only for
model
evaluation)
Fit Model
Forward
Stepwise
Regression
No
No
Yes
(continuous
response
models
only)
Yes
Fit Model
Logistic
Regression
No
No
No
Yes (only for
model
evaluation)
Fit Model
Generalized
Regression
No
Yes
Yes
Yes
Fit Model
Partial
Least
Squares
No
Yes
Yes
Yes
Partition
Decision
Tree
Yes
Yes
Yes
Yes
Partition
Bootstrap
Forest
Yes
Yes
No
Yes
Partition
Boosted
Tree
Yes
Yes
No
Yes
Partition
K Nearest
Neighbors
Yes
Yes
No
Yes
Partition
Naive
Bayes
Yes
Yes
No
Yes
Neural
Neural
Yes
Yes
Yes
Yes
Appendix B
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Appendix B
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Index
Basic Analysis
Numerics
3D Correspondence Analysis option 220–229
5% Contours option 124
A
Agreement Statistic option 213, 222, 236
All fits 62–89
All Graphs option 144–145
All Pairs, Tukey HSD test 156, 158–180
Analysis of Means
charts 154–175
for Proportions option 213, 219–220
options 155
Analysis of Means Methods 142, 152–155
Analysis of Variance
See also Oneway
report 110–111, 149–150
ANOM
See also Analysis of Means
for Variances 175–176
Transformed Ranks 153
ANOMV display options 155
B
Bartlett test 168
Beta Binomial fit 91–92
Beta fit 62, 86
Binomial fit 91
Bivariate Normal Ellipse menu 122–124
Bivariate platform 99
example 100
launching 100
options 102–105, 122–125
report window 101
Block button 97
Block Means report 150
bootstrapping 311–325
Box plots 144
Brown-Forsythe test 168
C
Capability Analysis option 59–61, 79–83
CDF plot 143–144, 173
Cell Labeling option 215
Change Format option 265
Change Item Label 273
Change Plot Scale option 265
Cluster Documents, Text Explorer 349
Cluster Terms, Text Explorer 349
Cochran Armitage Trend Test 214, 225
Cochran Mantel Haenszel test 213, 221
Coefficient of Variation 268
Color by Density Quantile option 125
Color Theme option 216
column tables 272
Compare Densities option 173
Compare Means options 142, 156–161
comparison circles 145, 157–158, 202–203
Composition of Densities option 173
Confid Curves Fit option 123
Confid Curves Indiv option 123
Confid Quantile option 160
Confid Shaded Fit option 124
Confid Shaded Indiv option 124
Confidence Interval options 47, 56
Confidence Limits option 65
Connect Means option 145–146
Connecting Letters Report 160
Contingency platform 209
example 210–211
launching 211
options 213–214
report window 212
368
Contingency Table 213, 216–218
Continuous Fit options 61–89
Contour Fill option 124
Contour Lines option 124
Correlation report 120
Correspondence Analysis option 213, 220
Count Axis option 46, 49
Covariance of Estimates report 247
D
Densities options 143, 173
Density Axis option 46, 49
Density Curve option 63
Density Ellipse option 119–120
density estimation fits 104
Detailed Comparisons Report 161
Details report 221
Diagnostic Plot option 63–65
Difference Matrix option 160
Discrete Fit options 62–92
Display Options 144–145, 155, 220
for categorical variables 46
for continuous variables 47–48
Distribution platform 33
categorical variables in 34
continuous variables in 34
example 34–36
launching 36
options 45–58
report window 37–39
Document Term Matrix, Text Explorer 328
E
Each Pair, Student’s t test 156, 158–178
Equal Variances option 117
Equivalence Test option 55, 143, 170–188
Exact Test 214, 225
See also Fisher’s Exact Test
options 162
Exponential fit 61, 85
Extreme Value fit 85
F
Fisher’s Exact Test 219
Index
Basic Analysis
Fit Each Value
command 117
menu 122–123
report 117
Fit Line command 103, 106–113, 133
Fit Mean
command 103, 105–106
menu 122–123
report 106
Fit Orthogonal command 117–118, 133
Fit Polynomial command 103, 106–136
Fit Special command 104, 113–114
Fit Spline command 115–116, 133
Fit X to Y option 117
Fit Y by X platform 95
Fix Parameters option 63
Frequencies
option 46
report 42
G
Gamma fit 62, 85
Gamma Poisson fit 90–91
GLog fit 62, 88–89
Goodness of Fit
option 63
tests 65–94
Gradient Between Ends option 216
Gradient Between Selected Points option 216
Grand Mean option 145
Graph in Variance Scale option 155
Group By option 102, 121–122
H
Histogram Options 48
histograms 39–66
borders option 102, 105
color option 46, 48
creating subsets 39
highlighting data 39, 41
option in Oneway 145
options for continuous variables 48–50
rescaling 39
resizing 39–40
selecting data 41
Index
Basic Analysis
specifying data 39
Horizontal Layout option 46, 48
I
Include missing for grouping columns
option 265
Inverse Prediction option 248
Iterations report 245
J
jitter 251
Johnson fits 62, 87–88
K
Kernel Control option 124
Kolmogorov Smirnov Test 162
Kruskal-Wallis test, see Wilcoxon Test
L
Lack of Fit report 109–110
language processing, Text Explorer 334
Latent Class Analysis option, Text
Explorer 349, 354
Latent Semantic Analysis, SVD in Text
Explorer 349
Levene test 168
Lift Curve option 248
Line Color option 123
Line of Fit option 65, 123, 172
Line Style option 123
Line Width option 123
Linear Fit
menu 122–124
report 107–136
Logistic platform 239
See also logistic regression
Covariance of Estimates report 247
examples 241–242, 250–254
Iterations report 245
launching 242
logistic plot 244, 248
options 247–250
Parameter Estimates report 247
report window 243–247
369
Whole Model Test report 245–247
logistic regression 239–240
LogNormal fit 61, 84
LSD Threshold Matrix option 160
M
Macros option 216
Make Into Data Table option 274
Mann-Whitney test, see Wilcoxon Test
Matching Column option 143, 173–174
Matching Dotted Lines option 145
Matching Lines option 145, 174
Max (summary statistics) 267
Mean CI Lines option 145
Mean Diamonds option 144, 150–151
Mean Error Bars option 145, 151
Mean Lines option 145, 151
Mean of Means option 145
Means and Std Dev option 142
Means for Oneway Anova report 150
Means/Anova option 142, 147
Means/Anova/Pooled t option 142, 147
Measures of Association option 214, 223–224
Median (summary statistics) 268
Median Reference Line option 65
Median Test 161
Mesh Plot option 125
Min (summary statistics) 267
Model Clustering option 125
Mosaic Plot
in Contingency 213–216
option in Distribution 47
N
N and N Missing (summary statistics) 267
N Categories (summary statistics) 267
Nonpar Density command 120–121
Nonparametric
options 143
tests 161
Nonparametric Bivariate Density report 121
Normal fit 61, 83
Normal Mixtures fits 62, 86
Normal Quantile Plot 49–50, 143, 172
Number of Cases, Text Explorer 341
370
Index
Basic Analysis
Number of Non-empty Cases, Text
Explorer 341
O
O’Brien test 168
Odds Ratios option 213, 248
Oneway platform 137–138
example 138–140
launching 140
options 141–145
plot 140
Order by count of grouping columns
option 265
Order By option 47
Ordered Differences Report 161
Orthogonal Fit Ratio menu 122–123
Orthogonal Regression report 118
Outlier Box Plot 51–52
P
Parameter Estimates report 112–136, 247
parametric resampling, Simulate platform 285
Parsing Options, Text Explorer 345
Plot Actual by Quantile option 172
Plot Options menu in Logistic 248
Plot Quantile by Actual option 172
Plot Residuals option 124
Points Jittered option 145
Points option 144
Points Spread option 145
Poisson fit 90
Polynomial Fit Degree menu 122
Polynomial Fit report 107–136
Portion of Non-empty Cases, Text Explorer 341
Power option 143, 171–172, 203–204
Prediction Interval option 58–70, 78
Prob Axis option 46, 49
Proportion of Densities option 173
Q
Quantile Box Plot 52
Quantile Density Contours menu 122–125
Quantiles
option 47, 63, 73, 142, 146–147
report 42
R
ranges, summary statistics 267
regression fits for Bivariate 104
regular expression editor, Text Explorer 335
Relative Risk option 213, 222
Remove Column Label 273
Remove Fit option 64, 123
Report option 123
Restore Column Label 273
Revert to Old Colors option 216
ROC Curve option 248
Rotate option 65
row profile 220
row tables 272
S
sample size, in tabulate 274
Save Coefficients option 124
Save Colors to Column option 216
Save commands in Distribution 57–58
Save Density Formula command 63
Save Density Grid option 125
Save Density Quantile option 125
Save Fitted Quantiles command 63
Save for Adobe Flash platform (.SWF)
option 45
Save options
in Oneway 144
Save Predicteds option 123
Save Probability Formula option 248
Save Residuals option 123
Save Spec Limits command 64
Save Value Ordering option 221
Saved Transformed command 64
Select Colors for Values window 215
Select Points by Density option 124
Select Points Inside option 124
Select Points Outside option 124
Separate Bars option 46
Set Alpha Level option
for Bivariate 124
for Contingency 213, 220
for Oneway 143, 155
Index
Basic Analysis
Set Bin Width option 48
Set Colors option 215
Set Spec Limits for K Sigma option 63
Shaded Contour option 124
Shadowgram option 48
Show Center Line option 155
Show Chart option 273
Show Control Panel option 274
Show Counts option 46, 49
Show Decision Limit Shading option 155
Show Decision Limits option 155
Show Delimiters, Text Explorer 344
Show Percents option 46, 49
Show Points option
for Bivariate 102
Show Shading option 274
Show Stem Report, Text Explorer 344
Show Summary Report option 155, 220
Show Table option 273
Show Test Build Panel option 274
Show tooltip option 273
Simulate platform data table results 309
Smooth Curve fit 62, 87
Smoothing Spline Fit
menu 122
report 115
Spec Limits option 63, 65
Specified Variance Ratio option 117
Specify Transformation or Constraint
window 113, 126
Stack option 45
Standard Deviation 267
Standard Error 268
Std Dev and Std Err (summary statistics) 267
Std Dev Lines option 145, 151
Std Error Bars option 46, 48
Steel With Control test 165
Steel-Dwass All Pairs test 165
Stem and Leaf plot 52
Stemming, Text Explorer 328, 334
stop words, Text Explorer 328, 347
Student’s t test 156, 158–178
Subset option 39
Summary of Fit report 108, 147
Summary Statistics
customize 48
371
options 45
report 42, 48
SVD Scatterplot Matrix, Text Explorer 349
Switch Response Level for Proportion
option 220
T
t test
option 142
report 148–149
Tabulate 259–283
term (token), Text Explorer 328
Term Options, Text Explorer 344
Test Mean option 53–54
Test Probabilities option 47–68
Test Std Dev option 54–55
Tests
option in Contingency 213
report 218–219
Text Explorer
Cluster Documents 349
Cluster Terms 349
editing regular expressions 335
language processing 334
Latent Class Analysis option 349, 354
Latent Semantic Analysis, SVD 349
Number of Cases 341
Number of Non-empty Cases 341
Parsing Options 345
platform options 343
Portion of Non-empty Cases 341
Show Delimiters 344
Show Stem Report 344
Stemming 328, 334
stop words 328, 347
SVD Scatterplot Matrix 349
term (token) 328
Term Options 344
Tokenizing 334
Tokens per Case 341
Topic Analysis, Rotated SVD 349
Topic Scatterplot Matrix 350
Total Tokens 341
Text Explorer, Word Separator List 338
Tokenizing, Text Explorer 334
Tokens per Case, Text Explorer 341
372
Tolerance Interval option 59–71, 78–79
Topic Analysis, Rotated SVD in Text
Explorer 349
Topic Scatterplot Matrix, Text Explorer 350
Total Tokens, Text Explorer 341
Transformed Fit
menu 122
report 114
Treat Numbers as Words, Text Explorer 335
Two Sample Test for Proportions 213, 223
U
Unequal Variances 143, 167–168
Uniform plot scale option 265
Uniform Scaling option 45
Univariate Variances, Prin Comp option 117
V
van der Waerden Test 161
Variance (summary statistics) 268
Vertical option 46, 48
W-Z
Weibull fits 61, 84
Welch’s test 168
Whole Model Test report 245–247
Wilcoxon
Each Pair test 165
Test 161
With Best, Hsu MCB test 156, 159–181
With Control, Dunnett’s test 156, 160
Word Separator List, Text Explorer 338
X Axis proportional option 145, 150–151
X, Continuous Regressor button 97
X, Grouping button 97
X, Grouping Category button 97
X, Regressor button 97
Y, Categorical Response button 97
Y, Response Category button 97
Index
Basic Analysis
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