SAS Structural Equation Modeling 1.1 for JMP · Advert. is the company’s advertising spending in millions of dollars, LastS. is last year’s sales in millions of dollars, and

SAS® Structural EquationModeling 1.1 for JMP®

SAS® Documentation

The correct bibliographic citation for this manual is as follows: SAS Institute Inc. 2011. SAS® Structural Equation Modeling 1.1 for JMP®. Cary, NC: SAS Institute Inc. SAS® Structural Equation Modeling 1.1 for JMP®

Copyright © 2011, SAS Institute Inc., Cary, NC, USA

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ContentsCredits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vChapter 1. About This Book . . . . . . . . . . . . . . . . . . . . . . . . . 1Chapter 2. Introduction to SAS Structural Equation Modeling for JMP . . . . . . . . . . 3Chapter 3. Getting Started with SAS Structural Equation Modeling for JMP . . . . . . . . 5Chapter 4. Linear Regression Analysis . . . . . . . . . . . . . . . . . . . . . 23Chapter 5. Path Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 39Chapter 6. Confirmatory Factor Analysis . . . . . . . . . . . . . . . . . . . . 61Chapter 7. Structural Equation Model . . . . . . . . . . . . . . . . . . . . . . 93Chapter 8. Latent Growth Curve Model . . . . . . . . . . . . . . . . . . . . . 115Chapter 9. Single Group Analysis Window . . . . . . . . . . . . . . . . . . . . 147Chapter 10. Model Library Window . . . . . . . . . . . . . . . . . . . . . . . 175Chapter 11. User Profile Window . . . . . . . . . . . . . . . . . . . . . . . . 187Chapter 12. Properties Windows . . . . . . . . . . . . . . . . . . . . . . . . 197Appendix A. Frequently Asked Questions . . . . . . . . . . . . . . . . . . . . 207

iv

Credits

Documentation

Writing Ruth Baldasaro

Editing Anne Baxter

Documentation Support Tim Arnold, Sharad Prabhu

Technical Review Yiu-Fai Yung, Wayne Watson

Software

JMP Wayne Watson

PROC CALIS Yiu-Fai Yung

Support Groups

Software Testing Wei Zhang, Ruth Baldasaro

Technical Support Duane Hayes

vi

Chapter 1

About This Book

This book describes the features of SAS Structural Equation Modeling for JMP and includes several exam-ples that show how to use them. The example topics include multiple regression, path analysis, confirmatoryfactor analysis, structural equation modeling, and latent growth curve modeling. Each example describeshow to specify the data, create the path diagram, analyze the path diagram, and view the results.

This book does not contain a comprehensive treatment of structural equation modeling (SEM) or more tech-nical details regarding the SAS/STAT® CALIS procedure. For more information about SEM, see Bollen(1989). For more information about the SAS/STAT CALIS procedure, see Chapter 26, “The CALIS Proce-dure” (SAS/STAT User’s Guide).

This book is organized as follows:

� This chapter describes the organization of the book.

� Chapter 2 provides a brief description of SAS Structural Equation Modeling for JMP, describes itsbenefits, and compares it to the SAS/STAT CALIS procedure.

� Chapter 3 gets you started with a simple example of how to us SAS Structural Equation Modeling forJMP.

� Chapter 4 through Chapter 8 show examples of other analyses which build on the getting startedexample.

� Chapter 9 through Chapter 12 describe features of the various SAS Structural Equation Modeling forJMP windows.

� Appendix A provides answers to frequently asked questions.

ReferencesBollen, K. A. (1989), Structural Equations with Latent Variables, New York: John Wiley & Sons.

2

Chapter 2

Introduction to SAS Structural EquationModeling for JMP

ContentsOverview of SAS Structural Equation Modeling for JMP . . . . . . . . . . . . . . . . . . . 3Benefits of SAS Structural Equation Modeling for JMP . . . . . . . . . . . . . . . . . . . . 3Comparing SAS Structural Equation Modeling for JMP to the SAS/STAT CALIS Procedure 4

Overview of SAS Structural Equation Modeling for JMP

SAS Structural Equation Modeling for JMP is a graphical user interface that provides easy access to struc-tural equation modeling (SEM) techniques. The interface enables you to quickly specify a path diagram torepresent the hypothesized relationships among the variables. You can specify models with only observedvariables (including multiple regression and path analysis models) and models that have both observed andlatent variables (including factor analysis and latent growth curve models). You access SAS StructuralEquation Modeling for JMP from the JMP interface. SAS Structural Equation Modeling for JMP uses theSAS/STAT CALIS procedure to perform the model estimation and then performs the remaining compu-tations. For more information about PROC CALIS, see Chapter 26, “The CALIS Procedure” (SAS/STATUser’s Guide).

Benefits of SAS Structural Equation Modeling for JMP

SAS Structural Equation Modeling for JMP provides the following benefits:

� An intuitive and efficient path diagram interface: You can quickly specify and modify diagrams.

� A customizable diagram view: You can choose to view the variable variances, errors, or defaultcovariances among exogenous variables. You can also choose to view either unstandardized or stan-dardized parameter estimates.

� A diagram to save and print: You can easily print a diagram or copy a diagram to a document orpresentation. You can also save the diagram to use with other data sets or to modify in future analyses.

4 F Chapter 2: Introduction to SAS Structural Equation Modeling for JMP

� A model comparison view: You can fit and save the results from multiple models in one project fileand view model fit statistics from multiple models in one table for easy model comparison.

� A flexible system for data handing: You can easily analyze raw, correlation, or covariance data. JMPcan read in many data file types in addition to JMP data tables. SAS Structural Equation Modeling forJMP can modify a table of correlation or covariance data to contain the appropriate SAS correlationor covariance table format for analysis.

Comparing SAS Structural Equation Modeling for JMP to theSAS/STAT CALIS Procedure

Table 2.1 compares the interface, estimation, and analysis features of SAS Structural Equation Modelingfor JMP to the SAS/STAT CALIS procedure (PROC CALIS).

Table 2.1 Program Features

SAS Structural EquationFeature Modeling for JMP PROC CALIS

Interface FeaturesGraphical user interface Yes NoPath diagram input and output Yes NoSaving and loading projects Yes NoScripting language No Yes

Estimation FeaturesMaximum likelihood Yes YesGeneralized least squares Yes YesWeighted least squares Yes YesUnweighted least squares Yes YesDiagonally weighted least squares Yes YesFull-information maximum likelihood Yes Yes

Analysis FeaturesModel fit Information Yes YesUnstandardized solution Yes YesStandardized solution Yes YesMean structure analysis Yes YesEquality constraints Yes YesModel comparison Yes NoMultiple-group analysis No YesDirect and indirect effects Yes YesBoundary and linear constraints No YesNonlinear constraints No YesModification indices Yes Yes

Chapter 3

Getting Started with SAS Structural EquationModeling for JMP

ContentsOverview of Getting Started Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Start a SAS Structural Equation Modeling for JMP Analysis . . . . . . . . . . . . . . . . . 7Create the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Specify the Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Add Variables to the Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Draw Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Label the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Specify Options for the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Perform the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15View the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

View Results in the Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Verify Accuracy of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18View Detailed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Save the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Save a Model Library File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Save a Single Group Analysis Project File . . . . . . . . . . . . . . . . . . . . . . . 22Save a Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Print a Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Overview of Getting Started Example

This getting started example uses a multiple regression model to show you how to use SAS StructuralEquation Modeling for JMP. This example begins with a description of the data and the example model;then it shows you how to start a new analysis, specify the data set, create the multiple regression model,analyze the model, view the results, and save the analysis. This example shows you how to create andperform an analysis in SAS Structural Equation Modeling for JMP, but it does not describe all the waysyou can create a model or all of the features of SAS Structural Equation Modeling for JMP. Throughout

6 F Chapter 3: Getting Started with SAS Structural Equation Modeling for JMP

this example are directions about where to find more information about other features of SAS StructuralEquation Modeling for JMP.

You can find the data file for this getting started example, Sales_Data.jmp, by going to the JMP Homewindow and selecting Analyze I Structural Equation Modeling I Sample Data I Sales Data. Thedata file Sales_Data.jmp contains raw data, which are observations for each variable for each unit includedin the study. The data file contains responses from 25 companies for four variables (N_emp is the numberof employees, Advert is the company’s advertising spending in millions of dollars, LastS is last year’s salesin millions of dollars, and CurrentS is the current year’s sales in millions of dollars). Table 3.1 shows thecontents of the data set.

Table 3.1 Raw Data from Sales_Data.jmp

N_emp Advert LastS CurrentS83 .140 2.350 2.52595 .325 2.430 2.91580 .265 2.030 2.26098 .240 2.575 2.60562 .220 1.735 2.02541 .110 1.075 1.075111 .350 2.915 3.25072 .215 1.690 1.780139 .385 3.590 3.79570 .175 1.700 1.66089 .190 2.285 2.55083 .160 2.105 2.285106 .200 2.595 2.69552 .150 1.215 1.26054 .220 1.740 1.95575 .120 2.010 2.19060 .165 1.600 1.710101 .245 2.670 2.53578 .190 2.015 2.06057 .205 1.535 1.92526 .145 0.785 0.97061 .100 1.585 1.64589 .265 2.260 2.76551 .115 1.305 1.555104 .290 2.550 2.935

This example shows you how to create the multiple regression model shown in Figure 3.1, where the numberof employees (N_emp), advertising spending (Advert), and last year’s sales (LastS) are all are used to predictthe current year’s sales (CurrentS).

Start a SAS Structural Equation Modeling for JMP Analysis F 7

Figure 3.1 Getting Started Example Path Diagram

Start a SAS Structural Equation Modeling for JMP Analysis

1 Open JMP 9.0.2 or later.

2 To open the Sales_Data.jmp data file, select Analyze I Structural Equation Modeling I SampleData I Sales Data.

3 Select Analyze I Structural Equation Modeling I Single Group Analysis. The Structural EquationModels for a Single Group window appears as shown in Figure 3.2.


Figure 3.2 Structural Equation Models for a Single Group Window

The Structural Equation Models for a Single Group window contains a set of project buttons and three tabs:Data, Analyses, and Comparisons. You use the Data tab to specify information about the data set. Youuse the Analyses tab to create, modify, and run analyses on one or more models. You use the Comparisonstab to examine the fit statistics from each analysis you run in the Analyses tab. The Comparisons tab isavailable only after you run an analysis on the Analyses tab.

Create the Model

In this section you learn how to specify the data set, add variables to the diagram, draw paths, and label theanalysis.

Specify the Data Set

On the Data tab, ensure that the following values are specified under Data Table Properties:

1 Verify that Name shows Sales_Data.

Add Variables to the Diagram F 9

2 Verify that Raw Data is selected from the Data Structure list.

The other properties of the Data tab are not used in this example. For more information about the otherfeatures of the Data tab, see the section “Data Tab” on page 151.

Add Variables to the Diagram

1 Click the Analyses tab. See Figure 3.3.

Figure 3.3 Analyses Tab

2 Click Palette to show the Variables area, which contains the shapes for latent (oval) and observed (rect-angle) variables.

3 On the Diagram tab, drag a variable from the Variables list to the desired location in the Diagram area.NOTE: If you drag more than one variable into the Diagram area at a time, a window appears and askswhether the variables should be arranged horizontally or vertically.


4 After all of the variables are in the diagram, drag the variables around until they look like the variables inFigure 3.4.

Figure 3.4 Getting Started Example Diagram Tab with Variables Only

Draw Paths

To draw paths from the independent variables N_emp, Advert, and LastS to the dependent variable Cur-rentS:

1 Rest the cursor on N_emp. A small palette appears that contains a single-headed and a double-headedarrow. Figure 3.5 shows an example of this palette. You use the double-headed arrow to representcovariances or correlations, and you use the single-headed arrow to represent unidirectional effects.

2 Select the single-headed arrow. It turns red when selected; see Figure 3.5.

Draw Paths F 11

Figure 3.5 Arrow Palette

3 Drag the cursor toward CurrentS. The black outline of CurrentS turns bold, indicating that the variable isa valid target for this path.

4 Release the mouse button. A single-headed path from N_emp to CurrentS appears in the diagram.

5 Repeat these steps for Advert and LastS. Figure 3.6 shows the resulting diagram.


Figure 3.6 Getting Started Example Diagram with Paths Connecting theVariables

Label the Analysis F 13

Label the Analysis

You can specify a label and notes for an analysis on the General tab. If a label is not specified, the analysisis automatically given a unique label Analysis 1, Analysis 2, and so on. Even though the label and notes areoptional, specifying a label is useful so that you can quickly locate a model when you are exploring severalmodels for the same data.

To specify a label and notes for an analysis:

1 Click the General tab.

2 In the Label box, type a title for the model.

3 In the Notes box, type a description of the model.

Figure 3.7 shows the General tab with model information in the Label and Notes boxes.

Figure 3.7 Getting Started Example General Tab Specifications

The title you type in the Label box is displayed in the Analyses list. Click the Analyses button to view thelist, as shown in Figure 3.8.


Figure 3.8 Getting Started Example Analyses List

Specify Options for the Analysis

1 Click the Methods tab.

2 In the Analyze area, select Covariances to specify a covariance matrix for the analysis of this example.A covariance matrix is the default.

3 In the Estimation area, select Maximum likelihood from the Method list. Maximum likelihood esti-mation is the default estimation method. For more details about the estimation options, see the section“Specify Estimation Method” on page 167.

4 In the Optimization area, select Default from the Method list. The Default optimization method usesthe optimization method best suited for the number of parameters the model is estimating. For moredetails about the optimization options, see the section “Specify Optimization Method” on page 168.

5 In the Optimization area, type the maximum number of iterations to be performed in this analysis in theMaximum iterations box. For this analysis, leave this box blank to use the default maximum number

Perform the Analysis F 15

of iterations, which is based on the optimization method. For more details about the maximum iterationsoption, see the section “Specify Maximum Iterations” on page 169.

Figure 3.9 shows these selections.

Figure 3.9 Getting Started Example Methods Tab Specifications

Perform the Analysis

After you have specified the desired path diagram and analysis options, click Run in the Perform Analysisarea to fit the model and generate the results output.

View the Results

In this section you learn what results can be viewed in the diagram and on the Results tab, and you learnhow to check that the model converged to a proper solution.


View Results in the Diagram

The results of the model appear in the diagram with parameter estimates above the paths and variables.Figure 3.10 shows the unstandardized estimate results when Unstd. Estimates is selected from the Viewlist.

Figure 3.10 Getting Started Example Diagram with Unstandardized Results

Figure 3.11 shows the standardized estimate results when Std. Estimates is selected from the View list.

View Results in the Diagram F 17

Figure 3.11 Getting Started Example Diagram with Standardized Results

NOTE: You cannot modify the model when Unstd. Estimates or Std. Estimates is selected from theView list. You can modify the model if you return to the diagram view that does not contain any parameterestimates. To return to the diagram view that does not contain any parameter estimates, select Input fromthe View list. Another way to modify a diagram after it has been run is to select Copy in the Analysis paneto copy the diagram into a new analysis.

In the Diagram area, the estimate of the variance of each variable is displayed above it, and the estimatedmultiple regression coefficients are displayed on the paths from the predictors to the outcome. Any param-eter estimate that differs significantly from 0 (based on t tests) is marked with an asterisk. Two asterisksindicate that p < 0.01; one asterisk indicates that p < 0.05.


Verify Accuracy of Results

Before you examine more detailed results on the Results tab, you should verify that the model convergedwithout any warning or error messages. After you click Run in the Perform Analysis area, a windowusually appears with a warning if any estimation problems occur. Even if a warning window does notappear, you should use the following steps to verify that the model has converged:

1 Click the SAS Log tab.

2 Check for model convergence and any error or warning messages in the SAS Log. If the model con-verges, the SAS Log contains the following message (or a similar message for convergence with anotherconvergence criterion):

Convergence criterion (ABSGCONV=0.00001) satisfied.

Because the model in this example converged without any errors or warnings, you can correctly interpretthe results in the Diagram area and on the Results tab.

NOTE: The JMP log is another place to check for potential problems with fitting a model. To open the JMPlog, go to the JMP Home window and double-click Log in the Window list.

View Detailed Results

The Diagram tab contains only some of the results. To view more detailed results, click the Results tab onthe Analyses tab.

The output in the Results tab is organized such that the modeling specifications and model fit are presentedfirst, followed by the parameter estimates for the model. The parameter estimates are organized to presentthe parameters for the single-headed arrow paths first, followed by the variance, covariance, and squaredmultiple correlations. Figure 3.12 and Figure 3.13 show the unstandardized results for this example.

View Detailed Results F 19

Figure 3.12 Getting Started Example Results Tab with Modeling Specifica-tions and Fit Results


Figure 3.13 Getting Started Example Results Tab with Maximum LikelihoodParameter Estimate Results

The parameter estimates under the PATH List heading are the unstandardized multiple regression coeffi-cients shown on the paths in the Diagram area in Figure 3.10. The parameter estimates under the VarianceParameters heading are the variance estimates shown above the variables in the Diagram area in Fig-ure 3.10. Although the Diagram area shows only the parameter estimates, the Results tab also contains thestandard error and t test values for each parameter estimate. The Results tab also contains some measuresof model fit (under the Fit Summary heading) and the R-square estimate for CurrentS (under the SquaredMultiple Correlations heading).

The gray triangles next to each section header collapse or expand the section of output. The red triangleat the top of the Results tab contains the options for what you can view in the output. These options arenot used in this example; for more detailed information about these options, see Chapter 9, “Single GroupAnalysis Window.”

In addition to the SAS Log and Results tabs, a SAS Code tab is produced after you click Run. This tabcontains the SAS syntax for the model and analysis information specified in the Diagram and Methods tabs.For more information about the syntax of the CALIS procedure, see Chapter 26, “The CALIS Procedure”(SAS/STAT User’s Guide).

Summary of Results F 21

Summary of Results

The parameter estimates in Figure 3.13 indicate that both Advert (the company’s advertising spending inmillions of dollars) and LastS (last year’s sales in millions of dollars) have positive and significant path esti-mates for their relationship with CurrentS (the current year’s sales in millions of dollars). This suggests thatthese variables explain a significant amount of the variance in CurrentS beyond the variance explained bythe other variables in the model. The positive path estimates indicate that higher levels of Advert and LastSare associated with higher levels of CurrentS. The only variable that is not significantly related with Cur-rentS is N_emp (the number of employees), which suggests that this variable does not explain a significantamount of variance in CurrentS when the variables Advert and LastS are included in the regression model.According the squared multiple correlation estimate for CurrentS, the three predictors explain 95.9% of thevariance in CurrentS, suggesting that these three predictors do a good job of explaining the variation inCurrentS.

Save the Results

After you build your model, you can save your work in any of the ways described in this section.

Save a Model Library File

A model library file contains a copy of a diagram for one model, but no other information about the model.This file type can be opened or modified in the Model Library window or in a Single Group Analysiswindow. To store a copy of your diagram in a model library file:

1 Click the Diagram tab.

2 Select Save the model from the Model Library list in the Actions area.

3 Specify the name of the model and where you want the model to be saved.

4 Click Save.


Save a Single Group Analysis Project File

For each analysis you create in a Single Group Analysis window, you can save a copy of the diagram, theparameter specifications, and the results (for each analysis that has results) together in one Single GroupAnalysis project file. This project file can be opened and modified only in a Single Group Analysis window.To save a Single Group Analysis project file with a copy of all the analysis information, including both thediagram and the results for each model analyzed:

1 In the Project area, click Save or Save As.


3 Click Save.

Save a Diagram

To store a copy of your diagram for immediate use in another document or application:

1 Right-click the white background of the Diagram area.

2 Select Copy diagram to the clipboard.

3 Paste the copy of the diagram into another document or application.

You can then save or print the document that contains a copy of the diagram.

Print a Diagram

1 In the Project area, click Print.

2 Specify the name of the printer and any printing options.

3 Click Print.

Chapter 4

Linear Regression Analysis

ContentsOverview of Linear Regression Analysis Example . . . . . . . . . . . . . . . . . . . . . . . 23Create the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Specify the Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Add Multiple Variables to the Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 25Draw Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Label Each Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Add Mean Structure to the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Add Mean Structure Using the Methods Tab . . . . . . . . . . . . . . . . . . . . . . 30Add Mean Structure Using the Variable Settings . . . . . . . . . . . . . . . . . . . . 31

Perform the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32View the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

View Results in the Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Verify Accuracy of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35View Detailed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Save Project File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Overview of Linear Regression Analysis Example

This linear regression analysis example is an extension of the example in Chapter 3, “Getting Started withSAS Structural Equation Modeling for JMP.” This example uses a multiple regression model to show youhow to label variables and add mean structure to a model in SAS Structural Equation Modeling for JMP.This example begins with a description of the data and the example model; then it shows you how to specifythe data set, create the multiple regression model, label model parameters, add mean structure, analyze themodel, view the results, and save the project.

You can find the data file for this multiple regression example, Sales_Data.jmp, by going to the JMP Homewindow and selecting Analyze I Structural Equation Modeling I Sample Data I Sales Data. Formore information about this data file, see the section“Overview of Getting Started Example” on page 5.

This linear regression analysis example shows you how to create the multiple regression model shown inFigure 4.1, where the number of employees (N_emp), advertising spending (Advert), and last year’s sales

24 F Chapter 4: Linear Regression Analysis

(LastS) are all used to predict the current year’s sales (CurrentS). Figure 4.1 shows a diagram of the modelwith labels on the paths from the predictors to the outcome. Unlike the getting started example, this modelestimates mean structure as part of the model. The diagram in Figure 4.1 shows the mean structure as thelabel Intercept, above the variable current year’s sales (CurrentS). The parameter estimate for Intercept isthe same as the intercept estimated in a multiple regression analysis in other programs.

Figure 4.1 Multiple Regression Path Diagram

Create the Model

In this section you learn how to specify the data set, add variables to the diagram, draw paths, label thepaths, and add mean structure to the analysis.


On the Data tab, ensure that the following values are specified under Data Table Properties:



Add Multiple Variables to the Diagram F 25

Add Multiple Variables to the Diagram

1 Click the Analyses tab.

2 Click Palette to open the Palette pane.

3 In the Variables list, select N_emp, Advert, LastS, and CurrentS by holding down the CTRL key andclicking each variable.

4 Drag the selected variables from the Variables list to the desired location in the Diagram area.

5 After you drop the variables in the Diagram area, the Arrange Variables window appears and askswhether you want to arrange the variables in a row or column. Figure 4.2 shows the Arrange Variableswindow.

Figure 4.2 Arrange Variables Window

6 Select Column.

7 Click OK.

8 Drag CurrentS to the right of the other variables so that the variables are arranged to look like the variablesin Figure 4.3.


Figure 4.3 Multiple Regression Diagram Tab with Variables Only

Draw Paths

To draw paths from the predictor variables, N_emp, Advert, and LastS, to the outcome variable, CurrentS:

1 Rest the cursor on N_emp. A small palette appears that contains a single-headed and a double-headedarrow. Figure 4.4 shows an example of this palette. You use the double-headed arrow to representcovariances, and you use the single-headed arrow to represent unidirectional effects.


Draw Paths F 27

Figure 4.4 Arrow Palette

3 Drag the cursor toward CurrentS. The black outline of CurrentS turns bold, indicating that the variable isa valid target for this path.

4 Release the mouse button. The single-headed path from N_emp to CurrentS appears.

5 Repeat these steps for Advert and LastS. Figure 4.5 shows the resulting diagram.


Figure 4.5 Multiple Regression Diagram Tab with Paths Connecting theVariables

Label Each Path F 29

Label Each Path

1 Right-click the path from N_emp to CurrentS.

2 Select Set variable properties. The Path Properties window appears. Figure 4.6 shows the Path Proper-ties window.

Figure 4.6 Path Properties Window

3 Select Free.

4 In the Name box, type path_N_emp.

5 Click OK.

6 Repeat these steps for the paths from Advert to CurrentS and from LastS to CurrentS. Give each path aunique name. Figure 4.7 shows the diagram with labels on all the paths that connect the variables.

NOTE: Any paths that are labeled with the same name for the path coefficient (or effect) parameter areconstrained to have the same parameter estimate.


Figure 4.7 Multiple Regression Diagram Tab with Labels on the PathsConnecting the Variables

Add Mean Structure to the Model

There are two ways to add mean structure to a model in the SAS Structural Equation Modeling for JMP:through the Methods tab and through the variable settings.

Add Mean Structure Using the Methods Tab


2 In the Analyze area, select Mean Structures. Figure 4.8 shows the Methods tab with Mean Structuresselected.

Add Mean Structure Using the Variable Settings F 31

Figure 4.8 Methods Tab with Mean Structure Selected

NOTE: When Mean Structures is specified, by default any latent variable mean in a model is specified tobe 0.

Add Mean Structure Using the Variable Settings

1 Right-click the variable CurrentS.

2 Select Set variable properties. The Variable Properties window appears.

3 Click the Mean/Intercept tab.

4 Select Perform means analysis. Selecting Perform means analysis for only one variable in the modelestimates the mean structure for all the variables in the model.

5 Select Free.

6 In the Name box, type Intercept.

7 Click OK.

Figure 4.9 shows the diagram with the Intercept above the variable CurrentS.


Figure 4.9 Multiple Regression Diagram Tab with Mean Structure Labelabove CurrentS


After you have specified the desired path diagram, click Run in the Perform Analysis area to fit the modeland generate the results output.

View the Results


View Results in the Diagram F 33

View Results in the Diagram

The results of the model appear in the diagram with parameter estimates above the paths and variables.Figure 4.10 shows the unstandardized parameter estimates when Unstd. Estimates is selected from theView list.

Figure 4.10 Multiple Regression Diagram Tab with Unstandardized Results

Figure 4.11 shows the standardized parameter estimates when Std. Estimates is selected from the Viewlist.


Figure 4.11 Multiple Regression Diagram Tab with Standardized Results

In the Diagram area, the parameter estimates of the mean/intercept and variance of each variable are dis-played above each variable separated by a comma, and the estimated multiple regression coefficients aredisplayed above the paths from the predictors to the outcome. Any parameter estimates that differ signifi-cantly from 0 (based on t tests) are marked with asterisks. Two asterisks indicate that p < 0.01; one asteriskindicates that p < 0.05.

Verify Accuracy of Results F 35








View Detailed Results

The Diagram tab contains only some of the results. To view more detailed results, click the Results tab onthe Analyses tab.

The output on the Results tab is organized such that the modeling specifications and model fit are presentedfirst, followed by the parameter estimates for the model. The parameter estimates are organized to presentthe parameters for the single-headed arrow paths first, followed by the estimates for the variance parameters,covariances, means and intercepts, and squared multiple correlations. Figure 4.12 and Figure 4.13 show theunstandardized results for this example.


Figure 4.12 Multiple Regression Results Tab with Modeling Specificationsand Fit Results

View Detailed Results F 37

Figure 4.13 Multiple Regression Results Tab with Maximum Likelihood Pa-rameter Estimate Results

The parameter estimates under the PATH List heading are the unstandardized multiple regression coeffi-cients shown on the paths in the Diagram area in Figure 4.10. The parameter estimates under the Meansand Intercepts and Variance Parameters headings are the mean and variance estimates shown above thevariables in the Diagram area in Figure 4.10. Although the Diagram area shows only the parameter esti-mates, the Results tab also contains the standard error and t test values for each parameter estimate. TheResults tab also contains some measures of model fit (under the Fit Summary heading) and the R-squareestimate for CurrentS (under the Squared Multiple Correlations heading).


Summary of Results

A full summary of the results for this model is found in Chapter 3, “Getting Started with SAS StructuralEquation Modeling for JMP.” The new results for this model are the estimates for the means of N_emp,Advert, and LastS, and the intercept of CurrentS. The results in Figure 4.13 indicate the following:

� The mean of N_emp is 77.48. This is the average number of employees for companies in this sample.

� The mean of Advert is 0.207. This is the average advertising spending in millions of dollars for thecompanies in this sample.

� The mean of LastS is 2.014. This is the average for last year’s sales in millions of dollars for thecompanies in this sample.

� The intercept of CurrentS, called Intercept, is 0.046. This number is the average current sales for acompany with no employees, no advertising spending, and no last year’s sales. This is not practicallymeaningful for this example, but is an example of how this parameter would be interpreted in othermultiple regression analyses.

Save Project File

To save a Single Group Analysis project file with a copy of all the analysis information, including both thediagram and the results:

1 In the Project area, click Save or Save As.


3 Click Save.

Chapter 5

Path Analysis

ContentsOverview of the Path Analysis Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Create Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Specify the Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Add Multiple Variables to the Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 41Draw Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Modify the Diagram to Show Error Variables . . . . . . . . . . . . . . . . . . . . . . 44Modify the Diagram to Show Variances . . . . . . . . . . . . . . . . . . . . . . . . . 44

Perform the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45View Model 1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

View Model 1 Results in the Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 46Verify Accuracy of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48View Detailed Model 1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Summary of Model 1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Create Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Copy Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Remove a Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Modify the Diagram to Show Default Covariance Path . . . . . . . . . . . . . . . . . 52

Perform the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53View Model 2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

View Model 2 Results in the Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 54Verify Accuracy of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56View Detailed Model 2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Summary of Model 2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Compare the Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Summary of Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Overview of the Path Analysis Example

Path analysis is a method for testing causal pathways among observed variables when there is more thanone outcome (endogenous) variable (Wright 1934). This example begins with a description of the data and

40 F Chapter 5: Path Analysis

the two example models; then it shows you how to specify the data set, create the path analysis models,analyze the models, and view the results. This example also shows you how to use the Comparisons tabto customize a table of the model fit statistics to compare model fit for multiple models, and how to modifythe diagram to view the error variables, parameter variances, and default covariances in the diagram.

You can find the data file for this path analysis example, Sales_Data.jmp, by going to the JMP Homewindow and selecting Analyze I Structural Equation Modeling I Sample Data I Sales Data. Formore information about this data file, see the section “Overview of Getting Started Example” on page 5.

This path analysis example shows you how to create the two path analysis models designed to predict cur-rent year’s sales (CurrentS). The models are shown in Figure 5.1 and Figure 5.2. In Figure 5.1, the numberof employees (N_emp) predicts last year’s sales (LastS); advertising spending (Advert) and last year’s sales(LastS) predict the current year’s sales (CurrentS); and last year’s sales(LastS) predicts advertising spend-ings (Advert).

Figure 5.1 Path Analysis Model 1 Diagram

The path analysis model in Figure 5.2 is the same as Figure 5.1, except for the removal of the path thatallows last year’s sales (LastS) to predict advertising spending (Advert).

Create Model 1 F 41


Create Model 1

In this section you learn how to specify the data set, add variables to the diagram, draw paths, and modifythe diagram to view the error variables and parameter variances in the diagram.


On the Data tab, verify that the following information is specified in the Data Table Properties area:



Add Multiple Variables to the Diagram

Now that the data have been specified, you can start building your model. To build the first path analysismodel:


2 Click Palette to open the Palette pane.


3 In the Variables list, select more than one variable at a time by holding down the CTRL key and clickingeach observed variable.

4 Drag the selected variables from the Variables list to the desired location in the Diagram area.

5 After you drop the variables in the Diagram area, the Arrange Variables window appears and askswhether the variables should be arranged in a row or column.

6 Select Row.

7 Click OK.

8 Repeat these steps for the variables LastS and CurrentS. Drag LastS and CurrentS above the other vari-ables so that the variables are arranged to look like the variables in Figure 5.3.

Figure 5.3 Path Analysis Model 1 Diagram Tab with Variables Only

Draw Paths F 43

Draw Paths

1 Rest the cursor on N_emp. A small palette appears that contains a single-headed and a double-headedarrow.

2 Select the single-headed arrow. (It turns red when selected.)

3 Drag the cursor toward LastS. The black outline of LastS turns bold, indicating that the variable is a validtarget for this path.

4 Release the mouse button. The single-headed path from N_emp to LastS appears.

5 Repeat these steps for the paths from Advert to CurrentS, from LastS to CurrentS, and from LastS toAdvert. Figure 5.4 shows the resulting diagram.

Figure 5.4 Path Analysis Model 1 Diagram Tab with Paths Connecting theVariables


Modify the Diagram to Show Error Variables

1 Right-click in the Diagram area.

2 Select Show error variables. Figure 5.5 shows the diagram with the error variables.

Figure 5.5 Path Analysis Model 1 with Error Variables

Modify the Diagram to Show Variances


2 Select Show variances. Figure 5.6 shows the diagram with the variance paths.


Figure 5.6 Path Analysis Model 1 with Error Variables and Variances



View Model 1 Results



View Model 1 Results in the Diagram

Figure 5.7 shows the unstandardized parameter estimates in the diagram when Unstd. Estimates is selectedfrom the View list.

Figure 5.7 Path Analysis Model 1 Diagram with Unstandardized Results

Figure 5.8 shows the standardized parameter estimates in the diagram when Std. Estimates is selected fromthe View list.

View Model 1 Results in the Diagram F 47

Figure 5.8 Path Analysis Model 1 Diagram with Standardized Results

In the Diagram area, the estimate of the variance of each variable is displayed below the variance anderror variance path for each variable. Each estimated path coefficient is displayed next to the path from onevariable to another. Any parameter estimates that differ significantly from 0 (based on t tests) are markedwith asterisks. Two asterisks indicate that p < 0.01; one asterisk indicates that p < 0.05.









View Detailed Model 1 Results

The Diagram area contains only some of the results. To view more detailed results, click the Results tabon the Analyses tab.

The output in the Results tab is organized such that the modeling specifications and model fit are presentedfirst, followed by the parameter estimates for the model. The parameter estimates are organized to presentthe parameters for the single-headed arrow paths first, followed by the estimates for the variances andsquared multiple correlations. Figure 5.9 and Figure 5.10 show the unstandardized results for this example.

View Detailed Model 1 Results F 49

Figure 5.9 Path Analysis Model 1 Results Tab with Modeling Specificationsand Fit Results


Figure 5.10 Path Analysis Model 1 Results Tab with Maximum LikelihoodParameter Estimate Results

Summary of Model 1 Results

Overall, Path Analysis Model 1 has excellent fit to the data according to most fit indices. The path parametersare all significant at the p < 0.05 level, and most are significant at the p < 0.01 level. These results indicatethat the paths in this model represent significant relationships among the variables. The path parameters areall positive values, indicating that higher values of the predictor are associated with higher values on theoutcome. The squared multiple correlations for both LastS and CurrentS are very high (0.959 and 0.970,respectively). These values indicate that most of the variance in these variables can be explained by thepredictors. Together, these results suggest that this model does a good job of capturing the relationshipsamong these variables.

Create Model 2 F 51

Create Model 2

Copy Model 1

Rather than creating a new model, you can copy and then modify Path Analysis Model 1 to create PathAnalysis Model 2. To copy the model:

1 Click Copy in the Analyses area.

A new analysis diagram appears with the same diagram that you selected to copy.

Remove a Path

To remove the path from LastS to Advert:

1 Right-click the path from LastS to Advert.

2 Select Delete. The path from LastS to Advert disappears. Figure 5.11 shows the modified diagram.



Modify the Diagram to Show Default Covariance Path

By default, when a model has more than one exogenous variable, SAS Structural Equation Modeling forJMP estimates a covariance for each pair of exogenous variables. To see the default covariances in thediagram:


2 Select Show default covariances. Figure 5.12 shows the diagram with the default covariance path.




After you have removed the path from LastS to Advert, click Run in the Perform Analysis area to fit themodel and generate the results output.


View Model 2 Results

View Model 2 Results in the Diagram

Figure 5.13 shows the unstandardized parameter estimates when Unstd. Estimates is selected from theView list.

Figure 5.13 Path Analysis Model 2 Diagram with Unstandardized Results

Figure 5.14 shows the Standardized estimate results when Std. Estimates is selected from the View list.

View Model 2 Results in the Diagram F 55

Figure 5.14 Path Analysis Model 2 Diagram with Standardized Results

In the Diagram area, the estimate of the variance of each variable is displayed below the variance or errorvariance path for each variable. Each estimated path coefficient is displayed next to the path from thepredictor to the outcome. Any parameter estimates that differ significantly from 0 (based on t tests) aremarked with asterisks. Two asterisks indicate that p < 0.01; on asterisk indicates that p < 0.05.









View Detailed Model 2 Results

The Diagram contains only some of the results. To view more detailed results, click the Results tab on theAnalyses tab. Figure 5.15 and Figure 5.16 show the unstandardized results for this example.

View Detailed Model 2 Results F 57

Figure 5.15 Path Analysis Model 2 Results Tab with Modeling Specificationsand Fit Results


Figure 5.16 Path Analysis Model 2 Results Tab with Maximum LikelihoodParameter Estimate Results

Summary of Model 2 Results

Overall, Path Analysis Model 2 also has excellent fit to the data according to most fit indices. The pathparameters are all significant at the p < 0.05 level, and most are significant at the p < 0.01 level. Theseresults indicate that the paths in this model represent significant relationships among the variables. Aswith the previous model, the path estimates are all positive, indicating that higher values of the predictorare associated with higher values on the outcome. The squared multiple correlations for both LastS andCurrentS are very high (0.970 and 0.959, respectively). These values indicate that most of the variance inthese variables can be explained by the predictors. Together, these results suggest that this model does agood job of capturing the relationships among these variables.

Summary of Model Comparison F 59

Compare the Models

To compare fit information for the two path analysis models:

1 Click the Comparisons tab.

2 In the Show area, select User-selected fit statistics, and then click Customize.

3 In the Fit Indices window, clear the fit indices that you do not want to compare. Click Clear All to clearall the fit indices.

4 Select the fit indices you want to compare. For this example, select Akaike Information Criterion,Bentler-Bonett NFI, Chi-Square, Chi-square DF, and Pr > Chi-Square. NOTE: At least one fit indexmust be selected.

5 Click OK to close the Fit Indices window. Figure 5.17 shows the Comparisons tab after the fit indicesare customized.

Figure 5.17 Comparisons Tab for the Two Path Analysis Model Results

Summary of Model Comparison

The fit statistics in the model comparison table indicate that having a path from LastS to Advert does improvemodel fit. However, both models have very good fit and the difference in model fit is not very large. Thus,both models appear to be a good representation of how the data were generated.


ReferencesWright, S. (1934), “The Method of Path Coefficients,” Annals of Mathematical Statistics, 5(3), 161–215.

Chapter 6

Confirmatory Factor Analysis

ContentsOverview of the Confirmatory Factor Analysis Example . . . . . . . . . . . . . . . . . . . 62Create the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Specify the Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Add Variables to the Diagram and Specify a Name for the Latent Variable . . . . . . . 65Modify the Diagram View to Show Default Covariances . . . . . . . . . . . . . . . . 70Specify Paths Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Perform the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74View Correlated CFA Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

View Correlated CFA Results in the Diagram . . . . . . . . . . . . . . . . . . . . . . 74Verify Accuracy of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77View Detailed Correlated CFA Results . . . . . . . . . . . . . . . . . . . . . . . . . 77Summary of Correlated CFA Results . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Create the Uncorrelated CFA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Copy the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Constrain the Latent Variables to Be Uncorrelated . . . . . . . . . . . . . . . . . . . 81

Perform the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83View Uncorrelated CFA Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

View Uncorrelated CFA Results in the Diagram . . . . . . . . . . . . . . . . . . . . 83Verify Accuracy of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86View Detailed Uncorrelated CFA Results . . . . . . . . . . . . . . . . . . . . . . . . 86Summary of Uncorrelated CFA Results . . . . . . . . . . . . . . . . . . . . . . . . . 89


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

62 F Chapter 6: Confirmatory Factor Analysis

Overview of the Confirmatory Factor Analysis Example

Confirmatory factor analysis (CFA) is a method for testing models in which one or more latent (unobserved)variables are hypothesized to predict (or explain) the correlations among several observed variables (Mulaik1972). This example begins with a description of the data and the two example models; then it shows youhow to specify the data set, create the CFA models, analyze the models, and view the results. This examplealso shows you how to use the Comparisons tab to create a table of the model fit statistics to compare thetwo models, and how to modify the diagram to view the default covariance parameters in the diagram.

You can find the data file for this confirmatory factor analysis example, CFA_Data.jmp, by going to the JMPHome window and selecting Analyze I Structural Equation Modeling I Sample Data I CFA Data.This data file CFA_Data.jmp contains covariance data, the variances for each variable, and covariances foreach pair of variables included in the study. The covariance data are based on responses to nine cognitiveabilities tests from 64 students. Three of the cognitive abilities tests measured reading skills (reading1, read-ing2, and reading3), three tests measured math skills (math1, math2, and math3), and three tests measuredwriting skills (writing1, writing2, and writing3).

This example shows you how to create the CFA models shown in Figure 6.1 and Figure 6.2. In both models,a latent variable called Read predicts the three reading ability variables, a latent variable called Math predictsthe three math ability variables, and a latent variable called Write predicts the three writing ability variables.In Figure 6.1, the latent variables have double headed arrows connecting each latent variable to the otherlatent variables. These double headed arrows indicate that the latent variables are correlated with each other.A covariance parameter is estimated for each pair of latent variables.

Overview of the Confirmatory Factor Analysis Example F 63

Figure 6.1 Correlated Confirmatory Factor Analysis Model Diagram

In addition to the correlated CFA model, you can also create a comparison model, shown in Figure 6.2. Inthe comparison model the covariance parameters have been fixed to be 0, so that the latent variables areuncorrelated with each other.


Figure 6.2 Uncorrelated Confirmatory Factor Analysis Model Diagram

Create the Model

In this section you learn how to specify the data set, add variables to the diagram, and modify the diagramto view the default covariance parameters in the diagram.

Add Variables to the Diagram and Specify a Name for the Latent Variable F 65


On the Data tab, verify (or specify) the following information:

1 In the Data Table Properties area, verify that Name shows CFA_Data.

2 In the Data Table Properties area, verify that Covariances, correlations is selected from the DataStructure list.

3 In the Number of Observations area, type 64 in the Number box.

Figure 6.3 shows the Data tab specifications for this example.

Figure 6.3 Data Tab Specifications

Add Variables to the Diagram and Specify a Name for the Latent Variable

Now that the data have been specified, you can start building your model. To build the correlated CFAmodel:


2 Click Palette to show the Palette pane.

3 Add a latent variable by selecting the oval shape and dragging it into the middle of the Diagram area.


4 Add multiple observed variables by selecting reading1, reading2, and reading3 in the Variables list (holddown the CTRL key and click each variable) and dragging the selected variables from the Variables listonto the oval in the Diagram area.

5 After you drop the variables in the Diagram area, a window appears and asks in what direction thevariables should be arranged. Figure 6.4 shows the Arrange Variables window.

Figure 6.4 Arrange Variables Window

6 Select To the right, and then click OK. The observed variables appear in the diagram to the right of thelatent variable, and paths are automatically drawn from the latent variable to the observed variables.

7 Specify a name for the latent variable by right-clicking the oval in the Diagram area and selecting Setvariable properties. The Latent Variable Properties window appears. See Figure 6.5.


Figure 6.5 Latent Variables Properties Window

8 Enter Read in the Variable box, and then click OK. Figure 6.6 shows the Diagram area with the readingvariables part of the correlated CFA model.


Figure 6.6 Confirmatory Factor Analysis Model with Reading Variables

9 Repeat the steps of adding a latent variable, adding multiple observed variables, and naming the latentvariable to create a latent variable Math to predict math1, math2 and math3 and a latent variable Write topredict writing1, writing2 and writing3. Arrange the variables to look like Figure 6.7.


Figure 6.7 Confirmatory Factor Analysis Model with All Variables


Modify the Diagram View to Show Default Covariances

The default covariances are the covariance parameters that are automatically estimated for all exogenousvariables. By default, these covariances are shown in the diagram as gray double-headed arrows. If thisdefault setting has been turned off, you can restore the default settings by using the following steps:

1 Right-click the Diagram area.

2 Select Show default covariances. Figure 6.8 shows the diagram with the gray double-headed arrowsrepresenting the default covariance paths.

Modify the Diagram View to Show Default Covariances F 71

Figure 6.8 Confirmatory Factor Analysis Model with Default Covariances


Specify Paths Constraints

Before you can run the analysis, you need to identify the model to set the scale for the latent variables inorder to obtain a unique solution. To identify this model:

1 Right-click the path from Read to Reading1.

2 Select Set path coefficient to 1. A 1 appears by the path. This constrains this path coefficient to be 1during model estimation.

3 Select the same option for the paths from Math to Math1 and Write to Writing1. Figure 6.9 shows themodel with all of the path constraints.

Specify Paths Constraints F 73

Figure 6.9 Confirmatory Factor Analysis Model with Path Constraints




View Correlated CFA Results


View Correlated CFA Results in the Diagram

The results of the model appear in the diagram with parameter estimates above the paths and variables. Bydefault, the results shown in the diagram are the unstandardized parameter estimates. Figure 6.10 showsthe unstandardized parameter estimates. You can also see the unstandardized parameter estimates when youselect Unstd. Estimates in the View list.

View Correlated CFA Results in the Diagram F 75

Figure 6.10 Correlated Confirmatory Factor Analysis Model Diagram withUnstandardized Results



Figure 6.11 Correlated Confirmatory Factor Analysis Model Diagram withStandardized Results

In the Diagram area, the estimate of the variance of each variable is displayed above it. Each estimated pathcoefficient is displayed next to the path from one variable to another. Any parameter estimates that differsignificantly from 0 (based on t tests) are marked with asterisks. Two asterisks indicate that p < 0.01; oneasterisk indicates that p < 0.05.









View Detailed Correlated CFA Results

The Diagram contains only some of the results. To view more detailed results, click the Results tab on theAnalyses tab.

The output in the Results tab is organized such that the modeling specifications and model fit are presentedfirst, followed by the parameter estimates for the model. The parameter estimates are organized to presentthe parameters for the single-headed arrow paths first, followed by the estimates for the variances, covari-ances, and squared multiple correlations. Figure 6.12, Figure 6.13, and Figure 6.14 show the unstandardizedresults for this example.


Figure 6.12 Correlated Confirmatory Factor Analysis Results Tab with Model-ing Specifications and Fit Results

View Detailed Correlated CFA Results F 79

Figure 6.13 Correlated Confirmatory Factor Analysis Results Tab with Maxi-mum Likelihood Parameter Estimate Results


Figure 6.14 Correlated Confirmatory Factor Analysis Results Tab with Maxi-mum Likelihood Parameter Estimate Results Continued

Summary of Correlated CFA Results

Overall, the correlated CFA model has good fit to the data according to most fit indices. The path parametersare all significant at the p < 0.01 level, and covariance parameters are significant at either the p < 0.05 orp < 0.01 level. These results indicate that the paths in this model represent significant relationships amongthe variables. Together, these results suggest that this model does a good job of capturing the relationshipsamong the variables in this data set.

Create the Uncorrelated CFA Model

By default, the exogenous variables in any model are correlated. However, you might want to examine analternative model in which the exogenous variables are not correlated. To create the alternative model inwhich the latent variables are uncorrelated, you need to replicate your original CFA model and then set thecovariances among the latent variables to be 0.

Copy the Model F 81

Copy the Model

Rather than creating a new model, you can copy and then modify your correlated CFA model. To copy themodel:

1 Click Copy in the Analyses pane.

A new analysis, Analysis 2, appears in the Analyses list, and a copy of the diagram from the previous modelappears in the Diagram area.

Constrain the Latent Variables to Be Uncorrelated

To create the uncorrelated CFA model, you fix the covariances among the latent variables to equal 0 whenthe model is estimated. To constrain the covariances to be 0:

1 Right-click the covariance path from Read to Write, and select set covariance to 0. A 0 appears by thecovariance path.

2 Select the same option for the covariance path between Read and Math and for the covariance pathbetween Write and Math. Figure 6.15 shows the diagram with the covariance path constraints. NOTE: Thecovariance paths become explicit (black, instead of gray) whenever they are constrained to a value.


Figure 6.15 Confirmatory Factor Analysis Model Diagram with Latent VariableCovariances Constrained to Zero



After you have fixed the covariance paths among the latent variables to be 0, click Run in the PerformAnalysis area to fit the uncorrelated CFA model and generate the results output.

View Uncorrelated CFA Results

View Uncorrelated CFA Results in the Diagram

Figure 6.16 shows the unstandardized parameter estimates when Unstd. Estimates is selected from theView list.


Figure 6.16 Uncorrelated Confirmatory Factor Analysis Model Diagram withUnstandardized Results


View Uncorrelated CFA Results in the Diagram F 85

Figure 6.17 Uncorrelated Confirmatory Factor Analysis Model Diagram withStandardized Results

In the Diagram area, the estimate of the variance of each variable is displayed above it, and each estimatedpath coefficient is displayed next to the path from the predictor to the outcome. Any parameter estimatesthat differ significantly from 0 (based on t tests) are marked with asterisks. Two asterisks indicate that p <0.01; one asterisk indicates that p < 0.05.









View Detailed Uncorrelated CFA Results

The Diagram area contains only some of the results. To view more detailed results, click the Results tabon the Analyses tab. Figure 6.18, Figure 6.19, and Figure 6.20 show the unstandardized results for thisexample.

View Detailed Uncorrelated CFA Results F 87

Figure 6.18 Uncorrelated Confirmatory Factor Analysis Results Tab withModeling Specifications and Fit Results


Figure 6.19 Uncorrelated Confirmatory Factor Analysis Results Tab withMaximum Likelihood Parameter Estimate Results

Summary of Uncorrelated CFA Results F 89

Figure 6.20 Uncorrelated Confirmatory Factor Analysis Results Tab withMaximum Likelihood Parameter Estimate Results Continued

Summary of Uncorrelated CFA Results

Overall, the uncorrelated CFA model has poor fit to the data according to most fit indices. The path param-eters are all significant at the p < 0.05 level. These path coefficient results indicate that the paths in thismodel represent significant relationships among the variables. Together, these results suggest that the latentvariables in the model are related to the indicators. However, the overall model does a poor job of capturingthe relationships among these variables.

Compare the Models

Now that you have fit two models to the data, you can compare the model fit for each model to decide whichmodel best represents the data. To compare the correlated CFA model to the uncorrelated CFA model:


2 In the Show area, select User-selected fit statistics, and then select Customize.

3 In the Fit Indices window, clear the fit statistics that you do not want to compare. For this example, clickClear All to clear all of the fit indices.


4 Now select the fit indices you want to compare. For this example, select Bentler Comparative FitIndex, RMSEA Estimate, Adjusted GFI (AGFI), Chi-Square, Chi-square DF, and Pr > Chi-Square.NOTE: At least one fit index must be selected.

5 Click OK to close the Fit Indices window.

6 In the Sort area, select Chi-Square, and then click to sort in descending order.

Figure 6.21 shows the Comparisons tab after the fit indices are customized.

Figure 6.21 Comparisons Tab with Customized Fit Indices

NOTE: By default, the analyses are listed in the order they were created, and the parsimony fit statisticsAkaike Information Criterion, Bozdogan CAIC, Schwarz Bayesian Criterion, and RMSEA are shown.

Based on the fit statistics, it appears that the correlated CFA model has a better model fit. Because thesemodels are nested models, you can use the chi-square difference test to determine whether the correlatedmodel is significantly different from the uncorrelated model:

1 Subtract the chi-square values (result = 19.9085).

2 Subtract the chi-square DF values (result = 3).

3 Obtain the probability that the chi-square difference value (19.9085) was observed in a chi-square distri-bution with degrees of freedom equal to the difference in degrees of freedom (3).

For this example, a �2.df D 3/ D 19:9085 has p < 0.001, so there is a significant difference (given˛ D 0:05) between the uncorrelated and correlated factor model. This result suggests that the correlatedmodel fits significantly better than the uncorrelated model.



The fit statistics in the model comparison table indicate that the correlated CFA model has a better model fitthan the uncorrelated CFA model. A chi-square difference test confirms that there is a significant differencebetween the correlated and uncorrelated models, suggesting that the model with correlated latent variablesdoes a significantly better job of describing the observed data.

ReferencesMulaik, S. A. (1972), The Foundations of Factor Analysis, New York: McGraw-Hill.

92

Chapter 7

Structural Equation Model

ContentsOverview of the Structural Equation Model Example . . . . . . . . . . . . . . . . . . . . . 93Create the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Specify the Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Add Variables to the Diagram and Specify a Name for the Latent Variable . . . . . . . 96Draw the Latent Variable Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Modify the Diagram to Show Error Variables . . . . . . . . . . . . . . . . . . . . . . 101Draw the Correlation Paths for Error Variables . . . . . . . . . . . . . . . . . . . . . 103Specify Parameter Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Perform the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109View SEM Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

View SEM Results in the Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Verify Accuracy of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111View Detailed SEM Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Summary of SEM Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Overview of the Structural Equation Model Example

Structural equation modeling (SEM) is a method for testing models that specify relationships among ob-served and latent variables that are believed to explain the covariance among the observed variables (Bollen1989). This SEM example begins with a description of the data and the example model; then it shows youhow to specify the data set, create the model, analyze the model, and view the results. This example alsoshows you how to modify the diagram to view the error variables, specify correlations among the errorvariables, and specify parameter constraints.

You can find the data file for this SEM example, Wheaton_Data.jmp, by going to the JMP Home windowand selecting Analyze I Structural Equation Modeling I Sample Data I Wheaton Data. This datafile Wheaton_Data.jmp contains covariance data, the variances for each variable, and covariances for eachpair of variables included in the study. The covariance data is created from 932 observations and wasoriginally presented in Wheaton, et al. (1977). The data set contains six variables: Anomie67, Powerless67,Anomie71, Powerless71, Education, and SEI.

94 F Chapter 7: Structural Equation Model

This example shows you how to create the structural equation model shown in Figure 7.1. In the model, alatent variable called Alien67 predicts Anomie67 and Powerless67, a latent variable called Alien71 predictsAnomie71 and Powerless71, and a latent variable called SES predicts Education and SEI.

Unlike the confirmatory factor analysis example in the previous chapter, this structural equation model ex-ample includes paths that predict one latent variable from another latent variable. In this example, the latentvariable relationships include SES predicting both Alien67 and Alien71, and Alien67 predicting Alien71.

In addition to the prediction paths, several correlation paths are also included in Figure 7.1. Since Anomie67and Anomie71 measure the same variable at different times, it is possible that the errors for these variablesare not only related to the latent variables Alien67 and Alien71, but are also correlated. To test this possibility,a correlation path is estimated for the error variables of Anomie67 and Anomie71. Similarly, Powerless67and Powerless71 also measure the same variable at different times; therefore a correlation path is alsoestimated for the error variables of Powerless67 and Powerless71.

Several parameters are fixed to specific values in this model. Specifically, the paths from Alien67 toAnomie67, Alien71 to Anomie71, and SES to Education are all set equal to 1, and the paths from Alien67to Powerless67 and Alien71 to Powerless71 are set to 0.833. Also, several parameters are constrained tobe equal. The variances for Anomie67 and Anomie71 are constrained to be equal as indicated by the labelev1 above the error variables that correspond to these variables. Similarly, the variances for Powerless67and Powerless71 are constrained to be equal as indicated by the label ev2. Finally, the correlation paths forthe error variables for Anomie67 with Anomie71 and Powerless67 with Powerless71 are constrained to beequal, as indicated by the label evc on the correlation paths.

Create the Model F 95

Figure 7.1 Structural Equation Model Diagram

Create the Model

In this section you learn how to specify the data set, add variables to the diagram, draw paths, and modifythe diagram to view the error variables in the diagram.




1 In the Data Table Properties area, verify that Name shows Wheaton_Data.

2 In the Data Table Properties area, verify that Covariances, correlations is selected from the DataStructure list.

3 In the Number of Observations area, type 932 into the Number box.

Figure 7.2 shows the Data tab specifications for this example.


Add Variables to the Diagram and Specify a Name for the Latent Variable

Now that the data have been specified, you can start building your model:


2 Click Palette to show the Palette pane.

3 Add a latent variable by selecting the oval shape and dragging it into the Diagram area.


4 Add a multiple observed variables by selecting Anomie67 and Powerless67 in the Variables list (holddown the CTRL key and click each variable).

5 Drag the selected variables from the Variables list onto the oval in the Diagram area.

6 After you drop the variables onto the oval in the Diagram area, the Arrange Variables window appearsand asks in what direction the variables should be arranged. Click Above, and then click OK.

7 To specify a name for the latent variable, double-click the oval in the Diagram area. The Latent VariableProperties window appears. Type Alien67 in the Variable box, and click OK. Figure 7.3 shows theDiagram area with only the variables related to the Alien67 part of the SEM example.

Figure 7.3 Structural Equation Model with Alien67 Variables

8 Repeat these steps to create a latent variable Alien71 to predict Anomie71 and Powerless71, and a latentvariable SES to predict Education and SEI. Arrange the variables to look like Figure 7.4.


Figure 7.4 Structural Equation Model with All Variables

Draw the Latent Variable Paths F 99

Draw the Latent Variable Paths

1 Rest the cursor on SES. A small palette appears that contains a single-headed and a double-headed arrow.


3 Drag the cursor toward Alien67. The black outline of Alien67 turns bold, indicating that the variable is avalid target for this path.

4 Release the mouse button. The single-headed path from SES to Alien67 appears.

5 Repeat these steps for the path from SES to Alien71, and the path from Alien67 to Alien71. Figure 7.5shows the resulting diagram.


Figure 7.5 Structural Equation Model Diagram with Latent Variable Paths

Modify the Diagram to Show Error Variables F 101



2 Select Show error variables. Figure 7.6 shows the diagram with the error variables.


Figure 7.6 Structural Equation Model Diagram with Correlated Error Variables

Draw the Correlation Paths for Error Variables F 103

Draw the Correlation Paths for Error Variables

1 Rest the cursor on error variable associated with Anomie67. A small palette appears that contains adouble-headed arrow.

2 Select the double-headed arrow. (It turns red when selected.)

3 Drag the cursor toward the error variable for Anomie71. A bold black outline around the error variableappears, indicating that the variable is a valid target for this path.

4 Release the mouse button. The double-headed path from Anomie67 to Anomie71 appears.

5 Repeat these steps for the correlation path for Powerless67 and Powerless71. Figure 7.7 shows theresulting diagram. Figure 7.7 shows the diagram with the correlated error variables.


Figure 7.7 Structural Equation Model Diagram with Correlated Variables

Specify Parameter Constraints F 105

Specify Parameter Constraints

Before you can run the analysis, you need to add some parameter constraints to identify the model.

Specify Path Parameters Constrained to 1

Before you can run the analysis, you need to identify the model to set the scale for the latent variables. Toidentify this model:

1 Right-click the path from Alien67 to Anomie67.

2 Select Set path coefficient to 1. A 1 appears by the path.

3 Select the same option for the paths from Alien71 to Anomie71 and from SES to Education.

Specify Parameters Constrained to a Fixed Value

To set parameters equal to a specific value:

1 Right-click the path from Alien67 to Powerless67.

2 Select Set path properties. A Path Properties window appears.

3 Select Fixed, and type 0.833 in the Value box. See Figure 7.8.

Figure 7.8 Path Properties window

4 Click OK.

5 Follow the same steps for the paths from Alien71 to Powerless71. Figure 7.9 shows Diagram area withthe path constraints.


Figure 7.9 Diagram Area with the Path Constraints

Specify Parameter Constraints F 107

Constrain Parameters to Be Equal

To set parameters to be equal, you must give the same name to all parameters:

1 Right-click the error variable for Alien67.

2 Select Set error properties. An Error Properties window appears.

3 Click the Variance tab.

4 Type ev1 in the Name box.

5 Click OK.

To specify similar constraints on the error variances for the variables Powerless67 and Powerless71, followthe same steps, except type ev2 in the Name box. To specify the error correlation constraints, follow thesame steps, except select Set covariance from the correlation path pop-up menu and type evc in the Namebox.

Figure 7.10 shows the model with all of the parameter constraints.


Figure 7.10 Structural Equation Model with Parameter Constraints




View SEM Results


View SEM Results in the Diagram

The results of the model appear in the diagram with parameter estimates next to the paths and above thevariables. Figure 7.11 shows the unstandardized parameter estimates when Unstd. Estimates is selectedfrom the View list.


Figure 7.11 Structural Equation Model Diagram with Unstandardized Results


In the Diagram area, the estimate of the variance of each variable is displayed above each variable orerror variable. Each estimated path coefficient is displayed next to the path from one variable to another.Any parameter estimates that differ significantly from 0 (based on t tests) are marked with asterisks. Twoasterisks indicate that p < 0.01; one asterisk indicates that p < 0.05.








View Detailed SEM Results

The Diagram contains only some of the results. To view more detailed results, click the Results tab on theAnalyses tab.

The output on the Results tab is organized such that the modeling specifications and model fit are pre-sented first, followed by the parameter estimates for the model. The parameter estimates are organized topresent the parameters for the single-headed arrow paths first, followed by the estimates for the variances,error covariances, and squared multiple correlations. Figure 7.12, Figure 7.13, and Figure 7.14 show theunstandardized results for this example.


Figure 7.12 Structural Equation Model Results Tab with Modeling Specifica-tions and Fit Results

View Detailed SEM Results F 113

Figure 7.13 Structural Equation Model Results Tab with Maximum LikelihoodParameter Estimate Results


Figure 7.14 Structural Equation Model Results Tab with Maximum LikelihoodParameter Estimate Results Continued

Summary of SEM Results

Overall, the structural equation model has excellent fit to the data according to most fit indices. For example,the RMSEA estimate (0.0231) is small, the Bentler comparative fit index (0.9979) is very close to 1, andthe chi-square probability (0.1419) is not significant. In addition to the overall fit indices, all the estimatedparameters are significant at either the p < 0.05 or p < 0.01 level. The parameter estimate results indicatethat the paths in this model represent significant relationships among the variables. Together, the overallfit and parameter estimate results suggest that this model does a good job of describing the relationshipsamong these variables.

ReferencesBollen, K. A. (1989), Structural Equations with Latent Variables, New York: John Wiley & Sons.

Wheaton, B., Muthèn, B., Alwin, D. F., and Summers, G. F. (1977), “Assessing Reliability and Stability inPanel Models,” in D. R. Heise, ed., Sociological Methodology, San Francisco: Jossey Bass.

Chapter 8

Latent Growth Curve Model

ContentsOverview of the Latent Growth Curve Model Example . . . . . . . . . . . . . . . . . . . . 116Create the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Specify the Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Add a Latent Variable to the Diagram and Specify Its Name . . . . . . . . . . . . . . 119Add Observed Variables to the Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 119Draw Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Draw a Covariance Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Specify Mean Structure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Specify Parameter Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Modify the Diagram to Show Error Variables . . . . . . . . . . . . . . . . . . . . . . 126Specify the Parameter Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Perform the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131View LGCM Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

View LGCM Results in the Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 131Verify Accuracy of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133View Detailed LGCM Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Summary of LGCM Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Alternative LGCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Copy the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Remove Error Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Perform the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139View Alternative LGCM Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

View Alternative LGCM Results in the Diagram . . . . . . . . . . . . . . . . . . . . 139Verify Accuracy of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141View Detailed Alternative LGCM Results . . . . . . . . . . . . . . . . . . . . . . . . 141Summary of Alternative LGCM Results . . . . . . . . . . . . . . . . . . . . . . . . . 144


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

116 F Chapter 8: Latent Growth Curve Model

Overview of the Latent Growth Curve Model Example

Latent growth curve modeling is a method that uses a structural equation model to estimate an unobservedgrowth function of a specific structure for a variable or set of variables that are observed over time (Meredithand Tisak 1990). This example begins with a description of the data and the example model; then it showsyou how to specify the data set, create the latent growth curve model, analyze the model, and view the results.This example also shows you how to estimate variable means, name parameters, and specify parameterconstraints. After fitting the initial model, this example shows how to fit an alternative model to the dataand then compare the results of the models.

You can find the data file for this latent growth curve model (LGCM) example, LGCM_Data.jmp, by goingto the JMP Home window and selecting Analyze I Structural Equation Modeling I Sample Data ILGCM Data. This data file LGCM_Data.jmp contains raw data, which are observations for each variablefor each unit included in the study. The data file contains responses from 16 individuals invited to a trainingprogram that was designed to boost self-confidence. During the training, each individual’s self-esteem wasmeasured initially (y1) and then yearly for four years at equally spaced intervals (y2–y5).

y1 y2 y3 y4 y517.6 21.4 25.6 32.1 37.713.2 14.3 18.9 20.3 25.411.6 13.5 17.4 22.1 39.610.7 11.1 13.2 18.2 21.418.7 23.7 28.6 31.5 34.018.3 19.2 20.5 23.2 25.99.2 13.5 17.8 19.2 21.118.3 23.5 27.9 30.2 34.611.2 15.6 20.8 22.7 30.417.0 22.9 26.9 31.9 35.610.4 13.6 18.0 25.6 29.317.7 19.0 22.5 28.5 30.714.5 19.4 21.1 28.8 31.520.0 21.4 28.9 30.2 35.614.6 19.3 21.7 28.5 32.011.7 15.2 19.1 23.7 28.7

This example shows you how to create the LGCM shown in Figure 8.1.

Overview of the Latent Growth Curve Model Example F 117

Figure 8.1 Latent Growth Curve Model Diagram

In Figure 8.1, the latent variables f_alpha and f_beta represent the intercept and slope of the latent growthstructure that represents the change in the observed variables (y1–y5) over time. The means and variancesof the intercept and slope are represented by the labels above the variables f_alpha and f_beta. m_alpharepresents the mean for the latent intercept variable, and v_alpha represents the variance for the latentintercept variable. Similarly, m_beta represents the mean for the latent slope variable, and v_beta representsthe variance for the latent slope variable. The model has a covariance path between the latent variablesf_alpha and f_beta, which means that the model estimates a covariance between the initial level of self-esteem and the linear growth of self-esteem.

The paths from the latent variables (f_alpha and f_beta) create a linear growth structure, with all the pathsfor the intercept (f_alpha), fixed to 1 to represent the constant influence of the intercept on the observedvariables. The path from the latent slope (f_beta), is fixed to be 0 for the initial self-esteem measure (y1),and subsequent paths from the latent slope to the observed variables are fixed to increase by 1 for eachadditional year that passes. The parameter constraints on the latent intercept variable (f_alpha) enablethe mean of the latent intercept (m_alpha) to be interpreted as the mean initial level of self-esteem and thevariance of the latent intercept (v_alpha) to be interpreted as the variability in the initial level of self-esteem.The parameter constraints on the latent slope variable (f_beta) enable the mean of the latent slope (m_beta)to be interpreted as the mean linear growth in the level of self-esteem and the variance of the latent slope(v_beta) to be interpreted as the variability in the linear growth of self-esteem.


Figure 8.1 shows several parameter constraints on the observed variables (y1–y5). First, the 0 above eachof the variables, y1 to y5, indicates that the observed variable intercepts have been set equal to 0. Theseintercepts must be fixed to 0 because in this model the random intercept is represented as a latent factor(f_alpha). Also, if the intercepts for the observed self-esteem variables (y1–y5) were not fixed to 0, theintercepts would be assumed to be free parameters, which would lead to a model with an overparameter-ized mean structure and would potentially lead to some identification problems. Besides constraining theobserved variable intercepts to 0, the ev1 below each error variable associated with the observed variablesindicates that the error variances associated with each of the error variables are fixed to be equal, becausethey have the same variance name. This parameter constraint represents the assumption that the mean andvariance of the error variables are normally distributed with a mean of 0 and have equal variance (ev1) overtime.

Create the Model

In this section you learn how to specify the data set, add variables to the diagram, draw paths, name param-eters, and specify parameter constraints.



1 In the Data Table Properties area, verify that Name shows LGCM_Data.

2 In the Data Table Properties area, verify that Raw data is selected from the Data Structure list. Fig-ure 8.2 shows the Data tab specifications for this example.

Add a Latent Variable to the Diagram and Specify Its Name F 119


Now that you have specified the data, you can start building your model.

Add a Latent Variable to the Diagram and Specify Its Name


2 Click Palette to show the Variables menu.

3 Add a latent variable by selecting the oval shape and dragging it into the Diagram area.

4 To specify a name for the latent variable, double-click the oval. The Latent Variable Properties windowappears.

5 Enter f_alpha in the Variable box.

6 Click OK.

7 Repeat these steps, except enter f_beta in the Variable box.

Add Observed Variables to the Diagram

1 In the Variables list, select variables y1–y5 by holding down the CTRL key and clicking each variable.


2 Drag the selected variables from the Variables list onto the oval called f_alpha in the Diagram area.

3 After you drop the variables in the Diagram area, the Arrange Variables window appears and asks inwhat direction the variables should be arranged. Select Below, and then click OK.

After you have added all of the variables to the diagram, arrange the variables to look like Figure 8.3

Figure 8.3 Structural Equation Model Diagram with Variables

Draw Paths F 121

Draw Paths

To draw paths from f_beta to the self-esteem variables (y1–y5):

1 Rest the cursor on f_beta. A small palette appears that contains a single-headed and a double-headedarrow.


3 Drag the cursor toward y1. The black outline of y1 turns bold, indicating that the variable is a valid targetfor this path.

4 Release the mouse button. The single-headed path from f_beta to y1 appears.

5 Repeat these steps for the paths from f_beta to the other self-esteem variables (y2–y5). Figure 8.4 showsthe resulting diagram.


Figure 8.4 Latent Growth Curve Model Diagram with Measurement PathsSpecified

Draw a Covariance Path

1 Rest the cursor on f_alpha. A small palette appears that contains a single-headed and a double-headedarrow.


3 Drag the cursor toward f_beta. The black outline of f_beta turns bold, indicating that the variable is avalid target for this path.

4 Release the mouse button. The double-headed path from f_alpha to f_beta appears. Figure 8.5 shows thediagram with all paths specified.

Specify Mean Structure Analysis F 123

Figure 8.5 Latent Growth Curve Model Diagram with All Paths Specified

Specify Mean Structure Analysis


2 In the Analyze area, select Mean Structures. Figure 8.6 shows the Methods tab specifications for thisexample.


Figure 8.6 Methods Tab Specifications

Specify Parameter Names

To make it easier to locate the parameters of interest in the results, you can specify the names of the param-eters.

Specify Mean Names

By default, latent variable means are estimated to be 0. For this example, you are interested in estimatingthe latent variable means. Specifying a name for a latent variable mean indicates that the mean should befreely estimated. To specify the mean names for f_alpha and f_beta:

1 In the Diagram area, right-click the f_alpha variable.

2 Select Set variable properties. The Latent Variable Properties window appears.


4 Select Free.

5 In the Name box, type m_alpha.

6 Click OK.

7 Repeat these steps for f_beta, except type m_beta in the Name box. Figure 8.7 shows the diagram withthe names of the latent variable means above f_alpha and f_beta.

Specify Parameter Names F 125

Figure 8.7 Latent Growth Curve Model Diagram with Mean Names

Specify Variance Names

To specify the variance names for f_alpha and f_beta:

1 In the Diagram area, right-click the f_alpha variable.

2 Select Set variable properties. The Latent Variable Properties window appears.


4 Select Free.

5 In the Name box, type v_alpha.

6 Click OK.

7 Repeat these steps for f_beta, except type v_beta in the Name box. Figure 8.8 shows the diagram withthe names of the latent variable means and variances above f_alpha and f_beta.


Figure 8.8 Latent Growth Curve Model Diagram with Mean and VarianceNames


For this model, you want to see the error variables in the model. To show the error variables:


2 Select Show error variables.

To position the error variables below the observed variables:

1 Select more than one variable at a time by holding down the CTRL key and clicking each observedvariable.

2 After all of the variables have been selected, right-click one of the variables and select Reposition errorvariables for selected variables.

Specify the Parameter Constraints F 127

3 A window appears and asks where you want to position the error variables. Select Below, and then clickOK. Figure 8.9 shows the diagram with the error variables.

Figure 8.9 Latent Growth Curve Model Diagram with Error Variables Show-ing

Specify the Parameter Constraints

To specify the growth structure of the model, the paths from the latent intercept variable (f_alpha) and latentslope variable (f_beta) to the observed variables (y1–y5) need to be constrained to have a specific structure,in this example a linear growth structure. In addition to the latent growth constraints, you also want toconstrain the intercepts for the observed variables to be 0 and constrain the error variances to be equal overtime.


Constrain Path Parameters to 1

To specify the latent intercept variable constraints:

1 Right-click the path from f_alpha to y1.

2 Select Set path coefficient to 1. A 1 appears by the path.

3 Select the same option for the paths from f_alpha to the other self-esteem variables (y2–y5).

NOTE: You can select more than one path at a time by holding down the SHIFT key and clicking each path.After all of the paths have been selected, right-click on one of the paths and select Set path coefficient to 1.

Constrain Parameters to a Fixed Value

To specify the linear latent slope variable constraints:

1 Right-click the path from f_beta to y1.

2 Select Set path properties. A Path window appears.

3 Select Fixed, and type 0 in the Value box.

4 Click OK.

5 Repeat these steps for the paths from f_beta to the other self-esteem variables (y2–y5), except for eachyear away from the initial measure (y1) increase by 1 the value you type in Step 3. For example, the pathfrom f_beta to y2 is 1, and the path from f_beta to y3 is 2.

Constrain Intercept Parameters to 0

To constrain the observed variable intercepts to be 0:

1 Right-click the variable y1.

2 Select Set mean/intercept to 0. A 0 appears above the variable.

3 Select the same option for the other self-esteem variables (y2–y5).

NOTE: You can select more than one variable at a time by holding down the CTRL key and clickingeach variable. After all of the variables have been selected, right-click one of the variables and select Setmean/intercept to 0.

Specify the Parameter Constraints F 129

Constrain Parameters to Be Equal

To constrain the observed error variables to be equal:

1 Right-click the error variable for y1.

2 Select Set error properties. The Error Properties window appears.

3 Click Variance.

4 Select Free, and type e1 in the Name box.

5 Click OK.

6 Repeat these steps for each of the error variables (y2–y5).

Figure 8.10 shows the diagram with all of the path constraints.


Figure 8.10 Latent Growth Curve Model Diagram with Path Constraints




View LGCM Results


View LGCM Results in the Diagram

The results of the model appear in the Diagram area with parameter estimates near the paths and thevariables. Figure 8.11 shows the unstandardized parameter estimates when Unstd. Estimates is selectedfrom the View list.


Figure 8.11 Latent Growth Curve Model Diagram with Unstandardized Results

In the Diagram area, the estimates of the means appear above each variable, and the variances appear aboveeach latent variable and below the error variables. Each estimated path coefficient is displayed near the pathfrom one variable to another. Any parameter estimates that differ significantly from 0 (based on t tests) aremarked with asterisks. Two asterisks indicate that p < 0.01; one asterisk indicates that p < 0.05.









View Detailed LGCM Results


The output in the Results tab is organized such that the modeling specifications and model fit are presentedfirst, followed by the parameter estimates for the model. Figure 8.12, Figure 8.13, and Figure 8.14 show theunstandardized results for this example.


Figure 8.12 Latent Growth Curve Model Results Tab with Modeling Specifica-tions and Fit Results

View Detailed LGCM Results F 135

Figure 8.13 Latent Growth Curve Model Results Tab with Maximum Likeli-hood Parameter Estimate Results


Figure 8.14 Latent Growth Curve Model Results Tab with Maximum Likeli-hood Parameter Estimate Results Continued

Summary of LGCM Results

Overall, the latent growth curve model has poor model fit according to several indices of model fit. However,examining the parameter estimates might still be useful. The variance estimates for the latent intercept(13.891) and equal error variances (3.322) are both statistically significant. However, the variance estimatefor the latent slope (0.807) is not statistically significant. These results indicate that there is a significantamount of individual variability in the intercept for self-esteem and significant variability in the observedmeasure of self-esteem that is not explained by the linear growth model. The covariance between thelatent intercept and slope variables (–0.353) is not statistically significant, indicating that there is not asignificant relationship between an individual’s initial level of self-esteem and their linear growth in self-esteem. Finally, the estimates for the mean latent intercept (14.159) and mean latent slope (4.048) aresignificant and imply that on average individuals had an initial level of self-esteem of 14.16 and self-esteemincreased by 4.05 for every additional year that passed.

Alternative LGCM F 137

Alternative LGCM

Because the LGCM did not fit well, you might want to try an alternative LGCM. One alternative would beto modify the model by relaxing the assumption of equal error variance over time. To do this, first copy themodel, and then remove the error variance constraints.

Copy the Model

Rather than creating a new model, you can copy your LGCM:

1 Click Copy in the Analyses area.

A new analysis, Analysis 2, appears in the Analyses list, and a copy of the diagram from the previous modelappears in the Diagram area.

Remove Error Constraints

To specify unique error variances at each time:

1 Right-click the error variable for y2.

2 Select Set error properties. An Error Properties window appears.


4 Select Free, and type e2 in the Name box.

5 Click OK.

6 Repeat these steps for each of the error variables (y3–y5), specifying a unique name for each error vari-ance.

Figure 8.15 shows the diagram with time-specific error variances, instead of equal error variances.

NOTE: Instead of specifying unique parameter names, you can leave the parameter name blank for the errorvariances. Then the program generates unique parameter names automatically.


Figure 8.15 Alternative Latent Growth Curve Model Diagram with Time-Specific Error Variances




View Alternative LGCM Results

View Alternative LGCM Results in the Diagram

The results of the model appear in the diagram with parameter estimates near the paths and the variables.Figure 8.16 shows the unstandardized parameter estimates when Unstd. Estimates is selected from theView list.


Figure 8.16 Alternative Latent Growth Curve Model Diagram with Unstan-dardized Results

In the Diagram area, the estimates of the means appear above each variable, and the variances appear aboveeach latent variable and below the error variables. Each estimated path coefficient is displayed near the pathfrom one variable to another. Any parameter estimates that differ significantly from 0 (based on t tests) aremarked with asterisks. Two asterisks indicate that p < 0.01; one asterisk indicates that p < 0.05.









View Detailed Alternative LGCM Results


The output in the Results tab is organized such that the modeling specifications and model fit are presentedfirst, followed by the parameter estimates for the model. Figure 8.17, Figure 8.18, and Figure 8.19 show theunstandardized results for this example.


Figure 8.17 Alternative Latent Growth Curve Model Results Tab with Model-ing Specifications and Fit Results

View Detailed Alternative LGCM Results F 143

Figure 8.18 Alternative Latent Growth Curve Model Results Tab with Maxi-mum Likelihood Parameter Estimate Results


Figure 8.19 Alternative Latent Growth Curve Model Results Tab with Maxi-mum Likelihood Parameter Estimate Results Continued

Summary of Alternative LGCM Results

Overall, the alternative latent growth curve model with time-specific error variances has good model fitaccording to several indices of model fit. The variance estimates for the latent intercept (14.701) and someof the error variances (e1=2.817, e3=1.944, e5=14.652) are statistically significant. However, the varianceestimate for the latent slope (0.451) and some of the error variances (e2=0.322 and e4=1.886) are notstatistically significant. These results indicate that there is a significant amount of individual variabilityin the intercept for self-esteem and significant variability in some of the observed measure of self-esteemthat is not explained by the linear growth model. The covariance between the latent intercept and slopevariables (0.353) is not statistically significant, indicating that there is not a significant relationship betweenan individual’s initial level of self-esteem and their linear growth in self-esteem. Finally, the estimates forthe mean latent intercept (14.030) and mean latent slope (3.968) are significant and imply that on averageindividuals had an initial level of self-esteem of 14.03 and self-esteem increased by 3.97 for every additionalyear that passed.

Compare the Models

Now that you have fit two models to the data, you want to compare the model fit for each model to decidewhich model best represents the data. To compare the LGCM with equal error variances to the LGCM withtime-specific error variances:


2 In the Show area, select User-selected fit statistics, and then click Customize. The Fit Indices windowappears.


3 In the Fit Indices window, clear the fit indices that you do not want to compare. For this example, clickClear All to clear all of the fit indices.

4 Now select the fit indices you want to compare. For this example, select Bentler Comparative Fit Index,RMSEA Estimate, Adjusted GFI (AGFI), Chi-Square, Chi-square DF, and Pr > Chi-Square.

5 Select OK to close the Fit Statistics window.

6 In the Sort area, select Chi-Square from the Sort By list, and then click the to sort in descendingorder.

Figure 8.20 shows the Comparisons tab after you have customized the fit indices.

Figure 8.20 Comparisons Tab for the Results of the Latent Growth CurveModels

NOTE: By default, the analyses are listed in the order in which they were created, and the parsimony fitindices Akaike Information Criterion, Bozdogan CAIC, Schwarz Bayesian Criterion, and RMSEA areshown.


The fit statistics in the model comparison table indicate that the LGCM with time-specific error variances hasa better model fit than the LGCM with equal error variances according to several fit indices. For example,the RMSEA value for the LGCM with time-specific error variances (0.104) is lower than the RMSEA valuefor the LGCM with equal error variances (0.288).


ReferencesMeredith, W. and Tisak, J. (1990), “Latent Curve Analysis,” Psychometrika, 55, 107–122.

Chapter 9

Single Group Analysis Window

ContentsIntroduction to the Single Group Analysis Window . . . . . . . . . . . . . . . . . . . . . . 148Project Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Create a New Project File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Select a Data Table to Analyze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Open a Project File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Save a Single Group Analysis Project File . . . . . . . . . . . . . . . . . . . . . . . 151Print Active Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Data Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Specify the Data Table Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Select the Data Set for Analysis . . . . . . . . . . . . . . . . . . . . . . . . 152Specify the Type of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Specify the Number of Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Specify Frequency Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Analyses Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Perform Analysis Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Analyses Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Select an Analysis for Viewing . . . . . . . . . . . . . . . . . . . . . . . . . 156Create a New Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Copy an Analysis Diagram into a New Diagram Window . . . . . . . . . . . 157Delete an Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Palette Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Define Latent Variables with the Palette Menu . . . . . . . . . . . . . . . . . 158Add Variables from the Palette Pane to the Diagram . . . . . . . . . . . . . . 160

Diagram Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Specify a Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Diagram Area Pop-Up Menu . . . . . . . . . . . . . . . . . . . . . . . . . . 162Variable Pop-Up Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Path Pop-Up Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Covariance Path Pop-Up Menu . . . . . . . . . . . . . . . . . . . . . . . . . 165Modify the Diagram Appearance . . . . . . . . . . . . . . . . . . . . . . . . 165Manage Model Library Files . . . . . . . . . . . . . . . . . . . . . . . . . . 165

General Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166Methods Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Specify Data Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

148 F Chapter 9: Single Group Analysis Window

Specify Estimation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Specify Optimization Method . . . . . . . . . . . . . . . . . . . . . . . . . 168Specify Maximum Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Results Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170SAS Log Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171SAS Code Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Comparisons Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Modify the Comparisons Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Specify Fit Indices Shown . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Specify Order of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Print the Comparisons Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Introduction to the Single Group Analysis Window

You can use the Single Group Analysis window to build, modify, analyze, and compare multiple structuralequation models that are fit to the same data set. To open this window, select Analyze I StructuralEquation Modeling I Single Group Analysis from the JMP Home window. The Structural EquationModels for a Single Group window appears. See Figure 9.1.

Project Area F 149

Figure 9.1 Structural Equation Models for a Single Group Window

Project Area

The Project area always appears in the upper left corner of the Structural Equation Models for a SingleGroup window. The Project area contains the tools for managing your Structural Equation Model (SEM)project files as described in the following sections.


Create a New Project File

Click New to start a new analysis either with a currently open data set or with a different data set:

� If you have not started a project and you want to switch data sets, the Select a Data Table to Analyzewindow appears.

� If you have started working on a project, a window appears and asks whether you want to save yourcurrent project.

– Click Yes to save the project you were working on as a structural equation model (SEM) projectfile. The Select a Data Table to Analyze window appears. For more information about projectfiles, see the section “Save a Single Group Analysis Project File” on page 151.

– Click No to delete the project you were working on. The Select a Data Table to Analyze windowappears.

– Click Cancel to return to the project you were working on.

Select a Data Table to Analyze

The Select a Data Table to Analyze window enables you to either select an open JMP data table or name orbrowse for a data set that is not already open. Figure 9.2 shows the Select a Data Table to Analyze window.

Figure 9.2 Select a Data Table to Analyze Window

� To select an open data table for analysis, select the JMP tables option, and then select the table youwant from the list.

� To select an unopened data set for analysis, select the Data set option, and then either type the locationand name of a data set or click Browse to search for a data set.

Open a Project File F 151

Open a Project File

Click Open in the Project area to open a previously saved SEM project file. If the previously saved SEMproject includes results, SAS log, and SAS code, this information is also opened. Also, if the previous SEMproject included multiple analyses, all of the analyses are opened.

Save a Single Group Analysis Project File

In the Project area, click Save or Save As to save a SEM project file. If the SEM project has not been run,only the information on the Diagram, General, and Methods tabs are saved. If the SEM project has beenrun, the Diagram, General, Methods, Results, SAS log, and SAS code tabs are all saved. Also, if the SEMproject includes multiple analyses, all of the analyses are saved.

Print Active Tab

To print a copy of the tab that is currently displayed:

1 Click Print in the Project area. The Print button is available when any of the Diagram, Results, SASLog, or SAS Code tabs is active on the Analyses tab. For more information, see the section “Analy-ses Tab” on page 154. The Print button is also available if the Comparisons tab is active. For moreinformation, see the section “Comparisons Tab” on page 171.

The Print window appears with a list of printers and printing options.

2 Select the appropriate printer.

3 Select printing options.

4 Click Print.

Data Tab

You specify information about the data set that you are analyzing on the Data tab. Figure 9.3 shows anexample Data tab.


Figure 9.3 Data Tab

Specify the Data Table Properties

Select the Data Set for Analysis

The name of the data set to be analyzed appears to the right of Name. SAS Structural Equation Modelingfor JMP automatically analyzes the most recently opened data set. If the data set name next to Name is notwhat you want to analyze, click New in the Project area to select another data set.

Specify the Type of Data

SAS Structural Equation Modeling for JMP automatically selects the type of data, either Raw Data orCovariances, Correlations. If the type is not correct, you can select the appropriate type from the DataStructure list. Alternatively, you can click Check. Then SAS Structural Equation Modeling for JMPexamines the format of the data set to determine the type of the data set.

Specify the Number of Observation F 153

Specify the Number of Observation

You can specify the number of observations in the data set in the Number of Observations area.

� If you are using correlation or covariance data for your analyses, you might need to specify the numberof observations.

� If you are using raw data, SAS Structural Equation Modeling for JMP automatically uses the numberof rows in the data set as the number of observations.

� If you are using raw data and you want to use a number other than the number of rows as the numberof observations, select the Use instead of the actual number of observations check box and thentype the number of observations you want in the Number box.

Specify Frequency Data

If Raw Data is selected from the Data Structure list, you can specify one variable in your data set torepresent the frequency of occurrence for the other values in the observation:

1 Select the variable name from the Select Columns list.

2 In the Assign Roles area, click Frequency. The variable name appears in the Frequency box.

SAS Structural Equation Modeling for JMP then treats the data set as if each observation appeared thenumber of times specified by the value of the frequency variable for that observation. Only the integerportion of the value is used. If the value of the frequency variable is less than 1 or is missing, thatobservation is not included in the analysis. The total number of observations is considered to be the sumof the frequency values. By default, if a variable in the JMP data table has been specified to have the Freqpreselected role, that variable is displayed in the Frequency box.

To remove a frequency variable:

1 From the Select Columns list, select the variable that matches the name in the Frequency box.

2 Click Remove. The variable name is deleted from the Frequency box. All observations for that variableare treated as if they were observed only one time.

NOTE: The Select Columns and Assign Roles areas are available only if Raw data is specified in the Datastructure box.


Analyses Tab

You can create, modify, and copy path diagrams, specify the analysis, run the analysis, and view the resultson the Analyses tab before an analysis is run. Figure 9.4 shows the Analyses tab. After the analysis is run,the Analyses tab contains additional tabs, as shown in Figure 9.5.

Figure 9.4 Analyses Tab

Perform Analysis Area F 155

Figure 9.5 Analyses Tab with Additional Tabs

Perform Analysis Area

The Perform Analysis area is located in the upper left corner of the Analyses tab and contains only theRun button. Click Run to run the model you specify in the Diagram, General, and Methods tabs.

Analyses Pane

When you click the Analyses button below the Perform Analysis area, the Analyses pane appears. TheAnalyses pane contains a list of the analyses in addition to options for creating new, modifying old, ordeleting analyses. Figure 9.6 shows the Analyses pane.


Figure 9.6 Analyses Pane

Select an Analysis for Viewing

The analysis currently open for viewing is highlighted in dark blue in the Analyses list. In Figure 9.6, Anal-ysis 1 is highlighted, so the Diagram, General, and Methods tabs contain the information from Analysis1. If Analysis 1 has been run, the Results, SAS Log, and SAS Code tabs also contain information fromAnalysis 1. To view a different analysis, select a different analysis name from the list.

Create a New Analysis

Click New to create a new analysis in the Analyses list. The analysis includes new Diagram, General, andMethods tabs. By default, the program names the new analysis Analysis n, where n is the number of theanalyses that have previously been created plus one.

Palette Pane F 157

Copy an Analysis Diagram into a New Diagram Window

Click Copy to create a new analysis by copying and modifying an analysis that is in the Analyses list.The new analysis includes copies of the Diagram, General, and Methods tabs of the copied analysis. Bydefault, the program names the new analysis Analysis n, where n is the number of the analyses that havepreviously been created plus one.

Delete an Analysis

To delete an analysis, select it from the Analyses list and click Delete.

Palette Pane

When you click the Palette button below the Perform Analysis area, the Palette pane appears. See Fig-ure 9.7. The Palette pane contains shapes that represent latent (circle or oval) and observed (square orrectangle) variables, and a list of observed or latent variables:

� When the Observed option is selected in the Type area, a list of the observed variables from the dataset appears.

� When the Latent option is selected in the Type area, a list of the latent variables which you definedappears.


Figure 9.7 Palette Pane

Define Latent Variables with the Palette Menu

In the Single Group Analysis window, only names for latent variables can be defined. Names for observedvariables come from the data set being analyzed.

1 Select Latent in the Type area of the Palette pane.

2 Click Define. The Define window appears. See Figure 9.8.

Palette Pane F 159

Figure 9.8 Define Variables Window for the Single Group Analysis

3 Create a list of latent variables in one of the following ways:

� Select Enter names, and type a list of names separated by commas, spaces, or new lines.

� Select Expand names, and type a prefix for the latent variable names in the Prefix box, a numberto start with in the Start box, and a number to end with in the End box.

� Select Copy names of numeric columns from a data set, and do one of the following:

– Type a data set name in the Data set box.– Click Browse to search for a data set.– If the data set is an open JMP table, select the data set from the Choose an open JMP table

list.

4 After you specify the names of the variables, click Add to move the list of the newly defined variables tothe Variables area on the right side of the Define window.

If you want to remove any variables from the list in the Variables area, click Clear to delete all variables,or click Remove to delete the selected variables in the list.


5 When you have finished creating your list of latent variables, click OK to add the new variables to theVariables list in the Palette area, or click Cancel to prevent the list in the Define window from beingadded to the Variables list in the Palette pane.

NOTE: The Variables list in the Palette pane is displayed only if Latent is selected in the Type area.

Add Variables from the Palette Pane to the Diagram

You can add either an unlabeled variable or a labeled variable to the Diagram area on the Analyses tab:

1 Click Palette to show the Palette pane, which contains the shapes for latent and observed variables.

2 Click the Diagram tab.

3 Do one of the following:

� To add an unlabeled variable, drag a shape from the Palette pane to the desired location in theDiagram area.

� To add a labeled variable, drag a variable from the Variables list to the desired location in theDiagram area.NOTE: If you drag more than one labeled variable into the Diagram area, a window appears andasks whether the variables should be arranged in a row or column.

Diagram Tab

You can build and modify a path diagram associated with a specific analysis on the Diagram tab. You canalso adjust some aspects of the diagram appearance and save or open a Model Library file. Figure 9.9 showsan example of the Diagram tab.

Diagram Tab F 161

Figure 9.9 Diagram Tab

Specify a Path

After you have added variables to the Diagram tab, you can draw a path from one variable to anothervariable:

1 Rest the cursor on one variable. A small palette appears that contains a single-headed and a double-headedarrow. Figure 9.10 shows an example of this palette.

2 Select the single-headed arrow to represent a unidirectional effect (or select the double-headed arrow torepresent a covariance or correlation). The arrow turns red in the palette when selected; see Figure 9.10.

3 Drag the cursor toward another variable. The black outline of the other variable turns bold, indicatingthat the variable is a valid target for this path.

4 Release the mouse. The single-headed (or double-headed) path from one variable to another variable iscreated.


Figure 9.10 Arrow Palette Example

Diagram Area Pop-Up Menu

You can right-click on the white background of the Diagram area and select any of the following commands:

� Add latent adds a latent variable to the Diagram area.

� Add observed adds an observed variable to the Diagram area.

� Selecting Show error variables causes error variables to appear next to all of the endogenous vari-ables in the Diagram area.

NOTE: If a covariance path (a double-headed arrow path) has been specified between two endogenousvariables, selecting Show error variables moves the covariance path from the variables to the errorvariables.

� Deselecting Show error variables causes error variables not to be displayed next to all of the en-dogenous variables in the Diagram area.

Diagram Tab F 163

� Selecting Show variance paths causes circular double-headed arrow paths, which represent vari-ances, to appear attached to the variables in the Diagram area.

NOTE: If Show error variables is selected when you select Show variance paths, the variance pathsfor the endogenous variables in the Diagram area appear attached to the error variables.

� Deselecting Show variance paths causes variance paths not to be displayed for all of the variables inthe Diagram area.

� Selecting Show default covariances causes gray double-headed arrow paths, which represent covari-ances or correlations, to appear between each pair of exogenous variables in the Diagram area.

� Deselecting Show default covariances causes the covariance paths that connect all pairs of exoge-nous variables not to be displayed in the Diagram area.

NOTE: Even when Show default covariances is not selected, covariances are estimated by defaultbetween all pairs of exogenous variables. Prior to running any model, you should select Show defaultcovariances to verify that the covariance paths that are estimated by default are correctly specifiedfor your model.

� Add area adds area in the diagram, providing space to add more variables.

� Remove excess area removes blank area from the periphery of the diagram.

� Copy the diagram to the clipboard copies the diagram to the clipboard. The copy of the diagramincludes any white space that has been added to the Diagram area, and the size of the variables isthe same as they were in the copied diagram (unless the program to which you copy the diagramautomatically resizes images to fit in the available area).

� Clear contents clears the contents of the Diagram area. All paths and variables are removed fromthe Diagram area.

Variable Pop-Up Menu

You can right-click a variable in the Diagram area and select any of the following commands:

� Set variable properties opens the Observed Variable Properties window or the Latent Variable Prop-erties window. See the section “Variable Properties Window” on page 197.

� Set variance coefficient to 1 fixes the variance parameter for the selected variable to be 1.

NOTE: Set variance coefficient to 1 does not appear in the variable pop-up menu if one of thefollowing occurs:

– If an error variable is attached to a variable, right-click on the error variable and select Set errorproperties to specify the variance properties.

– If a variance path is attached to a variable, right-click on the variance path and select Set vari-ance to 1 to fix the variance parameter for the variable to be 1.

� Set mean/intercept to 0 fixes the mean or intercept parameter for the selected variable to be 0.

� Set color specifies a fill color for the variable.


� Increase variable size increases the size of the selected variable.

� Decrease variable size decreases the size of the selected variable.

� Set as default variable size sets the size of the selected variable to be the default variable size.

� Apply default variable size to all variables changes the size of all of the variables in the Diagramarea to be the default variable size.

� Restore original default variable size to all variables changes the size of all of the variables in theDiagram area to be the original default variable size.

� Arrange selected variables horizontally moves two or more selected variables into a straight hor-izontal line of variables. (To select two or more variables, hold down the CTRL key and click eachvariable in the Diagram area.)

� Arrange selected variables vertically moves two or more selected variables into a straight verticalline of variables. (To select two or more variables, hold down the CTRL key and click each variablein the Diagram area.)

� Reposition error variables for selected variables opens the Arrange Error Variables window. SelectAbove, To the left, To the right or Below for the position of the error variable, and then click OK.The error variable (or set of error variables) in the diagram moves to the selected position.

� Select selects a variable from the diagram.

� Deselect deselects a variable from the diagram.

� Delete deletes a variable from the diagram. Any paths and error variable associated with the variableare also removed from the Diagram area.

Path Pop-Up Menu

You can right-click a path in the Diagram area and select any of the following commands:

� Set path properties opens the Path Properties window. See the section “Path Properties Window” onpage 202.

� Set path coefficient to 1 sets the path coefficient to 1 for the selected path.

� Reverse path reverses the direction of the arrow of the selected path.

� Set color specifies a color for the path.

� Select selects a path from the diagram area.

� Deselect deselects a path from the diagram area.

� Delete deletes a path from the diagram area.

Diagram Tab F 165

Covariance Path Pop-Up Menu

You can right-click on a covariance path in the Diagram area and select any of the following commands:

� Set covariance properties opens the Covariance Properties window. See the section “CovarianceProperties Window” on page 203.

� Set covariance to 0 fixes the covariance to be 0.

� Select selects the covariance path.

� Deselect deselects the covariance path.

� Delete deletes the covariance path. NOTE: The Delete option is not available for default covariances.

Modify the Diagram Appearance

You can modify the diagram appearance in any of the following ways:

� To adjust the diagram zoom, select the blue diamond in the Zoom area.

– To zoom in, slide the blue diamond to the right.

– To zoom out, slide the blue diamond to the left.

� To modify the diagram, select one of the following from the Diagram list in the Actions area:

– Select Add drawing area to add area to the diagram.

– Select Remove excess drawing area to remove excess area from the diagram.

– Select Clear the diagram to remove all paths and variables from the diagram.

� To view the input parameters or parameter estimates, select one of the following from the View list inthe Actions area:

– Select Input to view the input model in the Diagram area.

– Select Unstd. Estimates to view the unstandardized parameter estimate results in the Diagramarea.NOTE: Unstandardized estimates are available only after the model has been run.

– Select Std. Estimates to view the standardized parameter estimate results in the Diagram area.NOTE: Standardized estimates are available only after the model has been run.

Manage Model Library Files

You can manage Model Library files by selecting one of the following from the Model Library list in theActions area:

� Select Get a model to open a Model Library file. The Open window appears. Select the ModelLibrary file you want, and then select Open.


� Select Save the model to save a path diagram as a Model Library file. The Save window ap-pears. Specify a name for the file and location where you want to save the file, and then click Save.NOTE: This action saves a file with the diagram only. The diagram can be used either in the ModelLibrary window or in the Single Group Analysis window.

General Tab

You can specify a name and a description for an analysis on the General tab. Figure 9.11 shows an exampleGeneral tab.

Figure 9.11 General Tab

To specify a description of the model:


2 In the Label area, type a title for the model.

3 In the Notes area, type a description of the model.

Methods Tab F 167

Methods Tab

You can specify methods options for an analysis on the Methods tab. Figure 9.12 shows an exampleMethods tab.

Figure 9.12 Methods Tab

Specify Data Type

To modify the data type for model analysis:

1 Select Correlations or Covariances as the type for the data you want to analyze.

2 Select the Mean Structures check box to specify that Mean Structure be included in the analysis.

By default, Covariances is selected in the Analyze area and Mean Structures is not selected.

Specify Estimation Method

In the Estimation area, select the method you want from the Method list. Figure 9.13 shows all of theavailable estimation methods. By default, Maximum likelihood is selected.


Figure 9.13 Estimation Methods

Specify Optimization Method

In the Optimization area, select the method you want from the Method list. Figure 9.14 shows all of theavailable the optimization methods.

Methods Tab F 169

Figure 9.14 Optimization Methods

By default, the Default method of optimization is used. The Default method selects a method based onestimation type and the number of parameters included in the analysis as follows:

� For models with fewer than 500 parameters and FIML estimation, Newton-Raphson is the defaultoptimization technique.

� For models with fewer than 500 parameters and non-FIML estimation, Levenberg-Marquardt is thedefault method.

� For models with 500 to 1,000 parameters, quasi-Newton is the default method.

� For models with more than 1,000 parameters, conjugate-gradient is the default method.

Specify Maximum Iterations

By default, the maximum number of iterations depends on which optimization method is used:

� If Levenberg-Marquardt or Newton-Raphson optimization is used, then 50 is the default maximumnumber of iterations.

� If quasi-Newton optimization is used, then 200 is the default maximum number of iterations.

� If conjugate-gradient optimization is used, then 400 is the default maximum number of iterations.


For more information about the available estimation and optimization methods, see Chapter 26, “The CALISProcedure” (SAS/STAT User’s Guide).

Results Tab

The Results tab appears only after a model has been run. By default, the Results tab includes Model-ing Information, Fit Summary, and parameter estimates (in the ML Estimation section). The ModelingInformation area contains information about the data set that was analyzed and the type of analysis (cor-relations, covariances, or means and covariances). The Fit Summary section contains the fit indices thatare selected in the User Profile. For information about the default fit indices, see the section “Specify FitIndices Preferences” on page 189.

More results might be available for a model. To view additional results output, click the red triangle nextto the Results title. A list of optional output appears as shown in Figure 9.15. Select one of the followingresults options:

� Select View as html to open a new window with the output shown in the Results tab in HTML format.

� Select one of the following to add information to the Results tab:

– Select Covariances of latent variables to add a table with the parameter estimates for thecovariances of the latent variables.

– Select Covariances between latent and manifest variables to add a table with the parameterestimates for the covariances between latent and manifest variables.

– Select Modification indices to add a table with modification indices.

– Select Optimization to add a table with the optimization information.

– Select Residual covariance/correlation matrix to add a table with the residual covari-ance/correlation matrix.

– Select Standardized results to add a table with the standardized parameter estimates.

� Select Manage additional tables to open the Manage Tables window, where you can select or de-select tables you want to view on the Add/Remove tab. You can also select the default tables to beshown in the Results tab using the Set Defaults tab. After you have made your selections in theManage Tables window, click OK to implement them or click Cancel to revert to your previoussettings.

SAS Log Tab F 171

Figure 9.15 List of the Results Output Options

SAS Log Tab

The SAS Log tab appears only after a model has been run. Select the SAS Log tab to view the SAS log forthe model specified in the Diagram, General, and Methods tabs.

SAS Code Tab

The SAS code tab appears only after a model has been run. Select the SAS Code tab to view a copy of SASsyntax that was used to fit the model in SAS.

Comparisons Tab

The Comparisons tab contains a table of fit indices from all of the analyzed models in the Analyses list,with options for which fit indices to view and for ordering the table. By default, the analyses are listedin the order in which they were created, and the following parsimony fit statistics are shown: Akaike’sinformation criterion (AIC), Bozdogan corrected AIC, Schwarz Bayesian criterion, and root mean squareerror of approximation (RMSEA). Figure 9.16 shows an example of a Comparisons tab.


Figure 9.16 Example Comparisons Tab

Modify the Comparisons Tab

Specify Fit Indices Shown

To specify which fit statistics appear in the table, select one of the following options in the Show area:

� Select Default parsimony statistics to show Akaike’s information criterion, Bozdogan CAIC,Schwarz Bayesian criterion, and RMSEA.

� Select User-selected fit statistics to show a set of user-specified fit statistics that are assigned in theuser profile window (see the section “Specify Fit Indices Preferences” on page 189).

� Select User-selected fit statistics, and then click Customize to open the Fit Statistics window, whichlists all the potential fit statistics to show. In the Fit Statistics window, you can select the fit statisticsthat you want to see and clear any fit statistics you do not want to see in the Comparisons tab.

Specify Order of Results

1 In the Sort area, select a fit statistic to use to sort the results.

2 Select either ascending or descending order.

Print the Comparisons Table F 173

Print the Comparisons Table

1 Click Print in the Project area.



4 Click Print to print a copy of the table on the Comparisons tab.

See Chapter 6, “Confirmatory Factor Analysis,” for an example of how to select a customized list of fitstatistics.

174

Chapter 10

Model Library Window

ContentsIntroduction to Model Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Define Observed Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Project Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Create a New Model Library File . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Open a Model Library File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Save a Model Library Project File . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Print Active Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178Variables Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Add Variables from the Variables Area to the Diagram . . . . . . . . . . . . 180Diagram Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Draw Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Diagram Area Pop-up Menu . . . . . . . . . . . . . . . . . . . . . . . . . . 183Variable Pop-Up Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Path Pop-Up Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185Covariance Path Pop-Up Menu . . . . . . . . . . . . . . . . . . . . . . . . . 185Modify the Diagram Appearance . . . . . . . . . . . . . . . . . . . . . . . . 185

General Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Introduction to Model Library

You can use the Model Library window to build a structural equation model diagram that can be saved foruse in multiple analyses with different data sets. To open this window, select Analyze I Structural Equa-tion Modeling I Model Library from the JMP Home window. The Model Library window appears, andif a data table is open, the Define Observed Variables window appears on top of the Model Library win-dow. Figure 10.1 shows the Model Library window, and Figure 10.2 shows the Define Observed Variableswindow.

176 F Chapter 10: Model Library Window

Figure 10.1 Model Library Window

Figure 10.2 Define Observed Variables Window

Define Observed Variables F 177

Define Observed Variables

In the Define Observed Variables window, you have three options:

� Click Yes to use the observed variables from the data set named in the Define Observed Variableswindow as the observed variables for the model defined in the Model Library window.

� Click No to open a Model Library window with no observed variables defined.

� Click Cancel to exit the Define Observed Variables window. No diagram will appear in the ModelLibrary window until a new Model Library window is opened by clicking New in the Project area ora previous Model Library file is opened by clicking Open in the Project area.

Project Area

The Project area always appears in the upper left corner of the Model Library window. The Project areacontains the tools for managing your Model Library files. Figure 10.1 shows the Project area in the ModelLibrary window.

Create a New Model Library File

Click New to open a new Model Library window and a new Define Observed Variables window.

Open a Model Library File

Click Open to open a previously saved Model Library file.

Save a Model Library Project File

Click Save to save a Model Library file. A Model Library file contains only a diagram. Click Save As tosave a Model Library file under a new name.


Print Active Tab

To print a copy of the tab that is currently active:

1 Click Print in the Project area. The Print window appears and shows a list of printers and printingoptions.



4 Click Print.

Variables Area

The Variables area contains shapes that represent latent (circle or oval) and observed (square or rectangle)variables, Type options, a list of variables, and the Define button.

You can select one of the following Type options:

� When you select Observed, the list below the Type area contains one of the following:

– If a data set has been specified, the list contains the observed variables from the data set.

– If a data set has not been specified, the list contains observed variables that you have defined.

� When you select Latent, the list contains latent variables that you have defined.

To define new observed or latent variables, click Define. The Define window appears. See Figure 10.3.

Variables Area F 179

Figure 10.3 Define Variables Window for the Model Library

Create a list of variables in one of the following ways:

� Select Enter names, and enter a list of names separated by commas, spaces, or new lines.

� Select Expand names, and type a prefix for the variable names in the Prefix box, a number to startwith in the Start box, and a number to end with in the End box.

� Select Copy names of numeric columns from a data set, and do one of the following:

– Type a data set name in the Data set box.

– Click Browse to search for a data set.

– If the data set is an open JMP table, select the data set from the Choose an open JMP table list.

After you specify the names of the variables, click Add to create the list of the newly defined variables inthe Variables area on the right side of the Define window.

You can click Clear to delete all variables from the variables list or click Remove to delete the selectedvariables in the list.

When you have finished creating your list of variables, click OK to add the new variables to the list ofvariables in the Variables area of the Model Library window.


If you decide that you do not want to add any newly defined variables to the list of variables in the ModelLibrary window, click Cancel.

Add Variables from the Variables Area to the Diagram

You can add either an unlabeled variable or a labeled variable to the Diagram area of the Model Librarywindow:

� To add an unlabeled variable to a diagram, drag a shape from the Variables area to the desired locationin the Diagram area.

� To add an labeled variable to a diagram, drag a variable from the Variables list to the desired locationin the Diagram area.

NOTE: If you drag more than one variable at a time into the Diagram area, a window appears and askswhether the variables should be arranged in a row or column.

Diagram Tab

You can build and modify a path diagram on the Diagram tab. You can also adjust some aspects of thediagram appearance. Figure 10.4 shows an example of the Diagram tab in the Model Library window.

Diagram Tab F 181

Figure 10.4 Diagram Tab in Model Library Window

Draw Paths

To draw a path from an independent variable to a dependent variable:

1 Rest the cursor on an independent variable. A small palette appears that contains a single-headed and adouble-headed arrow. Figure 10.5 shows an example of this palette. You use the double-headed arrowto represent covariances or correlations, and you use the single-headed arrow to represent unidirectionaleffects.


3 Drag the cursor toward a dependent variable. The black outline of the dependent variable turns bold,indicating that the variable is a valid target for this path.


4 Release the mouse button. The single-headed path from an independent variable to a dependent variableis created.

Figure 10.5 Arrow Palette in the Diagram Tab

NOTE: To specify a correlation or covariance path, repeat these steps but select the double-headed arrowinstead of a single-headed arrow.

Diagram Tab F 183

Diagram Area Pop-up Menu

You can right-click on the white background of the Diagram area and select one of the following commands:

� Add latent adds a latent variable to the Diagram area.

� Add observed adds an observed variable to the Diagram area.

� Selecting Show error variables causes error variables to appear next to all of the endogenous vari-ables in the Diagram area.

NOTE: If a covariance path (a double-headed arrow path) has been specified between two endogenousvariables, selecting Show error variables moves the covariance path from the variables to the errorvariables.

� Deselecting Show error variables causes error variables not to be displayed next to all of the en-dogenous variables in the Diagram area.

� Selecting Show variance paths causes circular double-headed arrow paths, which represent vari-ances, to appear attached to the variables in the Diagram area.

NOTE: If Show error variables is selected when you select Show variance paths, the variance pathsfor the endogenous variables in the Diagram area appear attached to the error variables.

� Deselecting Show variance paths causes variance paths not to be displayed for all of the variables inthe Diagram area.

� Selecting Show default covariances causes gray double-headed arrow paths, which represent covari-ances or correlations, to appear between each pair of exogenous variables in the Diagram area.

� Deselecting Show default covariances causes the covariance paths that connect all pairs of exoge-nous variables not to be displayed in the Diagram area.

NOTE: Even when Show default covariances is not selected, covariances are estimated by defaultbetween all pairs of exogenous variables. Prior to running any model, you should select Show defaultcovariances to verify that the covariance paths that are estimated by default are correctly specifiedfor your model.

� Add area adds area in the diagram, providing space to add more variables.

� Remove excess area removes blank area from the periphery of the diagram.

� Copy the diagram to the clipboard copies the diagram to the clipboard. The copy of the diagramincludes any white space that has been added to the Diagram area, and the size of the variables isthe same as they were in the copied diagram (unless the program to which you copy the diagramautomatically resizes images to fit in the available area).

� Clear contents clears the contents of the Diagram area. All paths and variables are removed fromthe Diagram area.


Variable Pop-Up Menu

You can right-click a variable in the Diagram area and select any of the following commands:

� Set variable properties opens the Observed Variable Properties window or Latent Variable Propertieswindow. See the section “Variable Properties Window” on page 197.

� Set variance coefficient to 1 fixes the variance parameter for the selected variable to be 1.

NOTE: Set variance coefficient to 1 does not appear in the variable pop-up menu if one of thefollowing occurs:

– If an error variable is attached to a variable, right-click on the error variable and select Set errorproperties to specify the variance properties.

– If a variance path is attached to a variable, right-click on the variance path and select Set vari-ance to 1 to fix the variance parameter for the variable to be 1.

� Set mean/intercept to 0 fixes the mean or intercept parameter for the selected variable to be 0.

� Set color specifies a fill color for the variable.

� Increase variable size increases the size of the selected variable.

� Decrease variable size decreases the size of the selected variable.

� Set as default variable size sets the size of the selected variable to be the default variable size.

� Apply default variable size to all variables changes the size of all of the variables in the Diagramarea to be the default variable size.

� Restore original default variable size to all variables changes the size of all of the variables in theDiagram area to be the original default variable size.

� Arrange selected variables horizontally moves two or more selected variables into a straight hor-izontal line of variables. (To select two or more variables, hold down the CTRL key and click eachvariable in the Diagram area.)

� Arrange selected variables vertically moves two or more selected variables into a straight verticalline of variables. (To select two or more variables, hold down the CTRL key and click each variablein the Diagram area.)

� Reposition error variables for selected variables opens the Arrange Error Variables window. SelectAbove, To the left, To the right or Below for the position of the error variable, and then click OK.The error variable (or set of error variables) in the diagram moves to the selected position.

� Select selects a variable from the diagram.

� Deselect deselects a variable from the diagram.

� Delete deletes a variable from the diagram. Any paths and error variable associated with the variableare also removed from the Diagram area.

Diagram Tab F 185

Path Pop-Up Menu

You can right-click a path in the Diagram area and select any of the following commands:

� Set path properties opens the Path Properties window. See the section “Path Properties Window” onpage 202.

� Set path coefficient to 1 sets the path coefficient to 1 for the selected path.

� Reverse path reverses the direction of the arrow of the selected path.

� Set color specifies a color for the path.

� Select selects a path from the diagram area.

� Deselect deselects a path from the diagram area.

� Delete deletes a path from the diagram area.

Covariance Path Pop-Up Menu

You can right-click on a covariance path in the Diagram area and select any of the following commands:

� Set covariance properties opens the Covariance Properties window. See the section “CovarianceProperties Window” on page 203.

� Set covariance to 0 fixes the covariance to be 0.

� Select selects the covariance path.

� Deselect deselects the covariance path.

� Delete deletes the covariance path. NOTE: The Delete option is not available for default covariances.

Modify the Diagram Appearance

You can modify the diagram appearance in any of the following ways:

� To adjust the diagram zoom, select the blue diamond in the Zoom area.

– To zoom in, slide the blue diamond to the right.

– To zoom out, slide the blue diamond to the left.

� To modify the diagram, select one of the following from the Diagram list in the Actions area:

– Select Add drawing area to add area to the diagram.

– Select Remove excess drawing area to remove excess area from the diagram.

– Select Clear the diagram to remove all paths and variables from the diagram.


General Tab

Select the General tab to specify a name and a description for a Model Library file. Figure 10.6 shows anexample General tab.

Figure 10.6 General Tab

To specify a description of the model:


2 In the Label area, type a title for the model.

3 In the Notes area, type a description of the model.

Chapter 11

User Profile Window

ContentsOverview of the User Profile Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Specify Diagram Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Specify Fit Indices Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Fit Indices on the Results Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Specify Method Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Data Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Estimation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Overview of the User Profile Window

You can customize your SAS Structural Equation Modeling for JMP experience by changing settings in theUser Profile window. Specifically, you can customize the fit indices displayed in the results, the appearanceof objects in the diagram, and the default method options. To open this window, select User Profile fromthe main Structural Equation Modeling menu. After you make changes in the User Profile window, clickOK to save your new settings or click Cancel to revert to your previous settings.

Specify Diagram Preferences

On the Diagram tab you can specify how you want variables and arrows to appear in diagrams. Figure 11.1shows the Diagram tab of the User Profile window.

188 F Chapter 11: User Profile Window

Figure 11.1 Diagram Tab of the User Profile Window

Specify Fit Indices Preferences F 189

In the Variable Shape area, you can select the shape you want a variable to have in the diagram. By default,Oval is selected as the Latent option and Rectangle is selected as the Observed option.

In the Arrow Direction area, you can select the direction of the arrows of paths that are automaticallycreated when you drop new variables onto a variable already in the diagram. By default, Target variable–> New variable is selected so that when you drop new variables onto a variable that is already in thediagram, arrows go from the variable already in the diagram to the newly added variables.

Specify Fit Indices Preferences

The Fit Indices tab contains a comprehensive list of the model fit indices available in SAS Structural Equa-tion Modeling for JMP. Figure 11.2 shows the Fit Indices tab of the User Profile window.


Figure 11.2 Fit Indices Tab of the User Profile Window

Fit Indices on the Results Tab F 191

Fit Indices on the Results Tab

By default, the following fit information is included on the Results tab:

� N observations is the number of observations included in the analysis

� adjusted goodness of fit (AGFI)

� parsimonious GFI

� RMSEA estimate

� RMSEA estimate lower 90% confidence limit

� RMSEA estimate upper 90% confidence limit

� probability of close fit

� Bentler comparative fit index

� chi-square

� chi-square DF

� Pr > chi-Square

� standardized root mean square residual (SRMSR)

For more information about the available fit indices, see Chapter 26, “The CALIS Procedure” (SAS/STATUser’s Guide).

You can modify the default fit indices on the Results Tab:

� To include a fit index in the Results tab, select its check box.

� To remove a fit index from the Results tab, clear its check box.

� To see all of the model fit information in the Results tab, click Select All.

� To clear all of the selected model fit information, click Clear All. NOTE: You must specify at leastone fit index to appear in the Results tab.

Specify Method Preferences

The Methods tab contains the options for the default methods of model analysis. Figure 11.3 shows theMethods tab of the User Profile window.


Figure 11.3 Methods Tab of the User Profile Window

Data Type

By default, Covarances is selected in the Analyze area and Mean Structures is not selected.

You can modify the default data type for model analysis:

� Select the Correlations or Covariances option to set the default data type you want to analyze.

� Select the box next to Mean Structures to specify that mean structure be analyzed by default.

Estimation Method F 193

Estimation Method

By default, Maximum likelihood is selected for the estimation method.

You can modify the default estimation method by selecting a different estimation from the Method list inthe Estimation area. Figure 11.4 shows the estimation options.

Figure 11.4 Estimation Options


Optimization Methods

By default, the Default method of optimization is selected. The Default method selects a method based onthe number of parameters included in the analysis as follows:

� For fewer than 500 parameters and FIML estimation, Newton-Raphson is the default optimizationtechnique.

� For fewer than 500 parameters and non-FIML estimation, Levenberg-Marquardt is the default opti-mization technique.

� For models with 500 to 1,000 parameters, quasi-Newton is the default method,

� For models with more than 1,000 parameters, conjugate-gradient is the default method.

By default, the maximum number of iterations depends on which optimization method is used:

� If Levenberg-Marquardt or Newton-Raphson optimization is used, then 50 is the default maximumnumber of iterations.

� If quasi-Newton optimization is used, then 200 is the default maximum number of iterations.

� If conjugate-gradient optimization is used, then 400 is the default maximum number of iterations.

You can modify the default optimization method:

1 In the Optimization area, select the default optimization technique from the Method list. Figure 11.5shows the optimization options.

2 In the Maximum iterations box, type the default maximum number of iterations.

Optimization Methods F 195

Figure 11.5 Optimization Options

For more information about the estimation and optimization options, see Chapter 26, “The CALIS Proce-dure” (SAS/STAT User’s Guide).

196

Chapter 12

Properties Windows

ContentsIntroduction to Properties Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Variable Properties Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Error Properties Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Path Properties Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202Covariance Properties Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203Variance Properties Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204Variance Windows and Tabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Introduction to Properties Windows

You can use Property Windows to specify the properties of paths and variables. You can access thesewindows by right-clicking on a path or variable in the Diagram area and selecting the appropriate command.

Variable Properties Window

To set variable properties, right-click on a variable in the Diagram area and select Set variable properties.The Variable Properties window appears. Figure 12.1 shows an Observed Variable Properties window. TheLatent Variable Properties window looks the same as Figure 12.1, except it has a different title.

198 F Chapter 12: Properties Windows

Figure 12.1 General Tab of the Variable Properties Window

The Variable Properties window has the following tabs:

� You can specify a name and label for a variable on the General tab. Type a name for the variable inthe Variable box, and type a label for the variable in the Label box. The label appears in the Diagramarea. Figure 12.1 shows the General tab of the Variable Properties window.

� You can specify the variance properties of a variable on the Variance tab. See Figure 12.2. Select oneof the options described in the section “Variance Windows and Tabs” on page 205.

Variable Properties Window F 199

Figure 12.2 Variance Tab of the Variable Properties Window

The Variance tab does not appear if Show error variables or Show variance paths is selected in thediagram properties pop-up menu. If either of these is selected, do one of the following to specify thevariance properties for a variable:

– If Show error variables is selected, right-click the error variable associated with a variable,select Set error properties, and click the Variance tab.

– If Show variance paths is selected, right-click the variance path associated with a variable andselect Set variance.

� You can specify the mean or intercept properties of a variable on the Mean/Intercept tab. See Fig-ure 12.3. Select Perform means analysis if you want to perform means analysis. After Performmeans analysis is selected, select one of the following options:

– Select Free to specify that the mean/intercept be freely estimated. If you select Free, thenyou can specify a name for the mean/intercept in the Name box and an initial value for themean/intercept in the Initial value box.

� If you want a unique parameter estimate for both the variance and mean/intercept, youmust specify a unique name for a parameter (or specify no name if you want the programto generate a name automatically).

� If you want the parameter to be fixed to be equal to another parameter, then specify thesame name for each parameter you want to have the same parameter estimate.

– Select Fixed to specify that the mean/intercept be fixed to a value. If you select Fixed, then youmust specify the fixed value for the mean/intercept in the Value box. NOTE: By default, latentvariable means are fixed to be 0.


Figure 12.3 Mean/Intercept Tab of the Variable Properties Window

Any specified name or value for a parameter appears above the variable as (Mean, Variance) except in thefollowing cases:

� If Show error variables is selected, the name or value appears above the error variable associatedwith the variable of interest.

� If Show variance paths is selected, the name or value appears next to the variance path for thevariable of interest.

Error Properties Window F 201

Error Properties Window

To set error variable properties, right-click on an error variable in the Diagram area and select Set errorproperties. The Error Properties window appears. See Figure 12.4.

Figure 12.4 General Tab of the Error Properties Window

The Error Properties window has two tabs:

� You can type a label for the variable in the Label box on the General tab. The label appears in theDiagram area. Figure 12.4 shows the General tab of the Error Properties window.

� You can specify the variance properties of a variable on the Variance tab. See Figure 12.5. Select oneof the options described in the section “Variance Windows and Tabs” on page 205.


Figure 12.5 Variance Tab of the Error Properties Window

NOTE: The Variance tab does not appear if Show variance paths is selected in the diagram prop-erties pop-up menu. If Show variance paths is selected, right-click on the variance path associatedwith a variable and select Set variance to specify the variance properties for that variable.

Path Properties Window

To set path properties, right-click on a path in the Diagram area and select Set path properties. The PathProperties window appears. See Figure 12.6.

Covariance Properties Window F 203

Figure 12.6 Path Properties Window

The Path Properties window has two options:

� Select Free to specify that the path be freely estimated. You can specify a name for the path in theName box and an initial value for the path in the Initial value box.

– If you want a unique path parameter estimate, you must specify a unique name for the pathparameter (or specify no name if you want the program to generate a name automatically).

– If you want the path parameter to be fixed to be equal to another parameter, then specify thesame name for each parameter you want to have the same parameter estimate.

� Select Fixed to specify that the path be fixed to a value. If you select Fixed, then you must specifythe fixed value for the path in the Value box.

Covariance Properties Window

To set covariance properties, right-click on a path in the Diagram area and select Set covariance properties.The Covariance Properties window appears. See Figure 12.7.


Figure 12.7 Covariance Properties Window

The Covariance Properties window has two options:

� Select Free to specify that the covariance be freely estimated. You can specify a name for the covari-ance in the Name box and an initial value for the covariance in the Initial value box.

– If you want a unique covariance parameter estimate, you must specify a unique name for thecovariance parameter (or specify no name if you want the program to generate a name automat-ically).

– If you want the covariance parameter to be fixed to be equal to another parameter, then specifythe same name for each parameter whose estimate you want to be the same.

� Select Fixed to specify that the covariance be fixed to a value. If you select Fixed, then you mustspecify the fixed value for the covariance in the Value box.

Variance Properties Window

To set variance properties, right-click on a variance path in the Diagram area and select Set varianceproperties. The Variance Properties window appears. See Figure 12.8. Select one of the options describedin the section “Variance Windows and Tabs” on page 205.

Variance Windows and Tabs F 205

Figure 12.8 Variance Properties Window

Variance Windows and Tabs

You can specify variance properties in the Variance Properties window or on the Variance tabs of the ErrorProperties, Observed Variable Properties, and Latent Variable Properties windows. Each of these windowsand tabs has a similar appearance and contains the same options:

� Select Free to specify that the variance be freely estimated. You can specify a name for the variancein the Name box and an initial value for the variance in the Initial value box.

– If you want a unique variance parameter estimate, you must specify a unique name for the vari-ance parameter (or specify no name if you want the program to generate a name automatically).

– If you want the variance parameter to be fixed to be equal to another parameter, then specify thesame name for each parameter you want to have the same parameter estimate.

� Select Fixed to specify that the variance be fixed to a value. If you select Fixed, then you must specifythe fixed value for the variance in the Value box.

206

Appendix A

Frequently Asked Questions

ContentsQuestions by Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207Detailed Questions and Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Questions by Category

Error Variables

What are error variables?

How do I control the display of error variables?

How do I set an error variance to 0?

Variance Paths

What are variance paths?

How do I control the display of variance paths?

How do I set a variance to 0?

Default Covariances

What are default covariances?

How do I control the display of the default covariances?

How do I set the default covariances to 0?

Building Models

How do I add variables to the diagram?

Why is there a message when I try to add more than 15 variables to the diagram when show default covari-ances is selected?

How do I add a path to represent unidirectional effects from one variable to another variable?

How do I add a path to represent correlation or covariance paths between variables?

208 F Appendix A: Frequently Asked Questions

How do I change models that have been run?

Miscellaneous Questions

What are the endogenous and exogenous variables in a model?

What parameters in the path diagram are freely estimated by default?

How do I constrain parameters in the path diagram to be equal?

How do I set preferences for the diagram?

Detailed Questions and Answers

What are error variables?

SAS Structural Equation Modeling for JMP creates an error variable for each endogenous variable. You canturn on an option to display all error variables. You cannot create error variables individually or turn themoff individually. Instead of turning off an individual error variable, you can set its variance to 0.

Back

How do I control the display of error variables?

1 Right-click on the diagram.

2 Select Show error variables. When a check mark is displayed next to this command, error variablesappear in the diagram; when a check mark is not displayed next to this command, error variables do notappear in the diagram.

Back

How do I set an error variance to 0?

1 Right-click on the error variable.

2 Select Set error properties. The Error Properties window appears.

3 Select the Variance tab.

4 Select Fixed, and enter 0 in the box.

5 Click OK. A 0 appears in the diagram above the error variable.

Back

Detailed Questions and Answers F 209

What are variance paths?

A variance path is a circular double-headed path that goes to and from the same variable. A variance pathis created for each variable that does not have an error variable pointing to it. You cannot create variancepaths individually. Instead you turn on an option to display variance paths. You cannot turn variance pathsoff individually, but you can set a variance path to be 0.

Back

How do I control the display of variance paths?


2 Select Show variance paths. When a check mark is displayed next to this command, variance pathsappear in the diagram; when a check mark is not displayed next to this command, variance paths do notappear in the diagram.

Back

How do I set a variance to 0?

1 Right-click on the variance path.

2 Select Set variance path. The Variance Properties window appears.

3 Select Fixed, and enter 0 in the box.

4 Click OK. A 0 appears in the diagram next to the variance path.

NOTE: Setting the variance of a non-error exogenous variables to 0 is not recommended because it resultsin a predicted covariance matrix that is not positive definite.

Back

What are default covariances?

A default covariance is a covariance between exogenous variables that is calculated even if you do notspecify a covariance path between two variables. In SAS Structural Equation Modeling for JMP, the defaultcovariances are light gray. If you want a default covariance to be 0 instead of an estimated parameter, youcan set it to be 0.

Back


How do I control the display of the default covariances?


2 Select Show default covariances. When a check mark is displayed next to this command, default co-variances appear in the diagram; when a check mark is not displayed next to this command, defaultcovariances do not appear in the diagram.

NOTE: The default covariances are calculated whether or not they are displayed.

Back

How do I set the default covariances to 0?

1 Right-click on the default covariance path.

2 Select Set covariance to 0. A 0 appears in the diagram next to the covariance.

Back

How do I add variables to the diagram?

In the Single Group Analysis window, click Palette to open the Palette pane which contains a group ofvariable tools. (In the Model Library window, the options appear in the Variables area.) Choose one of thefollowing alternatives:

� Drag a shape from the Palette pane (or Variables area) to the desired location on the Diagram area.

� Right-click on the Diagram area and select Add observed or Add latent to create an observed orlatent variable at that location.

� Drag one or more variable names from the Variable list to the Diagram area. If you have selectedmore than one variable, a window appears and asks if you want to arrange the variables in a row orcolumn. Select Row or Column and then click OK.

� Drag one or more variable names from the Variable list to a variable in the Diagram area. A windowappears and asks you where you want the variables to appear relative to the variable already in theDiagram area. Select the relative position you want, and click OK.

Back

Why is there a message when I try to add more than 15 variables to the diagram when Showdefault covariances is selected?

A message appears when you try to add more than 15 variables to the Diagram area with the Show defaultcovariances option selected, because it takes a long time for the variables to appear in the Diagram area with


default covariance paths. To avoid waiting a long time, you can click Yes to turn off the default covarianceoption. Then only the selected variables appear in the Diagram area. It is recommended that you click Yesto turn the Show default covariances option off while you create a model in the Diagram area, and then turnthe option back on when the model is complete. To have both the selected variables and default covariancesappear in the Diagram area, click No. It might take a long time for both the selected variables and defaultcovariances to appear. To terminate the request to add variables to the Diagram area, click Cancel.

Back

How do I add a path to represent unidirectional effects from one variable to another variable?

1 Rest the cursor on a variable. A small palette appears that contains a single-headed and a double-headedarrow.


3 Drag the cursor toward the other variable. The black outline of the other variable turns bold, indicatingthat the other variable is a valid target for this path.

4 Release the mouse button. The single-headed path from one variable to another variable appears.

Back

How do I add a path to represent correlation or covariance paths between variables?

1 Rest the cursor on a variable. A small palette appears that contains a single-headed and a double-headedarrow.


3 Drag the cursor toward the other variable. The black outline of the other variable turns bold, indicatingthat the other variable is a valid target for this path.

4 Release the mouse button. The double-headed path from one variable to another variable appears.

Back

How do I change models that have been run?

Choose one of the following alternatives:

� (Recommended) Click Analyses to view the list of analyses, and then click Copy to create a copyof the model in a new analysis. This enables you to keep the results from the original analysis forcomparison, and also enables you to modify the model without recreating it from scratch.

� Click Analyses to view the list of analyses, and then click New to create a new analysis with an emptyDiagram area. This enables you to keep the results from the original analysis for comparison, butyou must respecify your model and then modify it.


� Select Input from list of views in the View area. This changes the current diagram from an outputview with parameter estimates to an input view with no parameter estimates. You can modify yourmodel in the input view, but you cannot compare your modified model to the original model.

Back

What are the endogenous and exogenous variables in a model?

Any variable is endogenous if it is predicted by at least one other variable. A variable is exogenous if itis not endogenous. In a path diagram, a variable is endogenous if it has at least one single-headed arrowpointing to it. SAS Structural Equation Modeling for JMP automatically generates error variables for anyendogenous variables. The error variables are displayed only when you select the display error variablesoption. These error variables are always exogenous. The endogenous or exogenous status of a variablechanges as you add, delete, or reverse the paths in the diagram.

Back

What parameters in the path diagram are freely estimated by default?

The following parameters are automatically estimated by SAS Structural Equation Modeling for JMP:

� the variances of all exogenous variables, including the error variables

� the covariances between all exogenous variables, excluding the error variables

� the means of all exogenous observed variables and the intercepts of all endogenous observed variables

Back

How do I constrain parameters in the path diagram to be equal?

To set parameters to be equal, the same name must be given to all parameters.

For equal paths:

1 Right-click on the path.

2 Select Set path properties. A Path Properties window appears.

3 Select Free, and type a parameter name in the Name box.

4 Click OK.

5 Repeat these steps, using the same name, for each path you want to be constrained to be equal.


NOTE: You can also specify one name for several paths simultaneously by selecting multiple paths. Toselect multiple paths, click on one path then hold the CTRL or SHIFT key, while you click the remainingpaths.

For equal means or intercepts:

1 Right-click on the variable (or error variable).

2 Select Set variable properties. A Variable Properties window appears.



5 Click OK.

6 Repeat these steps, using the same name, for each mean or intercept you want to be constrained to beequal.

For equal variances or error variances:

1 Right-click on the variable (or error variable).

2 Select Set variable properties. A Variable Properties window appears.



5 Click OK.

6 Repeat these steps, using the same name, for variance you want to be constrained to be equal.

Back

How do I set preferences for the diagram?

1 Select Analyze IStructural Equation Modeling IUser Profile from the JMP Home window.

2 Select the desired preference on the Diagram tab.

3 Click OK.

Back

Your Turn

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