SAS Structural Equation Modeling 1.3 for JMP

Transcription

1 SAS Structural Equation Modeling 1.3 for JMP SAS Documentation

2 The correct bibliographic citation for this manual is as follows: SAS Institute Inc SAS Structural Equation Modeling 1.3 for JMP. Cary, NC: SAS Institute Inc. SAS Structural Equation Modeling 1.3 for JMP Copyright 2012, SAS Institute Inc., Cary, NC, USA ISBN All rights reserved. Produced in the United States of America. For a hard-copy book: No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the publisher, SAS Institute Inc. For a Web download or e-book: Your use of this publication shall be governed by the terms established by the vendor at the time you acquire this publication. The scanning, uploading, and distribution of this book via the Internet or any other means without the permission of the publisher is illegal and punishable by law. Please purchase only authorized electronic editions and do not participate in or encourage electronic piracy of copyrighted materials. Your support of others rights is appreciated. U.S. Government Restricted Rights Notice: Use, duplication, or disclosure of this software and related documentation by the U.S. government is subject to the Agreement with SAS Institute and the restrictions set forth in FAR , Commercial Computer Software-Restricted Rights (June 1987). SAS Institute Inc., SAS Campus Drive, Cary, North Carolina st electronic book, May st printing, July 2012 SAS Publishing provides a complete selection of books and electronic products to help customers use SAS software to its fullest potential. For more information about our e-books, e-learning products, CDs, and hard-copy books, visit the SAS Publishing Web site at support.sas.com/publishing or call SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. indicates USA registration. Other brand and product names are registered trademarks or trademarks of their respective companies.

3 Contents Credits v Chapter 1. About This Book Chapter 2. Introduction to SAS Structural Equation Modeling for JMP Chapter 3. Getting Started with SAS Structural Equation Modeling for JMP Chapter 4. Linear Regression Analysis Chapter 5. Path Analysis Chapter 6. Confirmatory Factor Analysis Chapter 7. Structural Equation Model Chapter 8. Latent Growth Curve Model Chapter 9. Single Group Analysis Window Chapter 10. Model Library Window Chapter 11. User Profile Window Chapter 12. Properties Windows Appendix A. Frequently Asked Questions

4 iv

5 Credits Documentation Writing Editing Documentation Support Technical Review Ruth Baldasaro Anne Baxter Tim Arnold, Sharad Prabhu Yiu-Fai Yung, Wayne Watson Software JMP PROC CALIS Wayne Watson Yiu-Fai Yung Support Groups Software Testing Technical Support Usability Wei Zhang, Ruth Baldasaro Duane Hayes Todd Barlow, Kevin Hodge

6 vi

7 Chapter 1 About This Book This book describes the features of SAS Structural Equation Modeling for JMP and includes several examples that show how to use them. The example topics include multiple regression, path analysis, confirmatory factor analysis, structural equation modeling, and latent growth curve modeling. Each example describes how 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 technical 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 Procedure (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 its benefits, 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 for JMP. Chapter 4 through Chapter 8 show examples of other analyses which build on the getting started example. Chapter 9 through Chapter 12 describe features of the various SAS Structural Equation Modeling for JMP windows. Appendix A provides answers to frequently asked questions. References Bollen, K. A. (1989), Structural Equations with Latent Variables, New York: John Wiley & Sons.

8 2

9 Chapter 2 Introduction to SAS Structural Equation Modeling for JMP Contents Overview of SAS Structural Equation Modeling for JMP Benefits of SAS Structural Equation Modeling for JMP Comparing 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 structural equation modeling (SEM) techniques. The interface enables you to quickly specify a path diagram to represent the hypothesized relationships among the variables. You can specify models with only observed variables (including multiple regression and path analysis models) and models that have both observed and latent variables (including factor analysis and latent growth curve models). You access SAS Structural Equation Modeling for JMP from the JMP interface. SAS Structural Equation Modeling for JMP uses the SAS/STAT CALIS procedure to perform the model estimation and then performs the remaining computations. For more information about PROC CALIS, see Chapter 26, The CALIS Procedure (SAS/STAT User 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 default covariances among exogenous variables. You can also choose to view either unstandardized or standardized parameter estimates. A diagram to save and print: You can easily print a diagram or copy a diagram to a document or presentation. You can also save the diagram to use with other data sets or to modify in future analyses.

10 4 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 file and 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. JMP can read in many data file types in addition to JMP data tables. SAS Structural Equation Modeling for JMP can modify a table of correlation or covariance data to contain the appropriate SAS correlation or covariance table format for analysis. Comparing SAS Structural Equation Modeling for JMP to the SAS/STAT CALIS Procedure Table 2.1 compares the interface, estimation, and analysis features of SAS Structural Equation Modeling for JMP to the SAS/STAT CALIS procedure (PROC CALIS). Table 2.1 Program Features SAS Structural Equation Feature Modeling for JMP PROC CALIS Interface Features Graphical user interface Yes No Path diagram input and output Yes No Saving and loading projects Yes No Scripting language No Yes Estimation Features Maximum likelihood Yes Yes Generalized least squares Yes Yes Weighted least squares Yes Yes Unweighted least squares Yes Yes Diagonally weighted least squares Yes Yes Full-information maximum likelihood Yes Yes Analysis Features Model fit Information Yes Yes Unstandardized solution Yes Yes Standardized solution Yes Yes Mean structure analysis Yes Yes Equality constraints Yes Yes Model comparison Yes No Multiple-group analysis No Yes Direct and indirect effects Yes Yes Boundary and linear constraints No Yes Nonlinear constraints No Yes Modification indices Yes Yes

11 Chapter 3 Getting Started with SAS Structural Equation Modeling for JMP Contents Overview of Getting Started Example Start a SAS Structural Equation Modeling for JMP Analysis Create the Model Specify the Data Set Add Variables to the Diagram Draw Paths Label the Analysis Specify Options for the Analysis Perform the Analysis View the Results View Results in the Diagram Verify Accuracy of Results View Detailed Results Summary of Results Save the Results Save a Model Library File Save a Single Group Analysis Project File Save a Diagram Print a Diagram Overview of Getting Started Example This getting started example uses a multiple regression model to show you how to use 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 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 and perform an analysis in SAS Structural Equation Modeling for JMP, but it does not describe all the ways you can create a model or all of the features of SAS Structural Equation Modeling for JMP. Throughout

12 6 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 Structural Equation Modeling for JMP. You can find the data file for this getting started example, Sales_Data.jmp, by going to the JMP Home window and selecting SAS I Structural Equation Modeling I Sample Data I Sales Data. The data file Sales_Data.jmp contains raw data, which are observations for each variable for each unit included in the study. The data file contains responses from 25 companies for four variables (N_emp is the number of employees, Advert is the company s advertising spending in millions of dollars, LastS is last year s sales in millions of dollars, and CurrentS is the current year s sales in millions of dollars). Table 3.1 shows the contents of the data set. Table 3.1 Raw Data from Sales_Data.jmp N_emp Advert LastS CurrentS This example shows you how to create the multiple regression model shown in Figure 3.1, where the number of employees (N_emp), advertising spending (Advert), and last year s sales (LastS) are all are used to predict the current year s sales (CurrentS).

13 Start a SAS Structural Equation Modeling for JMP Analysis 7 Figure 3.1 Getting Started Example Path Diagram Start a SAS Structural Equation Modeling for JMP Analysis 1 Open JMP or later. 2 To open the Sales_Data.jmp data file, select SAS I Structural Equation Modeling I Sample Data I Sales Data. 3 Select SAS I Structural Equation Modeling I Single Group Analysis. The Structural Equation Models for a Single Group window appears as shown in Figure 3.2.

14 8 Chapter 3: Getting Started with SAS Structural Equation Modeling for JMP 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. You use the Analyses tab to create, modify, and run analyses on one or more models. You use the Comparisons tab to examine the fit statistics from each analysis you run in the Analyses tab. The Comparisons tab is available 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 the analysis. 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.

15 Add Variables to the Diagram 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 other features 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 (rectangle) 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 asks whether the variables should be arranged horizontally or vertically.

16 10 Chapter 3: Getting Started with SAS Structural Equation Modeling for JMP 4 After all of the variables are in the diagram, drag the variables around until they look like the variables in Figure 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 CurrentS: 1 Rest the cursor on N_emp. A small palette appears that contains a single-headed and a double-headed arrow. Figure 3.5 shows an example of this palette. You use the double-headed arrow to represent covariances 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.

17 Draw Paths 11 Figure 3.5 Arrow Palette 3 Drag the cursor toward CurrentS. The black outline of CurrentS turns bold, indicating that the variable is a 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.

18 12 Chapter 3: Getting Started with SAS Structural Equation Modeling for JMP Figure 3.6 Getting Started Example Diagram with Paths Connecting the Variables

19 Label the Analysis 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 analysis is automatically given a unique label Analysis 1, Analysis 2, and so on. Even though the label and notes are optional, specifying a label is useful so that you can quickly locate a model when you are exploring several models 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 the list, as shown in Figure 3.8.

20 14 Chapter 3: Getting Started with SAS Structural Equation Modeling for JMP 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 estimation is the default estimation method. For more details about the estimation options, see the section Specify Estimation Method on page In the Optimization area, select Default from the Method list. The Default optimization method uses the optimization method best suited for the number of parameters the model is estimating. For more details about the optimization options, see the section Specify Optimization Method on page In the Optimization area, type the maximum number of iterations to be performed in this analysis in the Maximum iterations box. For this analysis, leave this box blank to use the default maximum number

21 Perform the Analysis 15 of iterations, which is based on the optimization method. For more details about the maximum iterations option, 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 Analysis area 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 learn how to check that the model converged to a proper solution.

22 16 Chapter 3: Getting Started with SAS Structural Equation Modeling for JMP 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 View list. 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.

23 View Results in the Diagram 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 the View list. You can modify the model if you return to the diagram view that does not contain any parameter estimates. To return to the diagram view that does not contain any parameter estimates, select Input from the View list. Another way to modify a diagram after it has been run is to select Copy in the Analysis pane to 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 estimated multiple regression coefficients are displayed on the paths from the predictors to the outcome. Any parameter estimate that differs significantly from 0 (based on t tests) is marked with an asterisk. Two asterisks indicate that p < 0.01; one asterisk indicates that p < 0.05.

24 18 Chapter 3: Getting Started with SAS Structural Equation Modeling for JMP Verify Accuracy of Results Before you examine more detailed results on the Results tab, you should verify that the model converged without any warning or error messages. After you click Run in the Perform Analysis area, a window usually appears with a warning if any estimation problems occur. Even if a warning window does not appear, 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 converges, the SAS Log contains the following message (or a similar message for convergence with another convergence criterion): Convergence criterion (ABSGCONV= ) satisfied. Because the model in this example converged without any errors or warnings, you can correctly interpret the 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 JMP log, 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 on the Analyses tab. The output in the Results tab is organized such that the modeling specifications and model fit are presented first, followed by the parameter estimates for the model. The parameter estimates are organized to present the parameters for the single-headed arrow paths first, followed by the variance, covariance, and squared multiple correlations. Figure 3.12 and Figure 3.13 show the unstandardized results for this example.

25 Figure 3.12 Getting Started Example Results Tab with Modeling Specifications and Fit Results View Detailed Results 19

26 20 Chapter 3: Getting Started with SAS Structural Equation Modeling for JMP Figure 3.13 Getting Started Example Results Tab with Maximum Likelihood Parameter Estimate Results The parameter estimates under the PATH List heading are the unstandardized multiple regression coefficients shown on the paths in the Diagram area in Figure The parameter estimates under the Variance Parameters heading are the variance estimates shown above the variables in the Diagram area in Figure Although the Diagram area shows only the parameter estimates, the Results tab also contains the standard error and t test values for each parameter estimate. The Results tab also contains some measures of model fit (under the Fit Summary heading) and the R-square estimate for CurrentS (under the Squared Multiple Correlations heading). The gray triangles next to each section header collapse or expand the section of output. The red triangle at the top of the Results tab contains the options for what you can view in the output. These options are not used in this example; for more detailed information about these options, see Chapter 9, Single Group Analysis Window. In addition to the SAS Log and Results tabs, a SAS Code tab is produced after you click Run. This tab contains 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).

27 Summary of Results 21 Summary of Results The parameter estimates in Figure 3.13 indicate that both Advert (the company s advertising spending in millions of dollars) and LastS (last year s sales in millions of dollars) have positive and significant path estimates for their relationship with CurrentS (the current year s sales in millions of dollars). This suggests that these variables explain a significant amount of the variance in CurrentS beyond the variance explained by the other variables in the model. The positive path estimates indicate that higher levels of Advert and LastS are associated with higher levels of CurrentS. The only variable that is not significantly related with CurrentS is N_emp (the number of employees), which suggests that this variable does not explain a significant amount 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 the variance in CurrentS, suggesting that these three predictors do a good job of explaining the variation in CurrentS. 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 Analysis window. 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.

28 22 Chapter 3: Getting Started with SAS Structural Equation Modeling for JMP 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, the parameter specifications, and the results (for each analysis that has results) together in one Single Group Analysis 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 the diagram and the results for each model analyzed: 1 In the Project area, click Save or Save As. 2 Specify the name of the model and where you want the model to be saved. 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.

29 Chapter 4 Linear Regression Analysis Contents Overview of Linear Regression Analysis Example Create the Model Specify the Data Set Add Multiple Variables to the Diagram Draw Paths Label Each Path Add Mean Structure to the Model Add Mean Structure Using the Methods Tab Add Mean Structure Using the Variable Settings Perform the Analysis View the Results View Results in the Diagram Verify Accuracy of Results View Detailed Results Summary of Results Save Project File Overview of Linear Regression Analysis Example This linear regression analysis example is an extension of the example in Chapter 3, Getting Started with SAS Structural Equation Modeling for JMP. This example uses a multiple regression model to show you how 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 specify the data set, create the multiple regression model, label model parameters, add mean structure, analyze the model, 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 Home window and selecting SAS I Structural Equation Modeling I Sample Data I Sales Data. For more 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 in Figure 4.1, where the number of employees (N_emp), advertising spending (Advert), and last year s sales

30 24 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 model with labels on the paths from the predictors to the outcome. Unlike the getting started example, this model estimates mean structure as part of the model. The diagram in Figure 4.1 shows the mean structure as the label Intercept, above the variable current year s sales (CurrentS). The parameter estimate for Intercept is the 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 the paths, and add mean structure to the analysis. 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. 2 Verify that Raw Data is selected from the Data Structure list.

31 Add Multiple Variables to the Diagram 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 and clicking 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 asks whether you want to arrange the variables in a row or column. Figure 4.2 shows the Arrange Variables window. 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 variables in Figure 4.3.

32 26 Chapter 4: Linear Regression Analysis 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-headed arrow. Figure 4.4 shows an example of this palette. You use the double-headed arrow to represent covariances, and you use the single-headed arrow to represent unidirectional effects. 2 Select the single-headed arrow. It turns red when selected; see Figure 4.4.

33 Draw Paths 27 Figure 4.4 Arrow Palette 3 Drag the cursor toward CurrentS. The black outline of CurrentS turns bold, indicating that the variable is a 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.

34 28 Chapter 4: Linear Regression Analysis Figure 4.5 Multiple Regression Diagram Tab with Paths Connecting the Variables

35 Label Each Path 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 Properties 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 a unique 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 are constrained to have the same parameter estimate.

36 30 Chapter 4: Linear Regression Analysis Figure 4.7 Multiple Regression Diagram Tab with Labels on the Paths Connecting 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 1 Click the Methods tab. 2 In the Analyze area, select Mean Structures. Figure 4.8 shows the Methods tab with Mean Structures selected.

37 Add Mean Structure Using the Variable Settings 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 to be 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 model estimates 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.

38 32 Chapter 4: Linear Regression Analysis Figure 4.9 Multiple Regression Diagram Tab with Mean Structure Label above CurrentS Perform the Analysis After you have specified the desired path diagram, click Run in the Perform Analysis area 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 learn how to check that the model converged to a proper solution.

39 View Results in the Diagram 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 the View 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 View list.

40 34 Chapter 4: Linear Regression Analysis 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 displayed above each variable separated by a comma, and the estimated multiple regression coefficients are displayed above the paths from the predictors to the outcome. Any parameter estimates that 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.

41 Verify Accuracy of Results 35 Verify Accuracy of Results Before you examine more detailed results on the Results tab, you should verify that the model converged without any warning or error messages. After you click Run in the Perform Analysis area, a window usually appears with a warning if any estimation problems occur. Even if a warning window does not appear, 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 converges, the SAS Log contains the following message (or a similar message for convergence with another convergence criterion): Convergence criterion (ABSGCONV= ) satisfied. Because the model in this example converged without any errors or warnings, you can correctly interpret the 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 JMP log, 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 on the Analyses tab. The output on the Results tab is organized such that the modeling specifications and model fit are presented first, followed by the parameter estimates for the model. The parameter estimates are organized to present the 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 the unstandardized results for this example.

42 36 Chapter 4: Linear Regression Analysis Figure 4.12 Multiple Regression Results Tab with Modeling Specifications and Fit Results

43 View Detailed Results 37 Figure 4.13 Multiple Regression Results Tab with Maximum Likelihood Parameter Estimate Results The parameter estimates under the PATH List heading are the unstandardized multiple regression coefficients shown on the paths in the Diagram area in Figure The parameter estimates under the Means and Intercepts and Variance Parameters headings are the mean and variance estimates shown above the variables in the Diagram area in Figure Although the Diagram area shows only the parameter estimates, the Results tab also contains the standard error and t test values for each parameter estimate. The Results tab also contains some measures of model fit (under the Fit Summary heading) and the R-square estimate for CurrentS (under the Squared Multiple Correlations heading).

44 38 Chapter 4: Linear Regression Analysis Summary of Results A full summary of the results for this model is found in Chapter 3, Getting Started with SAS Structural Equation 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 This is the average number of employees for companies in this sample. The mean of Advert is This is the average advertising spending in millions of dollars for the companies in this sample. The mean of LastS is This is the average for last year s sales in millions of dollars for the companies in this sample. The intercept of CurrentS, called Intercept, is This number is the average current sales for a company with no employees, no advertising spending, and no last year s sales. This is not practically meaningful for this example, but is an example of how this parameter would be interpreted in other multiple regression analyses. Save Project File To save a Single Group Analysis project file with a copy of all the analysis information, including both the diagram and the results: 1 In the Project area, click Save or Save As. 2 Specify the name of the model and where you want the model to be saved. 3 Click Save.

45 Chapter 5 Path Analysis Contents Overview of the Path Analysis Example Create Model Specify the Data Set Add Multiple Variables to the Diagram Draw Paths Modify the Diagram to Show Error Variables Modify the Diagram to Show Variances Perform the Analysis View Model 1 Results View Model 1 Results in the Diagram Verify Accuracy of Results View Detailed Model 1 Results Summary of Model 1 Results Create Model Copy Model Remove a Path Modify the Diagram to Show Default Covariance Path Perform the Analysis View Model 2 Results View Model 2 Results in the Diagram Verify Accuracy of Results View Detailed Model 2 Results Summary of Model 2 Results Compare the Models Summary of Model Comparison References Overview of the Path Analysis Example Path analysis is a method for testing causal pathways among observed variables when there is more than one outcome (endogenous) variable (Wright 1934). This example begins with a description of the data and

46 40 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 tab to customize a table of the model fit statistics to compare model fit for multiple models, and how to modify the 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 Home window and selecting SAS I Structural Equation Modeling I Sample Data I Sales Data. For more 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 current year s sales (CurrentS). The models are shown in Figure 5.1 and Figure 5.2. In Figure 5.1, the number of 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 spendings (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 that allows last year s sales (LastS) to predict advertising spending (Advert).

47 Create Model 1 41 Figure 5.2 Path Analysis Model 2 Diagram Create Model 1 In this section you learn how to specify the data set, add variables to the diagram, draw paths, and modify the diagram to view the error variables and parameter variances in the diagram. Specify the Data Set On the Data tab, verify that the following information is specified in the Data Table Properties area: 1 Verify that Name shows Sales_Data. 2 Verify that Raw Data is selected from the Data Structure list. Add Multiple Variables to the Diagram Now that the data have been specified, you can start building your model. To build the first path analysis model: 1 Click the Analyses tab. 2 Click Palette to open the Palette pane.

48 42 Chapter 5: Path Analysis 3 In the Variables list, select more than one variable at a time by holding down the CTRL key and clicking each 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 asks whether 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 variables 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

49 Draw Paths 43 Draw Paths 1 Rest the cursor on N_emp. A small palette appears that contains a single-headed and a double-headed arrow. 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 valid target 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 to Advert. Figure 5.4 shows the resulting diagram. Figure 5.4 Path Analysis Model 1 Diagram Tab with Paths Connecting the Variables

50 44 Chapter 5: Path Analysis 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 1 Right-click in the Diagram area. 2 Select Show variances. Figure 5.6 shows the diagram with the variance paths.

51 Perform the Analysis 45 Figure 5.6 Path Analysis Model 1 with Error Variables and Variances Perform the Analysis After you have specified the desired path diagram, click Run in the Perform Analysis area to fit the model and generate the results output. View Model 1 Results In this section you learn what results can be viewed in the diagram and on the Results tab, and you learn how to check that the model converged to a proper solution.

52 46 Chapter 5: Path Analysis View Model 1 Results in the Diagram Figure 5.7 shows the unstandardized parameter estimates in the diagram when Unstd. Estimates is selected from 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 from the View list.

53 View Model 1 Results in the Diagram 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 and error variance path for each 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. Two asterisks indicate that p < 0.01; one asterisk indicates that p < 0.05.

54 48 Chapter 5: Path Analysis Verify Accuracy of Results Before you examine more detailed results on the Results tab, you should verify that the model converged without any warning or error messages. After you click Run in the Perform Analysis area, a window usually appears with a warning if any estimation problems occur. Even if a warning window does not appear, 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 converges, the SAS Log contains the following message (or a similar message for convergence with another convergence criterion): Convergence criterion (ABSGCONV= ) satisfied. Because the model in this example converged without any errors or warnings, you can correctly interpret the 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 JMP log, go to the JMP Home window and double-click Log in the Window list. View Detailed Model 1 Results The Diagram area contains only some of the results. To view more detailed results, click the Results tab on the Analyses tab. The output in the Results tab is organized such that the modeling specifications and model fit are presented first, followed by the parameter estimates for the model. The parameter estimates are organized to present the parameters for the single-headed arrow paths first, followed by the estimates for the variances and squared multiple correlations. Figure 5.9 and Figure 5.10 show the unstandardized results for this example.

55 Figure 5.9 Path Analysis Model 1 Results Tab with Modeling Specifications and Fit Results View Detailed Model 1 Results 49

56 50 Chapter 5: Path Analysis Figure 5.10 Path Analysis Model 1 Results Tab with Maximum Likelihood Parameter 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 parameters are all significant at the p < 0.05 level, and most are significant at the p < 0.01 level. These results indicate that the paths in this model represent significant relationships among the variables. The path parameters are all positive values, indicating that higher values of the predictor are associated with higher values on the outcome. 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 the predictors. Together, these results suggest that this model does a good job of capturing the relationships among these variables.

57 Create Model 2 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 Path Analysis 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.

58 52 Chapter 5: Path Analysis Figure 5.11 Path Analysis Model 2 Diagram Modify the Diagram to Show Default Covariance Path By default, when a model has more than one exogenous variable, SAS Structural Equation Modeling for JMP estimates a covariance for each pair of exogenous variables. To see the default covariances in the diagram: 1 Right-click in the Diagram area. 2 Select Show default covariances. Figure 5.12 shows the diagram with the default covariance path.

59 Perform the Analysis 53 Figure 5.12 Path Analysis Model 2 Diagram Perform the Analysis After you have removed the path from LastS to Advert, click Run in the Perform Analysis area to fit the model and generate the results output.

60 54 Chapter 5: Path Analysis 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 the View 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.

61 View Model 2 Results in the Diagram 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 error variance path for each variable. Each estimated path coefficient is displayed next to the path from the predictor to the outcome. Any parameter estimates that differ significantly from 0 (based on t tests) are marked with asterisks. Two asterisks indicate that p < 0.01; on asterisk indicates that p < 0.05.

62 56 Chapter 5: Path Analysis Verify Accuracy of Results Before you examine more detailed results on the Results tab, you should verify that the model converged without any warning or error messages. After you click Run in the Perform Analysis area, a window usually appears with a warning if any estimation problems occur. Even if a warning window does not appear, 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 converges, the SAS Log contains the following message (or a similar message for convergence with another convergence criterion): Convergence criterion (ABSGCONV= ) satisfied. Because the model in this example converged without any errors or warnings, you can correctly interpret the 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 JMP log, go to the JMP Home window and double-click Log in the Window list. View Detailed Model 2 Results The Diagram contains only some of the results. To view more detailed results, click the Results tab on the Analyses tab. Figure 5.15 and Figure 5.16 show the unstandardized results for this example.