GETTING STARTED WITH THE STUDENT EDITION OF LISREL 8.51 FOR WINDOWS

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1 GETTING STARTED WITH THE STUDENT EDITION OF LISREL 8.51 FOR WINDOWS Gerhard Mels, Ph.D. Senior Programmer Scientific Software International, Inc. 1. Introduction The Student Edition of LISREL 8.51 for Windows (Jöreskog & Sörbom 2001) is a Windows application for Structural Equation Modeling, Multilevel Structural Equation Modeling, Multilevel Linear and Nonlinear Modeling and Formal Inference-based Recursive Modeling. This application consists of a 32-bit Windows application LISWIN32.EXE that interfaces with the 32-bit applications LISREL85.EXE, PRELIS25.EXE, MULTILEV5.EXE, CATFIRM.EXE and CONFIRM.EXE. This edition is a free download that is available at The only limitation of the Student Edition of The Student Edition of LISREL 8.51 for Windows is that it only allows for models based on 12 observed variables. PRELIS25.EXE is a 32-bit application for manipulating data, transforming data, generating data, computing moment matrices, computing asymptotic covariance matrices, performing regression analyses, performing exploratory factor analyses, etc. The 32-bit application LISREL85.EXE is intended for Standard and Multilevel Structural Equation Modeling. The Full Information Maximum Likelihood (FIML) method for missing data is also available for both Standard and Multilevel Structural Equation Modeling. MULTILEV5.EXE fits multilevel linear and nonlinear models to raw data while CATFIRM.EXE and CONFIRM.EXE allow Formal Inference-based Recursive Modeling for raw categorical and continuous data respectively. This document is intended as a tutorial to familiarize new or potential users of LISREL 8.51 for Windows with the features of the application. Section 2 describes the various files used and generated by the Student Edition of LISREL 8.51 for Windows. A step-by-step procedure to fit a measurement model to an SPSS data set with the Student Edition of LISREL 8.51 for Windows is described in Section 3. A procedure to fit a structural equation model with latent variables is outlined in Section 4. The Robust Maximum Likelihood (RML) and Weighted Least Squares (WLS) methods of the Student Edition of LISREL 8.51 for Windows are illustrated in Sections 5 and 6 respectively. A Multilevel Confirmatory Factor Analysis model is fitted to an SPSS data set in Section 7. Section 8 illustrates how the Student Edition of LISREL 8.51 for Windows can be used to compute latent variable scores. An illustrative example of using latent variable scores is provided in section 9. The final section illustrates the use of the multiple group analysis feature of the Student Edition of LISREL 8.51 for Windows to assess the cross validation of a measurement for retail experience. 1

2 2. Files The Student Edition of LISREL 8.51 for Windows uses a PRELIS System File (PSF) to store raw data. Whenever PRELIS25.EXE or LISREL85.EXE processes a PSF, a Data System File (DSF), which has the same file name as the PSF, is created. This DSF contains all the data information that LISREL85.EXE requires to fit structural equation models to the data. A structural equation model can be specified by means of a path diagram, a SIMPLIS project file, a LISREL project file, a SIMPLIS syntax file or a LISREL syntax file. The Student Edition of LISREL 8.51 for Windows uses a graphics file with the default extension PTH to capture a path diagram. The extensions SPJ and LPJ are used for SIMPLIS and LISREL project files respectively. SIMPLIS and LISREL syntax files are text files with the default extensions SPL and LS8 respectively. These five file types can access the data from the PSF or the DSF. If a user has prepared any of these files, then the Student Edition of LISREL 8.51 for Windows can be used to fit the specified model to the data specified in the corresponding PSF or DSF. Path diagram, SIMPLIS project and LISREL project files are described in Du Toit & Du Toit (2001). SIMPLIS syntax files are described in Jöreskog & Sörbom (1996c) while the LISREL syntax files are outlined in Jöreskog & Sörbom (1996b). All these files are also described in the online Help File, which can be accessed by using the Contents option from the Help menu of the Student Edition of LISREL 8.51 for Windows. Whenever PRELIS25.EXE processes a PSF interactively, a PRELIS syntax file with the same file name as the PSF is created. A PRELIS syntax file is a text file with default extension PR2. PRELIS syntax files are described by Jöreskog & Sörbom (1996a) as well as in the online Help File of the Student Edition of LISREL 8.51 for Windows. Multilevel Modeling and Formal Inference-based Recursive Modeling syntax files are also text files with default extension PR2. Multilevel Modeling syntax files are described in Jöreskog et al. (1999) while Formal Inference-based Recursive Modeling syntax files are described in Du Toit & Du Toit (2001). Both these types of syntax files are also described in the online Help file of the Student Edition of LISREL 8.51 for Windows. LISWIN32.EXE starts up by opening a root window with three menus. The File menu can then be used to open existing or new PSF s and PTH files in PSF and PTH windows respectively. It can also be used to open new or existing LISREL and SIMPLIS project files in LISREL and SIMPLIS project (LPJ and SPJ) windows. Text editor windows are used to display new or existing syntax files and output files. PTH, PSF, LPJ, SPJ and text editor windows have window-specific menus. 3. Fitting a Measurement model to SPSS data The SPSS for Windows data file DEPRESS.SAV contains 204 observations of 12 continuous indicators of three latent variables. More specifically, the first 5 indicators (SELF1 to SELF5) are indicators of the latent variable Self-esteem, DEPRES1 to DEPRES4 are indicators of the latent variable Depressiveness and IMPULS1 to IMPULS3 are indicators of the latent variable Impulsiveness. The theoretical measurement model is a CFA model that specifies that the 12 continuous indicators are indeed indicators of Self- 2

3 esteem, Depressiveness and Impulsiveness. measurement model to the SPSS data set follows. A step-by-step procedure to fit this Actions Use the Import Data in Free Format option from the File menu of the root window of the Student Edition of LISREL 8.51 for Windows to load the Open Data File dialog box. Select SPSS for Windows (*.sav) from the Files of type drop-down list box. Browse to the folder that contains the SPSS for Windows data file Depress.sav. Double-click on the file name Depress.sav to open the following PSF window. Actions Click on the variable label SELF1 to highlight the entire column. Right-click to open the variable menu as shown in the following PSF window. 3

4 Actions Select the Define Variables option to load the Define Variables dialog box. Select the label SELF1 to produce the following Define Variables dialog box. Actions Click on the Variable Type push button to load the Variable Types for SELF1 dialog box. Select the Continuous radio button. Check the Apply to all checkbox to produce the Variable Types for SELF1 dialog box below. Actions Click on the OK push button to reload the Variable Types dialog box. Click on the OK push button to return to the PSF window. Save your changes to the PSF by using the Save option from the File menu. Select the Open option from the File menu to load the Open dialog box. Enter the name depress0.spl in the File name string field. Click on the Open push button to open the following text editor window. 4

5 The file above is a SIMPLIS syntax file to generate a blank path diagram that contains the observed and latent variables of the model. Line 1 specifies the data source. Lines 2 and 3 provide labels for the latent variables Self-esteem, Depressiveness and Impulsiveness respectively. Line 4 requests the generation of a path diagram. Line 5 indicates that there are no more SIMPLIS commands to be processed. Action Click on the Run LISREL push button to produce the following path diagram window. Actions 5

6 Activate the Arrow icon on the Draw toolbar. Use your mouse to click and drag 5 indicator paths from the latent variable selfest to SELF1 to SELF5. Use your mouse to click and drag 4 indicator paths from the latent variable depress to DEPRES1 to DEPRES4. Use your mouse to click and drag 3 indicator paths from the latent variable impuls to IMPULS1 to IMPULS3. Activate the Double-headed Arrow icon on the Draw toolbar. Use your mouse to click and drag 3 covariance paths between the three latent variables Self-esteem, Depressiveness and Impulsiveness to produce the following path diagram window. Action Click on the Run LISREL push button to produce the following path diagram window. 6

7 The requested results are listed in the output window for Depress0.out. A SIMPLIS syntax file may also be used to generate the previous path diagram window. The corresponding SIMPLIS syntax file is shown in the following text editor window Lines 4 to 7 are used to specify the measurement model for the latent variables Selfesteem, Depressiveness and Impulsiveness. 7

8 4. Fitting a Structural Equation Model to SPSS data The SPSS for Windows data file DEPRESS.SAV contains 204 observations of 12 continuous indicators of three latent variables. More specifically, the first 5 indicators (SELF1 to SELF5) are indicators of the latent variable Self-esteem, DEPRES1 to DEPRES4 are indicators of the latent variable Depressiveness and IMPULS1 to IMPULS3 are indicators of the latent variable Impulsiveness. The theoretical measurement model is a CFA model that specifies that the 12 continuous indicators are indeed indicators of Selfesteem, Depressiveness and Impulsiveness. One possible structural model for the three latent variables is a model that suggests that Depressiveness and Impulsiveness are correlated antecedents of Self-esteem. A step-by-step procedure to fit this latent variable model to the SPSS data set follows. Use the steps in section 3 to import the SPSS for Windows data file DEPRESS.SAV as the Prelis System File DEPRESS.PSF. Define all the variables as continuous as illustrated in section 3 and save the changes to DEPRESS.PSF by using the Save option on the File menu of the PSF window for DEPRESS.PSF. Select the Open option on the File menu to load the Open dialog box. Enter the name depress2.spl in the File name string field. Click on the Open push button to open the following text editor window. The file above is a SIMPLIS syntax file to generate a blank path diagram that contains the observed and latent variables of the model. Line 1 specifies the data source. Lines 2 and 3 provide labels to the latent variables Self-esteem, Depressiveness and Impulsiveness respectively. Line 4 specifies that the latent variable Self-esteem is an endogenous variable. Line 5 specifies that the indicators SELF1 to SELF5 are indicators of an endogenous latent variable. Line 6 requests the generation of a path diagram. Line 7 inicates that there are no more SIMPLIS commands to be processed. Action Click on the Run LISREL push button to produce the following path diagram window. 8

9 Actions Activate the Arrow icon on the Draw toolbar. Use your mouse to click and drag 5 indicator paths from the latent variable selfest to SELF1 to SELF5. Use your mouse to click and drag 4 indicator paths from the latent variable depress to DEPRES1 to DEPRES4. Use your mouse to click and drag 3 indicator paths from the latent variable impuls to IMPULS1 to IMPULS3. Use your mouse to click and drag 2 structural paths from the latent variables impuls and depress to selfest. Activate the Double-headed Arrow icon on the Draw toolbar. Use your mouse to drag 1 covariance path between the exogenous latent variables Depressiveness and Impulsiveness to produce the following path diagram window. 9

10 Action Click on the Run LISREL push button to produce the following path diagram window. The requested results are displayed in the output window for Depress3.out. 10

11 SIMPLIS syntax may also be used to generate the previous path diagram. The corresponding SIMPLIS syntax file is shown in the following text editor window Lines 4 to 9 are used to specify the structural equation model for the latent variables Selfesteem, Depressiveness and Impulsiveness. Action Click on the Run LISREL push button to produce the path diagram on page Robust Maximum Likelihood Browne (1987) formulated a Robust Maximum Likelihood (RML) method for factor analysis and related models. Satorra & Bentler (1988) extended this method by providing a correct Chi-square test statistic. This method is available in the Student Edition of LISREL 8.51 for Windows and the associated formulae are provided in Jöreskog et al (1999). To implement this method, the user needs to compute the Asymptotic Covariance Matrix (ACM) of the sample variances and covariances. A step-by-step procedure to implement the RML method in the Student Edition of LISREL 8.51 for Windows for the structural equation model of section 4 may be described as follows. Use the Open option from the File menu to open the PSF window for Depress.psf. Select the Output Options option from the Statistics menu to load the Output dialog box. Check the Save to File checkbox in the Asymptotic Covariance Matrix section. Enter the name depress.acm in the string field in the Asymptotic Covariance Matrix section to produce the following Output dialog box. 11

12 Actions Click on the OK push button to run PRELIS25.EXE to generate the text editor window containing the output file Depress.out. This action causes PRELIS25.EXE to generate the Data System File (DSF) Depress.dsf that will be needed to implement the RML method. Modify the existing SIMPLIS syntax file depress3.spl to produce the SIMPLIS syntax file depress4.spl shown in the following text editor window by using the Save As option on the File menu. Line 1 specifies the new data source. The option ME=ML has been added in the Lisrel Output command. 12

13 Action Click on the Run LISREL push button to produce the following path diagram window. 6. Weighted Least Squares Browne (1982, 1984) formulated an Asymptotically Distribution Free (ADF) method for covariance structures. This method is implemented in the Student Edition of LISREL 8.51 for Windows as Weighted Least Squares (WLS) and extended to correlation structures. To implement this method, the user needs to compute the Asymptotic Covariance Matrix (ACM) of the sample variances and covariances or sample correlations. A step-by-step procedure to implement the WLS method in the Student Edition of LISREL 8.51 for Windows for the structural equation model of sections 4 and 5 may be described as follows. Generate the Data System File (DSF) depress.dsf exactly as described in section 5. Modify the existing SIMPLIS syntax file depress3.spl to produce the SIMPLIS syntax file shown in the following text editor window. 13

14 The option ME=ML in the Lisrel Output command has been replaced by ME=WLS. Action Click on the Run LISREL push button to produce the following path diagram window. The requested results are displayed in the output window for Depress3.out. 14

15 7. Multilevel Confirmatory Factor Analysis The Student Edition of LISREL 8.51 for Windows includes a Multilevel Structural Equation Modeling module. This module is described in Du Toit & Du Toit (2001) and can be used to fit structural equation models to multilevel data. A step-by-step procedure to implement this module to fit a two-factor Confirmatory Factor Analysis (CFA) model to an SPSS data set may be outlined as follows. This data set forms part of the data library of the Multilevel Project at the University of London, and come from the Junior School Project (Mortimore et al, 1988). Mathematics and language tests were administered in three consecutive years to more than 1000 students from 50 primary schools, which were randomly selected from primary schools maintained by the Inner London Education Authority. The data for school 33 are provided in the TUTORIAL folder of the Student Edition of LISREL 8.51 for Windows as the SPSS for Windows file jsp2.sav. Use the Import External Data in Other Formats option from the File menu of the Student Edition of LISREL 8.51 for Windows to load the Input Database dialog box. Select SPSS for Windows (*.sav) from the Files of type drop-down list box and select the TUTORIAL folder of the Student Edition of LISREL 8.51 for Windows to produce the following dialog box. Actions Select the file name jsp2.sav. Click on the Open push button to load the following Save As dialog box. 15

16 Actions Enter the name jsp2.psf in the File name string field. Click on the Save push button to open the following PSF window. Action Select the Define Variables option from the Data menu of the PSF window to open the following Define Variables dialog box. 16

17 Actions Click on MATH1 to highlight (select) it. Click on the Missing Values push button to open the following Missing Values for MATH1 dialog box Actions Select the Missing Values radio button, enter the string in the Global Missing Value string field and check the Apply to all check box to produce the following Missing Values for MATH1 dialog box 17

18 Actions Click on the OK push buttons of the Missing Values and Define Variables dialog boxes to return to the PSF window. Use the Save option from the File menu to save the changes to the PSF. Use the Close option of the File menu of the PSF window to close jsp2.psf. Use the New option from the File menu to open the following New dialog box Action Select the Path Diagram option from the New menu to open the following Save As dialog box Action Enter the name jsp2.pth in the File name string field to open the following PTH window 18

19 Actions Select the Title and Comments option from the Setup menu to open the Title and Comments dialog box. Enter the title Multilevel CFA model for Numeric and Verbal Ability in the Title string field to produce the following Title and Comments dialog box Actions Click on the Next push button to load the Groups dialog box. Enter the label Group1: Between Schools in the first string field. Use the down arrow key to create a string field for the second group. 19

20 Enter the label Group2: Within Schools in the second string field to produce the following Group Names dialog box Action Click on the Next push button to load the following Labels dialog box. Actions Click on the Add/Read Variables push button to open the Add/Read Variables dialog box. Select the Prelis System File option from the Read from file drop-down list box. Use the Browse push button to select the file jsp2.psf in the TUTORIAL folder to produce the following Add/Read Variables dialog box 20

21 Actions Click on the Add Latent Variables push button to load the Latent Variables dialog box. Enter the label NABILITY for Numeric Ability to produce the following Add Latent Variables dialog box Actions Click on the OK push button. Enter the label VABILITY for Verbal Ability. Click on the OK push button to produce the following Labels dialog box. 21

22 Action Click on the OK push button to produce the following PTH window Actions Click and drag the observed variables MATH1, MATH2, MATH3, ENG1, ENG2 and ENG3 one by one into the PTH window. Click and drag the latent variables NABILITY and VABILITY one by one into the PTH window to produce the following PTH window 22

23 Actions Use the Arrow on the Drawing Toolbar to insert paths from NABILITY to MATH1, MATH2 and MATH3. Use the Arrow on the Drawing Toolbar to insert paths from VABILITY to ENG1, ENG2 and ENG3 to produce the following PTH window 23

24 Action Select the Build SIMPLIS Syntax option from the Setup menu to produce the following SPJ window 24

25 Actions Insert the command line $CLUSTER SCHOOL after the Raw Data from File command. Change the command line MATH1 = NABILITY to MATH1 = 1*NABILITY to set the scale of NABILITY. Change the command line ENG1 = VABILITY to ENG1 = 1*VABILITY to set the scale of VABILITY. In the Relationships paragraph of the Between Schools group, insert the following SIMPLIS commands Set the Variance of NABILITY Free Set the Variance of VABILITY Free Set the Covariance between NABILITY and VABILITY Free Copy and Paste the relationships of the Between Schools group to the Relationships paragraph of the Within Schools group. Click on the Run LISREL icon window. on the main toolbar to produce the following PTH 25

26 26

27 8. Computing Latent Variable Scores The Student Edition of LISREL 8.51 for Windows can compute latent variable scores for the latent variables of a structural equation model with latent variables. Karl Jöreskog describes the computation and the uses of these latent variable scores at The following step-by-step procedure may be used to compute latent variable scores for the latent variables Self-esteem, Depressiveness and Impulsiveness from the PRELIS System File (PSF) Depress.psf generated in section 3. Modify the existing SIMPLIS syntax file depress2.spl to produce the SIMPLIS syntax file depress.spl shown in the following text editor window by using the Save As option on the File menu. Line 1 specifies the data source. Lines 4 to 7 are used to specify the measurement model for the latent variables Selfesteem, Depressiveness and Impulsiveness. Line 8 instructs the Student Edition of LISREL 8.51 for Windows to add the latent variable scores as columns of the PRELIS System File depress.psf Action Click on the Run LISREL push button to produce the following path diagram window. 27

28 Action Use the Open option from the File menu to open the PRELIS System File depress.psf to produce the following PSF window. The final three columns shown in the PSF window above contain the latent variable scores for the latent variables Self-esteem, Depressiveness and Impulsiveness. 28

29 9. Using Latent Variable Scores Latent variable scores for the latent variables Self-esteem, Depressiveness and Impulsiveness were computed from the Data System File Depress.dsf and PRELIS System File Depress.psf in Section 8. The following step-by-step procedure may be used to fit a regression model with an interaction term to these latent variable scores. Use the Open option from the File menu to open the PRELIS System File depress.psf. Use the Compute option from the Transformation menu to load the Compute dialog box. Click on the Add push button to load the Add Variables dialog box. Enter the name dep_imp for the interaction between the latent variables scores of the latent variables Depressiveness and Impulsiveness in the string box. Click on the OK push button to return to the Compute dialog box. Click and drag the variable name dep_imp to the top left of the string box. Click on the = key on the keypad to add an = sign to the string box. Click and drag the variable name depress to the top right of the string box. Click on the * key on the keypad to insert the symbol * for multiplication in the string box. Click and drag the variable name impuls to the top right of the string box to produce the following Compute dialog box. Action Click on the OK push button to run PRELIS25.EXE to produce the following PSF window. 29

30 The final column of the PSF window above contains the interaction scores. Actions Select the Regressions option from the Statistics menu to load the Regression dialog box. Highlight the variable name selfest and click on the Y-variables Add push button. Highlight the variable names depress, impuls and dep_imp and click on the X- variables Add push button to generate the following Regression dialog box. Action Click on the Run push button to run PRELIS25.EXE to produce the desired regression results. 30

31 10. Cross Validation The cross validation of a structural equation model refers to the ability of the model to be invariant across two or more random samples from the same population. Boshoff & Terblanche (2002) consider the cross validation of a measurement model for retail experience. A variation of this measurement model will be used to illustrate how the multiple group feature of the Student Edition of LISREL 8.51 for Windows may be used to assess the cross validation of a structural equation model. This measurement model specifies COHAN1 to COHAN3 to be indicators of Complaint HANdling (COHAN), STENV1 to STENV5 to be indicators of STore ENVironment (STENV) and MEVAR1 to MEVAR4 to be indicators of MErchandise VARiety (MEVAR). Data on these 11 indicators for two samples of South African consumers are provided in the Prelis System Files SAMPLE1.PSF and SAMPLE2.PSF. Actions Use the Open option on the File menu to load the Open dialog box. Browse to the folder that contains the SIMPLIS syntax file RetailH0.spl. Double-click on the Open push button to open the following text editor window. The SIMPLIS syntax file above specifies the measurement models under the null hypothesis. This null hypothesis states that the measurement model parameters (factor loadings, factor variances, factor covariances and measurement error variances) are identical (invariant) across the two samples. The first line identifies the first sample as the first group. The second line provides the data source for the first sample. Line 3 provides labels for the 3 latent variables. Lines 4-10 specify the measurement model for the first sample. Line 11 identifies the second sample as the second group. Line 12 provides the data source for the second sample. Line 13 requests a path diagram in the form of a PTH file. Line 14 indicates the end of the syntax to be processed. Since no Relationships paragraph is used for the second sample, all parameters are assumed to be equal across the two samples. 31

32 Action Click on the Run LISREL push button to produce the following path diagram window. The alternative hypothesis states that at least two parameters of the measurement model are not identical across the two samples. The measurement models for the two samples under this alternative hypothesis are specified in the SIMPLIS syntax file RetailH1.spl. Actions Use the Open option on the File menu to load the Open dialog box. Browse to the folder that contains the SIMPLIS syntax file RetailH1.spl. Double-click on the Open push button to open the following text editor window. 32

33 The first line identifies the first sample as the first group. The second line provides the data source for the first sample. Line 3 provides labels for the 3 latent variables. Lines 4-10 specify the measurement model for the first sample. Line 11 identifies the second sample as the second group. Line 12 provides the data source for the second sample. Lines specify the measurement model for the second sample. Line 27 requests a path diagram in the form of a PTH file. Line 28 indicates the end of the syntax to be processed. Note that the Set commands for the second sample are required to ensure that the factor variances, factor covariances and measurement error variances are different across the two samples. Otherwise, these parameters are assumed to be equal across the two samples. Action Click on the Run LISREL push button to produce the following path diagram window. 33

34 A Chi-square difference test is used to assess the cross validation of the measurement model. In other words, a Chi-square difference test is used to test the null and alternative hypotheses. The test statistic value for the Chi-square difference test is merely the difference between the goodness-of-fit Chi-square test statistic values of the measurement models under the null and the alternative hypotheses. The associated degrees of freedom are merely the difference between the degrees of freedom of the measurement models under the null and the alternative hypotheses. The Chi-square difference test results for the measurement model for retail experience are summarized in the MS-Excel workbook Retail.xls. The contents of this file are shown below. The small P-values suggest that there is sufficient evidence that the null hypothesis is rejected. In other words, the cross validation of the measurement model for retail experience is not supported by the data of the two samples. 34

35 References Boshoff, H.C. & Terblanche, N. (2002). The Validation of Retail Experience Constructs. In preparation. Browne, M.W. (1982). Covariance Structures. In D.M. Hawkins (Ed.), Topics in Applied Multivariate Analysis, pp Cambridge: Cambridge University Press. Browne, M.W. (1984). Asymptotically Distribution-free Methods in the Analysis of Covariance Structures. British Journal of Mathematical and Statistical Psychology, 37, Browne, M.W. (1987). Robustness in Statistical Inference in Factor Analysis and Related Models. Biometrika, 74, Du Toit, S.H.C. & Du Toit, M. (2001). Multilevel Structural Equation Modeling. In I.G.G. Kreft, & J. de Leeuw (Eds.), Multilevel Modeling, in preparation. Du Toit, M. & Du Toit, S.H.C. (2001). Interactive LISREL User s Guide. Lincolnwood, IL: Scientific Software International, Inc. Jöreskog, K. & Sörbom, D. (1996a). PRELIS 2: User s Reference Guide. Chicago, IL: Scientific Software International, Inc. Jöreskog, K. & Sörbom, D. (1996b). LISREL 8: User s Reference Guide. Chicago, IL: Scientific Software International, Inc. Jöreskog, K. & Sörbom, D. (1996c). Structural Equation Modeling with the SIMPLIS Command Language. Chicago, IL: Scientific Software International, Inc. Jöreskog, K. & Sörbom, D. (2001). The Student Edition of LISREL 8.51 for Windows [Computer Software]. Lincolnwood, IL: Scientific Software International, Inc. Jöreskog, K., Sörbom, D., Du Toit, S.H.C. & Du Toit, M. (1999). LISREL 8: New Statistical Features. Chicago, Illinois: Scientific Software International, Inc. Mortimore, P., Sammons, P., Stoll, L., Lewis, D. & Ecob, R. (1988). School Matters: The Junior Years. Wells: Open Books. Satorra, A. & Bentler, P.M. (1988). Scaling Corrections for Chi-square Statistics in Covariance Structure Analysis. Proceedings of the Business and Economic Statistics Section of the American Statistical Association, 1988,