# Structural Equation Modeling Using LISREL

This page describes how to fit a full structural equation model in LISREL. Details on the model, and a link to download the data, can be found here. Estimation was done using the free student version of LISREL, which can be downloaded here.

The previous page on confirmatory factor analysis showed how to read in the raw data from an SPSS file (click here to review).

The LISREL or SIMPLIS syntax used to estimate the model can be created by drawing a path diagram. To do this, go to **File → New** and scroll down to choose **Path Diagram**.

When prompted, name the path diagram **Full SEM**. An empty drawing area opens.

To establish the variable names for the model, go to **Setup → Title and Comments**.

The **Title and Comments** dialog box opens. Enter the title **Full SEM**.

Click **Next**. A new dialog box titled **Group Names** opens.

Because the purpose is not to compare different groups, this can be left blank. Click **Next**.

The **Labels** dialog box opens, which is used to identify the names of the variables in the model.

To specify the observed variables from the raw data file, click on **Add/Read Variables**. A new box opens. Change the **Read from file** option to **PRELIS System File**, then browse to find the PRELIS file that was created from the PASW file.

Click **OK**. The names of the variables from the data file will appear in the **Observed Variables** column. To name the latent variables, click **Add Latent Variables**. The first latent variable will be named **Intelligence**.

The full SEM will also contain a latent variable for humor, so click **Add Latent Variables** a second time to create an additional variable named **Humor**.

Click **OK**. A final dialog box opens to describe the data.

Because LISREL will be reading from a raw data file, it is not necessary to include any information about the number of observations. Click **OK** one more time.

The drawing pad appears again. In the model, humor is treated as endogenous (that is, it is caused by intelligence). Thus, it is necessary to identify humor as an Eta variable by clicking in the box next to the variable name on the left side of the screen. In addition, it is necessary to define the observed variables which correspond to humor as Y variables in the same manner.

Drag the variables onto the drawing pad. (Note: If you get error messages, change the **Models: ** drop-down menu to **Basic Model**.) Then draw arrows from **Intelligence** to its indicators, **Humor** to its indicators, and from **Intelligence** to **Humor**. If Humor is not declared to be endogenous (an Eta variable), LISREL will not let you draw the path.

The model is not identified unless additional assumptions are introduced. By default, LISREL will assume that the variances for the two latent variables equal one. In this example, the model will be identified by assuming the variance of Intelligence is one, but the scale for Humor will be set by constraining one of the factor loadings. This can be done by clicking on the loading for Simpsons and changing it to one. LISREL won’t recognize this change, however, until the user then right-clicks on the constraint and chooses **Fix**.

To build the SIMPLIS syntax that will estimate the model, go to **Setup → Build SIMPLIS Syntax**. LISREL’s syntax editor opens and displays the SIMPLIS syntax.

Click the **Run LISREL** button to run the model.

An output file is created (.OUT extension) which contains the model estimates, standard errors, and a number of model fit indices. In addition, the results now display in the path diagram. It is possible to toggle between different types of estimates using the **Estimates** pull down menu.

The results look like the following:

More information on fit indices and the specific model parameters can be found in the model’s output file (.OUT extension) written to the working directory. The output for this model looks like the following:

```
DATE: 2/11/2010
TIME: 13:18
L I S R E L 8.80
BY
Karl G. Joreskog & Dag Sorbom
This program is published exclusively by
Scientific Software International, Inc.
7383 N. Lincoln Avenue, Suite 100
Lincolnwood, IL 60712, U.S.A.
Phone: (800)247-6113, (847)675-0720, Fax: (847)675-2140
Copyright by Scientific Software International, Inc., 1981-2006
Use of this program is subject to the terms specified in the
Universal Copyright Convention.
Website: www.ssicentral.com
The following lines were read from file D:My DocumentsSEMfullsem.SPJ:
Full SEM
Raw Data from file 'D:My DocumentsSEMINTELLIGENCE.psf'
Latent Variables humor intelligence
Relationships
SIMPSONS = 1.00*humor
FAMILYGU = humor
AMERDAD = humor
READING = intelligence
WRITING = intelligence
MATH = intelligence
ANALYTIC = intelligence
humor = intelligence
Path Diagram
End of Problem
Sample Size = 100
Full SEM
Covariance Matrix
SIMPSONS FAMILYGU AMERDAD READING WRITING MATH
-------- -------- -------- -------- -------- --------
SIMPSONS 0.69
FAMILYGU 0.58 0.85
AMERDAD 0.63 0.64 0.84
READING 0.21 0.24 0.19 1.12
WRITING 0.14 0.18 0.15 0.82 0.93
MATH 0.28 0.27 0.29 0.84 0.77 1.00
ANALYTIC 0.23 0.24 0.21 0.82 0.71 0.80
Covariance Matrix
ANALYTIC
--------
ANALYTIC 0.95
Full SEM
Number of Iterations = 5
LISREL Estimates (Maximum Likelihood)
Measurement Equations
SIMPSONS = 1.00*humor, Errorvar.= 0.12 , Rý = 0.83
Standerr (0.032)
Z-values 3.66
P-values 0.000
FAMILYGU = 1.02*humor, Errorvar.= 0.26 , Rý = 0.69
Standerr (0.091) (0.047)
Z-values 11.21 5.60
P-values 0.000 0.000
AMERDAD = 1.10*humor, Errorvar.= 0.15 , Rý = 0.82
Standerr (0.086) (0.039)
Z-values 12.81 3.85
P-values 0.000 0.000
READING = 0.94*intellig, Errorvar.= 0.23 , Rý = 0.80
Standerr (0.084) (0.044)
Z-values 11.24 5.24
P-values 0.000 0.000
WRITING = 0.84*intellig, Errorvar.= 0.22 , Rý = 0.76
Standerr (0.078) (0.040)
Z-values 10.85 5.57
P-values 0.000 0.000
MATH = 0.91*intellig, Errorvar.= 0.16 , Rý = 0.84
Standerr (0.078) (0.035)
Z-values 11.74 4.66
P-values 0.000 0.000
ANALYTIC = 0.86*intellig, Errorvar.= 0.20 , Rý = 0.79
Standerr (0.078) (0.038)
Z-values 11.13 5.35
P-values 0.000 0.000
Structural Equations
humor = 0.24*intellig, Errorvar.= 0.51 , Rý = 0.10
Standerr (0.080) (0.091)
Z-values 3.01 5.63
P-values 0.003 0.000
Correlation Matrix of Independent Variables
intellig
--------
1.00
Covariance Matrix of Latent Variables
humor intellig
-------- --------
humor 0.57
intellig 0.24 1.00
Goodness of Fit Statistics
Degrees of Freedom = 13
Minimum Fit Function Chi-Square = 13.68 (P = 0.40)
Normal Theory Weighted Least Squares Chi-Square = 13.93 (P = 0.38)
Estimated Non-centrality Parameter (NCP) = 0.93
90 Percent Confidence Interval for NCP = (0.0 ; 14.21)
Minimum Fit Function Value = 0.14
Population Discrepancy Function Value (F0) = 0.0094
90 Percent Confidence Interval for F0 = (0.0 ; 0.14)
Root Mean Square Error of Approximation (RMSEA) = 0.027
90 Percent Confidence Interval for RMSEA = (0.0 ; 0.11)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.60
Expected Cross-Validation Index (ECVI) = 0.44
90 Percent Confidence Interval for ECVI = (0.43 ; 0.58)
ECVI for Saturated Model = 0.57
ECVI for Independence Model = 6.53
Chi-Square for Independence Model with 21 Degrees of Freedom = 632.67
Independence AIC = 646.67
Model AIC = 43.93
Saturated AIC = 56.00
Independence CAIC = 671.90
Model CAIC = 98.00
Saturated CAIC = 156.94
Normed Fit Index (NFI) = 0.98
Non-Normed Fit Index (NNFI) = 1.00
Parsimony Normed Fit Index (PNFI) = 0.61
Comparative Fit Index (CFI) = 1.00
Incremental Fit Index (IFI) = 1.00
Relative Fit Index (RFI) = 0.96
Critical N (CN) = 197.84
Root Mean Square Residual (RMR) = 0.030
Standardized RMR = 0.034
Goodness of Fit Index (GFI) = 0.96
Adjusted Goodness of Fit Index (AGFI) = 0.92
Parsimony Goodness of Fit Index (PGFI) = 0.45
Time used: 0.016 Seconds
```

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