4.2 CFA with Categorical Indicators using LISREL
Launch LISREL and load the data by choosing File → Import External Data in Other Formats. Navigate to the C:\temp\CFA folder, choose the file values_ord.sav, and click Open. When prompted, enter the name values_ord in the File Name field and click Save. The data will then be displayed in spreadsheet form. It is possible to define the variables in the dataset as ordinal by going to Data → Define Variables, highlighting all the variables, and opening the Variable Type menu.
The next step is to estimate the polychoric correlation and asymptotic covariance matrices. Go to Statistics → Output Options. In the dialog box that opens, choose Correlations from the Moment Matrix drop-down menu and click in the empty box next to LISREL system file. Place a check in the box next to Save to file under Asymptotic Covariance Matrix and enter the name values.acm.
Click OK to save both the LISREL system file and the asymptotic covariance matrix to the working directory, in this case C:\temp\CFA.
The next step is to create the path diagram of the confirmatory factor model. Click on File → New and choose Path diagram. When prompted, save the diagram. Give it the name values_ord and click Save. An empty drawing space will open.
Go to Setup → Titles and Comments to give the model a name. In the box that appears, enter “CFA of Ordinal Data” in the Title field.
Click Next. The Group Names box opens but can be ignored because the model will be tested on only a single sample. Click Next again. The Labels box now displays.
The names of the observed variables are saved in the LISREL system file created previously. Click the Add/Read Variable button to access them.
In the Add/Read Variables box make sure the Read from file radio button is selected and LISREL System File is chosen from the drop-down menu. Browse to the location where the LISREL System File was saved, choose values_ord.dsf, and click OK. The observed variable names will then appear in the Labels box. To add the names of the latent variables click on Add Latent Variables. In the box that appears enter Economic and click OK. Do the same to add the label Morals.
When finished, choose Next. The Data dialog box then appears. Choose Correlations in the Statistics from drop-down menu and make sure LISREL System Data is chosen under File type.
Click OK. Next, create the path diagram by dragging the names of each of the observed and latent variables to the drawing space. As in the previous examples, add single-headed arrows pointing from the latent variables to the observed variables and a double-headed arrow between ECONOMIC and MORALS. Finally, set the scale of the latent variables by constraining the path from ECONOMIC to PRIVTOWN and from MORALS to HOMOSEX to one. This is done by double-clicking the weight, changing it to 1.00, right-clicking, and choosing Fix. When finished the path diagram should look like the following.
To request the weighted least squares estimator, go to Output → SIMPLIS Outputs. Choose Diagonally Weighted Least Squares.
Click OK. To create the SIMPLIS syntax based on the path diagram, go to Setup → Build SIMPLIS syntax. The SIMPLIS syntax editor will open displaying the commands needed for estimation.
Click the Run LISREL
. The unstandardized solution appears in the path diagram along with χ2
and RMSEA statistics to assess model fit. To view the standardized estimates, choose Standardized Solution
from the Estimates
Additional information can be read from the text output file values_ord.out automatically saved in the working directory.
Under the Measurement Equations heading appear the unstandardized estimates, standard errors, t-values, and R2 statistics. Statistical significance can be assessed by looking at the size of the standard errors (in parentheses) relative to the unstandardized parameter estimates. When the latter are twice as large as the former the estimates can be considered significant at the .05 level. For this example all of the unconstrained path coefficients are significant. In addition, the (Satorra-Bentler Scaled) χ2 statistic of 6.23 (df=7) has a corresponding p-value of .51. This is not large enough to reject the null of a good fit. The RMSEA is 0.00, indicating a very good model fit.
The GOVTRESP variable has a relatively weak standardized loading on both ECONOMIC (.25) and MORALS (.20). The remaining variables, however, have moderate to strong loadings on the respective common factor. The standardized loading for PRIVTOWN is .65; for COMPETE it is .80; for HOMOSEX it is .69; for ABORTION it is .84; and for EUTHANAS it is .70. The R2 statistics listed in the output file are interpreted as the amount of variance in the observed variables accounted for by the latent variables. Despite receiving a path from both ECONOMIC and MORALS, the GOVTRESP has the smallest R2 (.10). The other observed variables have moderate to high R2 statistics. The multiple correlation coefficient for PRIVTOWN is .42; for COMPETE it is .63; for HOMOSEX it is .48; for ABORTION it is .71; and for EUTHANAS it is .50. Finally, the correlation between the two factors is -.03. Comparing the covariance estimate to its standard error in the output, this is not significant.