2.3 CFA using LISREL
This section demonstrates how to estimate a confirmatory factor model using LISREL. Because the previous subsection revealed that the single common factor model was a poor fit to the data, this section will begin with the two factor model.
LISREL can be launched from any computer running Windows in the UITS Student Technology Centers by going to Start → All Programs → Statistical Software → LISREL 8.80 → LISREL 8.80.
The easiest way to get LISREL to analyze raw data is to import the data file and save it as a .psf (PRELIS system) file. PRELIS, the pre-processor to LISREL, can read data files from a number of statistical programs, including SPSS. To open the SPSS file values.sav saved in the C:\temp\CFA folder, go to File → Import External Data in Other Formats. The Open dialog box displays. Change Files of Type to SPSS Data File(*.sav), navigate to the correct folder, and click on values.sav.
Click Open. You will then be prompted to save the data as a .psf file. Name the file values and click Save to store it in the working directory (currently C:/temp/CFA). A spreadsheet with the raw data will display.
Like Amos, LISREL allows you to construct a path diagram of the model you wish to estimate. Go to File → New and choose Path Diagram. You will be immediately prompted to save the path diagram. Name the file values and click Save (the .pth extension will differentiate this file from the other files named values in the working directory). An empty window opens where you will eventually draw the diagram.
The next step is to name the variables that will be in the model. Go to Setup → Title and Comments to open the Title and Comments dialog box. Enter Two Factor Model in the Title field and click Next.
The Group Names box opens, which is used to label different groups when comparing models for multiple independent samples. Because we are interested only in the single sample of American respondents, we can skip this box by clicking Next.
The Labels dialog box then opens, which is used to identify the latent and observed variables to be analyzed.
Currently no variables have been selected. To choose variable names, click on Add/Read Variables. This opens a new dialog box used to locate the PRELIS system file. Verify that the Read from file radio button is chosen and pick PRELIS System File from the drop-down menu. Then click Browse to choose the PRELIS system file created earlier.
The names of the observed variables are now listed in the Labels box. To add the names of the latent variables, click Add Latent Variables. Enter ECONOMIC in the box that opens. Repeat to enter the name MORALS for the second common factor. Click OK.
Click Next, and a final dialog box opens.
Raw data from a PRELIS system file will be analyzed. If desired, the data can be viewed and edited by clicking on the Edit button. Because this system file already contains information about the sample size, it is not necessary to make further changes. Click OK.
It is now possible to begin drawing the path diagram. The names of the observed and latent variables appear on the left side of the screen. Drag all of the observed variables to the drawing pad along with the latent variables ECONOMIC and MORALS.
Next, click on the single-headed arrow on the tool bar and connect the ECONOMIC factor to PRIVTOWN, GOVTRESP, and COMPETE. Also draw arrows from MORALS to HOMOSEX, ABORTION, and EUTHANAS. Because the usual assumption is that the latent variables “cause” the observed variables, the arrows should point towards the six indicators. Finally, draw a two-headed arrow connecting each latent variable.
Unlike Amos, it is not necessary to draw the unique factors representing measurement error for each of the observed variables. LISREL includes these by default and automatically sets their scales by constraining the loadings to one. To set the scale of ECONOMIC, constrain the regression weight of the PRIVTOWN variable to one. Double-click on the line at the point where 0.00 appears and change the loading to 1.00. LISREL will not recognize this constraint, however, unless you then right-click on the loading and choose Fix.
Do the same for the path connecting MORALS to HOMOSEX to set the metric for the second common factor.
The final step before estimation is to build from the path diagram the corresponding syntax LISREL uses for estimation. There are actually two languages that LISREL understands: LISREL syntax and SIMPLIS syntax. As its name suggests, SIMPLIS is more straightforward and easy to read than LISREL syntax. A SIMPLIS syntax file can be built from the path diagram by choosing Setup → Build SIMPLIS syntax. This opens an editor displaying the SIMPLIS commands needed to estimate the model.
To begin estimation, click the Run LISREL
. The unstandardized estimates will then appear in the path diagram by default. To view the standardized estimates, choose Standardized Solution
from the Estimates
The path diagram then will look like the following:
The unstandardized estimates, standardized estimates, t-values, and modification index information can all be obtained by choosing the appropriate option from the Estimates drop-down menu. Alternatively, each time the Run LISREL button is clicked a text output file is written to the working directory (extension .out) which contains additional information. It is always a good idea to inspect the output file for any error messages and, in some cases, warnings that a model may not be identified. For this model the output file is the following:
The χ2 statistic for model fit is 35.3 (df=8), which is large enough to reject the null that the model is a good fit to the data (the path diagram displays the Normal Theory Weighted Least Squares χ2; to be consistent with the output from Amos this example considers the Minimum Fit Function χ2 printed in the output). In addition, the Root Mean Square Error of Approximation is .054. Using a cut-off rule of .05, the RMSEA is too high to indicate a good fit.
Because each variable receives only a single common factor loading, the standardized loadings represent the correlation between each observed variable and the corresponding factor. Considering first the indicators of ECONOMIC, the standardized loadings are .59 for PRIVOWN, .15 for GOVTRESP, and .71 for COMPETE. Considering the indicators of MORALS, the standardized loadings are .63 for HOMOSEX, .79 for ABORTION, and .66 for EUTHANAS. It is possible to ascertain the statistical significance of the estimates by comparing the unstandardized loadings displayed in the equations under the Measurement Equations heading in the output file with their standard errors displayed in parentheses. When the unstandardized loadings are at least twice the size of the standard errors, the estimates are significant at the .05 level. In this case each of the unconstrained estimates is significant.
A good deal of the variance in each observed variable, with the exception of GOVTRESP, is accounted for. The R2 for PRIVTOWN is .35; for COMPETE it is .50; for HOMOSEX it is .40; for ABORTION it is .63; and for EUTHANAS it is .44. Only GOVTRESP, with its R2 of .022, does not fit in well with the model. It may be the case that this survey question taps some kind of value dimension distinct from the economic dimension measured by the PRIVTOWN and COMPETE variables.
LISREL reports modification indices, both in the path diagram (by choosing Modification Indices from the Estimation menu) and in the output. These numbers offer suggestions for improving the overall model fit. Two recommendations are given in the output: add an error covariance between HOMOSEX and GOVTRESP or add a path from GOVTRESP to MORALS. Both of these suggest that the GOVTRESP item has something in common with the morality dimension, either by sharing measurement error with the HOMOSEX variable or as a direct indicator of the latent morality dimension.
Because the standardized loading of GOVTRESP on ECONOMIC was low, it is possible that the item is actually tapping a different values dimension. One final model is therefore estimated adding a path from MORALS to GOVTRESP. This results in the following standardized solution:
The (abbreviated) output is the following:
This model fits the data well. The χ2 measure of model fit is 7.93 (df=7), which is too small to reject the null of a good fit (p=.34). Additionally, the RMSEA has declined to .011, which is small enough (below .05) to indicate a good fit.
The unconstrained loadings are all statistically significant at the .05 level, having estimates that are more than twice the size of their standard errors. GOVTRESP continues to have a low standardized loading on the ECONOMIC factor (.15) and has a similarly low standardized loading on MORALS (.18). However, the remaining standardized loadings range from .62 (PRIVTOWN) to .79 (ABORTION). In between are HOMOSEX (.64), EUTHANAS (.67), and COMPETE (.68).
Despite receiving a path from both common factors, GOVTRESP continues to have by far the smallest R2 (.053). The remaining variables are moderately well accounted for by the corresponding factors. The R2 values are, in order of increasing magnitude, .38 for PRIVTOWN, .41 for HOMOSEX, .44 for EUTHANAS, .46 for COMPETE, and .62 for ABORTION. Finally, the correlation between ECONOMIC and MORALS is a negligible -.01 and, for this sample, is statistically indistinguishable from zero (covariance = -.03, std. error = .13).
The conclusion from this analysis is that two nearly orthogonal dimensions underlie the economic and moral values of American citizens. Additionally, it is unclear whether the GOVTRESP item is tapping either dimension. Future surveys should incorporate more reliable measures of economic values.