4.3 CFA with Categorical Indicators using Mplus
Mplus cannot directly read data from an SPSS system file. Raw data must come from a free or fixed format text file. See sections 2.4 and 3.4 for syntax to translate an SPSS file to an ASCII file. For this example the data is saved as the tab-delimited file values_ord.dat in the C:\temp\CFA folder.
After launching Mplus the syntax editor appears. The following commands are used to estimate the confirmatory factor model with ordinal observed variables.
TITLE: Factor Analysis with Categorical Outcome Variables;
DATA FILE IS values_ord.dat;
VARIABLE: NAMES ARE privtown govtresp compete
homosex abortion euthanas;
CATEGORICAL ARE privtown govtresp
compete homosex abortion euthanas;
MODEL: economic BY privtown govtresp compete;
morals BY homosex abortion euthanas govtresp;
The syntax is similar to previous sections except that an extra line is added to the VARIABLE statement to define the observed variables as categorical. The default in Mplus is to assume all observed variables are continuous unless specified otherwise. When categorical indicators are declared Mplus employs by default a robust weighted least squares estimator similar to the Diagonally Weighted Least Squares estimator in LISREL. After clicking on Run
to carry out the estimation, a text output file is produced. A selection of the output file is the following:
Under the Model Results heading the unstandardized loadings appear along with standard errors, the ratio of the estimates to their standard errors, and two standardized estimates. The Est./S.E. column can be used to evaluate significance. If the absolute value of the number in this column is greater than 1.96 the estimate can be interpreted as significant at the .05 level. In this case all of the unconstrained loadings estimates are significant.
The column StdYX is comparable to the standardized estimates provided by LISREL. GOVTRESP has relatively low standardized loadings on both factors (.252 for ECONOMIC and .203 for MORALS). For PRIVTOWN the loading is .645, for COMPETE it is .797, for HOMOSEX it is .690, for ABORTION it is .844, and for EUTHANAS it is .706.
The squared multiple correlations provide information on how much variance the factors account for in the observed variables. Despite receiving a path from both latent variables, GOVTRESP has a low R2 of only .101. The remaining R2 statistics are, in order of increasing magnitude, PRIVTOWN (.416), HOMOSEX (.476), EUTHANAS (.498), COMPETE (.636), and ABORTION (.713). Finally, the correlation between the two common factors is a very small -.034, and the covariance estimate of -.015 is not statistically distinguishable from zero.