4. Confirmatory Factor Analysis with Categorical Indicators

The maximum likelihood estimation approaches used in the previous two sections relied on the rather strong assumption of multivariate normality. In practice a substantial amount of social science data is non-normal. Survey responses are often coded as yes/no or as scores on an ordered scale (e.g. strongly disagree, disagree, neutral, agree, strongly agree). In the presence of categorical or ordinal data alternative estimators are more appropriate.

This section continues to use the political values example developed above, but with scores recoded on a three point scale. Original responses ranging from 1 to 3 were recoded as 1; those ranging from 4 to 7 were recoded as 2; and those ranging from 8 to 10 were recoded as 3. Initially observations with missing data are dropped (40 of the original 1200) in order to focus only on the problem of categorical outcome variables in the confirmatory factor model. However, the final subsection will show how to request pairwise rather than listwise deletion in Mplus so as to use as much information in the raw data file as possible. This section does not consider Amos, which lacks a means for estimating the polychoric correlation matrix used by the other two programs in the presence of categorical data. The assumption in this section continues to be, as in the previous examples, that the latent variables represent continuous (not categorical) concepts.

Mplus and LISREL employ a multi-step method for ordinal outcome variables that analyzes a matrix of polychoric correlations rather than covariances. This approach works as follows: 1) thresholds are estimated by maximum likelihood, 2) these estimates are used to estimate a polychoric correlation matrix, which in turn is used to 3) estimate parameters through weighted least squares using the inverse of the asymptotic covariance matrix as the weight matrix (Muthén, 1984; Jöreskog, 1990). In Mplus these steps take place automatically when the syntax includes a line identifying outcomes as categorical. In LISREL the polychoric correlation matrix and asymptotic covariance matrix must first be estimated using PRELIS. This will produce a LISREL system file (.dsf) containing both the polychoric correlations and information about where the covariance matrix is saved. The information in the .dsf file is then used by LISREL when the weighted least squares estimator is requested.