Contextualized Personality Questionnaires: A Case for Copulas in Structural Equation Models for Categorical Data

For structural equation models (SEMs) with categorical data, correlated measurement residuals are not easily implemented. The problem lies mainly in the absence of a categorical analogue to the multivariate normal distribution and the absence of closed form formulas in SEMs for categorical data. We present a novel technique to handle measurement residuals that keeps the attractive SEM mainframe intact yet adds flexibility in dependence modeling without excessive computational burden. The technique is based upon the concept of copula functions and is introduced with a data set of ordinal responses originating from a contextualized personality study on aggression. Focus is on models arising in a multitrait-multimethod context, where the flexibility in dependence structures allows for method effects that can vary across the latent trait dimension. The empirical application illustrates that ignoring design-implied correlated measurement residuals can potentially influence study results and conclusions in both a quantitative as well as a qualitative way. Model parameter estimates can be biased, but more important, model inferences can be heavily distorted.