Semi-Nonparametric Methods for Detecting Latent Non-normality: A Fusion of Latent Trait and Ordered Latent Class Modeling.

Ordered latent class analysis (OLCA) can be used to approximate unidimensional latent distributions. The main objective of this study is to evaluate the method of OLCA in detecting non-normality of an unobserved continuous variable (i.e., a common factor) used to explain the covariation between dichotomous item-level responses. Using simulation, we compared a model in which probabilities of class membership were estimated to a restricted submodel in which class memberships were fixed to normal Gauss-Hermite quadrature values. Our results indicate that the likelihood ratio statistic follows a predictable chi-square distribution for a wide range of sample sizes (N = 500-12,000) and test instrument characteristics, and has reasonable power to detect non-normality in cases of moderate effect sizes. Furthermore, under situations of large sample sizes, large numbers of items, or centrally located item difficulties, simulations suggest that it may be possible to describe the shape of latent trait distributions. Application to data on the symptoms of major depression, assessed in the National Comorbidity Survey, suggests that the latent trait does not depart from normality in men but does so to a small but significant extent in women.