Pernicious Polychorics: The Impact and Detection of Underlying Non-normality

Abstract

Ordinal data in social science statistics are often modeled as discretizations of a multivariate normal vector. In contrast to the continuous case, where SEM estimation is also consistent under non-normality, violation of underlying normality in ordinal SEM may lead to inconsistent estimation. In this article, we illustrate how underlying non-normality induces bias in polychoric estimates and their standard errors. This bias is strongly affected by how we discretize. It is therefore important to consider tests of underlying multivariate normality. In this study we propose a parametric bootstrap test for this purpose. Its performance relative to the test of Maydeu-Olivares is evaluated in a Monte Carlo study. At realistic sample sizes, the bootstrap exhibited substantively better Type I error control and power than the Maydeu-Olivares test in ordinal data with ten dimensions or higher. R code for the bootstrap test is provided.