Testing Categorized Bivariate Normality With Two-Stage Polychoric Correlation Estimates

Structural equation modeling (SEM) with ordinal indicators rely on an assumption of categorized normality. This assumption may be tested for pairs of variables using the likelihood ratio G 2 or Pearson’s X 2 statistics. For increased computational efficiency, SEM programs usually estimate polychoric correlations in two stages. However, two-stage polychoric estimates are not asymptotically efficient and G 2 and X 2 need not be asymptotically chi-square when the estimator is not efficient. Recently, Maydeu-Olivares and Joe (2005) have introduced a new statistic, M n , that is asymptotically chi-square even for estimators that are not efficient. We investigate the behavior of G 2 , X 2 , and M n when testing underlying bivariate normality with polychoric correlations estimated in two stages.