A Dimension-Reduction Method for Detecting Non-Multinormality in Two-Level Structural Equation Models

Abstract

Testing multinormality in two-level structural equation models (SEMs) presents a fundamental challenge because observations from the same level-2 unit are correlated, violating the independence assumption required by classical normality tests. In this paper, we develop a novel generalized Shapiro–Wilk (GW) test that explicitly accounts for this dependence. The proposed method rearranges the dependent observations into a random matrix and employs principal component analysis (PCA) to project this matrix onto a set of principal directions, achieving effective dimension reduction. On each projected direction, the scale-invariant Shapiro–Wilk statistic is applied to test for sphericity, leveraging the property that spherical distributions preserve the null distribution of such statistics. The Johnson SB-transform is then used to approximate the null distribution of the combined test statistic. A Monte Carlo study demonstrates that the proposed GW test controls type I error rates satisfactorily and exhibits strong power against a range of non-normal alternatives, including heavy-tailed and asymmetric distributions. The method is further illustrated using real alcohol use data from nested families, highlighting its practical utility. Comparative evaluation indicates that the GW test performs favorably relative to a recently proposed approach. The procedure is applicable to balanced level-1 designs and provides researchers with a necessary diagnostic tool for assessing multinormality assumptions in two-level SEMs.