COMPARATIVE EVALUATION OF SPHERICITY TESTS UNDER NORMAL AND HEAVY-TAILED MULTIVARIATE DISTRIBUTIONS

Abstract

This study systematically evaluates the performance of four sphericity tests—Mauchly’s Test, the Traditional Likelihood Ratio Test (LRT), John’s Invariant Test, and the Quasi-LRT—across varying dimensionalities (p = 2 to 5), sample sizes, and underlying data distributions. Through extensive simulation under both multivariate normal and multivariate t-distributions, we assess the empirical Type I error rates and statistical power of each test to provide comprehensive insights into their practical reliability and robustness.

Under multivariate normal conditions, Mauchly’s test and the Traditional LRT generally maintain nominal Type I error rates and achieve high power for moderate-to-large samples and low dimensions. However, both exhibit inflated Type I error and instability in small samples and higher dimensions, with the LRT particularly vulnerable when eigenvalues approach zero. John’s Invariant Test consistently demonstrates strong power and controlled Type I error across most scenarios, outperforming others under deviations from normality. The Quasi-LRT shows promising power in large samples and high-dimensional contexts but suffers from substantial Type I error inflation in small samples, especially under heavy-tailed distributions.

When applied to heavy-tailed multivariate t-distributed data, all tests experience degradation in Type I error control, with Mauchly’s and the Traditional LRT exhibiting increased liberalness in small samples. In contrast, John’s Test and the Quasi-LRT display relative robustness, though none fully maintain nominal error rates. Power analyses reveal that John’s and the Quasi-LRT tests retain strong sensitivity across distributions, while the LRT’s performance is notably erratic under non-normal conditions.

Our findings highlight the nuanced trade-offs between Type I error control and power across testing procedures, emphasizing that no single test is universally optimal. Practitioners are advised to consider sample size, dimensionality, and distributional assumptions when selecting sphericity tests, favoring John’s Invariant Test or the Quasi-LRT under non-normal or small-sample conditions. Future research should explore bootstrap and permutation methodologies to enhance reliability, particularly in challenging scenarios.