Score test for a separable covariance structure with the first component as compound symmetric correlation matrix Score test for a separable covariance structure with the first component as compound symmetric correlation matrix

Likelihood ratio tests (LRTs) for separability of a covariance structure for doubly multivariate data are widely studied in the literature. There are three types of LRT: biased tests based on an asymptotic chi-square null distribution; unbiased/unmodified tests based on an empirical null distribution; and unbiased/modified tests with a test statistic modified to follow a theoretical chi-square null distribution. The Rao’s score test (RST) statistic, an alternative for both biased and unbiased/unmodified versions of the corresponding LRT test statistics are derived for a common case. The separability of a covariance structure with the first component as a compound symmetric correlation matrix under the assumption of multivariate normality is tested. Simulation studies compare the biased LRT to biased RST, and unbiased/unmodified LRT to unbiased/unmodified RST. The RSTs outperform their corresponding LRTs in general. Three examples are presented. Since the RST does not require estimation of a general variance-covariance matrix (the alternative hypothesis), this test can be performed for small sample sizes, where the variance-covariance matrix could not be estimated for the corresponding LRT, making the LRT infeasible. In cases where both LRT and RST are feasible, the RST outperforms a comparable LRT.