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

Abstract

a b s t r a c tLikelihood 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, is derived for a common case. In this paper the separability of a covariance structure with the first component as a compound symmetric correlation matrix under the assumption of multivariate normality is tested. For this purpose Monte Carlo simulation studies, which compare the biased LRT to biased RST, and unbiased/unmodified LRT to unbiased/unmodified RST, are conducted. It is shown that the RSTs outperform their corresponding LRTs in the sense of empirical Type I error as well as empirical null distribution. Moreover, since the RST does not require estimation of a general variance-covariance matrix (the alternative hypothesis), RST can be performed for small sample sizes, where the variance-covariance matrix could not be estimated for the corresponding LRT, making the LRT infeasible. Three examples are presented to illustrate and compare statistical inference based on LRT and RST.

A comparison of likelihood ratio tests and Rao's score test for three separable covariance matrix structures

“…After differentiating the log‐likelihood function and plugging in MLEs, we obtain the RST statistic to test the null hypothesis , , in , , or as which has an asymptotic distribution, where the df ν values are given in and when Ψ has UN and AR(1) or CS structure, respectively; see Filipiak et al. () for details. Remark Observe that the above RST statistic does not depend on the MLE of the unstructured variance‐covariance matrix Ω, as the LRT statistic does.…”

“…Roy and Khattree () and Filipiak et al. (). Hypothesis testing on a UN ⊗ UN structure has been studied by many authors using unmodified LRT‐based tests (Lu and Zimmerman, ; Roy, ; Srivastava et al., ), or modified LRT‐based tests (Mitchell et al., ; Simpson, ) to match the theoretical distribution with ν degrees of freedom (df).…”

“…In turn, Filipiak et al. () showed that, for testing the hypothesis of CS ⊗ UN, the RST is much less biased and good to use for small samples also, even with sample size smaller than (which is not possible for the LRT). Thus, one of the aims of this paper is to verify the advantages of using the RST when testing UN ⊗ UN and AR(1) ⊗ UN hypotheses.…”

“…Lu and Zimmerman (), Roy and Leiva (), and very recently Filipiak et al. () derived ENDs of the LRT statistic for testing 2 − separable covariance structure with both unstructured, both structured, and half‐structured and half‐unstructured components, respectively. One can see clearly from these papers that the ENDs of the LRT statistic are far more different (biased) from their limiting χ 2 distributions for samples that are small but of size greater than .…”

“…All of the results of simulation studies, except the empirical percentiles of the null distribution of the LRT and RST statistics with 38 df for testing separability with the first component as the CS correlation matrix, which can be found in Filipiak et al. (), are presented in this paper in Appendix A and Supplementary Material A. In Section both test procedures are implemented on the rhesus monkey data.…”