In this paper test criteria are developed for testing hypotheses of "compound symmetry" in a normal multivariate population of $t$ variates $(t \geq 3)$ on basis of samples. A feature common to the twelve hypotheses considered is that the set of $t$ variates is partitioned into mutually exclusive subsets of variates. In regard to the partitioning, the twelve hypotheses can be divided into two contrasting but very similar types, and the six in one type can be paired off in a natural way with the six in the other type. Three of the hypotheses within a given type are associated with the case of a single sample and moreover are simple modifications of one another; the remaining three are direct extensions of the first three, respectively, to the case of $k$ samples $(k \geq 2)$. The gist of any of the hypotheses is indicated in the following statement of one, denoted by $H_1(mvc)$: within each subset of variates the means are equal, the variances are equal and the covariances are equal and between any two distinct subsets the covariances are equal. The twelve sample criteria for testing the hypotheses are developed by the Neyman-Pearson likelihood-ratio method. The following results are obtained for each criterion (assuming that the respective null hypotheses are true) for any admissible partition of the $t$ variates into subsets and for any sample size, $N$, for which the criterion's distribution exists: (i) the exact moments; (ii) an identification of the exact distribution as the distribution of a product of independent beta variates; (iii) the approximate distribution for large $N$. Exact distributions of the single-sample criteria are given explicitly for special values of $t$ and special partitionings. Certain psychometric and medical research problems in which hypotheses of compound symmetry are relevant are discussed in section 1. Sections 2-6 give statements of the hypotheses and an illustration, for $H_1(mvc)$, of the technique of obtaining the moments and identifying the distributions. Results for the other criteria are given in sections 7-8. Illustrative examples showing applications of the results are given in section 9.
"Testing Compound Symmetry in a Normal Multivariate Distribution." Ann. Math. Statist. 19 (4) 447 - 473, December, 1948. https://doi.org/10.1214/aoms/1177730145