#StackBounty: #repeated-measures #covariance-matrix #sphericity What is the origin of the Winer's compound symmetry test of the var…

Bounty: 100

In his famous book, Winer (Statistical principles in experimental design 1971; reedited Winer, Brown, Michels, 1991, p. 517) introduced a test of compound symmetry. The test expands on the likelihood ratio test of Wilks (1934) with a correction factor $C$ brought to the likelihood ratio. The whole test is

$M = -(N-1) lnBig(frac{vert S_1vert}{vert S_0vert} Big) $, $C = frac{q(q+1)^2(2q-3)}{6(N-1)(q-1)(q^2+q-4)}$ and $nu = q(q+1)/2-2$

in which $N$ is the sample size, $q$ is the number of repeated measures, $S_0$ is the observed covariance matrix, $S_1$ is the predicted matrix under compound symmetry and $vert cdot vert$ denotes the determinant of a matrix, such that $(1-C) times M sim chi^2(nu) $. The value $M$ is in fact the likelihood ratio under a multinormal distribution of the observed mean of the hypothesized covariance matrix onto the observed covariance matrix.

Sadly, Winer did not provide any derivation of the correction factor and no reference to published work. So my question is: how did he derived the correction factor? Was it published in previous work and if so, what is the reference?

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