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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) ln\Big(\frac{\vert S_1\vert}{\vert S_0\vert} \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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    $\begingroup$ In order to answer this question (without borrowing the referenced book) you will probably need to be more specific about the derivation of the likelihood ratio statistic $M$. You say that it arises "under a multinormal distribution of the observed mean of the hypothesized covariance matrix onto the observed covariance matrix". Can you show specifically what this model is ---i.e., by specifying the sampling distributions? $\endgroup$
    – Ben
    Commented Dec 6, 2018 at 1:48
  • $\begingroup$ @Ben: the underlying distribution of the population is multinormal; you can check Votaw, 1948, who also used the log-likelihood ratio. Link to paper is projecteuclid.org/euclid.aoms/1177730145 $\endgroup$
    – Denis Cousineau
    Commented Dec 6, 2018 at 2:55
  • $\begingroup$ @DenisCousineau is the null hypothesis that the data are independent, heteroscedastic? $\endgroup$
    – AdamO
    Commented Dec 6, 2018 at 14:59
  • $\begingroup$ The null is that covariance matrix is "compound symmetry" which means that the variances are all equal and that the covariances are all equal. The mean is not part of the hypothesis and is taken to be the observed mean, $\endgroup$
    – Denis Cousineau
    Commented Dec 7, 2018 at 13:08

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