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I read a few texts about the multivariate approach for repeated measures where k repeated measures are transformed to (k-1) "difference scores". All texts say that using this approach, the assumption of sphericity is not required. But none of those texts explains why this is. Can anyone explain why?

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  • $\begingroup$ Multivariate (MANOVA) approach to RM is not univariate (split-plot) ANOVA approach which, like simple, simple ANOVA, requires homogeneity of variances. Sphericity in the end amounts to the homogeneity. Sphericity implies that "difference variables" (i.e. with 3 RM levels these are: RM1-RM2, RM1-RM3, RM2-RM3) have the identity covariance matrix. If that is so, univariate ANOVA can cope. $\endgroup$
    – ttnphns
    Commented May 5, 2014 at 11:53
  • $\begingroup$ why does the multivariate approach does not need the assumption of sphericity but the univariate (ANOVA) approach does? $\endgroup$
    – beginneR
    Commented May 5, 2014 at 13:39
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    $\begingroup$ Because MANOVA explicitly accounts for any covariances among its dependent variables, since it is multivariate by definition. Univariate ANOVA cannot cope (without special corrections). Because the "difference variables" are all interrelated, their nonzero covariances automatically mean their nonequal variances, which is contra ANOVA assumption of homogeneity of variance across levels of a factor. $\endgroup$
    – ttnphns
    Commented May 5, 2014 at 13:52
  • $\begingroup$ @ttnphns: I believe it is not true that the "sphericity assumption" implies that the difference variables have the identity covariance matrix. First, their variances are not equal to 1. Second, their covariances are not equal to 0. See my answer here for an example. $\endgroup$
    – amoeba
    Commented May 12, 2016 at 12:43

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