My current understanding is that sphericity violations inflate the F-value and therefore type I error. I'd like to know how exactly this manifests itself mathematically.

To follow-up, why does sphericity only concern itself with repeated measures within a given factor? In other words, how is it that differences in the variance of the differences among all possible groups in, say, a 2x2 design not impact the F-value estimates? Intuitively, one might think that it would affect the F-value of the interaction in a 2x2 design.

I am looking for simple mathematical illustration of the way in which sphericity violations affect RM-ANOVA results.


1 Answer 1


It inflates the df rather than the F. The assumption is that all the contrasts are independent which means that V'CV is a diagonal matrix where V is a matrix of "orthogonal" contrasts and C is the population covariance matrix. This will be true for some patterns of C such as compound symmetry but is generally not true. The df in ANOVA are computed assuming the contrasts pooled for an effect are independent. If they are not independent, the df will be too high. With tests based on 1 df such as each of the three tests in a 2x2 design, there is no pooling of contrasts because only one contrast is tested at a time. When there is only one contrast, there is no problem of independence of contrasts.


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