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manual:sphericity

# Sphericity

ANOVAs with repeated measures (within-subject factors) are susceptible to the violation of the assumption of sphericity. Sphericity is the condition where the variances of the differences between all combinations of related groups (levels) are equal. Violation of sphericity is when the variances of the differences between all combinations of related groups are not equal. Violation of sphericity can cause the F-test too liberal (i.e., an increase in the Type I error rate).

### Mauchly's Test of Sphericity

Mauchly's Test of Sphericity is a formal way of testing the assumption of sphericity. It is commonly used although it can fail to detect departures from sphericity in small samples and over-detecting them in large samples.

### Estimating Sphericity ($\epsilon$)

The degree of non-sphericity can be quantified a statistic called epsilon ($\epsilon$). An epsilon of 1 means sphericity is met. $\epsilon < 1$ indicates the violation of sphericity. The lowest value of $\epsilon$ is 1/(T-1) where T is the total number of measurements.

$\epsilon$ can be used to correct the degrees of freedom in the F test for within-subject effect or within-between interactions. The new degrees of the freedom is $\epsilon$ times the old ones. 