Next section: ANOVA with one within-subject variable

Since real data rarely meet the assumption of sphericity, this assumption cannot be safely ignored. In 1954 a statistician named Box developed an index of the degree to which the assumption of sphericity is violated. The index, called epsilon (ε) ranges from 1.0, meaning no violation to 1/df where df is the degrees of freedom in the numerator of the F ratio. In the Gesiser-Greenhouse correction, the sample value of epsilon can be used to correct the probability value for violations of sphericity by multiplying both the degrees of freedom numerator and denominator by the sample value of ε. (Note that the corrected df are used only to compute the p value, not to divide the sum of squares in order to find the mean square.) The corrected probability value will always be higher (less significant) than the uncorrected value except when the effect has one degree of freedom in which case ε will be 1.0 and the corrected and uncorrected probability values will be the same. Statistical packages often refer to this correction as the Geiser-Greenhouse correction because of an article of theirs in 1957 that discussed this issue.

The "Geiser-Greenhouse correction" is known to be somewhat conservative. An alternative correction developed by Huynh and Feldt is less conservative and is often computed by standard statistical packages.

Although the assumption of sphericity has been discussed for many years, it is still often ignored in practice. This is unfortunate since an uncorrected probability value from an analysis variance with within-subject variables is very rarely valid.

Next section: ANOVA with one within-subject variable