Assumptions of Within-Subject Designs (2 of 2)
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