ANOVA Designs, Multi-Factor
Between Means, Correlated Pairs
Consequences of Violating the Assumption of Sphericity
Although ANOVA is robust to most violations of
its assumptions, the assumption of sphericity is an exception:
Violating the assumption of sphericity leads to a substantial
increase in the Type I error rate. Moreover, this assumption is
rarely met inpractice. Although violations of this assumption
had received little attention in the past, the current consensus
of data analysts is that it is no longer considered acceptable
to ignore them.
Approaches to Dealing with Violations of Sphericity
If an effect is highly significant, there is a
conservative test that can be used to protect against an inflated
Type I error rate. This test consists of adjusting the degrees
of freedom for all within subject variables as follows: The degrees
of freedom numerator and
denominator are divided by the number of scores per subject minus
one. Consider the effect of Task shown in Table
3. There are three scores per subject and therefore the degrees
of freedom should be divided by two. The adjusted degrees of freedom
(2)(1/2) = 1 for the numerator and
(90)(1/2) = 45 for the
The probability value is obtained using the F
probability calculator with the new degrees of freedom parameters.
The probability of an F of 228.06 or larger with 1 and 45 degrees
of freedom is less than 0.001. Therefore, there is no need to
worry about the assumption violation in this case.
Possible violation of sphericity does make a difference
in the interpreation of the analysis shown in Table 2. The probability
value of an F or 5.18 with 1 and 23 degrees of freedom is 0.032,
a value that would lead to a more cautios conclusion than the
p value of 0.003 shown in Table 2.
The correction described above is very conservative
and should only be used when, as in Table 3, the probability value
is very low. A better correction, but one that is very complicated
to calculate is to multiply by a quantity called ε. There
are two methods of calculating ε. The correction called
the Huynh-Feldt (or H-F) is slightly preferred to the called the
Geisser Greenhouse (or G-G) although both work well. The G-G correction
is generally considered a little too conservative.
A final method for dealing with violations of sphericity
is to use a multivariate approach to within-subjects variables.
This method has much to recommend it, but it is beyond the score
of this text.