Within-Subjects ANOVA


David M. Lane


Introduction to ANOVA, ANOVA Designs, Multi-Factor ANOVA, Difference 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 are:

(2)(1/2) = 1 for the numerator and
(90)(1/2) = 45 for the denominator

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.


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