From Table 14.3b you can see that is calculated in the usual manner. Similarly,
and are calculated just as main effects always are (take the sum of the
squared deviations from the grand mean and multiply by the appropriate constant [i.e., the
number of observations contributing to each mean]). Finally, the error term is obtained by
subtracting and from.
The summary table is shown in Table 14.3c and looks a bit different from ones you
have seen before. In this table I have made a deliberate split into Between-Subject factors
and Within-Subject factors. The terms for Weeks and Error are parts of the Within-Subject
term, and so are indented under it. (In this design the Between-Subject factor is not further
broken down, which is why nothing is indented under it. But wait a few pages and you will
see that happen too.) Notice that I have computed an Ffor Weeks but not for subjects, for
the reasons given earlier. The Fvalue for Weeks is based on 4 and 32 degrees of freedom,
and. We can therefore reject H 0 : and conclude
that the relaxation program led to a reduction in the duration per week of headaches
reported by subjects. Examination of the means in Table 14.3 reveals that during the last
three weeks of training, the amount of time per week involving headaches was about one-
third of what it was during baseline.
You may have noticed that no Subject 3 Weeks interaction is shown in the summary
table. With only one score per cell, the interaction term isthe error term, and in fact some
people prefer to label it S 3 Winstead of error. To put this differently, in the design discussed
here it is impossible to separate error from any possible Subject 3 Weeks interaction, because
they are completely confounded. As we saw in the discussion of structural models, both of
these effects, if present, are combined in the expected mean square for error.
I spoke earlier of the assumption of sphericity, or compound symmetry. For the data in
the example, the variance-covariance matrix follows, represented by the notation , where
the is used to indicate that this is an estimate of the population variance-covariance
matrix.
21.000 11.750 9.250 7.833 7.333
11.750 28.500 13.750 16.375 13.375
9.250 13.750 11.500 8.583 8.208
7.833 16.375 8.583 11.694 10.819
7.333 13.375 8.208 10.819 16.945
Visual inspection of this matrix suggests that the assumption of sphericity is reason-
able. The variances on the diagonal range from 11.5 to 28.5, whereas the covariances off
the diagonal range from 7.333 to 16.375. Considering that we have only nine subjects,
these values represent an acceptable level of constancy. (Keep in mind that the variances
do not need to be equal to the covariances; in fact, they seldom are.) A statistical test of this
assumption of sphericity was developed by Mauchly (1940) and is given in Winer (1971,
p. 596). It would in fact show that we have no basis for rejecting the sphericity hypothesis.
Box (1954b), however, showed that regardless of the form of , a conservative test on null
hypotheses in the repeated-measures analysis of variance is given by comparing
against —that is, by acting as though we had only two treatment levels. This
test is exceedingly conservative, however, and for most situations you will be better ad-
vised to evaluate Fin the usual way. We will return to this problem later when we consider
a much better solution found in Greenhouse and Geisser’s (1959) extension of Box’s work.
As already mentioned, one of the major advantages of the repeated-measures design is
that it allows us to reduce the error term by using the same subject for all treatments. Sup-
pose for a moment that the data illustrated in Table 14.3 had actually been produced by five
independent groups of subjects. For such an analysis, would equal 717.11. In this
case, we would not be able to pull out a subject term because SSbetween subjwould beSSerrorF.05(1, n 2 1)FobtgaN
=gNgNF.05(4,32)=2.68 m 1 = m 2 = Á =m 5SSsubjects SSweeks SStotalSSsubjects SSweeksSStotal468 Chapter 14 Repeated-Measures Designs