Two-Way Within-Subjects Design - Analysis of Variance Designs - page 298

For example, because Factor B has three treatments we could examine a
pairwise comparison with the following formula:

( n )( ˆ B at a j ) 2

2

SS B comp at a j = . (11.18)

Here we compute a difference between two means (from the AB matrix
of means in Table 11.2) at one of the levels of Factor A and square it. This
value is multiplied by the sample size and divided by 2 because there are
two means being compared:

SS A comp at a j
1

MS B comp at a j = (11.19)

MS A comp at a j
MS A × B × S .

F B comp at a j = (11.20)

Main Comparisons
If the interaction effect is not statistically significant, but one or more of
the main effects are statistically significant and the effect comprises three
or more levels, then an analysis of the main comparisons is in order. We
would begin by isolating a pairwise comparison of marginal means; in
the present example, we would focus on the marginal means for Factor B
(1.17, 4.25, 6.25) in the AB matrix of means in Table 11.2. Recall that we
subtract one mean from another to create our difference value, which we
call ˆ (read “psi hat”). The formula is as follows:

( a )( n )( ˆ B ) 2

SS B comp =

2

df B comp = 1 (11.21)

SS B comp
1

MS B comp = (11.22)

MS B comp
MS B × S .

F B comp = (11.23)

We have purposely omitted some of the computational details involved
in factorial within-subjects simple effects analyses, simple comparisons,
and main comparisons. However, many of the relevant computational
details are covered in Chapters 7 and 8 (between-subjects designs) and
can be readily applied with the present (within-subjects) formulas. For
more details on these analyses, we recommend Keppel et al. (1992) and
Keppel and Wickens (2004).