How SPSS works our standard errors for simple effects in a WW design
Suppose we have a repeated measures ANOVA with two two level repeated measures factors, W1 and W2. Supppose further that we are interested in comparing the difference between the two levels of W2 at each level of W1.
/EMMEANS does this for us but what does it do? There are two other ways that we can do the equivalent test which illustrate what is going on more clearly. Firstly, we can compute the difference between the two columns being compared, W12 and W11 where Wij is the combination of factors with W2=i and W1=j.
If we then perform a one sample t-test on this difference (ie a paired t-test) the standard error of the mean will correspond to the standard error of the simple effect and the mean will be the difference being tested by the simple effect so the 95% confidence interval for the W12-W11 difference would be the mean difference +/- [t( on n-1 df, 2.5%) times the se of the mean difference for n subjects].
Another equivalent way to compute the simple effect of W1 on W2 is to perform a repeated measures ANOVA in SPSS on the two columns, W11 and W12. Then the standard error of the difference W12-W11 equals sqrt(2 MSE/n) where MSE is the Mean Square Error in the repeated measures ANOVA only containing the two columns, W11 and W12.
It follows, further from this, that the equivalence of the above two tests, firstly a paired t-test on the difference W12-W11 and secondly the RM ANOVA on W11 and W12, follows from
sqrt[ (2/n) SS(W12 versus W11) / sqrt[ (2/n) MSE(W11 versus W12)]
= average W12 - W11 difference / standard error of the average W12 - W11 difference
= t on n-1 df for the simple effect represented by the difference, W12 - W11, outputted by /EMMEANS in RM ANOVA.
This follows since SS(W1) = n[mean W11 - mean(W11+W12)/2]2 + n[mean W12 - mean(W11+W12)/2]2
= 2n [mean of W12 - mean W11)/2]2 since (mean of W11 + mean of W12)/2 is half the difference between the means of W11 and W12
= (n/2) (mean of W11 - mean of W12)2 = A.
Now the paired t-test is identical to the RM ANOVA F test. The paired t-test uses the (mean of W11 - mean of W12) in its numerator so it follows that since (2/n) sqrt(A) = (mean of W11 - mean of W12), the numerator of the paired t-test, then its accompanying denominator must be (2/n) sqrt[MSE for SS(W1)].
In the above SS and MSE are the sums of squares and Mean Square Error respectively on n subjects used to compute the F ratio in the RM ANOVA using only W11 and W12.
Other equivalent computations of simple effects for repeated measures factors to that used by /EMMEANS may be obtained similarly. In using this approach SPSS is using the standard errors only associated with the particular combinations of factors being differenced to comprise the simple effect hence each simple effect will have a different standard error. In effect SPSS uses separate ANOVAs to obtain Mean Square Errors to compare different simple effect contrasts as advocated by amongst others, Howell (2002), p.490.
Howell (2002, p.492) and Boniface (1996, p.156) suggest an improved MSE for testing the simple effect of differences between levels of factor W1 at a particular level of W2 in a RM ANOVA should alternatively take the form
[SS(subjects x W1) + SS(subjects x W1 x W2)] / [df(subjects x W1) + df(subjects x W1 x W2)] with degrees of freedom, [df(subjects x W1) + df(subjects x W1 x W2)]/2.
Boniface (1995) Experiment design and statistical methods for behavioural and social research. Chapman and Hall:London.
Howell DC (2002) Statistical Methods for Psychologists. Fifth Edition. Duxbury Press:Pacific Grove, CA.