Manual computation for pairwise comparisons involving three or more groups in a repeated measures ANOVA

From Boniface (p.42-3) using what he calls 'reliability'

SS(subjects x group) = $$ \sum_text{ij} (X_text{ij} - (\bar{X}_text{group} - \bar{X}) - \bar{X}_text{sub})^text{2} $$

= $$ \sum_text{ij} (X_text{ij} - \bar{X}_text{group} - \bar{X}_text{sub} + \bar{X})^text{2} $$

MS(error) = SS(subjects x group) / ((Nsub-1)(Ngroup-1))

The (uncorrected) t statistic on (Nsub-1)(Ngroup-1) degrees of freedom between a pair of means is $$\frac{\mbox{difference in means}}{\sqrt{\frac{\mbox{2MS(error)}}{\mbox{n}}}}$$.

Using Tukey's test for all pairwise comparisons we can use

$$\frac{\mbox{difference in means}}{\sqrt{\frac{\mbox{MS(error)}}{\mbox{n}}}}$$ = $$sqrt{2}t$$

and compare to the studentised range statistic, q(Ngroup,(Nsub-1)(Ngroup-1)), at 0.05 and 0.01 levels (Tables in Howell, 1997).

References

Boniface DR (1995) Experiment design and statistical methods for behavioural and social research. Chapman and Hall:London.

Howell DC (1997) Statistical methods for psychologists. Fourth Edition. Wadsworth, Belmont,CA. the studenised range table are also probably in Howell's 2002 edition.