Using zero-order correlations of subject adjusted differences to obtain a one-way repeated measures covariate effect
It is possible to reconstruct the sums of squares in repeated measures ANOVA using multiple regression using the GLM parameterisation (see the GLM Graduate Statistics talk) or univariate (between subjects) ANOVA but these are more cumbersome approaches than using the repeated measures ANOVA. One special case occurs when the effect of a covariate in the presence of a single within subjects factor is assessed using a zero-order Pearson correlation.
To illustrate this approach suppose we wish to assess the correlation between a measure, ECAP, and outcome, PERF, which are each recorded on eight subjects at the same four different rates (100, 200, 300 and 400). This can be done by correlating the 8 x 4 differences obtained by subtracting each subject's respective ECAP and PERF means from each of that subject's four ECAP and PERF observations. Subtracting the different subject rate means removes the within subject factor of rate (see Boniface (1995)) and allows us to then perform a 'zero-order' Pearson correlation on the obtained ECAP and PERF difference scores.
This resultant correlation is equivalent to the signed square root of the partial eta-squared for ECAP entered as a covariate in a one-way repeated measures ANOVA with rate as a four category within subjects factor.
The correlation has (8x4)-3-2 degrees of freedom. In general the degrees of freedom equal N - (r-1) - 2 for a total sample size of N and r levels of the repeated measures factor.
The above assumes the rate means vary within subject. If there is no difference between the within subjects factor (e.g. rate) then the within subjects variance may be regarded as random white noise and it is, therefore, not necessary to adjust the ECAP and PERF scores, as described above, by subtracting respective ECAP and PERF subject means from each observation.
Reference
Boniface DR (1995) Experiment design and statistical methods for behvioural and social research. Chapman and Hall:London. (In CBSU library).