This document shows how subtracting out \(SS_{subject}\) from \(SS_{within}\) when computing the F-statistic for the repeated measures ANOVA results in subtracting out \(SS_{subject}\) from \(SS_{total}\) in the denominator of the formula for eta-squared (\(\eta^2\)). This produces what’s called partial eta-squared (\(\eta_p^2\)).
Here’s the table from the book showing the algebraic relation between cohen’s f \(\eta^2\), F and its degrees of freedom:
| \(\eta^{2}\) | \(f\) | F | |
|---|---|---|---|
| \(\eta^{2}\) | \(\eta^{2} = \frac{SS_{between}}{SS_{total}}\) | \(\eta^{2} = \frac{f^{2}}{1+f^{2}}\) | \(\eta^{2} = \frac{F\frac{df_{between}}{df_{within}}}{1+F \frac{df_{between}}{df_{within}}}\) |
| \(f\) | \(f = \sqrt{\frac{\eta^{2}}{1-\eta^{2}}}\) | \(f = \sqrt{\frac{SS_{between}}{SS_{within}}}\) | \(f=\sqrt{F\frac{df_{between}}{df_{within}}}\) |
For repeated measures ANOVA we replace \(SS_{within}\) with \(SS_{error}\) where
\[ SS_{error} = SS_{within} - SS_{subject} \] and
\[ df_{error} = df_{within} - df_{subject} \]
The effect size, Cohen’s \(f\), is replaced by partial Cohen’s f (\(f_p\)) by adapting the formula from the table above:
\[ f_p =\sqrt{F\frac{df_{between}}{df_{error}}} \]
or
\[ f_p^2 =F\frac{df_{between}}{df_{error}} \]
Now, since
\[ F = \frac{MS_{between}}{MS_{error}} = \frac{\frac{SS_{beween}}{df_{between}}}{\frac{SS_{error}}{df_{error}} } \] \[ f_p^2 =F\frac{df_{between}}{df_{error}} = \frac{\frac{SS_{beween}}{df_{between}}}{\frac{SS_{error}}{df_{error}} } \frac{df_{between}}{df_{error}} = \frac{SS_{between}}{SS_{error}} \]
Using the formula above relating \(\eta^2\) to \(f^2\),
\[ \eta_p^2 = \frac{f_p^2}{1+f_p^2} \] So
\[ \eta_p^2 = \frac{\frac{SS_{between}}{SS_{error}}}{1+ \frac{SS_{between}}{SS_{error}}} = \frac{\frac{SS_{between}}{SS_{error}}}{\frac{SS_{error} + SS_{between}}{SS_{error}}} = \frac{SS_{between}}{SS_{error}+SS_{between}} \] Now, since
\[ SS_{total} = SS_{between} + SS_{subject} + SS_{error} \] \[ SS_{error} + SS_{between} = SS_{total} - SS_{subject} \] you can see that
\[ \eta_p^2 = \frac{SS_{betwen}}{SS_{total} - SS_{subject}} \]
Compare this to the formula for the ‘regular’ \(\eta^2\) in the table above:
\[ \eta^2 = \frac{SS_{betwen}}{SS_{total}} \] Just as we subtract out \(SS_{subject}\) from \(SS_{within}\) to compute the repeated measures ANOVA, we also subtract out \(SS_{subject}\) from \(SS_{total}\) to compute partial eta-squared (\(\eta_p^2\)), which is the adjusted measure of \(\eta^2\) for the repeated measures ANOVA.