# Single degree of freedom effect sizes in repeated measures anova

Field (2005) suggests for repeated measures, including those with more than one factor, only comparing the main effect of two groups as it is more difficult to interpret more complex effect sizes. For example, interactions usually have to be decomposed for interpretability using simple effects into single degree of freedom t-tests representing pairs of group differences or linear trends. In the special case of two group comparisons Field notes that

$$\mbox{Effect size} = \sqrt{ \frac{F(1,df)}{F(1,df)+df}} = \sqrt{ \frac{t(df)^{text{2}}{t(df)}text{2}+df}} $$ = the point biserial correlation, r(pb).

df in the above is the degree of freedom of the *error* term associated with the effect of interest.

Since the effect has a single degree of freedom, r(pb), may be rewritten as

$$ \sqrt{\frac{\frac{\mbox{df SS(effect)}}{\mbox{SS(error)}}}{\frac{\mbox{df SS(effect)}}{\mbox{SS(error)}} + df}}$$ = $$ \sqrt{\frac{\mbox{SS(effect)}}{\mbox{SS(effect) + SS(error)}}}$$

which tells us that r(pb) is the square root of partial $$\eta^text{2}$$.

r(pb) is also the point-biserial correlation resulting from correlating a dichotomous group variable with the response (see Calculating, Interpreting and Reporting Estimates of "Effect Size"). r(pb) can also be transformed into effect sizes which compare a pair of groups such as Cohen's d (Rosenthal(1994)). The above formula for the square of r(pb), partial $$eta^text{2}$$, can be rewritten to involve the regression estimate which corresponds to the group difference. This further parameterisation may be found in the SPSS Univariate and MANOVA algorithm pages which are part of a list of SPSS algorithms. These pages present the formulae used by the listed SPSS procedures. To access login using *guest* as login and password. Field (2005, (pages 478-480)) further applies these single degree of freedom tests to an example featuring two within subject factors (drink and imagery) and their interaction.

It follows from Rosenthal(1994) that

$$d = sqrt{ \frac{4r(pb)^{text{2}}{1-r(pb)}text{2}}} $$.

Since the point-biserial correlation is a special case of the Pearson correlation (Howell, 2002) one could justify using the rules of thumb from Cohen (1988) for r(pb), namely 0.1 (small), 0.3 (medium) and 0.5 (large). The square of these is 0.01, 0.09 and 0.25 which can be interpreted as rules of thumb for magnitudes of partial $$\eta^{text{2}$$. These are, as expected, higher than those suggested by Cohen for $$\eta}text{2}$$ since the denominator is smaller for partial $$eta^text{2}$$.

References

Field A (2005) Discovering statistics using SPSS. 2nd Edition. Sage:London.

Howell DC (2002) Statistical methods for psychology. %th Edition. Duxbury Press:Pacific Grove, CA.

Rosenthal, R. (1994) Parametric measures of effect size. In H.Cooper and L.V. Hedges (Eds.) *The handbook of research synthesis.* New York:Russell Sage Foundation.