Inputs for the power.exe program which may be downloaded from the CP Doncaster website

Computing the degrees of freedom and obtaining the program are described [:FAQ/power/rmPowN: here.]

In addition to the above you can also need to compute a F ratio from a pilot study. This can be achieved using partial- $$\eta^text{2}$$ values (see [:FAQ/effectSize: here]) since

$$ \mbox{F(df1, df2) =} \frac{\mbox{df2 }\eta^{text{2}}{\mbox{df1 }(1- \eta}text{2})}

where df2 is the df for the error term and df1 is the df for the term of interest.

df(error) = (df(of any within subject factors in term of interest) or 1 otherwise) multiplied by (total number of subjects - 1 - sum of df for all between subject terms in model or 0 otherwise).

Alternatively, to inputting a F ratio and sample sizes, you can input what is called the ratio of treatment to error effect sizes which is the ratio of the standard deviations of variability due to the effect of interest and its error term. Kirk (1968) amongst others states an unbiased estimator of this ratio is the square root of the F ratio (above) minus 1 divided by sample size, Ns. For a few further details also see [http://www.soton.ac.uk/~cpd/anovas/datasets/#_Computer_programs_for_planning_ANOV here.]

It follows sd(effect of interest)/sd(error) = $$\sqrt{\frac{\frac{\mbox{df2 }\eta^{text{2}}{\mbox{df1 }(1- \eta}text{2})} -1}{Ns} }

Example

Suppose we wish to obtain power for an anova involving a repeated measures factor with three levels, 10 in each group, with a R-squared of 27% and type I error of 5%.

df1 = number of levels of the repeated measures factor -1 = 3 - 1 = 2.

df2 = df(error) = (number in each group -1)df1 = (10-1)(3-1) = 18. $$F = \frac{\mbox{18 x 0.27}}{\mbox{2 x 0.73}} = 3.32$$

$$ \mbox{sd(factor)/sd(error)} = \sqrt{\frac{\frac{\mbox{18 x 0.27}}{\mbox{2 x 0.73}}-1}{10}} = 0.48$$

Using a sd(factor)/sd(error) of 0.48 in the Doncaster program we get a power of 41% which agrees with that produced by SPSS. Alternatively we could have put into the Doncaster program a F of 3.32 based on 10 replicates per sample (group) also gives a power of 0.41.

Note The G*POWER version 2 manual, available with free software here, [http://www.psycho.uni-duesseldorf.de/aap/projects/gpower/ here] gives ways of computing R-squared for repeated measures designs which incorporate the average correlation within subject.

Reference

Kirk, R. E. (1968, 1982, 1994) Experimental Design: Procedures for the Behavioral Sciences. Brooks/Cole, Belmont, CA.