# 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 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 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. Doncaster, on his website which lists his power programs, cites Kirk (1968) amongst others as stating an unbiased estimator of this ratio is the square root of the F ratio (above) minus 1 divided by (common) group sample size, N. For a few further details also see here.

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

Example

Suppose we wish to obtain power for an anova involving a repeated measures factor with three levels, each with 32 subjects, with a R-squared of 15.2% and type I error of 5%.

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

df2 = df(error) = (total number in sample -1)df1 = (32-1)(3-1) = 62. $$F = \frac{\mbox{62 x 0.152}}{\mbox{2 x 0.848}} = 5.54$$

$$ \mbox{sd(factor)/sd(error)} = \sqrt{\frac{\frac{\mbox{62 x 0.152}}{\mbox{2 x 0.848}}-1}{32}} = 0.38$$

Using a sd(factor)/sd(error) of 0.38 in the Doncaster program we get a power of 77% which agrees with that produced by SPSS. Alternatively we could have put into the Doncaster program a F of 5.54 based on 32 replicates (all subjects are in each group) which gives a power of 0.76.

Note The G*POWER versions 2 & 3 manuals, available with free software here, gives ways of computing R-squared for repeated measures designs which additionally to Doncaster's program incorporates the average correlation within subject. This involves computing scaling factors for $$\eta^text{2}$$ used to compute the F statistic computed in a between subjects ANOVA. The new F value can then be entered into the Doncaster program using the same inputs effectively treating the repeated measures term as a between subjects ANOVA with an inflated effect size. For example (m/(1+(m-1)*rho)) is used for scaling up the between subjects 'only' F for the between subjects factor and m/(1-rho) for the within subject factor in an ANOVA of one between and one within subjects factor, where there are m levels of the repeated measures factor with an average inter-factor level correlation of rho (equals zero for a between subjects ANOVA). Further details are on the G*POWER website in the on-line tutorial.

Examples comparing Doncaster program to G*POWER3 for a repeated measures one-way factor

- G*POWER appears to be less conservative than the Doncaster program needing a total of just 20 (as opposed to 32) in the 3 group repeated measures example for 80% power using an analogous effect size (f) = $$\sqrt{\frac{\mbox{df1 F}}{\mbox{df2}}} = \sqrt{\frac{\mbox{2 x 5.54}}{\mbox{62}}} = \sqrt{0.152/(1-0.152)}$$ = 0.42 and making the conservative assumption of zero correlation within subject.
For a repeated measures on 36 subjects with 4 levels of a repeated measures factor with an average inter-factor level correlation of 0.3 and a medium sized $$eta

^{text{2}$$ of 0.06 (ignoring inter-factor level correlation) the rescaled $$eta}text{2}$$ = 0.06 x m/(1-rho) = 0.06 x [4/(1-0.3)] = 0.06 x 5.71 = 0.34 giving an F of (32 x 0.34) / (3 x 0.66) = 5.5 taking inter-factor level correlation into account. Putting 5.5 as the F ratio on 36 replicates with 3, 32 df and a proposed sample size of 36 gives a power of 0.84 compared to 0.80 using G*POWER 3.

Reference

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