- alpha is likelihood of making a type I error (usually = 0.05)
- etasq is partial $$\eta^text{2}$$/100 so, for example, 5.9% = 0.059
Partial $$\eta^text{2}$$ = $$ \frac{\mbox{SS(effect)}}{\mbox{SS(effect) + SS(its error)}}$$
or, in other words, the proportion of variance in outcome predicted by the effect after adjusting for other terms in the anova. [attachment:etasqrp.pdf Click here for further details on partial $$\eta^text{2}$$] and [attachment:etasq.pdf here.]
If the Sums of Squares are not available you can[:FAQ/power/rsqform: construct eta-squared].
For B between subject factors in term of interest with levels $$b_{i}$$, i=1, ..., B and W with subject factors in term of interest with levels $$w_{j}$$, j=1, ..., W
- num(erator) = df of term of interest= $$ \prod_{\mbox{factors}} $$ (number of levels of factor -1) in term of interest
bsum = sum of dfs involving only between subject factors in anova or zero otherwise. df = Product of number of levels minus 1 of each between subject factor in term of interest. e.g. For a three way interaction involving three between subject factors, abc, we sum the dfs of the six lower order combinations: ab, ac and bc, a, b and c to that of abc.
- wdf = df of within subject effect if in term of interest or 1 otherwise. df = Product of number of levels minus 1 of each within subject factor in term of interest
- corr is the average correlation between levels of the repeated measures (=0 if no within subjects factors)
- ntot is the total sample size
[:FAQ/power/powexampleN: Example input]
Power can be computed using an EXCEL [attachment:aov.xls spreadsheet] or the SPSS syntax below.
[ COPY AND PASTE THE BOXED BELOW SYNTAX BELOW INTO A SPSS SYNTAX WINDOW AND RUN; ADJUST INPUT DATA AS REQUIRED]
DATA LIST free /alpha num bsum wdf corr ntot rsq. BEGIN DATA. .05 2 1 2 0.0 60 0.0588 .05 2 1 2 0.3 67 0.0588 END DATA. set errors=none. matrix. get m /variables=alpha num bsum wdf corr ntot rsq /missing=omit. compute alpha=make(1,1,0). compute num=make(1,1,0). compute bsum=make(1,1,0). compute wdf=make(1,1,0). compute corr=make(1,1,0). compute ntot=make(1,1,0). compute rsq=make(1,1,0). compute alpha=m(:,1). compute num=m(:,2). compute bsum=m(:,3). compute wdf=m(:,4). compute corr=m(:,5). compute ntot=m(:,6). compute rsq=(m:,7). end matrix. compute denom = (ntot-1-bsum)*wdf. COMPUTE power = 1 - NCDF.F(IDF.F(1-ALPHA,num,denom),num,denom,((NTOT-1-bsum)*wdf*RSQ/(1-RSQ))/(1-corr)). EXE. formats ntot (f7.0) alpha (f5.2) num (f5.2) denom (f5.2) corr (f5.2)rsq (f5.2) power (f5.2). variable labels ntot 'Total Sample Size' /alpha 'Alpha' /num 'Numerator F' /denom 'Denominator F' /corr 'Correlation' /rsq 'R-squared' /power 'Power'. report format=list automatic align(center) /variables=ntot alpha num denom corr rsq power /title "ANOVA power, between subjects factor possibly in a mixed design for given total sample size" .