- 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.]
For B between subject factors with levels $$b_{i}$$, i=1, ..., B and W with subject factors with levels $$w_{i}$$, j=1, ..., W
- num(erator) = $$ \prod_{\mbox{factors}} $$ (number of levels of factor -1) in term of interest
d1 = $$\sum_{i}^{B} (b_{i} - 1) $$ if B > 0 in anova
- = 0 otherwise
d2 = $$ \prod_{j} (w_{j} - 1) $$ if W > 0 in term of interest
- = 1 otherwise
- prod = number of combinations of within subject 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 d1 d2 prod ntot rsq. BEGIN DATA. .05 2 1 2 3 60 0.0588 .05 2 1 2 3 67 0.0588 END DATA. set errors=none. matrix. get m /variables=alpha num d1 d2 prod ntot rsq /missing=omit. compute alpha=make(1,1,0). compute num=make(1,1,0). compute d1=make(1,1,0). compute d2=make(1,1,0). compute prod=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 d1=m(:,3). compute d2=m(:,4). compute prod=m(:,5). compute ntot=m(:,6). compute rsq=(m:,7). end matrix. compute denom = (ntot-1-d1)*d2. COMPUTE power = 1 - NCDF.F(IDF.F(1-ALPHA,num,denom),num,denom,NTOT*prod*RSQ/(1-RSQ)). EXE. formats ntot (f7.0) alpha (f5.2) num (f5.2) denom (f5.2) rsq (f5.2) power (f5.2). variable labels ntot 'Total Sample Size' /alpha 'Alpha' /num 'Numerator F' /denom 'Denominator F' /rsq 'R-squared' /power 'Power'. report format=list automatic align(center) /variables=ntot alpha num denom rsq power /title "ANOVA power, between subjects factor possibly in a mixed design for given total sample size" .