- 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) = $$ \prod_{\mbox{factors}} $$ (number of levels of factor -1) in term of interest
d1 = $$\sum_{i}^{B} (b_{i} - 1) + \prod_{\mbox{combinations}} (b_{i} - 1)$$ if B > 0 in anova
- = 0 otherwise
where combinations means all lower order combinations of at least two between subject factors making up the factor of interest. e.g. abc has lower order combinations combinations ab, ac and bc.
d2 = $$ \prod_{j}^{W} (w_{j} - 1) $$ if W > 0 in term of interest
- = 1 otherwise
- prod = number of combinations of levels of within subject factors in term of interest (= 1 if no within subjects factors in term of interest)
- 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" .