• 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" .