• 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 SS are not available you can form [:FAQ/power/rsqform: 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 within subject levels
• power is the power of the test

[:FAQ/power/powexample: Example input]

Computation may be performed using an EXCEL [attachment:anovan.xls spreadsheet] or the below SPSS syntax.

[ COPY AND PASTE THE BOXED BELOW SYNTAX BELOW INTO A SPSS SYNTAX WINDOW AND RUN; ADJUST INPUT DATA AS REQUIRED]

DATA LIST free
/alpha etasq num d1 d2 prod power. 
BEGIN DATA. 
.05 0.059 2 1 2 3 0.85
.05 0.059 2 1 2 3 0.85
END DATA.

matrix.
get m /variables=alpha etasq num d1 d2 prod power  /missing=omit.
compute alpha=make(1,1,0).
compute etasq=make(1,1,0).
compute num=make(1,1,0).
compute d1=make(1,1,0).
compute d2=make(1,1,0).
compute power=make(1,1,0).
compute prod=make(1,1,0).
compute alpha=m(:,1).
compute etasq=m(:,2).
compute num=m(:,3).
compute d1=m(:,4).
compute d2=m(:,5).
compute power=m(:,6).
end matrix.


define apow   (!pos !tokens(1)
                   / !pos !tokens(1)
                   / !pos !tokens(1)
                   / !pos !tokens(1)
                   / !pos !tokens(1)
                   / !pos !tokens(1)
                   / !pos !tokens(1)).

COMPUTE #POW = !6.

compute #conf = (1-!1).
compute #lc3 = 1.
compute #ind=0.
compute prod=!7.
compute ntot = 700.000.
comment COMPUTE #LC1 = 2.000.
compute denom=(ntot-1-!4)*!5.
COMPUTE #CUMF2 = 1 - NCDF.F(IDF.F(#conf,!3,denom),!3,denom,ntot*prod**!2/(1-!2)).
COMPUTE #DIFF=1.
SET MXLOOPS=10000.
LOOP IF (#DIFF GT .00005) .
+       DO IF (#CUMF2 GT #pow) .
+               COMPUTE #LC3 = ntot.
+               COMPUTE ntot = (Ntot - rnd(1)).
+                      COMPUTE denom=(ntot-1-!4)*!5.                      
+               COMPUTE #CUMF2 = 1 - NCDF.F(IDF.F(#conf,!3,denom),!3,denom,ntot*prod*!2/(1-!2)).
+       ELSE .
+               COMPUTE #LC1 = ntot.
+               COMPUTE ntot = (ntot + #LC3)/2.
+                      COMPUTE denom=(ntot-1-!4)*!5.
+               COMPUTE #CUMF2 = 1 - NCDF.F(IDF.F(#conf,!3,denom),!3,denom,ntot*prod*!2/(1-!2)).
+          END IF. 
+       COMPUTE #DIFF = ABS(#CUMF2 - #pow) .
END LOOP .
compute pow2 = 1 - NCDF.F(IDF.F(#conf,!3,denom),!3,denom,ntot*prod*!2/(1-!2)).
if (ntot-trunc(ntot) gt 0.5) #ind=1.
if (#ind eq 0) ntot=trunc(ntot)+1.
if (#ind eq 1) ntot=rnd(ntot).
EXECUTE .
compute alpha=!1.
compute etasq=!2.
compute num=!3.
compute d1=!4.
compute d2=!5.
compute power=!6.
compute denom=(ntot-1-d1)*d2.
formats ntot (f7.0) alpha (f5.2) num (f5.2) denom (f5.2) etasq (f5.2) power (f5.2).
variable labels ntot 'Total Sample Size Required' /alpha 'Alpha' /num 'Numerator' /denom 'Denominator' /etasq 'Partial Eta-Sq' /power 'Power'.
report format=list automatic align(center)
  /variables=ntot alpha num denom etasq power 
  /title "Anova term sample size for given power (any anova)" .
!enddefine.
matrix.
get m /variables=alpha etasq num d1 d2 power  /missing=omit.
compute alpha=make(1,1,0).
compute etasq=make(1,1,0).
compute num=make(1,1,0).
compute d1=make(1,1,0).
compute d2=make(1,1,0).
compute power=make(1,1,0).
compute alpha=m(:,1).
compute etasq=m(:,2).
compute num=m(:,3).
compute d1=m(:,4).
compute d2=m(:,5).
compute power=m(:,6).
end matrix.
apow alpha etasq num d1 d2 power prod.