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• alpha is the 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 $$\eta^text{2}$$ = $$\frac{\mbox{SS(effect)}}{\mbox{SS(effect) + SS(its error)}}$$ where SS is the sum of squares associated with a particular term in the anova.

In other words, partial $$\eta^text(2)$$ represents the proportion of variance in outcome predicted by the effect after adjusting for other terms in the anova. Click here for further details on partial $$\eta^text{2}$$ and here. If SS are not available you can construct eta-squared.

• correlation = average correlation between repeated measures levels
• num(erator) = df of term of interest = the product of the (number of levels of each factor -1) in term of interest
• sum (B-1) = 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.

• Prod (W-1) = 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
• Prod W = product of all within subject levels or one if there are none.
• power is the power of the test

Computation may be performed using an EXCEL spreadsheet or the below SPSS syntax. Power analysis software using Winer (1991, pp 136-138) for balanced anovas may be downloaded from here with details of how to compute inputs here.

[ 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 bsum wdf corr power.
BEGIN DATA.
.05 0.059 2 1 2 0.3 0.85
.05 0.059 2 1 2 0.0 0.85
END DATA.

matrix.
get m /variables=alpha etasq num bsum wdf corr power  /missing=omit.
compute alpha=make(1,1,0).
compute etasq=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 power=make(1,1,0).
compute alpha=m(:,1).
compute etasq=m(:,2).
compute num=m(:,3).
compute bsum=m(:,4).
compute wdf=m(:,5).
compute corr=m(:,6).
compute power=m(:,7).
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 = !7.

compute #conf = (1-!1).
compute #lc3 = 1.
compute #ind=0.
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,(denom*(!2/(1-!2)))/(1-!6)).
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,(denom*(!2/(1-!2)))/(1-!6)).
+       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,(denom*(!2/(1-!2)))/(1-!6)).
+          END IF.
+       COMPUTE #DIFF = ABS(#CUMF2 - #pow) .
END LOOP .
compute pow2 = 1 - NCDF.F(IDF.F(#conf,!3,denom),!3,denom,(denom*(!2/(1-!2)))/(1-!6)).
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 bsum=!4.
compute wdf=!5.
compute corr=!6.
compute power=!7.
compute denom=(ntot-1-bsum)*wdf.
formats ntot (f7.0) alpha (f5.2) num (f5.2) denom (f5.2) etasq (f5.2) corr (f5.2) power (f5.2).
variable labels ntot 'Total Sample Size Required' /alpha 'Alpha' /num 'Numerator' /denom 'Denominator' /etasq 'Partial Eta-Sq' /corr 'Correlation' /power 'Power'.
report format=list automatic align(center)
/variables=ntot alpha num denom etasq corr power
/title "Anova term sample size for given power (any anova)" .
!enddefine.
matrix.
get m /variables=alpha etasq num bsum wdf corr power  /missing=omit.
compute alpha=make(1,1,0).
compute etasq=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 power=make(1,1,0).
compute alpha=m(:,1).
compute etasq=m(:,2).
compute num=m(:,3).
compute bsum=m(:,4).
compute wdf=m(:,5).
compute corr=m(:,6).
compute power=m(:,7).
end matrix.
apow alpha etasq num bsum wdf corr power.

Reference

Doncaster CP and Davey AJH (2007) Analysis of covariance. How to choose and construct models for the life sciences. CUP:Cambridge.

Winer BJ, Brown DR and Michels KM (1991) Statistical principles in experimental design, 3rd edition. McGraw-Hill:New York.

None: FAQ/power/rmPowN (last edited 2016-01-05 11:49:58 by PeterWatson)