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| 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) = df of term of interest= $$ \prod_{\mbox{factors}} $$ (number of levels of factor -1) in term of interest |
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| * num(erator) = $$ \prod_{\mbox{factors}} $$ (number of levels of factor -1) in term of interest | * bsum = sum of dfs involving '''only''' between subject factors in the 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 |
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| * bsum = sum of dfs of between subject factors = $$\sum_{i}^{B} (b_{i} - 1) + \prod_{\mbox{combinations}} (b_{i} - 1)$$ 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. * wdf = df of the within subject factor term (if any) in term of interest = $$ \prod_{j}^{W} (w_{j} - 1) $$ if W > 0 in term of interest = 1 otherwise |
* wdf = 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 |
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| * corr is the average correlation between levels of repeated measures (=0 if no within subjects factors) | * corr is the average correlation between levels of the repeated measures (=0 if no within subjects factors) |
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| Power can be computed using an EXCEL [attachment:aov.xls spreadsheet] or the SPSS syntax below. | Power can be computed using an EXCEL [attachment:aov.xls spreadsheet] or the SPSS syntax below. Power analysis software using Winer (1991, pp 136-138) for balanced anovas may be downloaded from [http://www.soton.ac.uk/~cpd/anovas/datasets/ here.] |
- 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].
- num(erator) = df of term of interest= $$ \prod_{\mbox{factors}} $$ (number of levels of factor -1) in term of interest
bsum = sum of dfs involving only between subject factors in the 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
- wdf = 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
- corr is the average correlation between levels of the repeated measures (=0 if no within subjects 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. Power analysis software using Winer (1991, pp 136-138) for balanced anovas may be downloaded from [http://www.soton.ac.uk/~cpd/anovas/datasets/ 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 num bsum wdf corr ntot rsq. BEGIN DATA. .05 2 1 2 0.0 60 0.0588 .05 2 1 2 0.3 67 0.0588 END DATA. set errors=none. matrix. get m /variables=alpha num bsum wdf corr ntot rsq /missing=omit. compute alpha=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 ntot=make(1,1,0). compute rsq=make(1,1,0). compute alpha=m(:,1). compute num=m(:,2). compute bsum=m(:,3). compute wdf=m(:,4). compute corr=m(:,5). compute ntot=m(:,6). compute rsq=(m:,7). end matrix. compute denom = (ntot-1-bsum)*wdf. COMPUTE power = 1 - NCDF.F(IDF.F(1-ALPHA,num,denom),num,denom,((NTOT-1-bsum)*wdf*RSQ/(1-RSQ))/(1-corr)). EXE. formats ntot (f7.0) alpha (f5.2) num (f5.2) denom (f5.2) corr (f5.2)rsq (f5.2) power (f5.2). variable labels ntot 'Total Sample Size' /alpha 'Alpha' /num 'Numerator F' /denom 'Denominator F' /corr 'Correlation' /rsq 'R-squared' /power 'Power'. report format=list automatic align(center) /variables=ntot alpha num denom corr rsq power /title "ANOVA power, between subjects factor possibly in a mixed design for given total sample size" .