Analysing Binomial Data in a Two‐Factor Experiment with Unequal Observations Per Cell

Binomial data were generated for a two‐factor experiment with both factors at three levels using unequal numbers of observations per cell. Three n‐patterns were used. The number of observations per cell ranged from 30 to 90 for two n‐patterns and from 38 to 98 for the third n‐pattern. The mean number of observations per cell was 60 for all n‐patterns. Samples were generated for values of μp of 0·05, 0·10, 0·15, 0·20, 0·30, 0·40 and 0·50 to determine the effects of n‐pattern and of the binomial parameter (μp) on α‐level for four methods of analysis. Methods of analysis considered were: two analyses of variance, one using percentage data; the other, the angular transformation for percentage data; a three‐factor χ2 analysis; and Friedman's (1937) non‐parametric two‐way analysis of variance. Because Friedman's test generally showed erratic α‐levels, no comparisons of power were made with Friedman's test. Of the remaining tests, the three‐factor χ2 was more powerful than the parametric analysis of variance procedures. There was no apparent difference between the parametric analyses of variance with respect to power, raising some doubt as to the usefulness of the arc‐sine transformation for percentage data.