Analysis of Variance by Randomization when Variances are Unequal

If there are significant factor and interaction effects with analysis of variance using ran-domization inference, they can be detected by tests that compare the F -statistics for the real data with the distributions of these statistics obtained by randomly allocating either the original observations or the residuals to the various factor combinations. Such tests involve the assumption that the effect of factors or interactions is to shift the observations for a factor combination by a fixed amount, without changing the amount of variation at that combination. In reality the expected amount of variation at each factor combination, as measured by the variance, may not be constant, which may upset the properties of the tests for the effects of factors and interactions. This paper discusses several possible methods for adjusting the randomization procedure to allow for this type of problem, including generalizations of methods that have been proposed for comparing the means of several samples when there is unequal variance but no factor structure. A simulation study shows that the best of the methods examined is one for which the randomized sets of data are designed to approximate the distributions of F -statistics when unequal variance is present.