Thanks to all contributors for the fruitfulness of this discussion. I
am speculating about a simpler solution: to use a non-parametric
approach. To avoid the requirement of having normal residuals, Frank
Harrell has suggested here the following non-parametric procedure:

library(Design)  # also requires library(Hmisc)
f <- lrm(y ~ a*b*c*d)
f
anova(f)

Could someone please tell me whether that also works when there is no
homoscedasticity? What are the assumptions of that method?

Paul

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