Multiple comparisons between two groups on multiple Bernoulli outcomes while accounting for covariates

The problem of adjusting for multiplicity when one has multiple outcome variables can be handled quite nicely by step-down permutation tests. More difficult is the problem when one wants an analysis of each outcome variable to be adjusted for some covariates and the outcome variables are Bernoulli. Special permutations can be used where the outcome vectors are permuted within each strata of the data defined by the levels of the (made discrete) covariates. This method is described and shown to control the familywise error rate at any prespecified level. The method is compared through simulation to a vector bootstrap approach, also using a step-down testing procedure. It is seen that the method using permutations within strata is superior to the vector bootstrap in terms of error control and power. The method is illustrated on a data set of 55 minor malformations of babies of diabetic and non-diabetic mothers.