Multiple testing corrections for p-values and confidence intervals from generalised linear mixed models

In this article, we describe and propose methods to derive \textit{p}-values and sets of confidence intervals with strong control of the family-wise error rates and coverage for tests of parameters from multiple generalised linear mixed models. We examine in particular analysis of a cluster randomised trial with multiple outcomes. While the need for corrections for multiple testing is debated, the justification for doing so is that, without correction, the probability of rejecting at least one of a set of null hypotheses is greater than the nominal rate of any single test, and hence the coverage of a confidence set is lower than the nominal rate of any single interval. There are few methods \textit{p}-value corrections for GLMMs and no efficient methods for deriving confidence intervals, limiting their application in this setting. We adapt the Bonferroni and Holm methods, and the randomisation-test approach of Romano \& Wolf (2005) to a generalised linear model framework. A search procedure for confidence interval limits using randomisation tests is developed to produce a set of confidence intervals under each method of correction. We show that the Romano-Wolf type procedure has nominal error rates and coverage under non-independent correlation structures in a simulation-based study, but the other methods only have nominal rates when outcomes are independent. We also compare results from the analysis of a real-world trial.