Choosing between methods of combining $p$-values

Abstract

Combining p -values from independent statistical tests is a popular approach to meta-analysis, particularly when the data underlying the tests are either no longer available or are difficult to combine. Numerous p -value combination methods appear in the literature, each with different statistical properties, yet often the final choice used in a meta-analysis can seem arbitrary, as if all effort has been expended in building the models that gave rise to the p -values. Birnbaum (1954) showed that any reasonable p -value combiner must be optimal against some alternative hypothesis. Starting from this perspective and recasting each method of combining p -values as a likelihood ratio test, we present theoretical results for some standard combiners that provide guidance on how a powerful combiner might be chosen in practice.