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. A diverse range of p-value
combination methods appear in the literature, each with different statistical
properties. Yet all too often the final choice used in a meta-analysis can
appear arbitrary, as if all effort has been expended 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 of the standard
combiners which provide guidance about how a powerful combiner might be chosen
in practice.