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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.
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Heard et al. (2017) studied this question.
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