Choosing an optimal method to combine P-values.

Fisher (1925) was the first to suggest a method of combining the p-values obtained from several statistics and many other methods have been proposed since then. However, there is no agreement about what is the best method. Motivated by a situation that now often arises in genetic epidemiology, we consider the problem when it is possible to define a simple alternative hypothesis of interest for which the expected effect size of each test statistic is known and we determine the most powerful test for this simple alternative hypothesis. Based on the proposed method, we show that information about the effect sizes can be used to obtain the best weights for Liptak's method of combining p-values. We present extensive simulation results comparing methods of combining p-values and illustrate for a real example in genetic epidemiology how information about effect sizes can be deduced.