On the optimally weighted z-test for combining probabilities from independent studies

Researchers have shown that the optimally weighted z-test, where the weights are the standardized expected difference in means, is more powerful than other methods when combining p-values from independent studies. However, in practice the effect for each independent study is usually unknown, which makes the optimally weighted z-test not applicable. A new test similar to the optimally weighted z-test, but with the effects being estimated from data, is derived. This new test is another generalized Fisher test which can be very powerful under certain situations. The new test is compared with existing methods through simulated data. Some suggestions for choosing tests to combine p-values from independent studies are given. The use of the new test is also illustrated by a real data application.