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I am working on a multiple comparisons problem for several hypothesis tests and have come across the harmonic mean $p$-value (HMP) for combining the $p$-value of these tests. Specifically, I am using a two-sample $t$-test to compare means of several samples.

Extending the interpretation of a standard $p$-value, is the HMP interpreted as the probability of obtaining test results at least as extreme as the observation given that every null hypothesis is correct? In other words, does HMP suggest the amount of evidence that at least one null hypothesis is false? Or is there another interpretation?

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The null hypothesis is that all the $p$-values are drawn from a uniform distribution on the unit interval. I have outlined the alternative hypotheses in this answer Test for significant excess of significant p-values across multiple comparisons

You would have to experiment to see if the harmonic p method is one which is best against the alternative that the signal is concentrated in one experiment or across all of them.

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