whose density function can be writtenThe test is implemented by the p.hmp command of the harmonicmeanp R package; a tutorial is available online. Equivalently, one can compare the HMP to a table of critical values (Table 1). The table illustrates that the smaller the false positive rate, and the smaller the number of tests, the closer the critical value is to the false positive rate. Multiple testing via the multilevel test procedure If the HMP is significant at some level for a group of p-values, one may search all subsets of the p-values for the smallest significant group, while maintaining the strong-sense family-wise error rate. Formally, this constitutes a closed-testing procedure. When is small (e.g. ), the following multilevel test based on direct interpretation of the HMP controls the strong-sense family-wise error rate at level approximately Define the HMP of any subset of the p-values to be Reject the null hypothesis that none of the p-values in subset are significant if , where . (Recall that, by definition, .) An asymptotically exact version of the above replaces in step 2 with where gives the number of p-values, not just those in subset . Since direct interpretation of the HMP is faster, a two-pass procedure may be used to identify subsets of p-values that are likely to be significant using direct interpretation, subject to confirmation using the asymptotically exact formula. Properties of the HMP The HMP has a range of properties that arise from generalized central limit theorem. It is: Robust to positive dependency between the p-values. Insensitive to the exact number of tests, L. Robust to the distribution of weights, w. Most influenced by the smallest p-values. When the HMP is not significant, neither is any subset of the constituent tests. Conversely, when the multilevel test deems a subset of p-values to be significant, the HMP for all the p-values combined is likely to be significant; this is certain when the HMP is interpreted directly. When the goal is to assess the significance of individual p-values, so that combined tests concerning groups of p-values are of no interest, the HMP is equivalent to the Bonferroni procedure but subject to the more stringent significance threshold (Table 1). The HMP assumes the individual p-values have (not necessarily independent) standard uniform distributions when their null hypotheses are true. Large numbers of underpowered tests can therefore harm the power of the HMP. While the