  Meta-analysis combines results from multiple studies aiming to increase power
in finding their common effect. It would typically reject the null hypothesis
of no effect if any one of the studies shows strong significance. The partial
conjunction null hypothesis is rejected only when at least $r$ of $n$ component
hypotheses are non-null with $r = 1$ corresponding to a usual meta-analysis.
Compared with meta-analysis, it can encourage replicable findings across
studies. A by-product of it when applied to different $r$ values is a
confidence interval of $r$ quantifying the proportion of non-null studies.
Benjamini and Heller (2008) provided a valid test for the partial conjunction
null by ignoring the $r - 1$ smallest p-values and applying a valid
meta-analysis p-value to the remaining $n - r + 1$ p-values. We provide
sufficient and necessary conditions of admissible combined p-value for the
partial conjunction hypothesis among monotone tests. Non-monotone tests always
dominate monotone tests but are usually too unreasonable to be used in
practice. Based on these findings, we propose a generalized form of Benjamini
and Heller's test which allows usage of various types of meta-analysis
p-values, and apply our method to an example in assessing replicable benefit of
new anticoagulants across subgroups of patients for stroke prevention.
