  Given a family of null hypotheses $H_{1},\ldots,H_{s}$, we are interested in
the hypothesis $H_{s}^{\gamma}$ that at most $\gamma-1$ of these null
hypotheses are false. Assuming that the corresponding $p$-values are
independent, we are investigating combined $p$-values that are valid for
testing $H_{s}^{\gamma}$. In various settings in which $H_{s}^{\gamma}$ is
false, we determine which combined $p$-value works well in which setting. Via
simulations, we find that the Stouffer method works well if the null $p$-values
are uniformly distributed and the signal strength is low, and the Fisher method
works better if the null $p$-values are conservative, i.e. stochastically
larger than the uniform distribution. The minimum method works well if the
evidence for the rejection of $H_{s}^{\gamma}$ is focused on only a few
non-null $p$-values, especially if the null $p$-values are conservative.
Methods that incorporate the combination of $e$-values work well if the null
hypotheses $H_{1},\ldots,H_{s}$ are simple.
