# Eleisha's Segment 16: Multiple Hypotheses

To Calculate:

1. Simulate the following: You have ${\displaystyle M=50}$ p-values, none actually causal, so that they are drawn from a uniform distribution. Not knowing this sad fact, you apply the Benjamini-Hochberg prescription with ${\displaystyle \alpha =0.05}$ and possibly call some discoveries as true. By repeated simulation, estimate the probability of thus getting N wrongly-called discoveries, for N=0, 1, 2, and 3.

2. Does the distribution that you found in problem 1 depend on M? On ${\displaystyle \alpha }$? Derive its form analytically for the usual case of ${\displaystyle \alpha \ll 1}$?

1. Suppose you have M independent trials of an experiment, each of which yields an independent p-value. Fisher proposed combining them by forming the statistic ${\displaystyle S=-2\sum _{i=0}^{i=M}\log(p_{i})}$ Show that, under the null hypothesis, S is distributed as ${\displaystyle {\text{Chisquare}}(2M)}$ and describe how you would obtain a combined p-value for this statistic.