# 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.