# Segment 16. Multiple Hypotheses

## Contents

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### Problems

#### To Calculate

1. Simulate the following: You have 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 $\alpha=0.05$ 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 $\alpha$? Derive its form analytically for the usual case of $\alpha \ll 1$?

$S = -2\sum_{i=0}^{i=M}\log(p_i)$
Show that, under the null hypothesis, S is distributed as $\text{Chisquare}(2M)$ and describe how you would obtain a combined p-value for this statistic.