Eleisha's Segment 16: Multiple Hypotheses

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 $\alpha \ll 1$?

To Think About

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 $S=-2\sum _{i=0}^{i=M}\log(p_{i})$ 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.

2. Fisher is sometimes credited, on the basis of problem 1, with having invented "meta-analysis", whereby results from multiple investigations can be combined to get an overall more significant result. Can you see any pitfalls in this?

Class Activity for February 21:

I was in a group with Andrea Hall, Ellen Le, and Sanmit Narvekar. Our work can be found over at Segment 16...Multiple Hypotheses

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