# Segment 16 Sanmit Narvekar

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## Segment 16

#### 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 $\displaystyle \alpha=0.05$ 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.

Here is the Matlab code:


M = 50;
alpha = 0.05;
nIters = 1000000;
discoveries = zeros(10,1);

for iter=1:nIters

% Draw 50 pvalues, uniformly distributed
pvals = rand(M,1);

% Benjamini & Hochberg FDR "prescription"
sortedPvals = sort(pvals);
nDiscoveries = sum((sortedPvals ./ ((1:M)' .* alpha ./ M)) < 1);
discoveries(nDiscoveries+1) = discoveries(nDiscoveries+1) + 1;
end

discoveries ./ nIters



I actually calculated it for N = 0 to 9 since some of the other values were found as well. Here are the results (starting with N=0 at the top):

   0.950169
0.046188
0.003375
0.000250
0.000017
0.000001
0
0
0
0


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 $\displaystyle \alpha \ll 1$ ?

The distribution does not depend on M. I ran the code above with M = 1000, and received the following results, which are very similar to the ones above:

   0.950202
0.046139
0.003332
0.000297
0.000025
0.000005
0
0
0
0


Obviously, it does depend on alpha, since changing alpha will (at the very least) affect the first bin N = 0, with changes propagating to the other ones. Here is alpha = 0.10:

   0.899984
0.086055
0.011699
0.001868
0.000321
0.000056
0.000014
0.000003
0
0


#### 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

$\displaystyle S = -2\sum_{i=0}^{i=M}\log(p_i)$

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.

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?