# Eleisha's Segment 16: Multiple Hypotheses

** To Calculate: **

1. Simulate the following: You have **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha}**
? Derive its form analytically for the usual case of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = -2\sum_{i=0}^{i=M}\log(p_i) }**
Show that, under the null hypothesis, S is distributed as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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?

** 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

** Back To: ** Eleisha Jackson