Eleisha's Segment 16: Multiple Hypotheses

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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?

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