A multiple-testing procedure makes R rejections, of which V are rejections of true null…

A multiple-testing procedure makes \(R\) rejections, of which \(V\) are rejections of true null hypotheses. Define the false discovery proportion and false discovery rate by \[ \operatorname{FDP}=\frac{V}{\max(R,1)}, \qquad\operatorname{FDR}=E(\operatorname{FDP}). \] The familywise error rate is \(\operatorname{FWER}=P(V\ge1)\). Prove that \(\operatorname{FDR}\le\operatorname{FWER}\), and show that they agree when every tested null is true. First consider ten true null hypotheses with independent uniform p-values. Reject each hypothesis whenever its own p-value is at most \(0.05\). Calculate both error rates. Next consider two hypotheses: one true null has a uniform p-value, while the other null is false and its p-value is zero with certainty. For a fixed \(\alpha\in(0,1)\), reject each whenever its p-value is at most \(\alpha/2\). Find the FWER, the FDR, and the ratio \(E(V)/E(R)\). Evaluate all three at \(\alpha=0.1\). Explain why controlling the expected fraction of false discoveries is not the same as bounding the probability of making any false discovery, and why the expectation of a ratio cannot generally be replaced by a ratio of expectations.

Compare the false discovery proportion with the indicator that at least one false rejection occurs.