The Annals of Statistics
- Ann. Statist.
- Volume 30, Number 1 (2002), 220-238.
Multiple hypotheses testing and expected number of type I. errors
The performance of multiple test procedures with respect to error control is an old issue. Assuming that all hypotheses are true we investigate the behavior of the expected number of type I errors (ENE) as a characteristic of certain multiple tests controlling the familywise error rate (FWER) or the false discovery rate (FDR) at a prespecified level. We derive explicit formulas for the distribution of the number of false rejections as well as for the ENE for single-step, step-down and step-up procedures based on independent $p$-values. Moreover, we determine the corresponding asymptotic distributions of the number of false rejections as well as explicit formulae for the ENE if the number of hypotheses tends to infinity. In case of FWER-control we mostly obtain Poisson distributions and in one case a geometric distribution as limiting distributions; in case of FDR control we obtain limiting distributions which are apparently not named in the literature. Surprisingly, the ENE is bounded by a small number regardless of the number of hypotheses under consideration. Finally, it turns out that in case of dependent test statistics the ENE behaves completely differently compared to the case of independent test statistics.
Ann. Statist., Volume 30, Number 1 (2002), 220-238.
First available in Project Euclid: 5 March 2002
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Asymptotic critical value behavior Ballot theorem Bolshev’s recursion Bonferroni test procedure Dempster’s formula, DKW inequality empirical distribution function familywise error rate false discovery rate independent $p$ -values Lagrange–Bürmann theorem multiple comparisons multiple level multiple test procedure order statistics Schur–Jabotinski theorem step-down test step-up test
Finner, H.; Roters, M. Multiple hypotheses testing and expected number of type I. errors. Ann. Statist. 30 (2002), no. 1, 220--238. doi:10.1214/aos/1015362191. https://projecteuclid.org/euclid.aos/1015362191