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The problem of multiple hypotheses testing often arises in sequential trials, e.g., in clinical trials where patients are collected sequentially to answer multiple questions about safety and efficacy of a new treatment. Typically, safety and efficacy are assessed based on a number of endpoints/questions, e.g., intensity of pain, blood pressure, pulse rate, or other health measurements. In such experiments, it is necessary to find a statistical answer to each posed question by testing each hypothesis while controlling some overall error probabilities of making wrong decisions. In this talk, we develop stopping rules and decision rules for testing multiple hypotheses simultaneously such that some desired error rates are controlled at pre-specified levels.
In the first part of the talk, we will consider controlling generalized familywise error rates, namely GFWER-I and GFWER-II, defined as the probabilities of making at least k (> 1) Type I errors and at least m (> 1) Type II errors respectively. We will describe our proposed stepwise methods of multiple testing and prove that, for testing simple versus simple hypotheses and for certain composite hypotheses, these methods can control, unlike fixed-sample-size procedures, both GFWER-I and II at pre-specified levels.
Next, we will consider controlling the tail probabilities of both False Discovery Proportion (FDP) and False Non-discovery Proportion (FNP) that are very popular in large-scale multiple testing problems. Extending the test procedures controlling generalized familywise error rates, we will show that our proposed method controls the probabilities of FDP and FNP being more than g1 and g2 respectively, where g1 and g2 are some fixed numbers between 0 and 1. |