Consecutive Sequential Probability Ratio Tests of Multiple Statistical Hypotheses

In this paper, we develop a simple approach for testing multiple statistical hypotheses based on the observations of a number of probability ratios enumerated consecutively with respect to the index of hypotheses. Explicit and tight bounds for the probability of making wrong decisions are obtained for choosing appropriate parameters for the proposed tests. In the special case of testing two hypotheses, our tests reduce to Wald's sequential probability ratio tests.Comment: 29 pages, no figure; The main results of this paper have appeared in Proceedings of SPIE Conferences, Baltimore, Maryland, April 24-27, 201

Continuous Monitoring of A/B Tests without Pain: Optional Stopping in Bayesian Testing

A/B testing is one of the most successful applications of statistical theory in modern Internet age. One problem of Null Hypothesis Statistical Testing (NHST), the backbone of A/B testing methodology, is that experimenters are not allowed to continuously monitor the result and make decision in real time. Many people see this restriction as a setback against the trend in the technology toward real time data analytics. Recently, Bayesian Hypothesis Testing, which intuitively is more suitable for real time decision making, attracted growing interest as an alternative to NHST. While corrections of NHST for the continuous monitoring setting are well established in the existing literature and known in A/B testing community, the debate over the issue of whether continuous monitoring is a proper practice in Bayesian testing exists among both academic researchers and general practitioners. In this paper, we formally prove the validity of Bayesian testing with continuous monitoring when proper stopping rules are used, and illustrate the theoretical results with concrete simulation illustrations. We point out common bad practices where stopping rules are not proper and also compare our methodology to NHST corrections. General guidelines for researchers and practitioners are also provided

A Rejection Principle for Sequential Tests of Multiple Hypotheses Controlling Familywise Error Rates

We present a unifying approach to multiple testing procedures for sequential (or streaming) data by giving sufficient conditions for a sequential multiple testing procedure to control the familywise error rate (FWER), extending to the sequential domain the work of Goeman and Solari (2010) who accomplished this for fixed sample size procedures. Together we call these conditions the "rejection principle for sequential tests," which we then apply to some existing sequential multiple testing procedures to give simplified understanding of their FWER control. Next the principle is applied to derive two new sequential multiple testing procedures with provable FWER control, one for testing hypotheses in order and another for closed testing. Examples of these new procedures are given by applying them to a chromosome aberration data set and to finding the maximum safe dose of a treatment