Abstract
To detect a change in the probability of a sequence of independent binomial random variables, a variety of asymptotic and exact testing procedures have been proposed. Whenever the sample size or the event rate is small, asymptotic approximations of maximally selected test statistics have been shown to be inaccurate. Although exact methods control the type I error rate, they can be overly conservative due to the discreteness of the test statistics in these situations. We extend approaches by Worsley and Halpern to develop a test that is less discrete to increase the power. Building on ideas from binary segmentation, the proposed test utilizes unused information in the binomial sequences to add a new ordering to test statistics that are of equal value. The exact distributions are derived under side conditions that arise in hypothetical segmentation steps and do not depend on the type of test statistic used (e.g., log likelihood ratio, cumulative sum, or Fisher's exact test). Using the proposed exact segmentation procedure, we construct a change point test and prove that it controls the type-I-error rate at any given nominal level. Furthermore, we prove that the new test is uniformly at least as powerful as Worsley's exact test. In a Monte Carlo simulation study, the gain in power can be remarkable, especially in scenarios with small sample size. Giving a clinical database example about pin site infections and an example assessing publication bias in neuropsychiatric drug research, we demonstrate the wide-ranging applicability of thetest.
Highlights
The problem of change point detection in binomial sequences has been addressed by many authors over the past decades
Building on ideas from binary segmentation, the proposed test utilizes unused information in the binomial sequences to add a new ordering to test statistics that are of equal value
The exact distributions are derived under side conditions that arise in hypothetical segmentation steps and do not depend on the type of test statistic used
Summary
The problem of change point detection ( sometimes referred to as threshold, cutpoint, breakpoint, or “disorder” detection) in binomial sequences has been addressed by many authors over the past decades (see, e.g., Carlstein, 1988; Chen & Gupta, 2011; Halpern, 1999; Hinkley & Hinkley, 1970; Lausen & Schumacher, 1992; Miller & Siegmund, 1982; Pettitt, 1979, 1980; Smith, 1975; Worsley, 1983). Those statistics include the log likelihood ratio, the cumulative sum and variations thereof (Pettitt, 1980), and the p-value of Fisher’s exact test (Halpern, 1999); and statistics based on Doob’s martingale decomposition (Brostrom, 1997) as well as Bayesian statistics may be used (Assareh, Smith, & Mengersen, 2015; Smith, 1975) These maximally selected test statistics T1m∶aζx = maxk=1,..,ζ(T1k∶ζ) arise in change point detection and in various other applications. These are used to get exact p-values on both subsequences left and right of the potential change point κ = argmaxk(Tk) conditional on Tκ.
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