Binomial early stopping times

Abstract

Sequential analysis for the purposes of possibly stopping a trial early is important whenever a result must be obtained as quickly as possible for public health, economic, or other reasons. A dominant research stream since the middle of the 20th century has been the challenge of generalizing Wald’s sequential probability ratio test (SPRT) to include composite alternative hypotheses. This article offers an alternative for a binomial distribution by constructing a single-parameter family of triangular rejection regions for the null hypothesis using a generation function argument in the two-dimensional space of successes versus failures. The result is algebraically equivalent to one line of the SPRT with unit power and with the second value of the binomial parameter as the undetermined parameter. Rounding then discretizes the line to the grid of two-dimensional integers and classical results for arbitrary stopping conditions are used to give expressions for the estimator, $\widehat{p},$ of the binomial parameter and its variance $\text{var}(\widehat{p}).$ The choice can be made by the practitioner in terms of their appetite for risk or more formally via a power analysis. An example is given of an ecological study where a quick binary decision was required and this desire had to be weighed against robustness of the results.

KEYWORDS:

• Generating function
• random walk
• rattus rattus
• sequential probability ratio test
• SPRT

ACKNOWLEDGMENTS

Thanks to Al Bramley for conversations that lead this work; to Briar Cook for discussions about the example and for supplying the data; to Robina Brock, and all the ZIP field staff, for collecting the data; and especially to the Editor and an anonymous reviewer for their thorough and constructive comments.

DISCLOSURE

The authors have no conflicts of interest to report.

FUNDING

Zero Invasive Predators is funded by The NEXT Foundation and The New Zealand Department of Conservation.