A Stopped Negative Binomial Distribution

TL;DR

This paper introduces the stopped negative binomial distribution, a new discrete distribution modeling the number of Bernoulli trials until s successes or t failures, with applications in clinical trial monitoring.

Contribution

It derives the distribution's closed form, explores its properties, and demonstrates its use in clinical trial sequential monitoring and post-hoc analysis.

Findings

Distribution derived in closed form

Properties of the distribution analyzed

Applied to clinical trial monitoring

Abstract

This paper introduces a new discrete distribution suggested by curtailed sampling rules common in early-stage clinical trials. We derive the distribution of the smallest number of independent Bernoulli(p) trials needed in order to observe either s successes or t failures. The closed form expression for the distribution as well as the compound distribution are derived. Properties of the distribution are shown and discussed. A case study is presented showing how the distribution can be used to monitor sequential enrollment of clinical trials with binary outcomes as well as providing post-hoc analysis of completed trials.