A general formulation for a one-sided group sequential design

Abstract

A major contribution to the statistical literature on group sequential designs was provided by Pampallona and Tsiatis who developed closed form functions that can be used to iteratively calculate the boundary points of a family of popular group sequential designs. A related area of interest is the use of conditional probability calculations to make interim decisions in stochastic curtailment procedures. The purpose of the paper is to develop group sequential designs based on conditional probabilities, to compare our results to the general closed form family of designs developed by Pampallona and Tsiatis, and to relate these to commonly used stochastic curtailment procedures. The problem and its solution are formulated and derived mathematically. A graphical interpretation of the results provides the reader with an alternative mechanism to understand the results and their significance. One-sided group sequential design boundary points, as closed form functions, are derived from conditional probability statements. These conditional probability statements can be interpreted as the probability, at the final analysis, of reversing the conclusion reached at an interim state. Under mild constraints, these boundary points are identical to the Pampallona and Tsiatis boundary points. At any interim stage when a boundary point is attained or surpassed we suggest a graphical approach to examine the conditional probability of reversing the interim decision at the final stage versus a range of possible parameter values. For stochastic curtailment procedures, we recommend relaxing (increasing) the conditional probability levels to at least 0.50 so that early stopping is at least as likely as for the O'Brien-Fleming procedure. The results are limited to one-sided group sequential designs. Conditional probabilities of reversing interim decisions provides a useful concept to develop group sequential designs and to evaluate stochastic curtailment procedures.