Exact repeated confidence intervals for Bernoulli parameters in a group sequential clinical trial

MERIT: Controlling Monte-Carlo error rate in large-scale Monte-Carlo hypothesis testing

The use of Monte-Carlo (MC)p $$ p $$ -values when testing the significance of a large number of hypotheses is now commonplace. In large-scale hypothesis testing, we will typically encounter at least somep $$ p $$ -values near the threshold of significance, which require a larger number of MC replicates thanp $$ p $$ -values that are far from the threshold. As a result, some incorrect conclusions can be reached due to MC error alone; for hypotheses near the threshold, even a very large number (eg,1 0 6 $$ 1{0}^6 $$ ) of MC replicates may not be enough to guarantee conclusions reached using MCp $$ p $$ -values. Gandy and Hahn (GH)6-8 have developed the only method that directly addresses this problem. They defined a Monte-Carlo error rate (MCER) to be the probability that any decisions on accepting or rejecting a hypothesis based on MCp $$ p $$ -values are different from decisions based on idealp $$ p $$ -values; their method then makes decisions by controlling the MCER. Unfortunately, the GH method is frequently very conservative, often making no rejections at all and leaving a large number of hypotheses "undecided". In this article, we propose MERIT, a method for large-scale MC hypothesis testing that also controls the MCER but is more statistically efficient than the GH method. Through extensive simulation studies, we demonstrate that MERIT controls the MCER while making more decisions that agree with the idealp $$ p $$ -values than GH does. We also illustrate our method by an analysis of gene expression data from a prostate cancer study.

A general formulation for a one-sided group sequential design

BACKGROUND

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.

PURPOSE

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.

METHODS

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.

RESULTS

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.

LIMITATIONS

The results are limited to one-sided group sequential designs.

CONCLUSIONS

Conditional probabilities of reversing interim decisions provides a useful concept to develop group sequential designs and to evaluate stochastic curtailment procedures.

Group sequential procedures for repeated events data with frailty

In this article we consider the design aspects of group sequential trials with recurrent study endpoints, where the subjects are from a heterogeneous population. The usual procedures of sequential analysis based on the "independent increments" property are no longer valid due to the heterogeneity of the study subjects, as pointed out by Cook and Lawless. We focus on the aspect of study planning and consider a special model (Poisson process with frailty) to investigate the effects of frailty (or subject heterogeneity). We prescribe the valid procedures and provide convenient tables for computing stopping boundaries and sample sizes. The methodology is illustrated through an example of an animal experiment with recurrent tumors, using procedures such as group sequential tests and repeated confidence intervals. Minimal cost analysis is considered for determining the optimal combination of study duration and sample size.