Most Powerful Test Sequences with Early Stopping Options

TL;DR

This paper demonstrates that sequential likelihood ratio tests with early stopping are most powerful for one-parameter exponential family data, providing a detailed probability measure construction and analyzing the impact of early stopping on distribution support.

Contribution

It introduces a comprehensive probability measure framework for sequential tests with early stopping, revealing how support reduction affects distribution properties and asymptotic behavior.

Findings

Likelihood ratio tests are most powerful with early stopping in exponential family models.

Early stopping induces mixtures and truncation in the distribution of test statistics.

Asymptotic distributions become mixtures of truncated normals under local alternatives.

Abstract

Sequential likelihood ratio testing is found to be most powerful in sequential studies with early stopping rules when grouped data come from the one-parameter exponential family. First, to obtain this elusive result, the probability measure of a group sequential design is constructed with support for all possible outcome events, as is useful for designing an experiment prior to having data. This construction identifies impossible events that are not part of the support. The overall probability distribution is dissected into stage specific components. These components are sub-densities of interim test statistics first described by Armitage, McPherson and Rowe (1969) that are commonly used to create stopping boundaries given an $\alpha$-spending function and a set of interim analysis times. Likelihood expressions conditional on reaching a stage are given to connect pieces of the probability anatomy together. The reduction of the support caused by the adoption of an early stopping rule induces sequential truncation (not nesting) in the probability distributions of possible events. Multiple testing induces mixtures on the adapted support. Even asymptotic distributions of inferential statistics are mixtures of truncated distributions. In contrast to the classical result on local asymptotic normality (Le Cam 1960), statistics that are asymptotically normal without stopping options have asymptotic distributions that are mixtures of truncated normal distributions under local alternatives with stopping options; under fixed alternatives, asymptotic distributions of test statistics are degenerate.