Abstract
SummaryWe describe a general method for finding a confidence region for a parameter vector that is compatible with the decisions of a two-stage closed test procedure in an adaptive experiment. The closed test procedure is characterized by the fact that rejection or nonrejection of a null hypothesis may depend on the decisions for other hypotheses and the compatible confidence region will, in general, have a complex, nonrectangular shape. We find the smallest cross-product of simultaneous confidence intervals containing the region and provide computational shortcuts for calculating the lower bounds on parameters corresponding to the rejected null hypotheses. We illustrate the method with an adaptive phase II/III clinical trial.
Highlights
For experiments designed to make inference about a parameter vector θ = (θ1, . . . , θK ), it is common to find confidence intervals for all of the individual θk such that the simultaneous coverage probability is at least 1 − α
For experiments conducted in a single stage, Hayter & Hsu (1994) showed how simultaneous 100(1 − α)% confidence intervals can be constructed to be compatible with some commonly used closed test procedures, in the sense that a null hypothesis Hk is rejected at familywise level α if and only if the confidence interval for θk excludes all values for which Hk is true
For one-sided problems where larger parameter values are more beneficial, no 100(1 − α)% lower confidence bound for any individual θk can exceed δk unless all hypotheses H1, . . . , HK can be rejected at familywise level α
Summary
For experiments conducted in a single stage, Hayter & Hsu (1994) showed how simultaneous 100(1 − α)% confidence intervals can be constructed to be compatible with some commonly used closed test procedures, in the sense that a null hypothesis Hk is rejected at familywise level α if and only if the confidence interval for θk excludes all values for which Hk is true. Often, these intervals are scarcely more informative than the test decisions. We will show that this problem is mitigated to some extent by the adaptive nature of the experiment
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