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Most powerful test sequences with early stopping options

Author

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  • Sergey Tarima

    (Institute for Health and Equity, Medical College of Wisconsin)

  • Nancy Flournoy

    (University of Missouri, 600 S State St.)

Abstract

We extended the application of uniformly most powerful tests to sequential tests with different stage-specific sample sizes and critical regions. In the one parameter exponential family, likelihood ratio sequential tests are shown to be uniformly most powerful for any predetermined $$\alpha $$ α -spending function and stage-specific sample sizes. To obtain this 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 components determined by the stopping stage. These components are the sub-densities of interim test statistics first described by Armitage et al. (J R Stat Soc: Ser A 132:235–244, 1969) that are commonly used to create stopping boundaries given an $$\alpha $$ α -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 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 that are typically normal, are not. Rather they are derived from mixtures of truncated multivariate normal distributions.

Suggested Citation

  • Sergey Tarima & Nancy Flournoy, 2022. "Most powerful test sequences with early stopping options," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(4), pages 491-513, May.
    • Handle: RePEc:spr:metrik:v:85:y:2022:i:4:d:10.1007_s00184-021-00839-w
    DOI: 10.1007/s00184-021-00839-w
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    References listed on IDEAS

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    1. HaiYing Wang & Nancy Flournoy & Eloi Kpamegan, 2014. "A new bounded log-linear regression model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(5), pages 695-720, July.
    2. Christopher Jennison & Bruce W. Turnbull, 2006. "Adaptive and nonadaptive group sequential tests," Biometrika, Biometrika Trust, vol. 93(1), pages 1-21, March.
    3. Sergey Tarima & Nancy Flournoy, 2019. "Asymptotic properties of maximum likelihood estimators with sample size recalculation," Statistical Papers, Springer, vol. 60(2), pages 373-394, April.
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    Cited by:

    1. Alessandro Baldi Antognini & Rosamarie Frieri & Maroussa Zagoraiou, 2023. "New insights into adaptive enrichment designs," Statistical Papers, Springer, vol. 64(4), pages 1305-1328, August.
    2. Sergey Tarima & Nancy Flournoy, 2023. "Group sequential tests: beyond exponential family models," Statistical Papers, Springer, vol. 64(4), pages 1361-1372, August.

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