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 \(\alpha \)-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 \(\alpha \)-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.
Similar content being viewed by others
References
Armitage P, McPherson C, Rowe B (1969) Repeated significance tests on accumulating data. J R Stat Soc Ser A 132:235–244
Bartky W (1943) Multiple sampling with constant probability. Ann Math Stat 14:363–377
Blackwell D (1947) Conditional expectation and unbiased sequential estimation. Ann Math Stat 105–110
Chan HP, Lai TL (2000) Asymptotic approximations for error probabilities of sequential or fixed sample size tests in exponential families. Ann Stat 1638–1669
Crowder MJ (1976) Maximum likelihood estimation for dependent observations. J R Stat Soc Ser B (Methodol) 38(1):45–53
Dodge H, Romig H (1929) A method of sampling inspection. Bell Syst Tech J 8:613–631
Efron B et al (1975) Defining the curvature of a statistical problem (with applications to second order efficiency). Ann Stat 3(6):1189–1242
Eisenberg B, Ghosh BK (1980) Curtailed and uniformly most powerful sequential tests. Ann Stat 8(5):1123–1131
Ferguson T (1996) A course in large sample theory. Routledge, New York
Ferguson TS (2014) Mathematical statistics: a decision theoretic approach, vol 1. Academic Press, London
Gu MG, Lai TL (1991) Weak convergence of time-sequential censored rank statistics with applications to sequential testing in clinical trials. Ann Stat 1403–1433
Jennison C, Turnbull B (1999) Group sequential methods with applications to clinical trials. CRC Press, Chapman & Hall/CRC Interdisciplinary Statistics
Jennison C, Turnbull BW (2006) Adaptive and nonadaptive group sequential tests. Biometrika 93(1):1–21
Karlin S, Rubin H (1956) The theory of decision procedures for distributions with monotone likelihood ratio. Ann Math Stat 272–299
Lai TL, Shih M-C (2004) Power, sample size and adaptation considerations in the design of group sequential clinical trials. Biometrika 91(3):507–528
Liu A, Hall W (1999) Unbiased estimation following a group sequential test. Biometrika 86(1):71–78
Liu A, Hall W, Yu KF, Wu C (2006) Estimation following a group sequential test for distributions in the one-parameter exponential family. Stat Sin 16(1):165–181
Neyman J, Pearson ES (1933) On the problem of the most efficient tests of statistical hypotheses. Philos Trans R Soc Lond Ser A Contain Pap Math Phys Charact 231:289–337
Philippou AN, Roussas G et al (1973) Asymptotic distribution of the likelihood function in the independent not identically distributed case. Ann Stat 1(3):454–471
Pocock SJ (1977) Group sequential methods in the design and analysis of clinical trials. Biometrika 64(2):191–199
Proschan MA, Lan KKG, Wittes JT (2006) Statistical monitoring of clinical trials: a unified approach. Springer, New York
Schou IM, Marschner IC (2013) Meta-analysis of clinical trials with early stopping: an investigation of potential bias. Stat Med 32(28):4859–4874
Slud EV (1984) Sequential linear rank tests for two-sample censored survival data. Ann Stat 551–571
Tarima S, Flournoy N (2019) Asymptotic properties of maximum likelihood estimators with sample size recalculation. Stat Pap 60:23–44
Tsiatis AA (1982) Repeated significance testing for a general class of statistics used in censored survival analysis. J Am Stat Assoc 77(380):855–861
Wald A (1947) Sequential analysis. Wiley, New York
Wang H, Flournoy N, Kpamegan E (2014) A new bounded log-linear regression model. Metrika 77(5):695–720
Wetherill G, Glazebrook K (1986) Sequential methods in statistics, 3rd edn. Chapman and Hall, New York
Acknowledgements
We grateful to the reviewers and the associate editor whose penetrating queries led to significant improvements in this paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tarima, S., Flournoy, N. Most powerful test sequences with early stopping options. Metrika 85, 491–513 (2022). https://doi.org/10.1007/s00184-021-00839-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-021-00839-w