Abstract
This manuscript considers group sequential tests powered for multiple ordered alternative hypotheses with a predetermined \(\alpha \)-spending function. Theorem 1 shows that if a fixed-sample size likelihood ratio test is monotone with respect to a one-dimensional test statistic, then a group sequential test constructed by interim cumulative likelihood ratio tests is most powerful for a sequence of ordered alternatives, at a given \(\alpha \)-spending function. This theorem extends Tarima and Flournoy (Metrika 85: 491-513, 2022) from the exponential family to non-exponential distributions with monotone likelihood ratio. A three-stage design powered for three ordered alternatives shows how the theory applies to uniform data. When the likelihood ratio is not monotone for finite sample sizes, locally most powerful tests can be constructed if a test is locally most powerful for a fixed sample size against a local alternative. A two-stage Cauchy example shows how such tests can be built using either a likelihood ratio test statistic or its MLE. Overall, if a parametric distribution of the data is either known or assumed, MLE-based group sequential tests powered for multiple ordered alternatives are most powerful for this set of hypotheses in either finite or in local asymptotic settings.
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. https://doi.org/10.2307/2343787
Bening VE, Korolev VY (2005) On an application of the student distribution in the theory of probability and mathematical statistics. Theory Probab Appl 49(3):377–391
Bening VE, Korolev VY (2012) Generalized Poisson models and their applications in insurance and finance. Walter de Gruyter
FDA (2019) Adaptive designs for clinical trials of drugs and biologics: guidance for industry. U.S. Department of Health and Human Services: Food and Drug Administration, Center for Drug Evaluation and Research, Center for Biologics Evaluation and Research, https://www.fda.gov/regulatory-information/search-fda-guidance-documents/adaptive-design-clinical-trials-drugs-and-biologics-guidance-industry
Flournoy N, Tarima S (2021) Choosing interim sample sizes in group sequential designs. In: Röhrig R, et al (eds) German Medical Data Sciences: Bringing Data to Life: Proceedings of the Joint Annual Meeting of the German Association of Medical Informatics, Biometry and Epidemiology (GMDS EV) and the Central European Network-International Biometrics Society (CEN-IBS) 2020 in Berlin, Germany, vol 278. IOS Press, pp 11–16
Glaz J, Pozdnyakov V (2005) A repeated significance test for distributions with heavy tails. Sequ Anal 24(1):77–98
Gnedenko BV, Korolev VY (1996) Random summation: limit theorems and applications. CRC Press, London
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) Efficient group sequential designs when there are several effect sizes under consideration. Stat Med 25(6):917–932
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 Science & Business Media
Schäfer H, Müller HH (2004) Construction of group sequential designs in clinical trials on the basis of detectable treatment differences. Stat Med 23(9):1413–1424
Tarima S, Flournoy N (2022) Most powerful test sequences with early stopping options. Metrika 85(4):491–513
Acknowledgements
We thank the editor and two anonymous reviewers for their thoughtful comments that helped improve the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Tarima, S., Flournoy, N. Group sequential tests: beyond exponential family models. Stat Papers 64, 1361–1372 (2023). https://doi.org/10.1007/s00362-023-01432-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-023-01432-1