Optimal group sequential multihypothesis tests for exponential families of distributions

• Published in: Sequential analysis Vol. 44; no. 2; pp. 227 - 251
• Main Author: Novikov, Andrey
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 03.04.2025
• Subjects: Multihypothesis testing; optimal group sequential tests; optimal stopping; sequential analysis; two-sided hypotheses
• ISSN: 0747-4946; 1532-4176
• DOI: 10.1080/07474946.2024.2428798

This article deals with group sequential multihypothesis testing. We consider a model with a group-specific cost c ( m ) of obtaining a group of m observations (in particular, this includes the popular model c ( m ) = v + m , where v is a group overhead cost). In this way, an important characteristic of a group sequential testing is the corresponding expected sampling cost (ESC), depending on the value of the parameter. Our concern is construction of optimal group sequential tests taking, as a criterion of optimization, a weighted sum of the ESCs weighted over some values of the parameter. Based on predetermined group sizes m 1 , m 2 , ... , m N , where N is a maximum number of groups admitted to the analysis, we describe the structure of optimal group sequential tests. For observations following a distribution from a one-parameter exponential family, we develop computational algorithms for designing optimal plans and evaluating their performance characteristics. For a series of exponential families (normal, exponential, binomial, Poisson, negative binomial) we implement the algorithms in the R programming language. The program code is available in a public GitHub repository. Applications of the developed methods are presented in a series of practical examples. In particular, we evaluate the efficiency of the optimal group sequential tests for three hypotheses about the mean of a normal distribution with known variance. The effect of optimization over the group sizes is discussed. Various cost structures are considered, with or without overhead cost in the groups. A method of construction of two-sided group sequential tests for two hypotheses, based on the optimal tests for three hypotheses, is proposed. We exemplify the use of the method constructing two-sided tests for the binomial proportions with symmetric and nonsymmetric alternatives and compare its efficiency with some group sequential tests known in the literature. Also, we apply the method to construction of two-sided tests for the mean of a normal distribution and compare its efficiency with that of the "double triangular" by Whitehead.