title: Estimation After a Group Sequential Trial.
abstract: Group sequential trials are one important instance of studies for which the sample size is not fixeda prioribut rather takes one of a finite set of pre-specified values, dependent on the observed data. Much work has been devoted to the inferential consequences of this design feature. Molenberghset al(2012) and Milanziet al(2012) reviewed and extended the existing literature, focusing on a collection of seemingly disparate, but related, settings, namely completely random sample sizes, group sequential studies with deterministic and random stopping rules, incomplete data, and random cluster sizes. They showed that the ordinary sample average is a viable option for estimation following a group sequential trial, for a wide class of stopping rules and for random outcomes with a distribution in the exponential family. Their results are somewhat surprising in the sense that the sample average is not optimal, and further, there does not exist an optimal, or even, unbiased linear estimator. However, the sample average is asymptotically unbiased, both conditionally upon the observed sample size as well as marginalized over it. By exploiting ignorability they showed that the sample average is the conventional maximum likelihood estimator. They also showed that a conditional maximum likelihood estimator is finite sample unbiased, but is less efficient than the sample average and has the larger mean squared error. Asymptotically, the sample average and the conditional maximum likelihood estimator are equivalent. This previous work is restricted, however, to the situation in which the the random sample size can take only two values,N=norN= 2n. In this paper, we consider the more practically useful setting of sample sizes in a the finite set {n1,n2, …,nL}. It is shown that the sample average is then a justifiable estimator , in the sense that it follows from joint likelihood estimation, and it is consistent and asymptotically unbiased. We also show why simulations can give the false impression of bias in the sample average when considered conditional upon the sample size. The consequence is that no corrections need to be made to estimators following sequential trials. When small-sample bias is of concern, the conditional likelihood estimator provides a relatively straightforward modification to the sample average. Finally, it is shown that classical likelihood-based standard errors and confidence intervals can be applied, obviating the need for technical corrections.
authors.affiliation: I-BioStat, Universiteit Hasselt, B-3590 Diepenbeek, Belgium.
authors.forename: Elasma
authors.initials: E
authors.lastname: Milanzi
authors.affiliation: I-BioStat, Universiteit Hasselt, B-3590 Diepenbeek, Belgium ; I-BioStat, Katholieke Universiteit Leuven, B-3000 Leuven, Belgium.
authors.forename: Geert
authors.initials: G
authors.lastname: Molenberghs
authors.affiliation: Department of Methodology and Statistics, Maastricht University, the Netherlands.
authors.forename: Ariel
authors.initials: A
authors.lastname: Alonso
authors.affiliation: Department of Medical Statistics, London School of Hygiene and Tropical Medicine, London WC1E7HT, United Kingdom.
authors.forename: Michael G
authors.initials: MG
authors.lastname: Kenward
authors.affiliation: Department of Statistics, North Carolina State University, Raleigh, NC, U.S.A.
authors.forename: Anastasios A
authors.initials: AA
authors.lastname: Tsiatis
authors.affiliation: Department of Statistics, North Carolina State University, Raleigh, NC, U.S.A.
authors.forename: Marie
authors.initials: M
authors.lastname: Davidian
authors.affiliation: I-BioStat, Katholieke Universiteit Leuven, B-3000 Leuven, Belgium ; I-BioStat, Universiteit Hasselt, B-3590 Diepenbeek, Belgium.
authors.forename: Geert
authors.initials: G
authors.lastname: Verbeke
keywords: Exponential Family; Frequentist Inference; Generalized Sample Average; Joint Modeling; Likelihood Inference; Missing at Random; Sample Average