Title: On the sample mean after a group sequential trial,
Abstract: A popular setting in medical statistics is a group sequential trial with
independent and identically distributed normal outcomes, in which interim
analyses of the sum of the outcomes are performed. Based on a prescribed
stopping rule, one decides after each interim analysis whether the trial is
stopped or continued. Consequently, the actual length of the study is a random
variable. It is reported in the literature that the interim analyses may cause
bias if one uses the ordinary sample mean to estimate the location parameter.
For a generic stopping rule, which contains many classical stopping rules as a
special case, explicit formulas for the expected length of the trial, the bias,
and the mean squared error (MSE) are provided. It is deduced that, for a fixed
number of interim analyses, the bias and the MSE converge to zero if the first
interim analysis is performed not too early. In addition, optimal rates for
this convergence are provided. Furthermore, under a regularity condition,
asymptotic normality in total variation distance for the sample mean is
established. A conclusion for naive confidence intervals based on the sample
mean is derived. It is also shown how the developed theory naturally fits in
the broader framework of likelihood theory in a group sequential trial setting.
A simulation study underpins the theoretical findings.