Point estimation following a two-stage group sequential trial

Abstract

Repeated testing in a group sequential trial can result in bias in the maximum likelihood estimate of the unknown parameter of interest. Many authors have therefore proposed adjusted point estimation procedures, which attempt to reduce such bias. Here, we describe nine possible point estimators within a common general framework for a two-stage group sequential trial. We then contrast their performance in five example trial settings, examining their conditional and marginal biases and residual mean square error. By focusing on the case of a trial with a single interim analysis, additional new results aiding the determination of the estimators are given. Our findings demonstrate that the uniform minimum variance unbiased estimator, whilst being marginally unbiased, often has large conditional bias and residual mean square error. If one is concerned solely about inference on progression to the second trial stage, the conditional uniform minimum variance unbiased estimator may be preferred. Two estimators, termed mean adjusted estimators, which attempt to reduce the marginal bias, arguably perform best in terms of the marginal residual mean square error. In all, one should choose an estimator accounting for its conditional and marginal biases and residual mean square error; the most suitable estimator will depend on relative desires to minimise each of these factors. If one cares solely about the conditional and marginal biases, the conditional maximum likelihood estimate may be preferred provided lower and upper stopping boundaries are included. If the conditional and marginal residual mean square error are also of concern, two mean adjusted estimators perform well.

Point estimation for adaptive trial designs II: Practical considerations and guidance

“…Finally, a conditional MUE (CMUE), θCMUE , can be calculated following Koopmeiners et al 105 and Grayling and Wason 106 It is defined as the value of 𝜃 that gives a conditional median equal to the observed overall MLE. More formally, it is the solution to the following equation:…”

“…For further comprehensive simulation results comparing the different estimators in the context of two-stage group sequential trials, we refer the reader to Grayling and Wason. 106 For each value of θ, we simulated 10 5 trial replicates and calculated the mean values of the point estimators as well as their standard errors across the trial replicates. Note that the unconditional estimators are all equal to the stage 1 MLE for the trial realizations that stop at the interim analysis.…”

“…Finally, a conditional MUE (CMUE), $$$ {\hat{\theta}}_{CMUE} $$$, can be calculated following Koopmeiners et al 105 and Grayling and Wason 106 It is defined as the value of $$$ \theta $$$ that gives a conditional median equal to the observed overall MLE. More formally, it is the solution to the following equation: $$$$ 0.5={\int}_{-\infty}^{{\hat{\theta}}_{obs}}f\left(\hat{\theta}|T=2\right)d\hat{\theta}, $$$$ where $$$ f\left(\hat{\theta}\ |\ T=2\right) $$$ is the conditional density of the overall MLE (conditional on continuing to stage 2), see Appendix A.2 for further details.…”

“…To do so, we used the (asymptotic) canonical joint distribution of the standardized test statistics at each stage, 11 which is bivariate normal—for further details, see the R code included in the Supporting Information. For further comprehensive simulation results comparing the different estimators in the context of two‐stage group sequential trials, we refer the reader to Grayling and Wason 106 …”

“…Finally, a conditional MUE (CMUE), , can be calculated following Koopmeiners et al 105 and Grayling and Wason 106 It is defined as the value of that gives a conditional median equal to the observed overall MLE. More formally, it is the solution to the following equation: where is the conditional density of the overall MLE (conditional on continuing to stage 2), see Appendix A.2 for further details.…”