Abstract
The two-stage drop-the-loser design provides a framework for selecting the most promising of K experimental treatments in stage one, in order to test it against a control in a confirmatory analysis at stage two. The multistage drop-the-losers design is both a natural extension of the original two-stage design, and a special case of the more general framework of Stallard & Friede (2008) (Stat. Med. 27, 6209–6227). It may be a useful strategy if deselecting all but the best performing treatment after one interim analysis is thought to pose an unacceptable risk of dropping the truly best treatment. However, estimation has yet to be considered for this design. Building on the work of Cohen & Sackrowitz (1989) (Stat. Prob. Lett. 8, 273–278), we derive unbiased and near-unbiased estimates in the multistage setting. Complications caused by the multistage selection process are shown to hinder a simple identification of the multistage uniform minimum variance conditionally unbiased estimate (UMVCUE); two separate but related estimators are therefore proposed, each containing some of the UMVCUEs theoretical characteristics. For a specific example of a three-stage drop-the-losers trial, we compare their performance against several alternative estimators in terms of bias, mean squared error, confidence interval width and coverage.
Highlights
The maximum likelihood estimate (MLE) of the treatment effect is often reported as standard at the end of a multistage trial
Many bias adjusted estimation procedures have been proposed, and unbiasedness is certainly not the only characteristic by which an estimator can be judged, the only way to achieve an efficient and “purely” unbiased estimate is to execute the following procedure: (i) identify an unbiased estimate based on part of the data—Y say, (ii) identify complete, sufficient statistics for the parameter in question—Z say, and (iii) employ the Rao-Blackwell improvement formula to obtain E[Y |Z]—the uniform minimum variance unbiased estimate (UMVUE)
Given M, a sufficient statistic of the data, Z, and a truncation adaptive stopping rule (Liu & Hall, 1999) one calculates the expectation of the first stage data, Y1 say, given the pair (M, Z) to obtain the truncation adaptable UMVUE. We refer to this approach as unconditional because, in a three-stage trial, for example, it produces an estimate of the treatment effect regardless of whether the trial stops at stage one, two, or three and it is unbiased by definition when one averages across all possible realizations of the sequential trial
Summary
The maximum likelihood estimate (MLE) of the treatment effect is often reported as standard at the end of a multistage trial. Given M, a sufficient statistic of the data, Z, and a truncation adaptive stopping rule (Liu & Hall, 1999) one calculates the expectation of the first stage data, Y1 say, given the pair (M, Z) to obtain the truncation adaptable UMVUE We refer to this approach as unconditional because, in a three-stage trial, for example, it produces an estimate of the treatment effect regardless of whether the trial stops at stage one, two, or three and it is unbiased by definition when one averages across all possible realizations of the sequential trial.
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