Estimation of the selected treatment mean in two stage drop-the-losers design

ABSTRACT

A common problem faced in clinical studies is that of estimating the effect of the most effective (e.g. the one having the largest mean) treatment among $k(\ge 2)$ available treatments. The most effective treatment is adjudged based on numerical values of some statistic corresponding to the $k$ treatments. A proper design for such problems is the so-called “Drop-the-Losers Design (DLD)”. We consider two treatments whose effects are described by independent Gaussian distributions having different unknown means and a common known variance. To select the more effective treatment, the two treatments are independently administered to $n_1$ subjects each and the treatment corresponding to the larger sample mean is selected. To study the effect of the adjudged more effective treatment (i.e. estimating its mean), we consider the two-stage DLD in which $n_2$ subjects are further administered the adjudged more effective treatment in the second stage of the design. We obtain some admissibility and minimaxity results for estimating the mean effect of the adjudged more effective treatment. The maximum likelihood estimator is shown to be minimax and admissible. We show that the uniformly minimum variance conditionally unbiased estimator (UMVCUE) of the selected treatment mean is inadmissible and obtain an improved estimator. In this process, we also derive a sufficient condition for inadmissibility of an arbitrary location and permutation equivariant estimator and provide dominating estimators in cases, where this sufficient condition is satisfied. The mean squared error and the bias performances of various competing estimators are compared via a simulation study. A real data example is also provided for illustration purpose.

KEYWORDS:

• Two-stage drop-the-losers design
• adaptive trial
• mean squared error
• UMVCUE
• admissible
• minimax
• inadmissible estimator

Acknowledgments

The authors would like to thank the two anonymous referees and the Editor of the journal for their comments and suggestions that have greatly improved this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of the first author is supported by Department of Science & Technology (DST)- INSPIRE, Govt. of India.