Exact Group Sequential Methods for Estimating a Binomial Proportion

We first review existing sequential methods for estimating a binomial proportion. Afterward, we propose a new family of group sequential sampling schemes for estimating a binomial proportion with prescribed margin of error and confidence level. In particular, we establish the uniform controllability of coverage probability and the asymptotic optimality for such a family of sampling schemes. Our theoretical results establish the possibility that the parameters of this family of sampling schemes can be determined so that the prescribed level of confidence is guaranteed with little waste of samples. Analytic bounds for the cumulative distribution functions and expectations of sample numbers are derived. Moreover, we discuss the inherent connection of various sampling schemes. Numerical issues are addressed for improving the accuracy and efficiency of computation. Computational experiments are conducted for comparing sampling schemes. Illustrative examples are given for applications in clinical trials. 1. Introduction Estimating a binomial proportion is a problem of ubiquitous significance in many areas of engineering and sciences. For economical reasons and other concerns, it is important to use as fewer as possible samples to guarantee the required reliability of estimation. To achieve this goal, sequential sampling schemes can be very useful. In a sequential sampling scheme, the total number of observations is not fixed in advance. The sampling process is continued stage by stage until a prespecified stopping rule is satisfied. The stopping rule is evaluated with accumulated observations. In many applications, for administrative feasibility, the sampling experiment is performed in a group fashion. Similar to group sequential tests [1, 8], [2], an estimation method based on taking samples by groups and evaluating them sequentially is referred to as a group sequential estimation method. It should be noted that group sequential estimation methods are general enough to include fixed-sample-size and fully sequential procedures as special cases. Particularly, a fixed-sample-size method can be viewed as a group sequential procedure of only one stage. If the increment between the sample sizes of consecutive stages is equal to , then the group sequential method is actually a fully sequential method. It is a common contention that statistical inference, as a unique science to quantify the uncertainties of inferential statements, should avoid errors in the quantification of uncertainties, while minimizing the sampling cost. That is, a statistical inferential method