Early stopping in the two-sample problem for bounded random variables

In a fixed sample size clinical trial, the question often asked is whether the final outcome has been determined before the data has been completely collected. A particular situation occurs when the intake of subjects is slow relative to the time required for the outcome variable to be realized. We assume that the range of values the outcome variable may take can be specified in advance. We also assume that simple randomization has been used and thus, under the null hypothesis, Student's t test is a good approximation to the exact test involving the permutational distribution. In order to determine with partial data whether the outcome of the final test of hypothesis is already certain, certain nonlinear extremal problems with constraints must be solved. Analytic solutions may be expressed in the form of noniterative algorithms. Simulation studies suggest an approximation to the analytic solution. The results also provide a basis for assessing at the end of the trial, whether missing values are of consequence in the test of significance. An application based on an actual clinical trial is presented.