Bias calculations for adaptive urn designs

A clinical trial model is considered in which two treatments with immediate binary responses are to be compared. An adaptive urn design is used to assign patients to the treatments. The bias and variance of the maximum likelihood estimators of the probabilities of success are derived by differentiating the fundamental identity of sequential analysis. By embedding the design in a continuous-time process, probability generating functions are then calculated to obtain approximations for the bias and variance. Simulation is used to assess the accuracy of the approximations. It is shown that the bias cannot be ignored, and that the adaptive rules which are subcritical in nature have the most mathematically tractable bias and are the least variable. Methods for correcting for the bias are also addressed.