Adaptive optimal designs applied to the logistic model

In this paper we develop a sequential procedure to approach the D-optimal design given a logistic response function. Chiang (1990) discussed the asymptotic properties of some sequential procedures designed to approach the D-optimal design asymptotically, using the maximum likelihood approach. Chiang could not show asymptotic normality and consistency for these procedures. We introduce an alternative to the maximum likelihood estimate. We show that the estimate of the D-optimal design, for both the one parameter case and the two parameter case, is consistent and asymptotically efficient, in the sense of the D-optimality criterion. Under the assumption that the scale parameter for the logistic response cure is known, a simulation study is conducted to compare the behavior of the maximum likelihood estimator and the new estimator for small or moderate sample size. We found that the new procedure perform better than the maximum likelihood approach on most of the values of the parameter space. Also, we proved theoretically, under the assumptions given by Chiang (1990) that the two procedure are asymptotically equivalent. Furthermore, this result is supported by the simulation. We noticed that the new procedure and the maximum likelihood approach perform the same in terms of the simulated mean square error loss.