Abstract

We consider optimal design under a cost constraint, where a scalar coefficient L sets the compromise between information and cost. For suitable cost functions, one can force the support points of an optimal design measure to concentrate around points of minimum cost by increasing the value of L, which can be considered as a tuning parameter that specifies the importance given to the cost constraint.

An example of adaptive design in a dose-finding problem with a bivariate binary model will be presented. As usual in nonlinear situations, the optimal design for any arbitrary choice of L depends on the unknown value of the model parameters. The construction of this optimal design can be made adaptive, by using a steepest-ascent algorithm where the current estimated value of the parameters (by Maximum Likelihood) is substituted for their unknown value. Then, taking benefit of the fact that the design space (the set of available doses) is finite, one can prove the strong consistency and asymptotic normality of the ML estimator when L is kept constant. Since the cost is reduced when L is increased, it is tempting to let L increase with the number of observations (patients enroled in the trial). The strong consistency of the ML estimator is then preserved when L increases slowly enough.