Optimal designs and limiting optimal designs for a trinomial response

Abstract

Designs for the continuation-ratio model for a trinomial response will be described. For the situation where the three response categories are: “no response”, “efficacy” and “adverse reaction”, both D-optimal designs and c-optimal designs for estimating the dose with the maximum probability of efficacy are found. Optimal designs are not available in closed form but designs with closed form expression are found which are approximately optimal. Motivated by these designs, a new concept is defined, “limiting optimality”, where a sequence of designs is said to be optimal in an asymptotic sense for a sequence of prior distributions. A member of the sequence is approximately optimal for the corresponding prior distribution. Algebraic forms of limiting optimal designs are derived for a special case of the model where the slopes are equal. They are shown to be very efficient, provide insight, and also provide starting designs for numerical algorithms.