On the equivalence of optimality design criteria for the placebo-treatment problem

We consider a class of optimality criteria and show that each criterion has its unique and equivalent dual within the class. This property can be used to find a variety of optimal designs, including a class of compound optimal designs and their relationships. As an example, we show that one type of D-optimal design provides analytical formula for a class of compound optimal designs, while its dual, the more traditional criterion, cannot.