Adaptive Designs for Delayed Response Models

Adaptive designs are effective mechanisms for flexibly allocating experimental resources. In clinical trials particularly, such designs allow researchers to balance short and long term goals of improving patient well being. This tradeoff often takes the form of trying to minimize patient losses during the experiment while simultaneously gathering sufficient information to support decisions about the treatment of future patients. Unfortunately, fully sequential designs require immediate responses since each allocation depends upon the responses of all previous assignments. Since optimal fully sequential strategies cannot be applied when responses are delayed, we seek optimal designs for models that specifically incorporate delays. Of interest are loss of efficiency compared to non-delayed situations and improvement over suboptimal delayed response designs. Here we propose a delay model in which patients arrive according to a Poisson process and their response times are exponential. We examine three designs: the optimal bandit rule, a new hyperopic rule, and a randomized play-the-winner rule. The results show that, except when the delay rate is several orders of magnitude different than the patient arrival rate, the delayed response bandit is nearly as efficient as the immediate response bandit. The hyperopic design also performs extremely well throughout the range of delays, while the randomized play-the-winner rule is far less efficient than either of the other methods.