Janis Hardwick
Robert Oehmke
Quentin F. Stout

University of Michigan

**Abstract**:
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.
Unfortunately, *fully* sequential strategies require
outcomes from all previous allocations prior to the next allocation.
This can prolong an experiment unduly.
As a result, we seek designs for models that specifically incorporate delays.

We utilize a delay model in which patients arrive according to a Poisson process and their response times are exponential. Three designs are examined with an eye towards minimizing patient losses: a delayed two armed bandit rule which is optimal for the model and objective of interest; a newly proposed 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 delayed hyperopic design also performs extremely well throughout the range of delays, despite the fact that the rate of delay is not one of its design parameters. The delayed randomized play-the-winner rule is far less efficient than either of the other methods.

We developed a dynamic programming approach to create optimal designs for arbitrary objective functions. We implemented this on both serial and parallel computers. These programs can also be used to provide exact evaluations of arbitrary designs, such as the randomized play-the-winner.

The hyperopic design approach used here can also be utilized for far more general objective functions. Its superior performance in optimizing total successes gives hope that it can also do quite well on other objectives. Myopic, aka greedy, algorithms are based on looking at the next allocation as if it were the last, whilr hyperopic designs optimize the number of observations on each arm, assuming that no further updating will occur.

**Keywords**:
response adaptive sampling design,
sequential allocation, controlled clinical trial,
design of experiments,
bandit problem,
dynamic programming, hyperopic design

**Complete paper**.
This appears in *Journal of Statistical Planning and Inference* 136 (2006),
pp. 1940-1955.

- Adaptive Allocation:
- Here is an explanation of this topic, and here are our relevant publications.
- Dynamic Programming (also known as backward induction):
- Here is an
overview of our work.

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