We first examine the optimal fully sequential allocation, where the results of all previous tests are known before we decide which population to sample next. This is compared to various ad-hoc rules which are easier to determine. We then consider optimal 1-, 2- and 3-stage designs, where we are only given 0, 1, or 2 midpoints in which to examine the results and adjust our allocation for the next stage. Such designs are often prefered in practice, and may be required if there is significant delay in obtaining results. We find that optimal 2- and 3-stage designs are very close to the fully sequential ones in terms of minimizing the objective function. However, the optimal stage sizes are poorly predicted by the asymptotic theory, and instead need to be found computationally. Robustness of the designs is also examined computationally, and it is shown that the adaptive designs are especially robust.
Throughout, efficient algorithms play an important role in enabling us to perform the required optimizations and evaluations for problems of useful size. Algorithmic techniques employed include variations of dynamic programming and forward induction.
While our focus is on evaluating polynomials, the techniques and algorithms can be applied to other problems involving a few populations with discrete outcomes. Examples of such problems include clinical trials and experiments to estimate the prevelance of disease.
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