Convergence of Nonparametric Long-Memory Phase I Designs

We examine Phase I cancer clinical trial designs that use toxicity estimates based on all available data at each dose-allocation decision, but refrain from employing parametric models or Bayesian decision rules. We show that one such design family, called here "interval designs", converges almost surely to the maximum tolerated dose under fairly general conditions. Another family called "point designs" does not converge. These results suggest that existing Bayesian designs, which are closer in spirit to the latter family, have to be substantially modified if a general convergence proof for them is desired.