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.