The geometric up and down design for allocating dosage levels

This paper deals with an experimental design for sequentially allocating dose levels to patients in a toxicity experiment by rules depending on prior treatments. The goal of this design is to provide requisite information for appropriate estimation and inference while minimizing patient exposure to highly toxic dose levels. Let $Y\ (n),\ n > 0,$ be a series of Bernoulli trials with probability of response $Q(x) = P(Y(n) = 1\ \vert\ X(n) = x),$ where $X(n)$ is the treatment dosage given to the nth subject. Let $X(n)$ take values from a finite set $\Omega\sb{x} = \{x\sb1,{\...},x\sb{K}\}$ according to the following rule. If $X(n) = x\sb{k}$ and $Y(n) = 1,$ treat the next subject at $x\sb{k-1}$. However, if $Y(n) = 0,$ treat at the same level $x\sb{k}$ until e consecutive successes are observed. If e consecutive successes are observed, treat the next subject at $x\sb{k+1}.$ We derive the asymptotic treatment distribution generated by this rule and show how to select e to center the treatment distribution around an arbitrary unknown quantile of the response function $Q(x).$ We describe maximum likelihood estimation for $\mu$. We also derive the finite means and variances of the treatment allocation proportions.