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SEQUENTIAL DESIGNS FOR BINARY RANDOM VARIABLES
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{\bf Nancy Flournoy \\
American University}
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Let $X(n)$, $n\geq 1,$ be a sequence of treatments for which the outcomes are
binary random variables, $Y(n),\,n\geq 1$. \ \ We take the treatment space to
be a finite set of $\,K$ points: \ $\Omega_X=\left\{x_1<\cdots<x_K\right\}$.
\ Let the probability of response vary with some stimulus$\,x$. \ Designs are
presented for two scenarios. \ First assume that $P\left\{Y(n)=1|x\right\}$
increases with $x$, as is the case when $Y$ is toxicity and $x\,$is dose. \
For this scenario, the goal is to estimate the dose $\mu$ for which \
$P\left\{Y(n)=1|\mu\right\}=\Gamma$, where $\Gamma$ is prespecified. \
Another objective is to avoid treatment at highly toxic dosages. \ In the
second scenario $Y(n)$ indicates success, and we assume that the response
function $P\left\{Y(n)=1|x\right\}$ is a unimodal function of $x$. \ \ This
is the case when too little and too much treatment are both bad and the goal
is to estimate the dose $\mu$ with the maximum success probability. \ In this
scenario it is desirable to avoid treatments with high risk of failure. \ For
these scenarios, we describe up-and-down designs and designs based on urn
models that cause treatments to cluster around, or converge to, the unknown
target $\mu$. \ These designs are characterized using Markov chains and
branching processes. \ Of note, they make no parametric assumptions about the
response function. \ 

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