Sequential analysis of Durham and Flournoy's biased coin design for phase I clinical trials

In a phase I clinical trial, we are interested in finding a dose mu that will produce toxicity at an acceptable probability level Gamma in the target population. We investigate various estimators of the target dose mu to be used with the up-and-down Biased Coin Design (BCD) introduced by Durham and Flournoy (1994). These estimators of mu are derived using isotonic regression, maximum likelihood, weighted least squares, and the simple empirical mean. The best of these estimators, based on the mean square error criterion, will be used to compare different stopping rules based on the fixed width confidence interval and the observed dose frequency. Given a vector of probability of toxicity at the different doses, we show how to derive the exact distribution of these (and many other) estimators. However, due to computational limitations, for modest samples (n > 15), the exact method becomes infeasible and bootstrap methods are used. Asymptotic results will be derived, when possible, assuming the probability of toxicity follows a logistic function. We derive the exact covariance between the number of subjects at dose x j and the number of toxicities at dose xk. We also approximate the covariance between logit( Pj) and logit(Pk), where Pi = P(toxicity|dose xi). Finally the "best" of these estimators with the "best" stopping rule in the BCD setting will be compared to the Continual Reassessment Method (CRM) in terms of expected sample size, the distribution of allocated doses and the distribution of the recommended dose.