Exact distributional properties of Efron's biased coin design with applications to clinical trials

Randomization and balance in treatment assignments are the two of the most important factors in sequential designs used in clinical trials. The simplest sequential trials involve two experimental groups. Patients arrive sequentially and are assigned to one of the two treatments. Randomization mitigates possible biases, such as selection bias and accidental bias, due to the differences between the experimental groups, and enhances validity of statistical analysis. It also serves as the basis for a class of inferential procedures called randomization tests. Large imbalances in treatment assignments can negatively impact efficiency of many statistical procedures and may create imbalances on key covariates.;Unfortunately, balance and randomization are competing goals. Much of the research on randomization procedures can be described as finding a trade-off between balance of the design and degree of randomness of treatment assignments.;In 1971, Efron introduced a design, called the biased coin design (BCD), in an effort to find a design combining good balancing properties with low susceptibility to experimental biases. Efron derived several key asymptotic results on the BCD that allowed assessing asymptotic balancing and bias properties of the BCD. Despite the fact that the BCD has attracted much attention since the publication of Efron's article, most finite distributional properties, such as the distribution of the imbalance and covariance between treatment assignments, have been open problems until now. The goal of this dissertation is to fill this gap.;The author has obtained the exact distribution of the imbalance process, sample size, joint distribution of a pair of treatment assignments and covariance matrix of treatment assignments of the BCD. With the help of these results, exact formulas are obtained for finite-stage probabilities and the moments of the imbalance, selection bias and accidental bias. This will allow for a full and exact evaluation of balancing properties and susceptibility to experimental biases without resorting to approximations provided by asymtotics or simulation.