Biased coin design with imbalance tolerance

One of the main aspects of a sampling procedure is to determine how to collect samples. A completely random sampling scheme is free of any bias and provides a basis for valid statistical inferences. A balanced sampling scheme strengthens efficiency in statistical inference procedures. This paper presents a sampling scheme, biased coin design with imbalance tolerance, which enforces balance in treatment allocations in sequential clinical trials. The design synthesizes Efron's pioneering work on biased coin design and Soares and Wu's big stick design. The underlying structure of the design is a Markov chain. An explicit formula of the high-order transition probabilities of the chain is obtained. This paper applies the formula to give exact assessments of randomness, balance, and the trade-off between them under the design.