Exact properties of Efron's biased coin randomization procedure

TL;DR

This paper provides exact formulas for the properties of Efron's biased coin randomization, enhancing understanding of its bias, balance, and inference in clinical trial design.

Contribution

It derives closed-form expressions for the exact distribution of treatment imbalance and the variance-covariance matrix, improving analysis of the biased coin design.

Findings

Exact distribution of treatment imbalance derived

Variance-covariance matrix of assignments obtained

Clarifies role of bias probability in trial properties

Abstract

Efron [Biometrika 58 (1971) 403--417] developed a restricted randomization procedure to promote balance between two treatment groups in a sequential clinical trial. He called this the biased coin design. He also introduced the concept of accidental bias, and investigated properties of the procedure with respect to both accidental and selection bias, balance, and randomization-based inference using the steady-state properties of the induced Markov chain. In this paper we revisit this procedure, and derive closed-form expressions for the exact properties of the measures derived asymptotically in Efron's paper. In particular, we derive the exact distribution of the treatment imbalance and the variance-covariance matrix of the treatment assignments. These results have application in the design and analysis of clinical trials, by providing exact formulas to determine the role of the coin's bias probability in the context of selection and accidental bias, balancing properties and randomization-based inference.