## The Annals of Statistics

- Ann. Statist.
- Volume 38, Number 3 (2010), 1546-1567.

### Exact properties of Efron’s biased coin randomization procedure

Tigran Markaryan and William F. Rosenberger

#### 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.

#### Article information

**Source**

Ann. Statist. Volume 38, Number 3 (2010), 1546-1567.

**Dates**

First available in Project Euclid: 24 March 2010

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1269452646

**Digital Object Identifier**

doi:10.1214/09-AOS758

**Mathematical Reviews number (MathSciNet)**

MR2662351

**Zentralblatt MATH identifier**

1189.62130

**Subjects**

Primary: 62E15: Exact distribution theory 62K99: None of the above, but in this section

Secondary: 62L05: Sequential design 62J10: Analysis of variance and covariance

**Keywords**

Accidental bias exact distribution theory randomization test restricted randomization selection bias

#### Citation

Markaryan, Tigran; Rosenberger, William F. Exact properties of Efron’s biased coin randomization procedure. Ann. Statist. 38 (2010), no. 3, 1546--1567. doi:10.1214/09-AOS758. http://projecteuclid.org/euclid.aos/1269452646.