Statistics and Its Interface

Volume 13 (2020)

Number 3

Evaluation of Bias for Outcome Adaptive Randomization Designs with Binary Endpoints

Pages: 287 – 315



Yaping Wang (Division of Biometrics IX, Office of Biostatistics, Center for Drug Evaluation and Research, U.S. Food and Drug Administration, Silver Spring, Maryland, U.S.A.)

Hongjian Zhu (Department of Biostatistics, School of Public Health, University of Texas, Houston, Tx., U.S.A.)

J. Jack Lee (Department of Biostatistics and the MS Anderson Cancer Center, University of Texas, Houston, Tx., U.S.A.)


Clinical trial designs applying outcome adaptive randomization (OAR) sequentially change randomization probabilities basing on observed outcomes. Compared to the conventional equal randomization procedure, OAR has the feature to assign more patients to the better treatment arm and yield higher overall response rates for patients in the trial. However, the true response rates tend to be underestimated in OAR trials. Although the bias converges to zero asymptotically as the sample size increases, it is nonnegligible in small trials. In this paper, we evaluated the bias of OAR designs with binary endpoints, with the allocation probabilities implemented under two respective randomization procedures, namely, the sequential maximum likelihood procedure (SMLE) and the doubly adaptive biased coin design (DBCD). We found that the patterns of bias are similar between the two adaptive randomization procedures. When the true response rate is less than 10%, we discover that the bias can be as large as 20% of the true response rates if the sample size is less than 30; the absolute value of the bias, however, remains small. To better gauge the magnitude of the bias, we derived some large-sample strategies to approximate the bias for two target allocation proportions and two randomization procedures. In addition, we conducted simulation studies to quantify the magnitude of the bias in finite samples to assess the accuracy of the asymptotic approximations. We also provided an intuitive explanation for the cause of the underestimation under OAR, and discussed remedies to alleviate bias in the OAR design. A deeper understanding of this bias can help us design better OAR trials and provide more accurate estimates.


adaptive design, clinical trial, estimation, bias

J. Jack Lee’s work was supported in part by grant CA016672 from the National Cancer Institute.

Received 4 April 2019

Accepted 14 January 2020

Published 22 April 2020