Minimizing Selection Bias Under the Blackwell and Hodges Model with an Equal Allocation Procedure in a Symmetric Allocation Space

Abstract

The investigator in a randomized open-label single-center trial knows the treatment assignments of all randomized subjects and thus can introduce a selection bias in the study results by allocating subjects with a better prognosis when he guesses the next allocation is to the experimental group. When the allocation procedure is known to the investigator, Blackwell and Hodges demonstrated that the truncated binomial design of the size N, where subjects are allocated at random with probability ½ until one of the treatment arms has $N/$2 subjects, minimizes the selection bias among all 1:1 allocation procedures that assign $N/2$ subjects to each treatment. In this article we demonstrate that for any allocation space symmetric with respect to the two treatments, the selection bias is also minimized with the procedure that allocates subjects with probability ½ whenever allocation to both treatments is allowed—the big stick design. We propose to control both the balance in treatment assignments and the selection bias by specifying the allocation space with acceptable, possibly increasing with time, imbalance and using the big stick randomization within it. We show how the selection bias can be further reduced when the investigator is not aware of the allocation procedure.

KEYWORDS:

• Allocation procedure
• Allocation space
• Big stick design
• Guessing strategy
• Randomization
• Selection bias

Supplementary Materials

The supplementary material contains Appendix with the proof of Theorem 1.

Acknowledgments

The author thanks two anonymous reviewers and the editor for the valuable comments that helped improve the article.

Disclosure Statement

The author reports there are no competing interests to declare.

Additional information

Funding

The author(s) reported there is no funding associated with the work featured in this article.