The Annals of Statistics

Sequential monitoring with conditional randomization tests

Victoria Plamadeala and William F. Rosenberger

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Sequential monitoring in clinical trials is often employed to allow for early stopping and other interim decisions, while maintaining the type I error rate. However, sequential monitoring is typically described only in the context of a population model. We describe a computational method to implement sequential monitoring in a randomization-based context. In particular, we discuss a new technique for the computation of approximate conditional tests following restricted randomization procedures and then apply this technique to approximate the joint distribution of sequentially computed conditional randomization tests. We also describe the computation of a randomization-based analog of the information fraction. We apply these techniques to a restricted randomization procedure, Efron’s [Biometrika 58 (1971) 403–417] biased coin design. These techniques require derivation of certain conditional probabilities and conditional covariances of the randomization procedure. We employ combinatoric techniques to derive these for the biased coin design.

Article information

Ann. Statist., Volume 40, Number 1 (2012), 30-44.

First available in Project Euclid: 15 March 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E15: Exact distribution theory 62K99: None of the above, but in this section
Secondary: 62L05: Sequential design 62J10: Analysis of variance and covariance

Biased coin design conditional reference set random walk restricted randomization sequential analysis


Plamadeala, Victoria; Rosenberger, William F. Sequential monitoring with conditional randomization tests. Ann. Statist. 40 (2012), no. 1, 30--44. doi:10.1214/11-AOS941.

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Supplemental materials

  • Supplementary material: Supplement to “Sequential monitoring with conditional randomization tests”. The supplement contains Appendix A (proof of Theorem 2.2), Appendix B (proof of Lemma 3.1), Appendix C (proof of Theorem 3.1), and Appendix D (proof of Lemma 4.3).