The Annals of Statistics

Sequential monitoring of response-adaptive randomized clinical trials

Hongjian Zhu and Feifang Hu

Full-text: Open access


Clinical trials are complex and usually involve multiple objectives such as controlling type I error rate, increasing power to detect treatment difference, assigning more patients to better treatment, and more. In literature, both response-adaptive randomization (RAR) procedures (by changing randomization procedure sequentially) and sequential monitoring (by changing analysis procedure sequentially) have been proposed to achieve these objectives to some degree. In this paper, we propose to sequentially monitor response-adaptive randomized clinical trial and study it’s properties. We prove that the sequential test statistics of the new procedure converge to a Brownian motion in distribution. Further, we show that the sequential test statistics asymptotically satisfy the canonical joint distribution defined in Jennison and Turnbull (2000). Therefore, type I error and other objectives can be achieved theoretically by selecting appropriate boundaries. These results open a door to sequentially monitor response-adaptive randomized clinical trials in practice. We can also observe from the simulation studies that, the proposed procedure brings together the advantages of both techniques, in dealing with power, total sample size and total failure numbers, while keeps the type I error. In addition, we illustrate the characteristics of the proposed procedure by redesigning a well-known clinical trial of maternal-infant HIV transmission.

Article information

Ann. Statist., Volume 38, Number 4 (2010), 2218-2241.

First available in Project Euclid: 11 July 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems 62G10: Hypothesis testing
Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations

Asymptotic properties Brownian process response-adaptive randomization power sample size type I error


Zhu, Hongjian; Hu, Feifang. Sequential monitoring of response-adaptive randomized clinical trials. Ann. Statist. 38 (2010), no. 4, 2218--2241. doi:10.1214/10-AOS796.

Export citation


  • Andersen, J. (1996). Clinical trials designs—made to order. J. Biopharm. Statist. 6 515–522.
  • Andersen, K. M. (2007). Optimal spending functions for asymmetric group sequential designs. Biom. J. 49 337–345.
  • Armitage, P. (1957). Restricted sequential procedures. Biometrika 44 9–26.
  • Armitage, P. (1975). Sequential Medical Trials. Blackwell, Oxford.
  • Athreya, K. B. and Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist. 39 1801–1817.
  • Bai, Z. D. and Hu, F. (2005). Asymptotics in randomized urn models. Ann. Appl. Probab. 15 914–940.
  • Berry, D. A. (2005). Introduction to Bayesian methods III: Use and interpretation of Bayesian tools in design and analysis. Clinical Trials 2 295–300.
  • Coad, D. S. and Rosenberger, W. F. (1999). A comparison of the randomized play-the-winner rule and the triangular test for clinical trials with binary responses. Stat. Med. 18 761–769.
  • Cheng, Y. and Shen, Y. (2005). Bayesian adaptive designs for clinical trials. Biometrika 92 633–646.
  • Connor, E. M., Sperling, R. S., Gelber, R., Kiselev, P., Scott, G., O’Sullivan, M. J., VanDyke, R., Bey, M., Shearer, W., Jacobson, R. L., Jimenez, E., O’Neill, E., Bazin, B., Delfraissy, J. F., Culname, M., Coombs, R., Elkins, M., Moye, J., Stratton, P. and Balsley, J. (1994). Reduction of maternal-infant transmission of human immunodeficiency virus type I with zidovudine treatment. The New England Journal of Medicine 331 1173–1180.
  • DeMets, D. L. (2006). Futility approaches to interim monitoirng by data monitoring committees. Clinical Trials 3 522–529.
  • Eisele, J. and Woodroofe, M. (1995). Central limit theorems for doubly adaptive biased coin designs. Ann. Statist. 23 234–254.
  • Ethier, S. N. and Kurts, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Gwise, T. E., Hu, J. and Hu, F. (2008). Optimal biased coins for two-arm clinical trials. Stat. Interface 1 125–135.
  • Hayre, L. S. (1979). Two-population sequential tests with three hypotheses. Biometrika 66 465–474.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • Hu, F. and Rosenberger, W. F. (2003). Optimality, variability, power: Evaluating response-adaptive randomization procedures for treatment comparisons. J. Amer. Statist. Assoc. 98 671–678.
  • Hu, F. and Rosenberger, W. F. (2006). The Theory of Response-Adaptive Randomization in Clinical Trials. Wiley, New York.
  • Hu, F., Rosenberger, W. F. and Zhang, L. (2006). Asymptotically best response-adaptive randomization procedures. J. Statist. Plann. Inference 136 1911–1922.
  • Hu, F. and Zhang, L. X. (2004). Asymptotic properties of doubly adaptive biased coin designs for multi-treatment clinical trials. Ann. Statist. 32 268–301.
  • Hu, F., Zhang, L. X. and He, X. (2009). Efficient randomized adaptive designs. Ann. Statist. 37 2543–2560.
  • Ivanova, A. V. (2003). A play-the-winner type urn model with reduced variability. Metrika 58 1–13.
  • Jennison, C. and Turnbull, B. W. (2000). Group Sequential Methods With Applications to Clinical Trials. Chapman and Hall, Boca Raton, FL..
  • Lan, K. and DeMets, D. L. (1983). Discrete sequential boundaries for clinical trials. Biometrika 70 659–663.
  • Lewis, R. J., Lipsky, A. M. and Berry, D. A. (2007). Bayesian decision-theoretic group sequential clinical trial design based on a quadratic loss function: A frequentist evaluation. Clinical Trials 4 5–14.
  • O’Brien, P. C. and Fleming, T. R. (1979). A multiple testing procedure for clinical trials. Biometrics 35 549–556.
  • Pocock, S. J. (1977). Group sequential methods in the design and analysis of clinical trials. Biometrika 64 191–199.
  • Pocock, S. J. (1982). Interim analyses for randomized clinical trials: The group sequential approach. Biometrics 38 153–162.
  • Proschan, M. A., Lan, K. and Wittes, J. T. (2006). Statistical Monitoring of Clinical Trials, A Unified Approach. Springer, New York.
  • Robbins, H. (1952). Some aspects of the sequential design of experiments. Bull. Amer. Math. Soc. 58 527–535.
  • Rosenberger, W. F. and Hu, F. (2004). Maximizing power and minimizing treatment failures in clinical trials. Clinical Trials 1 141–147.
  • Rosenberger, W. F. and Lachin, J. M. (2002). Randomization in Clinical Trials: Theory and Practice. Wiley, New York.
  • Rosenberger, W. F., Stallard, N., Ivanova, A., Harper, C. N. and Ricks, M. L. (2001). Optimal adaptive designs for binary response trials. Biometrics 57 909–913.
  • Rout, C. C., Rocke, D. A., Levin, L., Gouws, E. and Reddy, D. (1993). A reevaluation of the role of crystalloid preload in the prevention of hypotension associated with apinal anesthesia for elective cesarean section. Anesthesiology 79 262–269.
  • Tamura, R. N., Faries, D. E., Andersen, J. S. and Heiligenstein, J. H. (1994). A case study of an adaptive clinical trial in the treatment of out-patients with depressive disorder. J. Amer. Statist. Assoc. 89 768–776.
  • Thompson, W. R. (1933). On the likelihood that one unknown probability exceeds another in the review of the evidence of the two samples. Biometrika 25 275–294.
  • Tymofyeyev, Y., Rosenberger, W. F. and Hu, F. (2007). Implementing optimal allocation in sequential binary response experiments. J. Amer. Statist. Assoc. 102 224–234.
  • Wald, A. (1947). Sequential Analysis. Wiley, New York.
  • Wathen, J. K. and Thall, P. F. (2008). Bayesian adaptive model selection for optimizing group sequential clinical trials. Statistics in Medicine 27 5586–5604.
  • Wei, L. J. and Durham, S. (1978). The randomized play-the-winner rule in medical trials. J. Amer. Statist. Assoc. 73 840–843.
  • Zelen, M. (1969). Play the winner and the controlled clinical trial. J. Amer. Statist. Assoc. 64 131–146.
  • Zhang, L. and Rosenberger, W. F. (2006). Response-adaptive randomization for clinical trials with continuous outcomes. Biometrics 62 562–569.
  • Zhang, L. X., Hu, F. and Cheung, S. H. (2006). Asymptotic theorems of sequential estimation-adjusted urn models. Ann. Appl. Probab. 16 340–369.