Statistical Science

Stochastic Approximation and Modern Model-Based Designs for Dose-Finding Clinical Trials

Ying Kuen Cheung

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In 1951 Robbins and Monro published the seminal article on stochastic approximation and made a specific reference to its application to the “estimation of a quantal using response, nonresponse data.” Since the 1990s, statistical methodology for dose-finding studies has grown into an active area of research. The dose-finding problem is at its core a percentile estimation problem and is in line with what the Robbins–Monro method sets out to solve. In this light, it is quite surprising that the dose-finding literature has developed rather independently of the older stochastic approximation literature. The fact that stochastic approximation has seldom been used in actual clinical studies stands in stark contrast with its constant application in engineering and finance. In this article, I explore similarities and differences between the dose-finding and the stochastic approximation literatures. This review also sheds light on the present and future relevance of stochastic approximation to dose-finding clinical trials. Such connections will in turn steer dose-finding methodology on a rigorous course and extend its ability to handle increasingly complex clinical situations.

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Statist. Sci., Volume 25, Number 2 (2010), 191-201.

First available in Project Euclid: 19 November 2010

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Coherence dichotomized data discrete barrier ethics indifference interval maximum likelihood recursion unbiasedness virtual observations


Cheung, Ying Kuen. Stochastic Approximation and Modern Model-Based Designs for Dose-Finding Clinical Trials. Statist. Sci. 25 (2010), no. 2, 191--201. doi:10.1214/10-STS334.

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  • Anbar, D. (1984). Stochastic approximation methods and their use in bioassay and phase I clinical trials. Commun. Statist. 13 2451–2467.
  • Babb, J., Rogatko, A. and Zacks, S. (1998). Cancer phase I clinical trials: Efficient dose escalation with overdose control. Statist. Med. 17 1103–1120.
  • Bartroff, J. and Lai, T. L. (2010). Approximate dynamic programming and its applications to the design of phase I cancer trials. Statist. Sci. 25 245–257.
  • Blum, J. R. (1954). Multidimensional stochastic approximation methods. Ann. Math. Statist. 25 737–744.
  • Cheung, Y. K. (2002). On the use of nonparametric curves in phase I trials with low toxicity tolerance. Biometrics 58 237–240.
  • Cheung, Y. K. (2005). Coherence principles in dose-finding studies. Biometrika 92 863–873.
  • Cheung, Y. K. (2007). Sequential implementation of stepwise procedures for identifying the maximum tolerated dose. J. Amer. Statist. Assoc. 102 1448–1461.
  • Cheung, Y. K. (2008). Dose-finding by the continual reassessment method (dfcrm). R package version 0.1-2. Available at
  • Cheung, Y. K. (2010). Dose Finding by the Continual Reassessment Method. Chapman and Hall, New York. To appear.
  • Cheung, Y. K. and Chappell, R. (2000). Sequential designs for phase I clinical trials with late-onset toxicities. Biometrics 56 1177–1182.
  • Cheung, Y. K. and Chappell, R. (2002). A simple technique to evaluate model sensitivity in the continual reassessment method. Biometrics 58 671–674.
  • Cheung, Y. K. and Elkind, M. S. V. (2010). Stochastic approximation with virtual observations for dose-finding on discrete levels. Biometrika 97 109–121.
  • Dixon, W. J. and Mood, A. M. (1948). A method for obtaining and analyzing sensitivity data. J. Amer. Statist. Assoc. 43 109–126.
  • Durham, S. D., Flournoy, N. and Rosenberger, W. F. (1997). A random walk rule for phase I clinical trials. Biometrics 53 745–760.
  • Eisenhauer, E. A., O’Dwyer, P. J., Christian, M. and Humphrey, J. S. (2000). Phase I clinical trial design in cancer drug development. J. Clin. Oncol. 18 684–692.
  • Faries, D. (1994). Practical modifications of the continual reassessment method for phase I cancer trials. J. Biopharm. Statist. 4 147–164.
  • Finney, D. J. (1978). Statistical Method in Biological Assay. Griffin, London.
  • Gasparini, M. and Eisele, J. (2000). A curve-free method for phase I clinical trials. Biometrics 56 609–615.
  • Geller, N. L. (1984). Design of phase I and II clinical trials in cancer: A statistician’s view. Cancer Investigation 2 483–491.
  • Haines, L. M., Perevozskaya, I. and Rosenberger, W. F. (2003). Bayesian optimal design for phase I clinical trials. Biometrics 59 591–600.
  • Han J., Lai, T. L. and Spivakovsky, V. (2006). Approximate policy optimization and adaptive control in regression models. Comput. Econom. 27 433–452.
  • Ji, Y., Li, Y. and Bekele, B. N. (2007). Dose-finding in phase I clinical trials based on toxicity probability intervals. Clin. Trials 4 235–244.
  • Kiefer, J. and Wolfowitz, J. (1952). Stochastic estimation of the maximum of a regression function. Ann. Math. Statist. 23 462–466.
  • Lai, T. L. and Robbins, H. (1979). Adaptive design and stochastic approximation. Ann. Statist. 7 1196–1221.
  • Lee, S. M. and Cheung, Y. K. (2001). Model calibration in the continual reassessment method. Clin. Trials 6 227–238.
  • Leonard, J. P., Furman, R. R., Cheung, Y. K. et al. (2005). Phase I/II trial of bortezomib plus CHOP-Rituximab in diffuse large B cell (DLBCL) and mantle cell lymphoma (MCL): Phase I results. Blood 106 147A.
  • Leung, D. H. Y. and Wang, Y. G. (2001). Isotonic designs for phase I trials. Control. Clin. Trials 22 126–138.
  • McLeish, D. L. and Tosh, D. (1990). Sequential designs in bioassay. Biometrics 46 103–116.
  • Morgan, B. J. T. (1992). Analysis of Quantal Response Data. Chapman and Hall, New York.
  • Muller, J. H., McGinn, C. J., Normolle, D., Lawrence, T., Brown, D., Hejna, G. and Zalupski, M. M. (2004). Phase I trial using a time-to-event continual reassessment strategy for dose escalation of cisplatin combined with gemcitabine and radiation therapy in pancreatic cancer. J. Clin. Oncol. 22 238–243.
  • Murphy, J. R. and Hall, D. L. (1997). A logistic dose-ranging method for phase I clinical investigations trials. J. Biopharm. Statist. 7 636–647.
  • O’Quigley, J. and Chevret, S. (1991). Methods for dose finding studies in cancer clinical trials: A review and results of a Monte Carlo study. Stat. Med. 10 1647–1664.
  • O’Quigley, J. and Conaway, M. (2010). Continual reassessment and related dose finding designs. Statist. Sci. 25 202–216.
  • O’Quigley, J., Pepe, M. and Fisher, L. (1990). Continual reassessment method: A practical design for phase I clinical trials in cancer. Biometrics 46 33–48.
  • Ratain, M. J., Mick, R., Schilsky, R. L. and Siegler, M. (1993). Statistical and ethical issues in the design and conduct of phase I and phase II clinical trials of new anticancer agents. J. Nat. Cancer Inst. 85 1637–1643.
  • Robbins, H. and Monro, S. (1951). A stochastic approximation method. Ann. Math. Statist. 22 400–407.
  • Sacks, J. (1958). Asymptotic distribution of stochastic approximation procedures. Ann. Math. Statist. 29 373–405.
  • Schneiderman, M. A. (1965). How can we find an optimal dose? Toxicol. Appl. Pharm. 7 44–53.
  • Shen, L. Z. and O’Quigley, J. (1996). Consistency of continual reassessment method under model misspecification. Biometrika 83 395–405.
  • Storer, B (1989). Design and analysis of phase I clinical trials. Biometrics 45 925–937.
  • Storer, B. and DeMets, D. (1987). Current phase I/II designs: Are they adequate? J. Clin. Res. Drug Develop. 1 121–130.
  • Thall, P. F. (2010). Bayesian models and decision algorithms for complex early phase clinical trials. Statist. Sci. 25 227– 244.
  • Thall, P. F., Millikan, R. E., Müller, P. and Lee, S. J. (2003). Dose-finding with two agents in phase I oncology trials. Biometrics 59 487–496.
  • Tighiouart, M. and Rogatko, A. (2010). Dose finding with escalation with overdose control (EWOC) in cancer clinical trials. Statist. Sci. 25 217–226.
  • Wasan, M. T. (1969). Stochastic Approximation. Cambridge Univ. Press.
  • Whitehead, J. and Brunier, H. (1995). Bayesian decision procedures for dose determining experiments. Stat. Med. 14 885–893.
  • Wu, C. F. J. (1985). Efficient sequential designs with binary data. J. Amer. Statist. Assoc. 80 974–984.
  • Wu, C. F. J. (1986). Maximum likelihood recursion and stochastic approximation in sequential designs. In Adaptive Statistical Procedures and Related Topics (J. Van Ryzin, ed.). IMS Monograph Series 8 298–314. IMS, Hayward, CA.
  • Ying, Z. and Wu, C. F. J. (1997). An asymptotic theory of sequential designs based on maximum likelihood recursion. Statist. Sinica 7 75–91.