Asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials. (English) Zbl 1105.62381

Hu, Feifang; Zhang, Li-Xin

Ann. Stat. 32, No. 1, 268-301 (2004).

Summary: A general doubly adaptive biased coin design is proposed for the allocation of subjects to K treatments in a clinical trial. This design follows the same spirit as Efron’s biased coin design and applies to the cases where the desired allocation proportions are unknown, but estimated sequentially. Strong consistency, a law of the iterated logarithm and asymptotic normality of this design are obtained under some widely satisfied conditions. For two treatments, a new family of designs is proposed and shown to be less variable than both the randomized play-the-winner rule and the adaptive randomized design. Also the proposed design tends toward a randomization scheme (with a fixed target proportion) as the size of the experiment increases.

MSC:

62P10	Applications of statistics to biology and medical sciences; meta analysis
62L05	Sequential statistical design

Keywords:

adaptive randomized design; asymptotic normality; randomized play-the-winner rule; urn model

References:

[1]	Bai, Z. D. and Hu, F. (1999). Asymptotic theorems for urn models with nonhomogeneous generating matrices. Stochastic Process. Appl. 80 87–101. · Zbl 0954.62014 · doi:10.1016/S0304-4149(98)00094-5
[2]	Bai, Z. D., Hu, F. and Rosenberger, W. F. (2002). Asymptotic properties of adaptive designs for clinical trials with delayed response. Ann. Statist. 30 122–139. · Zbl 1012.62087 · doi:10.1214/aos/1015362187
[3]	Doob, J. L. (1936). Note on probability. Ann. Math. 37 363–367. JSTOR: · Zbl 0013.40802 · doi:10.2307/1968449
[4]	Efron, B. (1971). Forcing a sequential experiment to be balanced. Biometrika 58 403–417. · Zbl 0226.62086 · doi:10.1093/biomet/58.3.403
[5]	Eisele, J. (1994). The doubly adaptive biased coin design for sequential clinical trials. J. Statist. Plann. Inference 38 249–262. · Zbl 0795.62066 · doi:10.1016/0378-3758(94)90038-8
[6]	Eisele, J. (1995). Biased coin designs: Some properties and applications. In Adaptive Designs (N. Flournoy and W. F. Rosenberger, eds.) 48–64. IMS, Hayward, CA. · Zbl 0926.62067
[7]	Eisele, J. and Woodroofe, M. (1995). Central limit theorems for doubly adaptive biased coin designs. Ann. Statist. 23 234–254. JSTOR: · Zbl 0835.62068 · doi:10.1214/aos/1176324465
[8]	Flournoy, N. and Rosenberger, W. F., eds. (1995). Adaptive Designs . IMS, Hayward, CA. · Zbl 0920.00040
[9]	Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Applications . Academic Press, London. · Zbl 0462.60045
[10]	Hayre, L. S. (1979). Two-population sequential tests with three hypotheses. Biometrika 66 465–474. · Zbl 0418.62062 · doi:10.1093/biomet/66.3.465
[11]	Hu, F. (2002). Sample size for response-adaptive randomization procedures. Unpublished manuscript.
[12]	Hu, F. and Rosenberger, W. F. (2003). Optimality, variability, power: Evaluating response-adaptive randomization procedures for treatment comparisons. J. Amer. Statist. Assoc. · Zbl 1040.62102 · doi:10.1198/016214503000000576
[13]	Ivanova, A. and Rosenberger, W. F. (2001). Adaptive designs for clinical trials with highly successful treatments. Drug Information J. 35 1087–1093.
[14]	Jennison, C. and Turnbull, B. W. (2000). Group Sequential Methods with Applications to Clinical Trials. Chapman and Hall/CRC Press, Boca Raton, FL. · Zbl 0934.62078
[15]	Matthews, P. C. and Rosenberger, W. F. (1997). Variance in randomized play-the-winner clinical trials. Statist. Probab. Lett. 35 223–240. · Zbl 0946.62074 · doi:10.1016/S0167-7152(97)00018-7
[16]	Melfi, V. and Page, C. (1998). Variability in adaptive designs for estimation of success probabilities. In New Developments and Applications in Experimental Design (N. Flournoy, W. F. Rosenberger and W. K. Wong, eds.) 106–114. IMS, Hayward, CA.
[17]	Melfi, V. and Page, C. (2000). Estimation after adaptive allocation. J. Statist. Plann. Inference 87 353–363. · Zbl 0969.62051 · doi:10.1016/S0378-3758(99)00198-6
[18]	Melfi, V., Page, C. and Geraldes, M. (2001). An adaptive randomized design with application to estimation. Canad. J. Statist. 29 107–116. · Zbl 1013.62082 · doi:10.2307/3316054
[19]	Robbins, H. (1952). Some aspects of the sequential design of experiments. Bull. Amer. Math. Soc. 58 527–535. · Zbl 0049.37009 · doi:10.1090/S0002-9904-1952-09620-8
[20]	Rosenberger, W. F. (1996). New directions in adaptive designs. Statist. Sci. 11 137–149.
[21]	Rosenberger, W. F. (2002). Randomized urn models and sequential design (with discussion). Sequential Anal. 21 1–41. · Zbl 0999.62061 · doi:10.1081/SQA-120004166
[22]	Rosenberger, W. F. and Lachin, J. M. (2002). Randomization in Clinical Trials . Wiley, New York. · Zbl 1007.62091
[23]	Rosenberger, W. F., Stallard, N., Ivanova, A., Harper, C. and Ricks, M. (2001). Optimal adaptive designs for binary response trials. Biometrics 57 909–913. · Zbl 1209.62181 · doi:10.1111/j.0006-341X.2001.00909.x
[24]	Smith, R. L. (1984). Properties of biased coin designs in sequential clinical trials. Ann. Statist. 12 1018–1034. JSTOR: · Zbl 0553.62068 · doi:10.1214/aos/1176346718
[25]	Smythe, R. T. (1996). Central limit theorems for urn models. Stochastic Process. Appl. 65 115–137. · Zbl 0889.60013 · doi:10.1016/S0304-4149(96)00094-4
[26]	Smythe, R. T. and Rosenberger, W. F. (1995). Play-the-winner designs, generalized Pólya urns, and Markov branching processes. In Adaptive Designs (N. Flournoy and W. F. Rosenberger, eds.) 13–22. IMS, Hayward, CA. · Zbl 0949.62578
[27]	Thompson, W. R. (1933). On the likelihood that one unknown probability exceeds another in view of the evidence of the two samples. Biometrika 25 285–294. · JFM 59.1159.03
[28]	Wei, L. J. (1978). The adaptive biased coin design for sequential experiments. Ann. Statist. 6 92–100. JSTOR: · Zbl 0374.62075 · doi:10.1214/aos/1176344068
[29]	Wei, L. J. (1979). The generalized Pólya’s urn design for sequential medical trials. Ann. Statist. 7 291–296. · Zbl 0399.62083 · doi:10.1214/aos/1176344614
[30]	Wei, L. J. (1988). Exact two-sample permutation tests based on the randomized play-the-winner rule. Biometrika 75 603–606. · Zbl 0657.62086 · doi:10.1093/biomet/75.3.603
[31]	Wei, L. J. and Durham, S. (1978). The randomized pay-the-winner rule in medical trials. J. Amer. Statist. Assoc. 73 840–843. · Zbl 0389.62067 · doi:10.2307/2286600
[32]	Wei, L. J., Smythe, R. T. and Smith, R. L. (1986). \(K\)-treatment comparisons with restricted randomization rules in clinical trials. Ann. Statist. 14 265–274. JSTOR: · Zbl 0587.62153 · doi:10.1214/aos/1176349854
[33]	Zelen, M. (1969). Play the winner rule and the controlled clinical trial. J. Amer. Statist. Assoc. 64 131–146.