Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T04:39:16.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Which design is better? Ehrenfest urn versus biased coin

Part of: Sequential methods Markov processes

Published online by Cambridge University Press:  19 February 2016

Yung-Pin Chen
Yung-Pin Chen*
Affiliation:
Smith College
*
∗ Postal address: Department of Mathematics, Smith College, Northampton, Massachusetts, 01063, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Two features are desired in designing a sequential clinical trial: randomness and balance. The former makes the ground for valid statistical inferences and the latter strengthens efficiency in inference procedures. Unfortunately randomness and balance can be in conflict with one another, and clinicians may be caught between the need for both of them. This paper raises an interesting question: can one design consistently achieve more balance than another when both designs own the same randomness? The Ehrenfest urn design is presented to allocate two treatments under a sequential clinical trial, and its balance and randomness properties are investigated. The design is compared with the biased coin design with imbalance tolerance.

Keywords

Sequential clinical trialEhrenfest urn designbiased coin designbig stick designimbalance toleranceselection biasMarkov chainhigh-order transition probability

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 62L05: Sequential design
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 738 - 749
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Atkinson, A. C. (1982). Optimum biased coin design for sequential clinical trials with prognostic factors. Biometrika 69, 61–67.Google Scholar
Billingsley, P. (1986). Probability and Measure. John Wiley, New York.Google Scholar
Bingham, N. H. (1991). Fluctuation theory for the Ehrenfest urn. Adv. Appl. Prob. 23, 598–611.Google Scholar
Blackwell, D. and Hodges, J. L. Jr. (1957). Design for the control of selection bias. Ann. Math. Statist. 28, 449–460.Google Scholar
Blom, G. (1989). Mean transition times for the Ehrenfest urn model. Adv. Appl. Prob. 21, 479–480.Google Scholar
Chen, Y. (1994). A study of the trade-offs between balance and randomness in various sequential sampling procedures. , Department of Statistics, Purdue University.Google Scholar
Chen, Y. (1999). Biased coin design with imbalance tolerance. Commun. Statist. Stoch. Models 15, 953–975.Google Scholar
Diaconis, P. (1988). Group Representations in Probability and Statistics. Lecture Notes—Monographs 11, Institute of Mathematical Statistics, Hayward.Google Scholar
Eagleson, G. K. (1969). A characterization theorem for positive definite sequences on the Krawtchouk polynomials. Austral. J. Statistics 11(1), 29–38.Google Scholar
Efron, B. (1971). Forcing a sequential experiment to be balanced. Biometrika 58, 403–417.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
Feinsilver, P. and Schott, R. (1991). Krawtchouk polynomials and finite probability theory. Probability Measures on Groups X edited by Heyer, H.. New York: Plenum Press.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Friedman, B. (1949). A simple urn model. Commun. Pure Appl. Math. 2, 59–70.Google Scholar
Goulden, I. and Jackson, D. (1986). Distributions, continued fractions, and the Ehrenfest urn model. Journal of Combinatorial Theory Series A 41, 21–31.Google Scholar
Hess, F. G. (1954). Alternative solution to the Ehrenfest problem. Amer. Math. Monthly 61, 323–327.Google Scholar
Hill, B., Lane, D., and Sudderth, W. (1980). A strong law for some generalized urn processes. The Annals of Probability Vol. 8, No. 2, 214–226.Google Scholar
Hoel, P., Port, S., and Stone, C. (1972). Introduction to Stochastic Processes. Boston, MA: Houghton Mifflin Company.Google Scholar
Iglehart, D. (1968). Limit theorems for the multi-urn Ehrenfest model. Ann. Math. Statist. Vol. 39, No. 3, 864–876.Google Scholar
Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. John Wiley, New York.Google Scholar
Kac, M. (1947). Random walk and the theory of Brownian motion. Amer. Math. Monthly 54, 369–391.CrossRefGoogle Scholar
Kac, M. (1959). Probability and Related Topics in Physical Sciences. Interscience, London.Google Scholar
Karlin, S. and McGregor, J. (1957). Random walks. Illinois J. Math. 3, 66–81.Google Scholar
Karlin, S. and McGregor, J. (1965). Ehrenfest urn models. J. Appl. Prob. 2, 352–376.CrossRefGoogle Scholar
Karlin, S. (1968). A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Karlin, S. and Taylor, M. H. (1975). A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Karlin, S. and Taylor, M. H. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Kemeny, J. G. and Snell, J. L. (1960). Finite Markov Chains. Van Nostrand-Reinhold, New York.Google Scholar
Kemeny, J. G. (1966). Representation theory for denumerable Markov chains. Trans. Amer. Math. Soc. 125, 47–62.Google Scholar
Melfi, V. F. (1992). Nonlinear Markov renewal theory with statistical applications. Ann. Prob. 20, 753–771.Google Scholar
Palacios, J. L. (1993). Fluctuation theory for the Ehrenfest urn via electric networks. Adv. Appl. Prob. 25, 472–476.Google Scholar
Palacios, J. L. (1994). Another look at the Ehrenfest urn via electric networks. Adv. Appl. Prob. 26, 820–824.Google Scholar
Pocock, S. J. and Simon, R. (1975). Sequential treatment assignment with balancing for prognostic factors in the controlled clinical trials. Biometrics 31, 103–115.Google Scholar
Pocock, S. J. (1979). Allocation of patients to treatment in clinical trials. Biometrics 35, 183–197.CrossRefGoogle ScholarPubMed
Pocock, S. J. (1983). Clinical Trials: A Practical Approach. John Wiley, New York.Google Scholar
Prescott, P., Ryan, J., and Shuttleworth, P. (1994). A comparative study of several antibiotic formulations using a design based on a combination of balanced incomplete blocks and Latin squares. Statistics in Medicine Vol. 13, 11–21.Google Scholar
Schach, S. (1971). Weak convergence results for a class of multivariate Markov processes. Ann. Math. Statist. Vol. 42, No. 2, 451–465.Google Scholar
Shapiro, S. H. and Louis, T. A. (1983). Clinical Trials: Issues and Approaches. Marcel Dekker, New York.Google Scholar
Siegert, A. J. F. (1949). On the approach to statistical equilibrium. Phys. Rev. 76, 1708–1714.Google Scholar
Simon, R. (1977). Adaptive treatment assignment methods and clinical trials. Biometrics 33, 743–749.Google Scholar
Simon, R. (1979). Restricted randomization designs in clinical trials. Biometrics 35, 503–512.Google Scholar
Smith, R. L. (1984a). Properties of biased coin designs in sequential clinical trials. Ann. Statist. 12, 1018–1034.Google Scholar
Smith, R. L. (1984b). Sequential treatment allocation using biased coin designs. J. R. Statist. Soc. B 46, 519–543.Google Scholar
Soares, J. F. and Wu, C. F. J. (1983). Some restricted randomization rules in sequential designs. Commun. Statist. Theory Meth. 12, 2017–2034.Google Scholar
Szegö, G., (1939). Orthogonal Polynomials. Providence, Rhode Island: American Mathematical Society.Google Scholar
Takács, L., (1960). Stochastic Processes: Problems and Solutions. London: Methuen and Co. Lt. John Wiley, New York.Google Scholar
Takács, L., (1979). On an urn problem of Paul and Tatiana Ehrenfest. Math. Proc. Camb. Phil. Soc. 86, 127–130.Google Scholar
Van Beek, K. W. H. and Stam, A. J. (1987). A variant of the Ehrenfest model. Adv. Appl. Prob. 19, 995–996.Google Scholar
Walters, P. (1982). An Introduction to Ergodic Theory. New York: Springer-Verlag.Google Scholar
Wei, L. J. (1978a). A class of designs for sequential clinical trials. J. Amer. Statist. Assoc. 72, 382–386.Google Scholar
Wei, L. J. (1978b). The adaptive biased coin designs for sequential experiments. Ann. Statist. 6, 92–99.CrossRefGoogle Scholar
Wei, L. J. (1978c). An application of an urn model to the design of sequential controlled clinical trials. J. Amer. Statist. Assoc. 73, 559–563.Google Scholar
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.Google Scholar