An Optimizing Up-and-Down Design

  • Euloge E. Kpamegan
  • Nancy Flournoy
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 51)


Assume that the probability of success is unimodal as a function of dose. Take the response to be binary and the possible treatment space to be a lattice. The Optimizing Up-and-Down Design allocates treatments to pairs of subjects in a way that causes the treatment distribution to cluster around the treatment with maximum success probability. This procedure is constructed to use accruing information to limit the number of patients that are exposed to doses with high probabilities of failure. The Optimizing Up-and-Down Design is motivated by Kiefer and Wolfowitz’s stochastic approximation procedure. In this paper, we compare its performance to stochastic approximation. As an estimator of the best dose, simulation studies demonstrate that the mode of the empirical treatment distribution using the Optimizing Up-and-Down Design converges faster than does the usual estimator using stochastic approximation.


constant gain stochastic algorithm Markov chains phase I/II clinical trial random walk designs stationary treatment distribution stochastic approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Chung, K. L. (1954). On a stochastic approximation method. Ann. Math. Statist 25, 463–483.zbMATHCrossRefGoogle Scholar
  2. Derman, C. (1956). Stochastic approximation. Ann. Math. Statist. 27, 879–886.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Derman, C. (1957). Non-parametric up and down experimentation. Ann. Math. Statist. 28, 795–799.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Dixon, W. J. and Mood, A. M. (1948). A method of obtaining and analyzing sensitivity data. JASA 43, 109–126.zbMATHCrossRefGoogle Scholar
  5. Dixon, W. J. (1965). The up-and-down method for small samples. JASA 60, 967–978.CrossRefGoogle Scholar
  6. Durham, S. D. and Flournoy, N. (1994). Random walks for quantiles estimation. In Statistical Decision Theory and Related Topics V Eds J. Berger and S. Gupta, pp. 467–476. New York: Springer-Verlag.CrossRefGoogle Scholar
  7. Durham, S. D. and Flournoy, N. (1995). Up-and-down designs I: stationary treatment distributions. In Adaptive Designs. IMS lecture Notes- Monograph Series 25, pp. 139–157. Hayward CA: IMS.CrossRefGoogle Scholar
  8. Durham, S. D. and Flournoy N. and Montazer-Haghighi, A. (1995). Up-and-down designs II: exact treatment moments. In Adaptive Designs. IMS lecture Notes- Monograph Series 25, pp. 158–178. Hayward CA: IMS.CrossRefGoogle Scholar
  9. Durham, S. D., Flournoy, N. and Li, W. (1998). A sequential design for maximizing the probability of a favorable response. Canadian J. Statist. 26, 479–495.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Flournoy, N., Durham, S. D. and Rosenberger, W. F. (1995). Toxicity in sequential dose-response experiments. Sequential Analysis 14, 217–228.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Giovagnoli, A. and Pintacuda, N. (1998). Properties of frequency distributions induced by general “up-and-down” methods for estimating quantiles. JSPI 25, 51–63.MathSciNetGoogle Scholar
  12. Gooley, T. A., Martin, P.J., Lloyd, D.F., Pettinger, M. (1994). Simulation as a design tool for Phase I/II clinical trials: An example from bone marrow transplantation. Controlled Clinical Trials 15, 450–460.CrossRefGoogle Scholar
  13. Hardwick, J. P. and Stout, Q. (1998). Personal communication. Google Scholar
  14. Harris, T. (1952). First passage and recurrence distributions. Transactions of the American Mathematical Society 73, 471–486.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes. New York:Academic Press.zbMATHGoogle Scholar
  16. Keilson, J. and Gerber, H. (1971). Some results for discrete unimodality. JASA 66, 386–389.zbMATHCrossRefGoogle Scholar
  17. Kiefer, J. and Wolfowitz, J. (1952). Stochastic approximation of the maximum of a regression function. Ann. Math. Statist. 25 , 529–532.MathSciNetGoogle Scholar
  18. Kpamegan, E. E. and Flournoy, N. (2000). Up-and-down designs for selecting the dose with maximum success probability. Submitted.Google Scholar
  19. Sacks, J. (1958). Asymptotic distribution of stochastic approximation procedures. Ann. Math. Statist. 29, 373–405.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Tsutakawa, R. K. (1967). Random walk design in bio-assay. JASA 62, 842–856.MathSciNetCrossRefGoogle Scholar
  21. Wasan, M. T. (1969). Stochastic Approximation. Cambridge: University Press.zbMATHGoogle Scholar
  22. Wetherwill, G. B. (1963). Sequential estimation of quantal response curves. Journal of the Royal Statistical Society B 25, 1–48.Google Scholar

Additional Reference

  1. Hardwick, J., Meyers, M. and Stout, Q.F. (2000). Directed walks for dose response problems with competing failure modes. (Submitted).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Euloge E. Kpamegan
    • 1
  • Nancy Flournoy
    • 1
  1. 1.Department of Mathematics and StatisticsAmerican UniversityUSA

Personalised recommendations