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Journal of Statistical Theory and Practice

, Volume 11, Issue 3, pp 361–374 | Cite as

Adaptive designs for quantal dose-response experiments with false answers

  • Norbert Benda
  • Paul-Christian Bürkner
  • Fritjof Freise
  • Heinz Holling
  • Rainer Schwabe
Article

Abstract

In psychophysical experiments with quantal dose-response a common problem is the occurrence of lapses due to inattention or, as in forced choice experiments, the occurrence of incorrect guesses—or both. For these situations an optimized sequential design is proposed based on the Fisher information evaluated at the maximum likelihood estimate. This sequential design is compared to the classical Robbins-Monro method of stochastic approximation and modifications thereof in the original nonparametric approach, as well as adapted to the current model with false answers and to Wu’s (1985) original method based on the maximum likelihood estimate by means of a simulation study. In these simulations the optimized sequential design turns out to perform substantially better than its nonparametric competitors, in particular, in the situation of starting points (initial guesses of the parameter of interest) that are far from the true value. Overall, the optimized version also outperforms the original maximum likelihood based method by Wu (1985).

Keywords

Psychophysics quantal response false answers forced choice experiments sequential design stochastic approximation 

AMS Subject Classification

62L05 62L20 62K05 91E30 

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References

  1. Abbott, W. S. 1925. A method of computing the effectiveness of an insecticide. Journal of Economic Entomology 18 (2):265–67.CrossRefGoogle Scholar
  2. Bather, J. A. 1989. Stochastic approximation: A generalisation of the Robbins–Monro procedure. In Proceedings of the Fourth Prague Symposium on Asymptotic Statistics, ed. P. Mandl and M. Hušková, 13–27. Prague, Czech Republic: Charles University.Google Scholar
  3. Chung, K. L. 1954. On a stochastic approximation method. Annals of Mathematical Statistics 25 (3):463–83.MathSciNetCrossRefGoogle Scholar
  4. Fechner, G. T. 1860. Elemente der Psychophysik. English translation: Howes, D. H., E. C. Boring, (eds.) and H. E. Adler (transl.). 1966. Elements of psychophysics. New York, NY: Holt Rinehart & Winston.Google Scholar
  5. Lai, T. L., and H. Robbins. 1979. Adaptive design and stochastic approximation. Annals of Statistics 7 (6):1196–221.MathSciNetCrossRefGoogle Scholar
  6. Lord, F. M. 1968. An analysis of the verbal scholastic aptitude test using Birnbaum’s three-parameter logistic model. Educational and Psychological Measurement 28 (4):989–1020.CrossRefGoogle Scholar
  7. McLeish, D. L., and D. H. Tosh. 1990. Sequential designs in bioassay. Biometrics 46 (1):103–16.MathSciNetCrossRefGoogle Scholar
  8. Morgan, B. J. T. 1992. Analysis of quantal response data. London, UK: Chapman & Hall.CrossRefGoogle Scholar
  9. Polyak, B. T. 1990. A new method of stochastic approximation type. Automation and Remote Control 51 (7):937–46.MathSciNetzbMATHGoogle Scholar
  10. Robbins, H., and S. Monro. 1951. A stochastic approximation method. Annals of Mathematical Statistics 22 (3):400–7.MathSciNetCrossRefGoogle Scholar
  11. Ruppert, D. 1988. Efficient estimators from a slowly convergent Robbins–Monro process. Technical report 781. School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY.Google Scholar
  12. Sacks, J. 1958. Asymptotic distribution of stochastic approximation procedures. Annals of Mathematical Statistics 29 (2):373–405.MathSciNetCrossRefGoogle Scholar
  13. Schwabe, R. 1986. Strong representation of an adaptive stochastic approximation procedure. Stochastic Processes and their Applications 23 (1):115–30.MathSciNetCrossRefGoogle Scholar
  14. Schwabe, R., and H. Walk. 1996. On a stochastic approximation procedure based on averaging. Metrika 44 (2):165–80.MathSciNetCrossRefGoogle Scholar
  15. Silvapulle, M. J. 1981. On the existence of maximum likelihood estimators for the binomial response models. Journal of the Royal Statistical Society B 43 (3):310–13.MathSciNetzbMATHGoogle Scholar
  16. Venter, J. H. 1968. An extension of the Robbins–Monro procedure. Annals of Mathematical Statistics 38 (1):181–90.MathSciNetCrossRefGoogle Scholar
  17. Wetherill, G. B., and K. D. Glazebrook. 1986. Sequential methods in statistics, 3rd ed. London, UK: Chapman & Hall.zbMATHGoogle Scholar
  18. Wu, C. F. J. 1985. Efficient sequential designs with binary data. Journal of the American Statistical Association 80 (392):974–84.MathSciNetCrossRefGoogle Scholar
  19. Ying, Z., and C. F. J. Wu. 1997. An asymptotic theory of sequential designs based on maximum likelihood recursions. Statistica Sinica 7 (1):75–91.MathSciNetzbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing, 20 Middlefield Ct, Greensboro, NC 27455 2017

Authors and Affiliations

  • Norbert Benda
    • 1
  • Paul-Christian Bürkner
    • 2
  • Fritjof Freise
    • 3
  • Heinz Holling
    • 2
  • Rainer Schwabe
    • 3
  1. 1.Federal Institute for Drugs and Medical DevicesBonnGermany
  2. 2.Institute for PsychologyUniversity of MünsterMünsterGermany
  3. 3.Institute for Mathematical StatisticsUniversity of MagdeburgMagdeburgGermany

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