Advertisement

Optimum Experiments with Sets of Treatment Combinations

  • Anthony C. AtkinsonEmail author
Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)

Abstract

Response surface designs are investigated in which points in the design region corresponds to single observations at each of s distinct settings of the explanatory variables. An extension of the “General Equivalence Theorem” for D-optimum designs is provided for experiments with such sets of treatment combinations. The motivation was an experiment in deep-brain therapy in which each patient receives a set of eight distinct treatment combinations and provides a response to each. The experimental region contains sixteen different sets of eight treatments.

Notes

Acknowledgements

I am grateful to Dr David Pedrosa of the Nuffield Department of Clinical Neurosciences, University of Oxford, for introducing me to the experimental design problem in deep-brain therapy that provided the motivation for this work.

I am also grateful to the referees whose comments strengthened and clarified the results of §3.

References

  1. 1.
    Atkinson, A.C.: Optimal model-based covariate-adaptive randomization designs. In: Sverdlov, O. (ed.) Modern Adaptive Randomized Clinical Trials: Statistical and Practical Aspects, pp. 131–154. Chapman and Hall/CRC Press, Boca Raton (2015)Google Scholar
  2. 2.
    Atkinson, A.C.: Optimum experiments for logistic models with sets of treatment combinations. In: Fackle-Fornius, E. (ed.) A Festschrift in Honor of Hans Nyquist on the Occasion of His 65th Birthday, pp. 44–58. Department of Statistics, Stockholm University, Stockholm (2015)Google Scholar
  3. 3.
    Atkinson, A.C., Donev, A.N., Tobias, R.D.: Optimum Experimental Designs, with SAS. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  4. 4.
    Atkinson, A.C., Fedorov, V.V., Herzberg, A.M., Zhang, R.: Elemental information matrices and optimal experimental design for generalized regression models. J. Stat. Plan. Inference 144, 81–91 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Atkinson, A.C., Woods, D.C.: Designs for generalized linear models. In: Dean, A., Morris, M., Stufken, J., Bingham, D. (eds.) Handbook of Design and Analysis of Experiments, pp. 471–514. Chapman and Hall/CRC Press, Boca Raton (2015)Google Scholar
  6. 6.
    Elfving, G.: Optimum allocation in linear regression theory. Ann. Math. Stat. 23, 255–262 (1952)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Fedorov, V.V., Leonov, S.L.: Optimal Design for Nonlinear Response Models. Chapman and Hall/CRC Press, Boca Raton (2014)zbMATHGoogle Scholar
  8. 8.
    Ford, I., Torsney, B., Wu, C.F.J.: The use of a canonical form in the construction of locally optimal designs for non-linear problems. J. R. Stat. Soc. Ser. B 54, 569–583 (1992)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Kiefer, J., Wolfowitz, J.: The equivalence of two extremum problems. Can. J. Math. 12, 363–366 (1960)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Pronzato, L., Pázman, A.: Design of Experiments in Nonlinear Models. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Verbeke, G., Molenberghs, G.: Linear Mixed Models for Longitudinal Data. Springer, New York (2000)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of StatisticsLondon School of EconomicsLondonUK

Personalised recommendations