Advertisement

Journal of Statistical Theory and Practice

, Volume 7, Issue 4, pp 658–673 | Cite as

Optimal Designs for Regression Models With a Constant Coefficient of Variation

Article

Abstract

In this article we consider the problem of constructing optimal designs for models with a constant coefficient of variation. We explore the special structure of the information matrix in these models and derive a characterization of optimal designs in the sense of Kiefer and Wolfowitz (1960). Besides locally optimal designs, Bayesian and standardized minimax optimal designs are also considered. Particular attention is spent on the problem of constructing D-optimal designs. The results are illustrated in several examples where optimal designs are calculated analytically and numerically.

Keywords

Optimal design Heteroscedasticity Constant coefficient of variation Polynomial regression 

AMS Subject Classification

62K05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Atkinson, A. C. 2008. Examples of the use of an equivalence theorem in constructing optimum experimental designs for random-effects nonlinear regression models. J. Stat. Plan. Inference, 138(9), 2595–2606.MathSciNetCrossRefGoogle Scholar
  2. Atkinson, A. C., and R. D. Cook. 1995. D-optimum designs for heteroscedastic linear models. Journal of the American Statistical Association, 90, 204–212.MathSciNetMATHGoogle Scholar
  3. Biedermann, S., H. Dette, and W. Zhu. 2006. Optimal designs for dose-response models with restricted design spaces. J. Am. Stat. Assoc., 101, 747–759.MathSciNetCrossRefGoogle Scholar
  4. Blake, K., R. Madabushi, H. Derendorf, and J. Lima. 2008. Population pharmacodynamic model of bronchodilator response to inhaled albuterol in children and adults with asthma. Chest, 134(5), 981–989.CrossRefGoogle Scholar
  5. Box, G. E. P., and H. L. Lucas. 1959. Design of experiments in non-linear situations. Biometrika, 46, 77–90.MathSciNetCrossRefGoogle Scholar
  6. Burridge, J., and P. Sebastiani. 1994. D-Optimal designs for generalised linear models with variance proportional to the square of the mean. Biometrika, 81(2), 295–304.MathSciNetCrossRefGoogle Scholar
  7. Chaloner, K., and K. Larntz. 1989. Optimal Bayesian designs applied to logistic regression experiments. J. Stat. Plan. Inference, 21, 191–208.MathSciNetCrossRefGoogle Scholar
  8. Chaloner, K., and I. Verdinelli. 1995. Bayesian experimental design: A review. Stat. Sci., 10(3), 273–304.MathSciNetCrossRefGoogle Scholar
  9. Chen, Y., E. R. Dougherty, and M. L. Bittner. 1997. Ratio-based decisions and the quantitative analysis of cDNA microarray images. J. Biomed. Optics, 2(4), 364–374.CrossRefGoogle Scholar
  10. Chernoff, H. 1953. Locally optimal designs for estimating parameters. Ann. Math. Stat., 24, 586–602.MathSciNetCrossRefGoogle Scholar
  11. Chien, J. Y., S. Friedrich, M. A. Heathman, D. P. de Alwis, and V. Sinha. 2005. Pharmacokinetics/pharmacodynamics and the stages of drug development: Role of modeling and simulation. AAPS J., 7(3), E544–E559.CrossRefGoogle Scholar
  12. Cornish-Browden, A. (1995). Fundamentals of enzyme kinetics (rev. ed.). London, UK, Portland Press.Google Scholar
  13. Danesi, R., F. Innocenti, S. Fogli, A. Gennari, E. Baldini, A. Di Paolo, B. Salvadori, G. Bocci, P. F. Conte, and M. Del Tacca. 2002. Pharmacokinetics and pharmacodynamics of combination chemotherapy with paclitaxel and epirubicin in breast cancer patients. B. J. Clin. Pharmacol., 53(5), 508–518.CrossRefGoogle Scholar
  14. Dette, H. 1997. Designing experiments with respect to “standardized” optimality criteria. J. R. Stat. Soc., Ser. B, 59, 97–110.MathSciNetCrossRefGoogle Scholar
  15. Dette, H., F. Bretz, A. Pepelyshev, and J. C. Pinheiro. 2008. Optimal designs for dose-finding studies. J. Am. Stat. Assoc., 103(483), 1225–1237.MathSciNetCrossRefGoogle Scholar
  16. Dette, H., L. Haines, and L. Imhof. 2007. Maximin and Bayesian optimal designs for regression models. Stat. Sin., 17, 463–480.MathSciNetMATHGoogle Scholar
  17. Dette, H., and T. Holland-Letz. 2009. A geometric characterization of c-optimal designs for heteroscedastic regression. Ann. Stat., 37(6B), 4088–4103.MathSciNetCrossRefGoogle Scholar
  18. Dette, H., C. Kiss, M. Bevanda, and F. Bretz. 2010. Optimal designs for the EMAX, log-linear and exponential models. Biometrika, 97(2), 513–518.MathSciNetCrossRefGoogle Scholar
  19. Dette, H., I. Martinez Lopez, I. Ortiz Rodriguez, and A. Pepelyshev. 2006. Efficient design of experiment for exponential regression models. J. Stat. Plan. Inference, 136, 4397–4418.MathSciNetCrossRefGoogle Scholar
  20. Dette, H., V. B. Melas, and A. Pepelyshev. 2004. Optimal designs for a class of nonlinear regression models. Ann. Stat., 32, 2142–2167.MathSciNetCrossRefGoogle Scholar
  21. Dette, H., and H. M. Neugebauer. 1997. Bayesian D-optimal designs for exponential regression models. J. Stat. Plan. Inference, 60, 331–349.MathSciNetCrossRefGoogle Scholar
  22. Dette, H., and W. K. Wong. 1996. Optimal Bayesian designs for models with partially specified heteroscedastic structure. Ann. Stat., 24, 2108–2127.MathSciNetCrossRefGoogle Scholar
  23. Fang, Z., and D. P. Wiens. 2000. Integer-valued, minimax robust designs for estimation and extrapolation in heteroscedastic, approximately linear models. J. Am. Stat. Assoc., 95, 807–818.MathSciNetCrossRefGoogle Scholar
  24. Firth, D. 1988. Multiplicative errors: Log-normal or gamma? J. R. Stat. Soc. Ser. B (Methodological), 50(2), 266–268.MathSciNetGoogle Scholar
  25. Ford, I., B. Torsney, and C. F. J. Wu. 1992. The use of canonical form in the construction of locally optimum designs for nonlinear problems. J. R. Stat. Soc., Ser. B, 54, 569–583.MATHGoogle Scholar
  26. Haines, L. M. 1993. Optimal design for nonlinear regression models. Commun. Stat. Theory Methods, 22, 1613–1627.CrossRefGoogle Scholar
  27. Haines, L. M. 1995. A geometric approach to optimal design for one-parameter non-linear models. J. R. Stat. Soc. Ser. B, 57, 575–598.MathSciNetMATHGoogle Scholar
  28. Han, C., and K. Chaloner. (2003). D and c-optimal designs for exponential regression models used in pharmacokinetics and viral dynamics. J. Stat. Plan. Inference, 115, 585–601.CrossRefGoogle Scholar
  29. Hoel, P. G. 1958. Efficiency problems in polynomial estimation. Ann. Math. Stat., 29, 1134–1145.MathSciNetCrossRefGoogle Scholar
  30. Johansen, S. 1984 Functional relations, random coefficients and nonlinear regression, with application to kinetic data. Lecture Notes Statist. 22.Google Scholar
  31. Kiefer, J. 1974. General equivalence theory for optimum designs (approximate theory). Ann. Stat., 2, 849–879.MathSciNetCrossRefGoogle Scholar
  32. Kiefer, J., J. Wolfowitz. 1960. The equivalence of two extremem problems. Can. J. Math., 12, 363–366.CrossRefGoogle Scholar
  33. López, S., J. France, W. J. Gerrits, M. S. Dhanoa, D. J. Humphries, and J. Dijkstra. 2000. A generalized Michaelis-Menten equation for the analysis of growth. J. Anim. Sci., 78(7), 1816–1828.CrossRefGoogle Scholar
  34. López-Fidalgo, J., and W. K. Wong. 2002. Design issues for the Michaelis-Menten model. J. Theoret. Biol., 215, 1–11.MathSciNetCrossRefGoogle Scholar
  35. Mukhopadhyaya, S., and L. M. Haines. 1995. Bayesian D-optimal designs for the exponential growth model. J. Stat. Plan. Inference, 44(3), 385–397.MathSciNetCrossRefGoogle Scholar
  36. Pukelsheim, F. 2006. Optimal design of experiments. Philadelphia, PA, SIAM.CrossRefGoogle Scholar
  37. Pukelsheim, F., and S. Rieder. 1992. Efficient rounding of approximate designs. Biometrika, 79, 763–770.MathSciNetCrossRefGoogle Scholar
  38. Ratkowsky, D. A. 1983. Nonlinear regression modeling: A unified practical approach. New York, NY, Marcel Dekker.MATHGoogle Scholar
  39. Ratkowsky, D. A. 1990. Handbook of nonlinear regression models. New York, NY, Marcel Dekker.MATHGoogle Scholar
  40. Seber, G. A. F., and C. J. Wild. 1989. Nonlinear regression. New York, NY, John Wiley and Sons.CrossRefGoogle Scholar
  41. Silvey, S. D. 1980. Optimal design. London, UK, Chapman and Hall.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2013

Authors and Affiliations

  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Institut für Angewandte StatistikJohannes Kepler Universität LinzLinzAustria

Personalised recommendations