The Annals of Statistics

A geometric characterization of c-optimal designs for heteroscedastic regression

Holger Dette and Tim Holland-Letz

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We consider the common nonlinear regression model where the variance, as well as the mean, is a parametric function of the explanatory variables. The c-optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest. A geometric characterization of c-optimal designs in this context is presented, which generalizes the classical result of Elfving [Ann. Math. Statist. 23 (1952) 255–262] for c-optimal designs. As in Elfving’s famous characterization, c-optimal designs can be described as representations of boundary points of a convex set. However, in the case where there appear parameters of interest in the variance, the structure of the Elfving set is different. Roughly speaking, the Elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space. The c-optimal designs are characterized as representations of the points where the line in direction of the vector c intersects the boundary of the new Elfving set. The theory is illustrated in several examples including pharmacokinetic models with random effects.

Article information

Ann. Statist., Volume 37, Number 6B (2009), 4088-4103.

First available in Project Euclid: 23 October 2009

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Primary: 62K05: Optimal designs

c-optimal design heteroscedastic regression Elfving’s theorem pharmacokinetic models random effects locally optimal design geometric characterization


Dette, Holger; Holland-Letz, Tim. A geometric characterization of c -optimal designs for heteroscedastic regression. Ann. Statist. 37 (2009), no. 6B, 4088--4103. doi:10.1214/09-AOS708.

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  • Atkinson, A. C. (2008). Examples of the use of an equivalence theorem in constructing optimum experimental designs for random-effects nonlinear regression models. J. Statist. Plann. Inference 138 2595–2606.
  • Atkinson, A. C. and Cook, R. D. (1995). D-optimum designs for heteroscedastic linear models. J. Amer. Statist. Assoc. 90 204–212.
  • Atkinson, A. C. and Donev, A. (1992). Optimum Experimental Designs. Clarendon Press, Oxford.
  • Beatty, D. A. and Pigeorsch, W. W. (1997). Optimal statistical design for toxicokinetic studies. Stat. Methods Med. Res. 6 359–376.
  • Biedermann, S., Dette, H. and Zhu, W. (2006). Optimal designs for dose-response models with restricted design spaces. J. Amer. Statist. Assoc. 101 747–759.
  • Box, G. E. P. and Lucas, H. L. (1959). Design of experiments in non-linear situations. Biometrika 46 77–90.
  • Brown, L. D. and Wong, W. K. (2000). An algorithmic construction of optimal minimax designs for heteroscedastic linear models. J. Statist. Plann. Inference 85 103–114.
  • Cayen, M. and Black, H. (1993). Role of toxicokinetics in dose selection for carcinogenicity studies. In Drug Toxicokinetics (P. Welling and F. de la Iglesia, eds.) 69–83. Dekker, New York.
  • Chaloner, K. and Verdinelli, I. (1995). Bayesian experimental design: A review. Statist. Sci. 10 273–304.
  • Chernoff, H. (1953). Locally optimal designs for estimating parameters. Ann. Math. Statist. 24 586–602.
  • Chernoff, H. and Haitovsky, Y. (1990). Locally optimal design for comparing two probabilities from binomial data subject to misclassification. Biometrika 77 797–805.
  • Dette, H. (1995). Designing of experiments with respect to “standardized” optimality criteria. J. Roy. Statist. Soc. Ser. B 59 97–110.
  • Dette, H., Bretz, F., Pepelyshev, A. and Pinheiro, J. C. (2008). Optimal designs for dose finding studies. J. Amer. Statist. Assoc. 103 1225–1237.
  • Dette, H., Melas, V. B. and Pepelyshev, A. (2004). Optimal designs for a class of nonlinear regression models. Ann. Statist. 32 2142–2167.
  • Elfving, G. (1952). Optimal allocation in linear regression theory. Ann. Math. Statist. 23 255–262.
  • Ette, E., Kelman, A., Howie, C. and Whiting, B. (1995). Analysis of animal pharmacokinetic data: Performance of the one point per animal design. J. Pharmacokinet. Biopharm. 23 551–566.
  • Fan, S. K. and Chaloner, K. (2003). A geometric method for singular c-optimal designs. J. Statist. Plann. Inference 113 249–257.
  • Fang, Z. and Wiens, D. P. (2000). Integer-valued, minimax robust designs for estimation and extrapolation in heteroscedastic, approximately linear models. J. Amer. Statist. Assoc. 95 807–818.
  • Fellmann, J. (1999). Gustav Elfving’s contribution to the emergence of the optimal experimental design theory. Statist. Sci. 14 197–200.
  • Ford, I., Torsney, B. and Wu, C. F. J. (1992). The use of canonical form in the construction of locally optimum designs for nonlinear problems. J. Roy. Statist. Soc. Ser. B 54 569–583.
  • Haines, L. M. (1993). Optimal design for nonlinear regression models. Comm. Statist. Theory Methods 22 1613–1627.
  • Haines, L. M. (1995). A geometric approach to optimal design for one-parameter non-linear models. J. Roy. Statist. Soc. Ser. B 57 575–598.
  • Han, C. and Chaloner, K. (2003). D- and c-optimal designs for exponential regression models used in pharmacokinetics and viral dynamics. J. Statist. Plann. Inference 115 585–601.
  • Jennrich, R. I. (1969). Asymptotic properties of non-linear least squares estimators. Ann. Math. Statist. 40 633–643.
  • King, J. and Wong, W. K. (1998). Optimal minimax designs for prediction in heteroscedastic models. J. Statist. Plann. Inference 69 371–383.
  • López-Fidalgo, J. and Wong, W. K. (2002). Design issues for the Michaelis−Menten model. J. Theoret. Biol. 215 1–11.
  • Montepiedra, G. and Wong, W. K. (2001). A new design criterion when heteroscedasticity is ignored. Ann. Inst. Statist. Math. 53 418–426.
  • Müller, C. H. and Pázman, A. (1998). Applications of necessary and sufficient conditions for maximum efficient designs. Metrika 48 1–19.
  • Ortiz, I. and Rodríguez, C. (1998). Optimal designs with respect to Elfving’s partial minimax criterion for heteroscedastic polynomial regression. Test 7 347–360.
  • Pázman, A. and Pronzato, L. (2009). Asymptotic normality of nonlinear least squares under singular experimental designs. In Optimal Design and Related Areas in Optimization and Statistics. Springer, Heidelberg.
  • Pukelsheim, F. (1981). On c-optimal design measures. Math. Operationsforsch. Statist. Ser. Statist. 12 13–20.
  • Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York.
  • Pukelsheim, F. and Rieder, S. (1992). Efficient rounding of approximate designs. Biometrika 79 763–770.
  • Randall, T., Donev, A. and Atkinson, A. C. (2007). Optimum Experimental Designs, with SAS. Oxford Univ. Press.
  • Ratkowsky, D. A. (1983). Nonlinear Regression Modeling: A Unified Practical Approach. Dekker, New York.
  • Ratkowsky, D. A. (1990). Handbook of Nonlinear Regression Models. Dekker, New York.
  • Retout, S. and Mentré, F. (2003). Further developments of the fisher information matrix in nonlinear mixed-effects models with evaluation in population pharmacokinetics. J. Biopharm. Statist. 13 209–227.
  • Rowland, M. (1993). Clinical Pharmacokinetics: Concepts and Applications. Williams and Wilkins, Baltimore.
  • Searle, S. R. (1982). Matrix Algebra Useful for Statistics. Wiley, New York.
  • Seber, G. A. F. and Wild, C. J. (1989). Nonlinear Regression. Wiley, New York.
  • Silvey, S. D. (1980). Optimal Design. Chapman and Hall, London.
  • Studden, W. J. (1968). Optimal designs on Tchebycheff points. Ann. Math. Statist. 39 1435–1447.
  • Studden, W. J. (2005). Elfving’s theorem revisited. J. Statist. Plann. Inference 130 85–94.
  • Wong, W. K. and Cook, R. D. (1993). Heteroscedastic G-optimal designs. J. Roy. Statist. Soc. 55 871–880.