The Annals of Statistics

On the de la Garza phenomenon

Min Yang

Full-text: Open access

Abstract

Deriving optimal designs for nonlinear models is, in general, challenging. One crucial step is to determine the number of support points needed. Current tools handle this on a case-by-case basis. Each combination of model, optimality criterion and objective requires its own proof. The celebrated de la Garza Phenomenon states that under a (p−1)th-degree polynomial regression model, any optimal design can be based on at most p design points, the minimum number of support points such that all parameters are estimable. Does this conclusion also hold for nonlinear models? If the answer is yes, it would be relatively easy to derive any optimal design, analytically or numerically. In this paper, a novel approach is developed to address this question. Using this new approach, it can be easily shown that the de la Garza phenomenon exists for many commonly studied nonlinear models, such as the Emax model, exponential model, three- and four-parameter log-linear models, Emax-PK1 model, as well as many classical polynomial regression models. The proposed approach unifies and extends many well-known results in the optimal design literature. It has four advantages over current tools: (i) it can be applied to many forms of nonlinear models; to continuous or discrete data; to data with homogeneous or nonhomogeneous errors; (ii) it can be applied to any design region; (iii) it can be applied to multiple-stage optimal design and (iv) it can be easily implemented.

Article information

Source
Ann. Statist., Volume 38, Number 4 (2010), 2499-2524.

Dates
First available in Project Euclid: 11 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1278861255

Digital Object Identifier
doi:10.1214/09-AOS787

Mathematical Reviews number (MathSciNet)
MR2676896

Zentralblatt MATH identifier
1202.62103

Subjects
Primary: 62K05: Optimal designs
Secondary: 62J12: Generalized linear models

Keywords
Locally optimal Loewner ordering support points

Citation

Yang, Min. On the de la Garza phenomenon. Ann. Statist. 38 (2010), no. 4, 2499--2524. doi:10.1214/09-AOS787. https://projecteuclid.org/euclid.aos/1278861255


Export citation

References

  • de la Garza, A. (1954). Spacing of information in polynomial regression. Ann. Math. Statist. 25 123–130.
  • Dette, H. (1997). Designing 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. (2008). Optimal designs for dose-finding studies. J. Amer. Statist. Assoc. 103 1225–1237.
  • Dette, H., Haines, L. M. and Imhof, L. (1999). Optimal designs for rational models and weighted polynomial regression. Ann. Statist. 27 1272–1293.
  • Dette, H., Melas, V. B. and Wong, W. K. (2005). Optimal design for goodness-of-fit of the Michaelis–Menten enzyme kinetic function. J. Amer. Statist. Assoc. 100 1370–1381.
  • Elfving, G. (1952). Optimum allocation in linear regression theory. Ann. Math. Statist. 23 255–262.
  • Fang, X. and Hedayat, A. S. (2008). Locally D-optimal designs based on a class of composed models resulted from blending Emax and one-compartment models. Ann. Statist. 36 428–444.
  • Ford, I., Torsney, B. and Wu, C. F. J. (1992). The use of a canonical form in the construction of locally optimal designs for non-linear problems. J. Roy. Statist. Soc. Ser. B 54 569–583.
  • Han, C. and Chaloner, K. (2003). D- and c-optimal designs for exponential regression models used in viral dynamics and other applications. J. Statist. Plann. Inference 115 585–601.
  • Karlin, S. and Studden, W. J. (1966). Optimal experimental designs. Ann. Math. Statist. 37 783–815.
  • Khuri, A. I., Mukherjee, B., Sinha, B. K. and Ghosh, M. (2006). Design issues for generalized linear models: A review. Statist. Sci. 21 376–399.
  • Kiefer, J. and Wolfowitz, J. (1960). The equivalence of two extremum problems. Canad. J. Math. 12 363–366.
  • Li, G. and Majumdar, D. (2008). D-optimal designs for logistic models with three and four parameters. J. Statist. Plann. Inference 138 1950–1959.
  • Pukelsheim, F. (2006). Optimal Design of Experiments. SIAM, Philadelphia, PA.
  • Stufken, J. and Yang, M. (2010). On locally optimal designs for generalized linear models with group effects. Technical report, Dept. Statistics, Univ. Missouri.
  • Wang, Y., Myers, R. H., Smith, E. P. and Ye, K. (2006). D-optimal designs for Poisson regression models. J. Statist. Plann. Inference 136 2831–2845.
  • Yang, M. and Stufken, J. (2009). Support points of locally optimal designs for nonlinear models with two parameters. Ann. Statist. 37 518–541.