## On the de la Garza phenomenon.(English)Zbl 1202.62103

Summary: 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 models, optimality criteria and objectives require their own proof. The celebrated A. de la Garza phenomenon [Ann. Math. Stat. 25, 123–130 (1954; Zbl 0055.13206)] 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.

### MSC:

 62K05 Optimal statistical designs 62J02 General nonlinear regression

### Keywords:

locally optimal; Loewner ordering; support points

Zbl 0055.13206
Full Text:

### References:

 [1] de la Garza, A. (1954). Spacing of information in polynomial regression. Ann. Math. Statist. 25 123-130. · Zbl 0055.13206 [2] Dette, H. (1997). Designing experiments with respect to “Standardized” optimality criteria. J. Roy. Statist. Soc. Ser. B 59 97-110. JSTOR: · Zbl 0884.62081 [3] Dette, H., Bretz, F., Pepelyshev, A. and Pinheiro, J. (2008). Optimal designs for dose-finding studies. J. Amer. Statist. Assoc. 103 1225-1237. · Zbl 1205.62165 [4] Dette, H., Haines, L. M. and Imhof, L. (1999). Optimal designs for rational models and weighted polynomial regression. Ann. Statist. 27 1272-1293. · Zbl 0957.62062 [5] 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. · Zbl 1117.62316 [6] Elfving, G. (1952). Optimum allocation in linear regression theory. Ann. Math. Statist. 23 255-262. · Zbl 0047.13403 [7] 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. · Zbl 1132.62056 [8] 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. JSTOR: · Zbl 0774.62080 [9] 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. · Zbl 1016.62088 [10] Karlin, S. and Studden, W. J. (1966). Optimal experimental designs. Ann. Math. Statist. 37 783-815. · Zbl 0151.23904 [11] 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. · Zbl 1246.62168 [12] Kiefer, J. and Wolfowitz, J. (1960). The equivalence of two extremum problems. Canad. J. Math. 12 363-366. · Zbl 0093.15602 [13] Li, G. and Majumdar, D. (2008). D-optimal designs for logistic models with three and four parameters. J. Statist. Plann. Inference 138 1950-1959. · Zbl 1134.62054 [14] Pukelsheim, F. (2006). Optimal Design of Experiments . SIAM, Philadelphia, PA. · Zbl 1101.62063 [15] Stufken, J. and Yang, M. (2010). On locally optimal designs for generalized linear models with group effects. Technical report, Dept. Statistics, Univ. Missouri. · Zbl 1253.62054 [16] 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. · Zbl 1090.62077 [17] Yang, M. and Stufken, J. (2009). Support points of locally optimal designs for nonlinear models with two parameters. Ann. Statist. 37 518-541. · Zbl 1155.62053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.