On the de la Garza Phenomenon

DOI: 10.1214/09-AOS787Deriving 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.Supported by NSF Grants DMS-07-07013 and DMS-07-48409. AMS 2000 subject classifications. Primary 62K05; secondary 62J12

Universal Optimality in Balanced Uniform Crossover Design

Kunert [Ann. Statist. 12 (1984) 1006-1017] proved that, in the class of repeated measurement designs based on t treatments, p = t periods and n = λt experimental units, a balanced uniform design is universally optimal for direct treatment effects if t ≥ 3 and λ = 1, or if t ≥ 6 and λ = 2. This result is generalized to t ≥ 3 as long as λ ≤ (t −1)/2.Primarily sponsored by NSF Grant DMS-01-03727, National Cancer Institute Grant P01-CA48112-08 and NIH Grant P50-AT00155 ( jointly supported by the National Center for Complementary and Alternative Medicine, the Office of Dietary Supplements, the Office for Research on Women's Health, and the National Institute of General Medicine). The contents are solely the responsibility of the authors and do not necessarily represent the official views of NIH