On the de la Garza phenomenon

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 non-homogeneous 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.

On Optimal Designs for Nonlinear Models: A General and Efficient Algorithm

“…Recently, Yang and Stufken (2009), Yang (2010), and Dette and Melas (2011) convincingly demonstrated that unifying results for multiple models, multiple optimality criteria and multiple objectives can be obtained in the context of nonlinear models. They show that we can focus on a subclass of designs with a simple form, no matter what type of optimal designs we are looking for, including optimal multi-stage designs.…”

Optimal designs for two-parameter nonlinear models with application to survival models

“…They obtain a series of excellent results, showing that under some conditions, for each given design there is always a design from a simple class which is better in the Statistica Sinica: Preprint doi: 10.5705/ss.2011.271 Loewner sense. These results were then generalised to models with more than two parameters by Yang (2010) and Dette and Melas (2011). Depending on the model, however, these conditions can be difficult to verify even using symbolic computational software.…”

Optimal Designs for Nonlinear Models with Random Block Effects

“…Even though the within-group information matrix does not share the desirable property of additivity as in traditional design problems, surprisingly it is still possible to identify complete class by the same way as in Theorem 1 in Yang (2010). This theorem shows that only a small number of support points are necessary to achieve optimal design under Model (2.5).…”