On the de la Garza Phenomenon

TL;DR

This paper develops a new approach to verify the de la Garza Phenomenon for nonlinear models, showing it holds for many common models and offering a versatile tool for optimal design analysis.

Contribution

A novel, unified method is introduced to establish the de la Garza Phenomenon for various nonlinear models, extending its applicability and simplifying optimal design derivations.

Findings

The de la Garza Phenomenon applies to many nonlinear models like Emax and exponential models.

The approach works for continuous/discrete data and homogeneous/non-homogeneous errors.

It is applicable to any design region and multiple-stage optimal design.

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.