Optimal designs for rational models

Zhuoqiong He, Purdue University

Abstract

Most of the design work has focused on the linear regression model due to its simplicity. However, as the amount of statistical analysis using nonlinear models increased in fields such as the chemical, biological and clinical sciences, people became aware of the need for optimal design for these nonlinear models. The nonlinear problem is intrinsically much harder than the linear problem as the information matrix and design efficiency of a design depends on the unknown parameters. A common approach is to design an experiment to be efficient for a best guess of the parameter values. This approach leads to what are called "locally optimal" designs. A natural generalization to the locally optimal design is to use a prior distribution on the parameters rather than a single guess. The optimal design is then called a Bayesian optimal design. Finding Bayesian D-optimal designs for a nonlinear model is usually quite difficult. For nondegenerate prior distributions, the examples of Bayesian D-optimal design considered so far for models with more than one parameter are found numerically and in general cannot be expressed in closed form. In this thesis, experimental designs for a rational model, $Y = P(x)/Q(x) +\varepsilon,$ are investigated, where $P(x) = \theta\sb0 + \theta\sb1x +\ \...\ + \theta\sb{p}x\sp{p}$ and $ Q(x) = 1 + \theta\sb{p+1}x +\ \...\ + \theta\sb{p+q}x\sp{q}$ are polynomials and $\varepsilon$ has a Normal distribution. Two approaches, the locally D-optimal design, and Bayesian D-optimal design which maximize the expected increase in Shannon information provided by the experiment asymptotically, are examined. Conditions are derived under which a $p + q + 1$ point design is the locally optimal design. The Bayesian optimal design is shown to be independent of parameters in $P(x)$ and to be equally weighted at each support point if the number of support points is the same as the number of parameters in the model. Many other properties of locally and Bayesian optimal designs are also found. The existence and uniqueness of the locally optimal design for extrapolation are shown. The number of support points for the locally optimal design for extrapolation are exact $p + q + 1.$ These $p + q + 1$ design points are proved to be independent of the extrapolation point $x\sb{e}$ and the parameters in $P(x).$ The corresponding weights are also independent of the parameters in $P(x),$ but depend on $x\sb{e}$ and are not equally weighted. In addition, the optimal Bayesian design for several one, two and three parameter models are characterized for a two point prior. Finally, the Michaelis-Menten model is examined extensively; local and Bayesian D-optimal designs and locally optimal designs for extrapolation are compared.

Degree

Ph.D.

Advisors

Studden, Purdue University.

Subject Area

Statistics

COinS