Optimal quadratic response surface designs for binomial data
by
K.G. Russell
University of Wollongong, N.S.W.

Consider the estimation of a probability of success, p(x₁, x₂), thought to be a function of two predictor variables, x₁ and x₂, that are both under the control of the experimenter. The aim is to find where p is maximised. If appropriate data are collected at an array of points (x₁, x₂), then ln[p/(1-p)] can be modelled in terms of a quadratic function in x₁ and x₂ using logistic regression, and the appropriate optimum point (x₁^*, x₂^*) can be estimated.

How do we select the experimental points? An extensive literature exists on optimal experimental designs for the estimation of response surfaces when the observations have a constant variance and the model is linear in the parameters. However, in the current situation, the model relating p to a quadratic function of x₁ and x₂ is non-linear, and the variance of each observation depends on the unknown value of p at that point. This makes the problem very difficult.

This paper describes the nature of the optimal design when the quadratic function for ln[p/(1-p)] is assumed known. It is hoped that this will assist in finding optimal designs in the general situation where the quadratic function is unknown.

Date received: April 1, 2002