D-optimal Approximate Design for Binary Regression and Quantal Response in Toxicology Studies

TL;DR

This paper develops methods for efficiently designing binary regression experiments in toxicology, providing analytical solutions for two-parameter models and extending to three-parameter models with a new formula.

Contribution

It introduces an analytical equation for D-optimal design in two-parameter models and extends the approach to three-parameter models with a new determinant formula.

Findings

Analytical solution (WC equation) for two-parameter D-optimal design.

Particle swarm optimization for cases without analytical solutions.

Neat formula for the determinant in three-parameter models.

Abstract

We provide a systematic treatment of $D$-optimal design for binary regression and quantal response models in toxicology studies. For the two-parameter case, we provide an analytical equation (WC equation) for computing the $D$-optimal design quickly and when analytical solution is not available, we apply particle swarm optimization to solve for the $D$-optimal design. Examples with various link functions are given as well as the sensitivity functions. We extend the two-parameter case to three-parameter case by providing a neat formula for the determinant of the information matrix. We also suggest practitioners to work with the neat formula to derive optimal designs for three-parameter binary regression models.