Analytic Solutions for D-optimal Factorial Designs under Generalized Linear Models

TL;DR

This paper introduces two analytic methods for finding D-optimal approximate designs in generalized linear models, including explicit solutions for models with multiple factors and conditions for boundary point designs.

Contribution

It provides novel analytic solutions for D-optimal designs in generalized linear models with multiple factors, bridging factorial and continuous factor designs.

Findings

Explicit solutions for two-factor models.

Necessary and sufficient conditions for boundary point designs.

Bridging factorial and continuous factor D-optimal designs.

Abstract

We develop two analytic approaches to solve D-optimal approximate designs under generalized linear models. The first approach provides analytic D-optimal allocations for generalized linear models with two factors, which include as a special case the $2^2$ main-effects model considered by Yang, Mandal and Majumdar (2012). The second approach leads to explicit solutions for a class of generalized linear models with more than two factors. With the aid of the analytic solutions, we provide a necessary and sufficient condition under which a D-optimal design with two quantitative factors could be constructed on the boundary points only. It bridges the gap between D-optimal factorial designs and D-optimal designs with continuous factors.