Abstract

There are theory-based methods for constructing model-based optimal designs when the sample size is large. The problem becomes challenging when the sample size is small. The theory may no longer apply, and even if it does, the optimal design may not be implementable. We provide examples and also show that a simple rounding procedure of the weights from an optimal approximate design to an optimal exact design can produce the wrong optimal exact design. To solve this longstanding, serious and practical problem, we propose a state-of-the-art nature-inspired metaheuristic algorithm to find efficient designs for an experiment with a small sample size. As an application, we use the algorithm to find an optimal design for a toxicology experiment to detect the existence of hormesis in a dose-response study and an optimal design to estimate the hormesis threshold. Being a metaheuristic algorithm, it can be used to find different types of optimal designs for various statistical models. We demonstrate its flexibility by finding locally D-optimal designs for estimating model parameters in logistic models for small experiments, along with user-friendly codes to produce all the designs.