An EM Algorithm for Fitting a Four-Parameter Logistic Model to Binary Dose-Response Data

Abstract

This article is motivated by the need of biological and environmental scientists to fit a popular nonlinear model to binary dose-response data. The four-parameter logistic model, also known as the Hill model, generalizes the usual logistic regression model to allow the lower and upper response asymptotes to be greater than zero and less than one, respectively. This article develops an EM algorithm, which is naturally suited for maximum likelihood estimation under the Hill model after conceptualizing the problem as a mixture of subpopulations in which some subjects respond regardless of dose, some fail to respond regardless of dose, and some respond with a probability that depends on dose. The EM algorithm leads to a pair of functionally independent two-parameter optimizations and is easy to program. Not only can this approach be computationally appealing compared to simultaneous optimization with respect to all four parameters, but it also facilitates estimating covariances, incorporating predictors, and imposing constraints. This article is motivated by, and the EM algorithm is illustrated with, data from a toxicology study of the dose effects of selenium on the death rates of flies. Other biological and environmental applications, as well as medical and agricultural applications, are also described briefly. Computer code for implementing the EM algorithm is available as supplemental material online.