A remark on small-sample properties of logistic regression in three-point designs

Nottingham and Birch (1) recently alleged that the maximum likelihood (ML) estimator beta of the steepness parameter in a logistic regression model could be seriously underestimated. They based their conclusion on a simulation study, investigating in particular a small-sample three-point design with a relatively large spacing between the doses. In the present work we study such situations in more detail and use complete enumeration to find the exact properties of the ML estimators. The result presented here show that the allegation by Nottingham and Birch was misleading. There is a substantial probability for an infinite outcome of beta, which appears to have been neglected by Nottingham and Birch. In fact, it will be demonstrated that the asymptotic normal approximation for beta fits quite well even with small samples, except in the upper tail where outcomes are infinite instead of large finite. The consequences for coverage probabilities of confidence intervals for both of the regression parameters are elucidated.