The distribution of the maximum likelihood estimator in up-and-down experiments for quantal dose-response data

Standard maximum likelihood logistic or probit regression has been used in biopharmaceutical practice for inference about tolerance threshold distributions in situations where subjects (patients) have been allocated doses according to an up-and-down design. For example, a steeper dose-response curve than expected was reported in one such study. This article demonstrates that the maximum likelihood estimator systematically and considerably exaggerates the regression parameter with moderately large sample sizes. Thus a probable explanation for finding a steeper curve than expected is the method used to analyze the experiment, that is, the bias in the maximum likelihood estimator. An additional consequence of this bias is that the mean/median/ED50 are estimated with a misleading precision. In particular, confidence intervals are much too narrow. As a conclusion, we warn against conventional logistic or probit regression in combination with up-and-down designs.