Interval estimation of inverse dose-response

The practical application of likelihood ratio test based interval estimates for LD50, ED50, and related quantities is considered. Our mathematical setting is that of a generalized linear model with a known scale parameter. We extend the results of Williams (1986, Biometrics 42, 641-645) by showing how Newton's method can be used to calculate the end points of the intervals. To accommodate epidemiologic applications we permit other explanatory variables besides those related to dose in our model. We illustrate the use of the methods in a case in which there are two sources of exposure, whose joint impact is of interest. We also discuss the computation of the confidence sets, when they consist of the whole real line or when they are unions of disjoint intervals. Special problems connected with the cases in which some of the maximum likelihood estimators do not exist are studied. Simulation is used to compare the adequacy of the likelihood ratio based approach to that of the classical Fieller limits. The Fieller limits frequently fail to exist in small samples. The likelihood ratio-based limits always exist, but they are sometimes slightly too narrow. The likelihood ratio-based limits appear not to be as often infinite as the Fieller limits are.