Dose-response curves and competing risks.

Points of the underlying dose-response curve of a lethal response or group of lethal responses induced by varying doses of a toxicant in a homogeneous population can be estimated from knowledge of the time of occurrence for all responses if the response(s) of interest is (are) statistically independent from the other competing responses (risks). In the case of statistical dependence, only tight upper and lower bounds can be established within which the points of the dose-response curve have to lie. These bounds for the response(s) of interest are far apart if the frequency of occurrence of the competing response(s) is large. In such situations, the shape of the underlying dose-response curve is only suggested by the imaginary band connecting the estimated bounds. The estimation procedures for both cases are illustrated with data from an experiment in which beagles received injections of 239Pu.