Nonlinear parameter estimation for toxicological threshold models

A number of toxicological models contain a threshold concentration, below which there is no (or background) response. The simplest of these is the `hockey stick' model, which contains three parameters (threshold, background response and slope), but a variety of more complex and physiologically realistic models containing thresholds have been developed. The threshold makes these models nonlinear in the parameters. Using a number of actual data sets, the paper shows that this nonlinearity may be quite strong, even leading to a confidence interval with multiple disjunct `compartments'. In this case, the traditional approaches of estimating confidence intervals by ignoring nonlinearity (linearization) or by transformation give unreliable estimates. As an alternative, a robust estimation method is shown to be able to deal with this kind of nonlinearities because it does not rely on an assumed shape of the confidence interval, but produces a sample of points that characterizes this (possibly irregularly shaped) set.