Nonparametric estimation and tests of fit for dose-response relations

Various parametric models have been used to describe the dependence of disease risks on continuous explanatory variables. The fit of such models is generally tested either against some larger but still parametric class of alternatives, or by making the covariates discrete in some arbitrary fashion. This communication introduces a more general concept of residuals which may be useful for visualizing the fit of the data. Subject only to the constraint of monotonicity, the maximum likelihood estimate (MLE) of the dose-response relation for a single covariate is shown to be a step function with discontinuities only at the covariate values of the failures. The MLE is found by a constrained Newton-Raphson search, and a procedure for reducing the dimensions of the search is suggested. Applications to the risk of lung cancer in relation to asbestos exposure are given.