Estimation of non-parametric multivariate risk functions in matched case-control studies with application to the assessment of interactions of risk factors in the study of cancer

In epidemiological studies one is interested in investigating the probability of disease depending on risk factors and in particular in detecting interactions of risk factors. Within the setting of parametric logistic regression, interactions can be modelled only in a clumsy and limited way. Modelling the risk function non‐parametrically, estimating it, for example, by a smoothing (thin plate) spline is attractive as a more explorative approach. For prospective studies this amounts to smoothing within the framework and distributional assumptions of generalized regression models (for binary observations). Case‐control studies as retrospective studies with exposure to risk factors being observed do not immediately fit into this setting. In the special case of one‐to‐one matched studies, however, there is an appropriate likelihood again within the range of generalized models. Inferences will be illustrated using simulated and real data. Copyright © 2001 John Wiley & Sons, Ltd.