Extending logistic regression to model diffuse interactions.

In an observational study focussed on association between a health outcome and numerous explanatory variables, the question of interactions can be problematic. Commonly, logistic regression of the outcome on the explanatory variables might be employed. Such modelling often includes an attempt to select some pairwise product interaction terms, from amongst the many such possible pairs. For several reasons, however, this can be unsatisfying. Here we consider a different approach based on a parsimonious extension of a logistic regression model without interaction terms. This extension permits an overall synergism or antagonism in how the explanatory variables combine to associate with the outcome, without any attempt to identify specific variables which give rise to interactive behaviour. We call this diffuse interaction. We elucidate some simple properties of the diffuse interaction model, and give an example of its application to epidemiological data. We also consider asymptotic behaviour in a restricted case of the model, to gain some insight into how well this kind of interaction can be detected from data. Copyright (c) 2005 John Wiley & Sons, Ltd.