On Average Predictive Comparisons and Interactions

In a regression context, consider the difference in expected outcome associated with a particular difference in one of the input variables. If the true regression relationship involves interactions, then this "predictive comparison" can depend on the values of the other input variables. Therefore, one may wish to consider an "average predictive comparison" as a target of inference, where the averaging is with respect to the population distribution of the input variables. We consider inferences about such targets, with emphasis on inferential performance when the regression model is misspecified. Particularly, in light of the difficulties in dealing with interaction terms in regression models, we examine inferences about average predictive comparisons when additive models are fitted to relationships truly involving pairwise interaction terms. We identify some circumstances where such inferences are consistent despite the model misspecification, notably when the input variables are independent, or have a multivariate normal distribution. Copyright (c) 2008 The Authors. Journal compilation (c) 2008 International Statistical Institute.