Introduction

The additive models of Chapters 8 and 11 have many attractive features. The joint effect of all the predictor variables upon the response is expressed as a sum of individual effects. These individual effects show how the expected response varies as any single predictor varies with the others held fixed at arbitrary values; because of the additivity, the effect of one predictor does not depend on the values at which the others are fixed. Thus, the individual component functions can be plotted separately to visualize the effect of each predictor, and these functions – taken together – allow us to understand the joint effects of all the predictors upon the expected response. If, for example, we wish to find conditions under which the expected response is maximized, then we need only maximize separately each of the component functions of the additive model. In summary, it is extremely convenient whenever an additive model provides an accurate summary of the data.

However, there are no guarantees that an additive model will provide a satisfactory fit in any given situation. Nonadditivity means that, as one predictor is varied, the effect on the expected response depends on the fixed values of the other predictors. A deviation from additivity is called an interaction.