statisticalrethinkin..

216 7. INTERACTIONSFIGURE 7.1. TOP: Dorsal scars for 5 adult Florida manatees. Rows of shortscars, for example on the individuals Africa and Flash, are indicative of propellerlaceration. BOTTOM: ree exemplars of damage on A.W.38 bombersreturning from missions.for each case i. Like the epicycles of the Ptolemaic and Copernican models (Chapters 4 and6), linear models give us a way to describe conditionality.Simple linear models frequently fail to provide enough conditioning, however. Everymodel so far in this book has assumed that each predictor has an independent associationwith the mean of the outcome. What if we want to allow the association to be conditional?For example, in the primate milk data from the previous chapters, suppose the relationshipbetween milk energy and brain size varies by taxonomic group (ape, monkey, prosimian).is is the same as suggesting that the influence of brain size on milk energy is conditionalon taxonomic group. e linear models of previous chapters cannot address this question.To model deeper conditionality—where the importance of one predictor depends uponanother predictor—we need INTERACTION. Interaction is a kind of conditioning, a way ofallowing parameters (really their posterior distributions) to be conditional on further aspectsof the data. e simplest kind of interaction, a linear interaction, is built by extendingthe linear modeling strategy to parameters within the linear model. So it is akin to placingepicycles on epicycles in the Ptolemaic and Copernican models. It is descriptive, but verypowerful.More generally, interactions are central to most statistical models beyond the cozy worldof Gaussian outcomes and linear models of the mean. In generalized linear models (GLMs,Chapter 9 and onwards), even when one does not explicitly define variables as interacting,they will always interact to some degree. Moreover, every variable essentially interacts withitself, as the impact of change in its value will depend upon its current value. Say goodbye to