statisticalrethinkin..

7.1. BUILDING AN INTERACTION 217simple constant slopes of linear regression. Say hello to impacts that may depend upon thecovariation of dozens of predictor variables.Multilevel models induce similar effects. Common sorts of multilevel models are essentiallymassive interaction models, in which estimates (intercepts and slopes) are conditionalon clusters (person, genus, village, city, galaxy) in the data. Multilevel interaction effects arecomplex. ey’re not just allowing the impact of a predictor variable to change dependingupon some other variable, but they are also estimating aspects of the distribution of thosechanges. is may sound like genius, or madness, or both. Regardless, you can’t have thepower of multilevel modeling without it.Models that allow for complex interactions are easy to fit to data. But they can be considerablyharder to understand. And so I spend this chapter reviewing simple interactioneffects: how to specify them, how to interpret them, and how to plot them. e chapterstarts with a case of an interaction between a single categorical (dummy) variable and a singlecontinuous variable. In this context, it is easy to appreciate the sort of hypothesis thatan interaction allows for. en the chapter moves on to show how manipulating the modelof the mean itself can help us understand the behavior of an interaction, before moving onto more complex interactions between multiple continuous predictor variables as well ashigher-order interactions among more than two variables. In every section of this chapter,the model predictions are visualized, averaging over uncertainty in parameters.My hope is that this chapter lays a solid foundation for interpreting generalized linearmodels and multilevel models in the later chapters.Rethinking: Statistics all-star, Abraham Wald. e World War II bombers story is the work of AbrahamWald (1902–1950). Wald was born in what is now Romania, but immigrated to the United Statesaer the Nazi invasion of Austria. Wald made many contributions over his short life. Perhaps mostgermane to the current material, Wald proved that for many types of rules for making statistical decisions,there will exist a Bayesian rule that is at least as good as any non-Bayesian one. Wald provedthis, remarkably, beginning with non-Bayesian premises, and so anti-Bayesians could not ignore it.is work was summarized in Wald’s 1950 book, published just before his death. 95 Wald died muchtoo young, from a plane crash while touring India.7.1. Building an interactionAfrica is special. e second largest continent, it is the most culturally and geneticallydiverse. Africa has about 3 billion fewer people than Asia, but it has just as many living languages.Africa is so genetically diverse that most of the genetic variation outside of Africa isjust a subset of the variation within Africa. Africa is also geographically special, in a puzzlingway: bad geography tends to be related to good economies within Africa, while the oppositeis true outside of Africa.To appreciate the puzzle, look at regressions of terrain ruggedness (a particular kind ofbad geography) against economic performance (log GDP per capita in the year 2000), bothinside and outside of Africa (FIGURE 7.2). Load the data used in this figure, and split it intoAfrica and non-Africa, with this code:library(rethinking)data(rugged)d