statisticalrethinkin..

228 7. INTERACTIONSthis model and these data, it’s highly implausible that the slope association ruggedness withlog-GDP is lower inside Africa than outside it.Rethinking: More on the meaning of posterior probability. Keep in mind that this number, 0.0036,is not the probability of any observable event. For example, in repeat resampling we do not expect0.36% of countries within Africa to suffer terrain ruggedness worse than a country outside of Africa.Instead, 0.0036 is the relative plausibility that your golem assigns to the question you asked of it, andonly for the data you presented to it. So the golem is telling that it is very very skeptical of the notionthat γ within Africa is lower than γ outside of Africa. How skeptical? Of all the possible states ofthe world it knows about, only a proportion 0.0036 of them are consistent with both the data and theclaim that γ in Africa is less than γ outside Africa. Your golem is skeptical, but it’s usually a good ideafor you to remain skeptical of your golem.7.2. Symmetry of the linear interactionBuridan’s ass is a toy philosophical problem in which an ass who always moves towardsthe closest pile of food will starve to death when he finds himself equidistant between twoidentical piles. e basic problem is one of symmetry: how can the ass decide between twoidentical options? Like many toy problems, you can’t take this one too seriously. Of coursethe ass will not starve. But thinking about how the symmetry is broken can be productive.Interactions are like Buridan’s ass. Like the two piles of identical food, the linear interactioncontains two symmetrical interpretations. Absent some other information, outside themodel, there’s no logical basis for preferring one over the other. Consider for example theGDP and terrain ruggedness problem. e interaction there has two equally valid phrasings.(1) How much does the influence of ruggedness (on GDP) depend upon whether thenation is in Africa?(2) How much does the influence of being in Africa (on GDP) depend upon ruggedness?While these two possibilities sound different to most humans, your golem thinks they areidentical.In this section, we’ll examine this fact analytically. en we’ll plot the ruggedness andGDP example again, but with the reverse phrasing—the influence of Africa depends uponruggedness.7.2.1. Buridan’s interaction. Consider yet again the mathematical form of the likelihood:Let’s expand γ i into the expression for µ i :y i ∼ Normal(µ i , σ)[likelihood]µ i = α + γ i r i + β A A i [linear model of µ]γ i = β r + β Ar A iµ i = α + (β r + β Ar A i )r i + β A A i= α + β r r i + β Ar A i r i + β A A iNow factor together the terms with A i in them:µ i = α + β r r i + (β A + β Ar r i )} {{ }GA i[linear model of slope]