statisticalrethinkin..

226 7. INTERACTIONSInteraction models ruin this paradise, however. Look at the interaction likelihood again: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 i[linear model of slope]Now the change in µ i that results from a unit change in r i is given by γ i . And since γ i is afunction of three things—β r , β Ar , and A i —we have to know all three in order to know theinfluence of r i on the outcome. e only time the slope β r has its old meaning is when A i = 0,which makes γ i = β r . Otherwise, to compute the influence of r i on the outcome, we have tosimultaneously consider two parameters and another predictor variable.e practical implication of this fact is that you can no longer read the influence of eitherpredictor from the table of estimates. Here are the parameter estimates:R code7.12precis(m7.5)Mean StdDev 2.5% 97.5%a 9.18 0.14 8.92 9.45br -0.18 0.08 -0.33 -0.04bA -1.85 0.22 -2.27 -1.42bAr 0.35 0.13 0.10 0.60sigma 0.93 0.05 0.83 1.03Since γ (gamma) doesn’t appear in this table—it wasn’t estimated—we have to compute itourselves. Its easy enough to do that at the MAP values (posterior means). For example, theMAP slope relating ruggedness to log-GDP within Africa is:And outside of Africa:γ = β r + β Ar (1) = −0.18 + 0.35 = 0.17γ = β r + β Ar (0) = −0.18So the relationship between ruggedness and log-GDP is essentially reversed inside and outsideof Africa.7.1.4.2. Incorporating uncertainty. But that’s only at the MAP values. To get some ideaof the uncertainty around those γ values, we’ll need to use the whole posterior. Since γdepends upon parameters, and those parameters have a posterior distribution, γ must alsohave a posterior distribution.To compute the posterior distribution of γ, you could do some integral calculus, or youcould just process the samples from the posterior:R code7.13post