statisticalrethinkin..

230 7. INTERACTIONSlog GDP year 20006 7 8 9 10 11otherContinentAfricaFIGURE 7.6. e other side of the interactionbetween ruggedness and continent. Bluepoints are nations with above median ruggedness.Black points are below the median.Dashed black line: relationship between continentand log-GDP, for an imaginary nationwith minimum observed ruggedness (0.003).Blue line: an imaginary nation with maximumobserved ruggedness (6.2).effect—the line does slope upwards a tiny amount, but the confidence bounds should preventus from getting excited about that fact. For a nation with very high ruggedness, there isalmost no negative effect on GDP of being in Africa.is perspective on the GDP and terrain ruggedness data is completely consistent withthe previous perspective. It’s simultaneously true in these data (and with this model) that(1) the influence of ruggedness depends upon continent and (2) the influence of continentdepends upon ruggedness. Indeed, something is gained by looking at the data in this symmetricalperspective. Just inspecting the first view of the interaction, back on page 224, it’snot obvious that African nations are on average nearly always worse off. It’s just at very highvalues of rugged that nations inside and outside of Africa have the same expected log-GDP.is second way of plotting the interaction makes it clear that terrain ruggedness doesn’tquite help African nations. Perhaps it just stops them from being hurt so badly.7.3. Continuous interactionse main point I want to convince the reader of is that interaction effects are difficultto interpret. ey are nearly impossible to interpret, using only posterior means and standarddeviations. We’ve already discussed two reasons for this (page 225). A third reason tobe wary of using only tables of numbers to interpret interactions is that interactions amongcontinuous variables are especially opaque. It’s one thing to make a slope conditional upon acategory, as in the previous example of ruggedness and being in Africa. In such a context, themodel reduces to estimating a different slope for each category. But it’s quite a lot harder tounderstand that a slope varies in a continuous fashion with a continuous variable. Interpretationis much harder in this case, even though the mathematics of the model are essentiallythe same as in the categorical case.In pursuit of clarifying the construction and interpretation of continuous interactionsamong two or more continuous predictor variables, in this section I develop a simple regressionexample and show you a way to plot the two-way interaction between two continuousvariables. e method I present for plotting this interaction is a triptych plot, a panel of threecomplementary figures that comprise a whole picture of the regression results. ere’s nothingmagic about having three figures—in other cases you might want more or less. Instead,the utility lies in making multiple figures that allow one to see how the interaction alters aslope, across changes in a chosen variable.