Statistical Rethinking by Richard McElreath

Author:Richard McElreath
Language: eng
Format: epub

FIGURE 7.6. The other side of the interaction between ruggedness and continent. Blue points are nations with above-median ruggedness. Black points are below the median. Dashed black line: relationship between continent and log-GDP, for an imaginary nation with minimum observed ruggedness (0.003). Blue line: an imaginary nation with maximum observed ruggedness (6.2).

This perspective on the GDP and terrain ruggedness data is completely consistent with the 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 continent depends upon ruggedness. Indeed, something is gained by looking at the data in this symmetrical perspective. Just inspecting the first view of the interaction, back on page 220, it’s not obvious that African nations are on average nearly always worse off. It’s just at very high values of rugged that nations inside and outside of Africa have the same expected log-GDP. This second way of plotting the interaction makes this clearer.

7.3. Continuous interactions

The main point I want to convince the reader of is that interaction effects are difficult to interpret. They are nearly impossible to interpret, using only posterior means and standard deviations. We’ve already discussed two reasons for this (page 219). A third reason to be wary of using only tables of numbers to interpret interactions is that interactions among continuous variables are especially opaque. It’s one thing to make a slope conditional upon a category, as in the previous example of ruggedness and being in Africa. In such a context, the model reduces to estimating a different slope for each category. But it’s quite a lot harder to understand that a slope varies in a continuous fashion with a continuous variable. Interpretation is much harder in this case, even though the mathematics of the model are essentially the same as in the categorical case.

In pursuit of clarifying the construction and interpretation of CONTINUOUS INTERACTIONS among two or more continuous predictor variables, in this section I develop a simple regression example and show you a way to plot the two-way interaction between two continuous variables. The method I present for plotting this interaction is a triptych plot, a panel of three complementary figures that comprise a whole picture of the regression results. There’s nothing magic 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 a slope, across changes in a chosen variable.

This example will also allow me to illustrate two benefits of CENTERING prediction variables. You met centering already (page 99). When a prediction variable is “centered,” it is just rescaled so that its mean is zero. Why would you ever do that to the data? There are two common benefits. First, centering the prediction variables can make it much easier to lean on the coefficients alone in understanding the model, especially when you want to compare the estimates from models with and without an interaction. Second, sometimes model fitting has a hard time with uncentered variables.

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