statisticalrethinkin..

5.1. SPURIOUS ASSOCIATION 139slowerfasteryoungerolderDivorce rate6 8 10 12 14Divorce rate6 8 10 12 14-1.5 -0.5 0.5 1.0 1.5Marriage rate residuals-1 0 1 2Age of marriage residualsFIGURE 5.5. Predictor residual plots for the divorce data. Le: States withfast marriage rates for their median age of marriage have about the samedivorce rates as do States with slow marriage rates. Right: States with oldmedian age of marriage for their marriage rate have lower divorce rates,while States with young median age of marriage have higher divorce rates.Linear regression models do all of this with a very specific additive model of how thepredictors relate to one another. is fact in embodied in the residual calculations you didabove, in which you used separate linear regressions to generate the residuals. But predictorvariables can be related to one another in non-additive ways. e basic logic of statisticalcontrol does not change in those cases, but the details definitely do.5.1.3.2. Counterfactual plots. A second sort of inferential plot displays the implied predictionsof the model. I call these plots counterfactual, because they can be produced for anyvalues of the predictor variables you like, even unobserved or impossible combinations likevery high median age of marriage and very high marriage rate. ere are no States with thiscombination, but in a counterfactual plot, you can ask the model for a prediction for such aState.e simplest use of a counterfactual plot is to see how the predictions change as youchange only one predictor at a time. is means holding the values of all predictors constant,except for a single predictor of interest. Such predictions will not necessarily look likeyour raw data—they are counterfactual aer all—but they will help you understand the implicationsof the model. Since it’s hard to interpret raw numbers in a table, plots that helpyou understand the model’s implications are priceless.Let’s draw a pair of counterfactual plots for the divorce model. Beginning with a plotshowing the impact of changes in Marriage.s on predictions:# prepare new counterfactual dataMAM.avg