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5.1. SPURIOUS ASSOCIATION 141MedianAgeMarriage.s = 0Marriage.s = 0Divorce6 8 10 12Divorce6 8 10 12-1 0 1 2Marriage.s-2 -1 0 1 2 3MedianAgeMarriage.sFIGURE 5.6. Counterfactual plots for the multivariate divorce model, m5.3.Each plot shows the change in predicted mean across values of a single predictor,holding the other predictor constant at its mean value (zero in bothcases). Shaded regions show 95% percentile intervals of the mean (dark,narrow) and 95% prediction intervals (light, wide).mtext( "Marriage.s = 0" )lines( MAM.seq , mu.mean )shade( mu.PI , MAM.seq )shade( MAM.PI , MAM.seq )ese plots have the same slopes as the residual plots in the previous section. But they don’tdisplay any data, raw or residual, because they are counterfactual. And they also show percentileintervals on the scale of the data, instead of on that weird residual scale. As a result,they are direct displays of the impact on prediction of a change in each variable.A tension with such plots, however, lies in their counterfactual nature. Is it really possible—in the large world rather than the small world of the model—to change median age of marriagewithout also changing the marriage rate? Probably not. Suppose for example that youpay young couples to postpone marriage until they are 35 years old. Surely this will alsodecrease the number of couples who ever get married—some people will die before turning35, among other reasons—decreasing the overall marriage rate. An extraordinary and evildegree of control over people would be necessary to really hold marriage rate constant whileforcing everyone to marry at a later age.In this example, the difficulty of separately manipulating marriage rate and marriage agedoesn’t impede inference much, only because marriage rate has almost no effect on prediction,once median age of marriage is taken into account. But in many problems, including alater one in this chapter, more than one predictor variable has a sizable impact on the outcome.In that case, while these counterfactual plots always help in understanding the model,they may also mislead by displaying predictions for impossible combinations of predictorvalues. If our goal is to intervene in the world, there may not be any realistic way to manipulateeach predictor without also manipulating the others.