statisticalrethinkin..

138 5. MULTIVARIATE LINEAR MODELSMarriage.s-1 0 1 2-2 -1 0 1 2 3MedianAgeMarriage.sFIGURE 5.4. Residual marriage rate in each State,aer accounting for the linear association withmedian age at marriage. Each gray line segmentis a residual, the distance of each observed marriagerate from the expected value, attempting topredict marriage rate with median age at marriagealone. So States that lie above the black regressionline have higher rates of marriage than expected,according to age at marriage. ose below the linehave lower rates than expected.}lines( c(x,x) , c(mu[i],y) , lwd=0.5 , col=col.alpha("black",0.7) )e result is shown as FIGURE 5.4. Notice that the residuals are variation in marriage ratethat is le over, aer taking out the purely linear relationship between the two variables.Now to use these residuals, let’s put them on a horizontal axis and plot them againstthe actual outcome of interest, divorce rate. I plot these residuals against divorce rate inFIGURE 5.5 (lehand plot), also overlaying the linear regression of the two variables. Youcan think of this plot as displaying the linear relationship between divorce and marriagerates, having statistically “controlled” for median age of marriage. e vertical dashed lineindicates marriage rate that exactly matches the expectation from median age at marriage.So States to the right of the line marry faster than expected. States to the le of the line marryslower than expected. Average divorce rate on both sides of the line is about the same, and sothe regression line demonstrates little relationship between divorce and marriage rates. eslope of the regression line is −0.13, exactly what we found in the multivariate model, m5.3.e righthand plot in FIGURE 5.5 displays the same kind of calculation, but now formedian age at marriage, “controlling” for marriage rate. So States to the right of the verticaldashed line have older than expected median age at marriage, while those to the le haveyounger than expected median age at marriage. Now we find that the average divorce rateon the right is lower than the rate on the le, as indicated by the regression line. States inwhich people marry older than expected for a given rate of marriage tend to have less divorce.e slope of the regression line here is −1.13, again the same as in the multivariate model,m5.3.So what’s the point of all of this? ere’s direct value in seeing the model-based predictionsdisplayed against the outcome, aer subtracting out the influence of other predictors.e plots in FIGURE 5.5 do this. But this procedure also brings home the message that regressionmodels answer with the remaining association of each predictor with the outcome,aer already knowing the other predictors. In computing the predictor residual plots, youhad to perform those calculations yourself. In the unified multivariate model, it all happensautomatically.