statisticalrethinkin..

362 15. MISSING DATA AND OTHER OPPORTUNITIESDivorce estimated - divorce observed-3 -2 -1 0 1 20.5 1.0 1.5 2.0 2.5Divorce observed standard errorDivorce rate (posterior)4 6 8 10 12 1423 24 25 26 27 28 29Median age marriageFIGURE 15.2. Le: Shrinkage resulting from modeling the measurementerror. Right: Comparison of regression that ignores measurement error(dashed line and gray shading) with regression that incorporates measurementerror (blue line and shading).divorce_est[1] 11.75 0.66 10.46 13.05...divorce_est[50] 11.53 1.09 9.48 13.73lp__ -57.99 6.53 -71.05 -45.50If you look back at Chapter 5, you’ll see that the former estimate for bMAM was about −1. Nowit’s almost half that, but still reliably negative. So compared to the original regression thatignores measurement error, the association between divorce and age at marriage has beenreduced.If you look again at FIGURE 15.1, you can see a hint of why this has happened. States withextremely low and high ages at marriage tend to also have more uncertain divorce rates. Asa result those rates have been shrunk towards the expected mean defined by the regressionline. FIGURE 15.2 displays this shrinkage phenomenon. On the le of the figure, the differencebetween the observed and estimated divorce rates are shown on the vertical axis, whilethe standard error of the observed is shown on the horizontal. e dashed line at zero indicatesno change from observed to estimated. Notice that States with more uncertain divorcerates—further right on the plot—have estimates more different from observed. is is yourfriend SHRINKAGE from before. Less certain estimates are improved by pooling informationfrom more certain estimates.is shrinkage results in pulling divorce rates towards the regression line, as seen in therighthand plot in the same figure. is plot shows the estimated divorce rates against theobserved ages at marriage, with the old no-error regression in gray and the new with-errorregression in blue.