On the Plot of Residuals Versus Observed Y's in Regression

In applied regression one often examines a proposed model through residual plots. When the residuals are plotted against the X-variables or predicted Y's, as generally recommended, the impression of a “horizontal band” in each plot is viewed as being in accord with the proposed model. Experimenters often plot the residuals against the observed Y's in addition and wrongly expect to see a “horizontal band” in this plot too, whereas these variables are correlated under the model. In fact, a regression of the residuals on the observed Y's is shown to have both slope, beY, and coefficient of determination, R2eY, equal to 1-R2, where R2 is the coefficient of determination for the proposed model. The plot of residuals versus observed Y's does give a visual impression of the degree to which the proposed model underpredicts for large Y and overpredicts for small Y.