SEEING A CURVE IN MULTIPLE-REGRESSION

Start with a multiple regression in which each predictor enters linear ly. How can we tell if there is a curve so that the model is not valid ? Possibly for one of the predictors an additional square or square-ro ot term is needed. We focus on the case in which an additional term is needed rather than the monotonic case in which a power transformation or logarithm might be sufficient Among the plots that have been used for diagnostic purposes, nine methods are applied here. All nine metho ds work fine when the predictors are not related to each other, but tw o of them are designed to work even when the predictors are arbitrary noisy functions of each other. These two are recent methods, Cook's CE RES plot and the plot for an additive model with nonparametric smoothi ng applied to one predictor. Even these plots, however, can miss a cur ve in some cases and show a false curve in others. To give a measure o f curve detection, the curve can be fitted nonparametrically, and this fit can be used in place of the predictor in the multiple regression. When a curve is detected, it can be approximated with a parametric cu rve such as a polynomial in an arbitrary power.