Generalized Additive Models and Nonparametric Regression

Abstract

Chapter Preview . Generalized additive models (GAMs) provide a further generalization of both linear regression and generalized linear models (GLM) by allowing the relationship between the response variable y and the individual predictor variables x j to be an additive but not necessarily a monomial function of the predictor variables x j . Also, as with the GLM, a nonlinear link function can connect the additive concatenation of the nonlinear functions of the predictors to the mean of the response variable, giving flexibility in distribution form, as discussed in Chapter 5. The key factors in creating the GAM are the determination and construction of the functions of the predictor variables (called smoothers). Different methods of fit and functional forms for the smoothers are discussed. The GAM can be considered as more data driven (to determine the smoothers) than model driven (the additive monomial functional form assumption in linear regression and GLM). Motivation for Generalized Additive Models and Nonparametric Regression Often for many statistical models there are two useful pieces of information that we would like to learn about the relationship between a response variable y and a set of possible available predictor variables x 1 , x 2 , …, x k for y : (1) the statistical strength or explanatory power of the predictors for influencing the response y (i.e., predictor variable worth) and (2) a formulation that gives us the ability to predict the value of the response variable y new that would arise under a given set of new observed predictor variables x 1, new , x 2, new , …, x k,new (the prediction problem).