Generalized Additive Models: Some Applications

Abstract

Generalized additive models have the form η(x) = α + σ f_j (x _j ), where η might be the regression function in a multiple regression or the logistic transformation of the posterior probability Pr(y = 1 | x) in a logistic regression. In fact, these models generalize the whole family of generalized linear models η(x) = β′x, where η(x) = g(μ(x)) is some transformation of the regression function. We use the local scoring algorithm to estimate the functions f_j (x_j ) nonparametrically, using a scatterplot smoother as a building block. We demonstrate the models in two different analyses: a nonparametric analysis of covariance and a logistic regression. The procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in a standard linear analysis. A variety of inferential tools have been developed to aid the analyst in assessing the relevance and significance of the estimated functions: these include confidence curves, degrees of freedom estimates, and approximate hypothesis tests.

The local scoring algorithm is analogous to the iterative reweighted least squares algorithm for solving likelihood and nonlinear regression equations. At each iteration, an adjusted dependent variable is formed and an additive regression model is fit using the backfitting algorithm. The backfitting algorithm cycles through the variables and estimates each coordinated function by smoothing the partial residuals.

Key Words:

• Generalized linear model
• Smooth
• Nonparametric regression
• Logistic regression