Generalized Additive Models

Likelihood-based regression models such as the normal linear regression model and the linear logistic model, assume a linear (or some other parametric) form for the covariates X₁, X₂, ⋯, Xₚ. We introduce the class of generalized additive models which replaces the linear form ∑ βⱼXⱼ by a sum of smooth functions ∑ sⱼ(Xⱼ). The sⱼ(·)'s are unspecified functions that are estimated using a scatterplot smoother, in an iterative procedure we call the local scoring algorithm. The technique is applicable to any likelihood-based regression model: the class of generalized linear models contains many of these. In this class the linear predictor η = Σ βⱼXⱼ is replaced by the additive predictor Σ sⱼ(Xⱼ); hence, the name generalized additive models. We illustrate the technique with binary response and survival data. In both cases, the method proves to be useful in uncovering nonlinear covariate effects. It has the advantage of being completely automatic, i.e., no "detective work" is needed on the part of the statistician. As a theoretical underpinning, the technique is viewed as an empirical method of maximizing the expected log likelihood, or equivalently, of minimizing the Kullback-Leibler distance to the true model.