Statistical Science

Generalized Additive Models

Trevor Hastie and Robert Tibshirani

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Abstract

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 $X_1, X_2, \cdots, X_p$. We introduce the class of generalized additive models which replaces the linear form $\sum \beta_jX_j$ by a sum of smooth functions $\sum s_j(X_j)$. The $s_j(\cdot)$'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 $\eta = \Sigma \beta_jX_j$ is replaced by the additive predictor $\Sigma s_j(X_j)$; 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.

Article information

Source
Statist. Sci. Volume 1, Number 3 (1986), 297-310.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.ss/1177013604

Digital Object Identifier
doi:10.1214/ss/1177013604

Mathematical Reviews number (MathSciNet)
MR858512

Zentralblatt MATH identifier
0645.62068

JSTOR
links.jstor.org

Keywords
Generalized linear models smoothing nonparametric regression partial residuals nonlinearity

Citation

Hastie, Trevor; Tibshirani, Robert. Generalized Additive Models. Statist. Sci. 1 (1986), no. 3, 297--310. doi:10.1214/ss/1177013604. http://projecteuclid.org/euclid.ss/1177013604.


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See also

  • See Comment: David R. Brillinger. [Generalized Additive Models]: Comment. Statist. Sci., Volume 1, Number 3 (1986), 310--312.
  • See Comment: J. A. Nelder. [Generalized Additive Models]: Comment. Statist. Sci., Volume 1, Number 3 (1986), 312--312.
  • See Comment: Charles J. Stone. [Generalized Additive Models]: Comment. Statist. Sci., Volume 1, Number 3 (1986), 312--314.
  • See Comment: Peter McCullagh. [Generalized Additive Models]: Comment. Statist. Sci., Volume 1, Number 3 (1986), 314--314.
  • See Comment: Trevor Hastie, Robert Tibshirani. [Generalized Additive Models]: Rejoinder. Statist. Sci., Volume 1, Number 3 (1986), 314--318.