Statistical Science
- Statist. Sci.
- Volume 1, Number 3 (1986), 297-310.
Generalized Additive Models
Trevor Hastie and Robert Tibshirani
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
https://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. https://projecteuclid.org/euclid.ss/1177013604
See also
- See Comment: David R. Brillinger. [Generalized Additive Models]: Comment. Statist. Sci., Volume 1, Number 3 (1986), 310--312.Project Euclid: euclid.ss/1177013605
- See Comment: J. A. Nelder. [Generalized Additive Models]: Comment. Statist. Sci., Volume 1, Number 3 (1986), 312--312.Project Euclid: euclid.ss/1177013606
- See Comment: Charles J. Stone. [Generalized Additive Models]: Comment. Statist. Sci., Volume 1, Number 3 (1986), 312--314.Project Euclid: euclid.ss/1177013607
- See Comment: Peter McCullagh. [Generalized Additive Models]: Comment. Statist. Sci., Volume 1, Number 3 (1986), 314--314.Project Euclid: euclid.ss/1177013608
- See Comment: Trevor Hastie, Robert Tibshirani. [Generalized Additive Models]: Rejoinder. Statist. Sci., Volume 1, Number 3 (1986), 314--318.Project Euclid: euclid.ss/1177013609