Generalized Additive Models - Program Overview

Methods. The Statistica Generalized Additive Models facilities are an implementation of methods developed and popularized by Hastie and Tibshirani (1990); additional detailed discussion of these methods can also be found in Schimek (2000). Statistica can use continuous and categorical predictor variables.

Distributions and link functions. With Statistica, you can choose from a wide variety of distributions for the dependent variable and link functions for the effects of the predictor variables on the dependent variable (see McCullagh and Nelder, 1989; Hastie and Tibshirani, 1990; see also GLZ Introductory Overview - Computational approach for a discussion of link functions and distributions):

Normal, Gamma, and Poisson distributions:

Log link: f(z) = log(z)

Inverse link: f(z) = 1/z

Identity link: f(z) = z

Binomial distributions:

Logit link: f(z)=log(z/(1-z))

Scatterplot smoother. Statistica uses the cubic spline smoother with user-defined degrees of freedom to find an optimum transformation (function) of the predictor variables. For details regarding this smoother, see Hastie and Tibshirani (1990; see also Schimek, 2000, for a discussion of scatterplot smoothers).

Output. STATISTICA reports a comprehensive set of results statistics to aid in the evaluation of the model-adequacy, model fit, and interpretation of results. Specifically, results include: the iteration history for the model fitting computations, summary statistics including the overall R-square value (computed from the deviance statistic) model degrees of freedom, and detailed observational statistics pertaining to the predicted response, residuals (see Hastie & Tibshirani, 1990; in particular formula 6.3), and the smoothing of the predictor variables. Results graphs include plots of observed responses vs. residual responses, predicted values vs. residuals, histograms of observed and residual values, normal probability plots of residual values, and partial residual plots for each predictor, indicating the cubic spline smoothing fit for the final solution.

Alternative Procedures. Generalized additive models are an extension of generalized linear models, which themselves are extensions of general linear models. As briefly noted in the Introductory Overview, the issue of over-fitting of the data should carefully be evaluated, and it is important to consider simpler models before accepting the more complex generalized additive model for final interpretation. Statistica includes comprehensive implementations of general linear models (GLM as well as generalized linear models modules (GLZ) and similar procedures (e.g., GRM, GDA for classification and categorical responses). Also, you can consider regression trees as an alternative to generalized additive models (see, for example, Hastie and Tibshirani, 1990, Chapter 4, for a discussion of regression trees in the context of generalizations of linear models).

Implementation of method in Statistica. The methods available in the Statistica Generalized Additive Models facilities are implementations of techniques developed and popularized by Hastie and Tibshirani (1990); see also, Nisbet, R., Elder, J., & Miner, G. (2009). Specifically, Statistica provides a convenient user interface to the popular GAMFIT program available at the StatLib library of the Department of Statistics at Carnegie Mellon University.