Generalized boosted additive models

Abstract

Regression analysis is a central method of statistical data analysis, but it is often inappropriate to model the relationship between the conditional distribution of a dependent variable as a function of one or more predictors when this relationship is characterized by complex nonlinear patterns. In such cases nonparametric regression methods are more suitable. Among nonparametric regression methods, generalized additive models have become very popular but they present a drawback when concurvity is present in the data. Concurvity can be defined as the presence of nonlinear dependencies among transformations of the explanatory variables considered in the model and often it directly follows from the presence of collinearity among untransformed predictors. In the context of generalized additive models the presence of concurvity leads to biased estimates of the model parameters and of their standard errors. For such reasons we focus on nonlinear categorical regression approach, applying the optimal scaling methodology as presented in the Gifi system. In the presence of collinearity among untransformed predictors, applying nonlinear transformations through optimal scaling implies that interdependence among these predictors decreases. Moreover, in the framework of nonlinear regression with optimal scaling we follow the approach proposed by Meulman (2003) of introducing in the model nonlinear prediction components, applying the basic idea of forward stagewise boosting procedure, with the aim of improving the prediction power of the model itself. We call this approach the Generalized Boosted Additive Model (GBAM).