Partly parametric generalized additive model

In many scientific studies, the response variable bears a generalized nonlinear regression relationship with a certain covariate of interest, which may, however, be confounded by other covariates with unknown functional form. We propose a new class of models, the partly parametric generalized additive model (PPGAM) for doing generalized nonlinear regression with the confounding covariate effects adjusted nonparametrically. To avoid the curse of dimensionality, the PPGAM specifies that, conditional on the covariates, the response distribution belongs to the exponential family with the mean linked to an additive predictor comprising a nonlinear parametric function that is of main interest, plus additive, smooth functions of other covariates. The PPGAM extends both the generalized additive model (GAM) and the generalized nonlinear regression model. We propose to estimate a PPGAM by the method of penalized likelihood. We derive some asymptotic properties of the penalized likelihood estimator, including consistency and asymptotic normality of the parametric estimator of the nonlinear regression component. We propose a model selection criterion for the PPGAM, which resembles the BIC. We illustrate the new methodologies by simulations and real applications. We have developed an R package PPGAM that implements the methodologies expounded herein.