Nonparametric Regression in R An Appendix to An R Companion to Applied Regression, Second Edition

In traditional parametric regression models, the functional form of the model is specified before the model is fit to data, and the object is to estimate the parameters of the model. In nonparametric regression, in contrast, the object is to estimate the regression function directly without specifying its form explicitly. In this appendix to Fox and Weisberg (2011), we describe how to fit several kinds of nonparametric-regression models in R, including scatterplot smoothers, where there is a single predictor; models for multiple regression; additive regression models; and generalized nonparametric-regression models that are analogs to generalized linear models. 1 Nonparametric Regression Models The traditional nonlinear regression model (described in the Appendix on nonlinear regression) fits the model y = m(x, ) + " where is a vector of parameters to be estimated, and x is a vector of predictors; the errors " are assumed to be normally and independently distributed with mean 0 and constant variance 2. The function m(x, ), relating the average value of the response y to the predictors, is specified in advance, as it is in a linear regression model. The general nonparametric regression model is written in a similar manner, but the function m is left unspecified: y = m(x) + " = m(x1, x2, . . . , xp) + " for the p predictors x = (x1, x2, . . . , xp) ′. Moreover, the object of nonparametric regression is to estimate the regression function m(x) directly, rather than to estimate parameters. Most methods of nonparametric regression implicitly assume that m is a smooth, continuous function.1 As in nonlinear regression, it is standard to assume that "i ∼ NID(0, 2). An important special case of the general model is nonparametric simple regression, where there is only one predictor: y = m(x) + " Nonparametric simple regression is often called “scatterplot smoothing” because an important application is to tracing a smooth curve through a scatterplot of y against x. We frequently use nonparametric regression in this manner in the body of the text. An exception to the implicit assumption of smoothness is wavelet regression, not discussed in this appendix, which is implemented in R, e.g., in the wavethresh package; see Nason and Silverman (1994, 2000); Nason (2008).