Multivariate Nonparametric Regression

As in many areas of biostatistics, oncological problems often have multivariate predictors. While assuming a linear additive model is convenient and straightforward, it is often not satisfactory when the relation between the outcome measure and the predictors is either nonlinear or nonadditive. In addition, when the number of predictors becomes (much) larger than the number of independent observations, as is the case for many new genomic technologies, it is impossible to fit standard linear models. In this chapter, we provide a brief overview of some multivariate nonparametric methods, such as regression trees and splines, and we describe how those methods are related to traditional linear models. Variable selection (discussed in Chapter 2) is a critical ingredient of the nonparametric regression methods discussed here; being able to compute accurate prediction errors (Chapter 4) is of critical importance in nonparametric regression; when the number of predictors increases substantially, approaches such as bagging and boosting (Chapter 5) are often essential. There are close connections between the methods discussed in Chapter 5 and some of the methods discussed in Section 3.8.2. In this chapter, we will briefly revisit those topics, but we refer to the respective chapters for more details. Support vector machines (Chapter 6), which are not discussed in this chapter, offer another approach to nonparametric regression. We start this chapter by discussing an example that we will use throughout the chapter. In Section 3.2 we discuss linear and additive models. In Section 3.3 we generalize these models by allowing for interaction effects. In Section 3.4 we discuss basis function expansions, which is a form in which many nonparametric regression methods, such as regression trees (Section 3.5), splines (Section 3.6) and logic regression (Section 3.7) can be written. In Section 3.8 we discuss the situation in which the predictor space is high dimensional. We conclude the chapter with discussing some issues pertinent to survival data (Section 3.9) and a brief general discussion (Section 3.10).