7 - Further Topics in Regression

Summary

Introduction

In Chapter 6 we showed how the basic bootstrap methods of earlier chapters extend to linear regression. The broad aim of this chapter is to extend the discussion further, to various forms of nonlinear regression models — especially generalized linear models and survival models — and to nonparametric regression, where the form of the mean response is not fully specified.

A particular feature of linear regression is the possibility of error-based resampling, when responses are expressible as means plus homoscedastic errors. This is particularly useful when our objective is prediction. For generalized linear models, especially for discrete data, responses cannot be described in terms of additive errors. Section 7.2 describes ways of generalizing error-based resampling for such models. The corresponding development for survival data is given in Section 7.3. Section 7.4 looks briefly at nonlinear regression with additive error, mainly to illustrate the useful contribution that resampling methods can make to analysis of such models. There is often a need to estimate the potential accuracy of predictions based on regression models, and Section 6.4 contained a general discussion of resampling methods for this. In Section 7.5 we focus on one type of application, the estimation of misclassification rates when a binary response y corresponds to a classification.

Not all relationships between a response y and covariates x can be readily modelled in terms of a parametric mean function of known form.