Regression Analysis by Example, 4th Edition

CHAPTER 13

FURTHER TOPICS

13.1 INTRODUCTION

In this chapter we discuss two topics that have come up several times earlier but we did not focus on them. We will be discussing generalized linear models (GLM), and robust regression. These are two vast topics, and would require full-length books. We will give brief descriptions of the topics and provide examples that illustrate the concepts. GLM unifies the concept of linear model building, a primary activity of statistical analysts.

The importance of robust models in any statistical analysis cannot be overemphasized. The earlier chapters have provided us with methods for constructing robust models. In Section 13.5 we discuss methods that exclusively aim at robustness. The discussion on these two topics will not be exhaustive but reflect our personal experience and preferences.

13.2 GENERALIZED LINEAR MODEL

As in Chapter 3, given a response variable Y and p predictor variables X₁, X₂,…, X_p, the linear regression model can be described as follows: an observation Y_i can be written as

where μ_i is called the linear predictor and ε_i is a random error assumed to have a Gaussian (normal) distribution.

The GLM extends the linear regression model in two ways. The β_i is assumed to have a distribution coming from the exponential family. The exponential family includes several standard distributions, in addition to the Gaussian. For example, it ...