Generalized linear models and logistic regression

Abstract

So far, most of the analyses we have described have been based around linear models that assume normally distributed populations of the response variable and of the error terms from the fitted models. Most linear models are robust to this assumption, although the extent of this robustness is hard to gauge, and transformations can be used to overcome problems with non-normal error terms. There are situations where transformations are not effective in making errors normal (e.g. when the response variable is categorical) and, in any case, it might be better to model the actual data rather than data that are transformed to meet assumptions. What we need is a technique for modeling that allows other types of distributions besides normal. Such a technique was introduced by Nelder & Wedderburn (1972) and further developed by McCullough & Nelder (1989) and is called generalized linear modeling (GLM). In this chapter, we will examine two common applications of GLMs: logistic regression, used when the response variable is binary, and Poisson regression, when the response variable represents counts. In the next chapter, we will describe log-linear models when both response and predictor variables are categorical and usually arranged in the form of a contingency table. Generalized linear models Generalized linear models (GLMs) have a number of characteristics that make them more generally applicable than the general linear models we have considered so far. One of the most important is that least squares estimation no longer applies and maximum likelihood methods must be used (Chapter 2).