Generalized Linear Models

So far, we’ve considered cases where we have some predictors \(X\in \R^p\) and a response \(Y \in \R\), and we want to learn something about the relationship or at least predict \(Y\) from \(X\). We started with linear regression as a simple tool, then moved to more flexible models.

But what if \(Y \in \{0, 1\}\) or some other restricted set, such as nonnegative integers?

In a generalized linear model, we make linear models for data where \(Y\) comes from some distribution parametrized by a function of \(\beta\T X\). We can think of generalized linear models as having two parts:

  1. The systematic part of the model relates the mean of \(Y\) to some function of \(\beta\T X\).
  2. The random part specifies the distribution of \(Y\) around that mean.

For example, in an ordinary linear model, the systematic part is simply \(\beta\T X\), and the random part specifies that \(Y\) has a normal distribution with variance \(\sigma^2\) around that mean.

We will begin with logistic regression, which is a (deceptively) simple method for modeling binary data.