5 The simple logistic regression model

When working with a binary response variable, we know a priori—assuming the standard numerical encoding—that the only possible values of the response are 0 and 1, but there’s no way to know which outcome will occur before making an observation. While this inherent randomness is built into the models that we considered in the last chapter, understanding the practical implications of all of this uncertainty requires a thorough theoretical foundation. That’s what we’ll establish in this chapter! After explaining the ins and outs of Bernoulli random variables, we’ll explain how to draw inferences about model parameters and predictions, and we’ll gain more insight into the role of the deviance in assessing the fit of a model. These tasks are similar to what we did in Chapter 3 with simple linear regression models, but the nonlinearity of the logistic function levels up the technical complexity, giving you plenty of new things to think about.

5.1 Bernoulli models for binary data

5.1.1 The theoretical simple logistic model

5.1.2 Inferences for logistic models

5.2 Binomial models for grouped responses

5.2.1 Binomial random variables and distributions

5.2.2 Binomial logistic regression and the deviance redux

5.2.3 Residual analysis and model checking

5.3 Examples

5.3.1 Intubation and throat pain

5.4 Exercises

5.5 Summary