Logistic Regression: Predicting Counts

For the most part, this book concerns itself with measurement data and the corresponding analyses based on normal distributions. In this chapter and the next we consider data that consist of counts. Elementary count data were introduced in Chapter 5. Frequently data are collected on whether or not a certain event occurs. A mouse dies when exposed to a dose of chloracetic acid or it does not. In the past, O-rings failed during a space shuttle launch or they did not. Men have coronary incidents or they do not. These are modeled as random events and we collect data on how often the event occurs. We also collect data on potential predictor (explanatory) variables. For example, we use the size of dose to estimate the probability that a mouse will die when exposed. We use the atmospheric temperature at launch time to estimate the probability that O-rings fail. We may use weight, cholesterol, and blood pressure to estimate the probability that men have coronary incidents. Once we have estimated the probability that these events will occur, we are ready to make predictions. In this chapter we investigate the use of logistic models to estimate probabilities. Logistic models (also known as logit models) are linear models for the log-odds that an event will occur. For a more complete discussion of logistic and logit models, see Christensen (1997). Section 1 introduces models for predicting count data. Section 2 presents a simple model with one predictor variable where the data are the proportions of trials that display the event. It also discusses the output one typically obtains from running a logistic regression program. Section 3 discusses how to perform model tests with count data. Section 4 discusses how logistic models are fitted. Section 5 introduces the important special case in which each observation is a separate trial that either displays the event or does not. Section 6 explores the use of multiple continuous predictors. Section 7 examines ANOVA type models with Section 8 examining ACOVA type models.