Chapter 4 described methods related to a central notion in inference, namely likelihood. This chapter and the next discuss how those ideas apply to some particular situations, beginning with the simplest model for the dependence of one variable on another, straight-line regression. There is then an account of exponential family distributions, which include many models commonly used in practice, such as the normal, exponential, gamma, Poisson and binomial densities, and which play a central role in statistical theory. We then briefly describe group transformation models, which are also important in statistical theory. This is followed by a description of models for data in the form of lifetimes, which are common in medical and industrial settings, and a discussion of missing data and the EM algorithm.

Straight-Line Regression

We have already met situations where we focus on how one variable depends on others. In such problems there are two or more variables, some of which are regarded as fixed, and others as random. The random quantities are known as responses and the fixed ones as explanatory variables. We shall suppose that only one variable is regarded as a response. Such models, known as regression models, are discussed extensively in Chapters 8, 9, and 10. Here we outline the basic results for the simplest regression model, where a single response depends linearly on a single covariate. We start with an example.