Chapter 1: Statistical Models

Abstract

This chapter introduces the concept of a statistical model. One particular type of statistical model—the generalized linear model—is the focus of this book, and so we begin with an introduction to statistical models in general. This allows us to introduce the necessary language, notation, and other important issues. We first discuss conventions for describing data mathematically (Sect. 1.2). We then highlight the importance of plotting data (Sect. 1.3), and explain how to numerically code non-numerical variables (Sect. 1.4) so that they can be used in mathematical models. We then introduce the two components of a statistical model used for understanding data (Sect. 1.5): the systematic and random components. The class of regression models is then introduced (Sect. 1.6), which includes all models in this book. Model interpretation is then considered (Sect. 1.7), followed by comparing physical models and statistical models (Sect. 1.8) to highlight the similarities and differences. The purpose of a statistical model is then given (Sect. 1.9), followed by a description of the two criteria for evaluating statistical models: accuracy and parsimony (Sect. 1.10). The importance of understanding the limitations of statistical models is then addressed (Sect. 1.11), including the differences between observational and experimental data. The generalizability of models is then discussed (Sect. 1.12). Finally, we make some introductory comments about using r for statistical modelling (Sect. 1.13).

…all models are approximations. Essentially, all models are wrong, but some are useful. However, the approximate nature of the model must always be borne in mind.

Box and Draper [2, p. 424]