Chapter 3: Linear Regression Models: Diagnostics and Model-Building

Abstract

As the previous two chapters have demonstrated, the process of building a linear regression model, or any regression model, is aided by exploratory plots of the data, by reflecting on the experimental design, and by considering the scientific relationships between the variables. This process should ensure that the model is broadly appropriate for the data. Once a candidate model has been fitted to the data, however, there are specialist measures and plots that can examine the model assumptions and diagnose possible problems in greater detail. This chapter describes these tools for detecting and highlighting violations of assumptions in linear regression models. The chapter goes on to discuss some possible courses of action that might alleviate the identified problems. The process of examining and identifying possible violations of model assumptions is called diagnostic analysis. The assumptions of linear regression models are first reviewed (Sect.{\thinspace}3.2), then residuals, the main tools of diagnostic analysis, are defined (Sect.{\thinspace}3.3). We follow with a discussion of the leverage, a measure of the location of an observation relative to the average observation location (Sect.{\thinspace}3.4). The various diagnostic tools for checking the model assumptions are then introduced (Sect.{\thinspace}3.5) followed by techniques for identifying unusual and influential observations (Sect.{\thinspace}3.6). The terminology of residuals is summarized in Sect.{\thinspace}3.7. Techniques for fixing any weaknesses in the models are summarised in Sect.{\thinspace}3.8, and explained in greater detail in Sects.{\thinspace}3.9 to 3.13. Finally, the issue of collinearity is discussed (Sect.{\thinspace}3.14).