ABSTRACT
In Chapter 1, we described experimental and observational studies as scientic enquiries in which an outcome (or response) is investigated with the objective of understanding how it is affected by the experimental conditions. In this context, statistical models are used to quantify relationships between the response variable and one or more explanatory variable(s) that dene the conditions. Two simple examples were illustrated in Section 1.3, where the single explanatory variable corresponded to either a qualitative variable (or factor, Example 1.1) or a quantitative variable (or variate, Example 1.2). In Chapter 4, we presented details of the analysis for data classied by a single explanatory factor, including the form of the underlying model, parameter estimation and statistical inference. This was mainly placed in the context of designed experiments. We now focus on the analysis of data where the single explanatory variable is quantitative, or a variate. This is usually known as regression analysis. However, the situation with either a qualitative or a quantitative explanatory variable results in the same basic form of linear model (Section 1.4). Both consist of a systematic component and a random component, with analysis based on the same underlying statistical theory to estimate parameters and predict from the tted model. In this chapter, we are concerned only with models including a single explanatory variate. More broadly, regression analysis refers to the more general approach with any number of quantitative (and qualitative) explanatory variables. In Chapter 14, we shall extend the model to incorporate two or more explanatory variates (multiple regression), and in Chapter 15, we consider models that also include qualitative explanatory variables (factors), sometimes called regression with groups. In Chapter 16, we use linear mixed models to take account of the structure in the observations, including blocking and pseudo-replication.
