Models for Variates and Factors

In the previous chapters, we developed models for one or more qualitative explanatory variables (factors; Chapters 4 to 11) and models for one or more quantitative explanatory variables (variates; Chapters 12 to 14). We now introduce models for a combination of qualitative and quantitative explanatory variables, i.e. one or more factors with one or more variates. Models for variates and factors arise in many situations, but they are simple extensions of the models discussed previously. We can think of them as either adding a variate to a model for factors, or vice versa. As an example of the rst, consider a eld trial set up as a CRD to study the effect of different types of fertilizers, where the linear model consists of a single factor to identify the response of each treatment group (fertilizer type). If differences in plant size between plots had been noticed (and measured) before the fertilizers were applied, the single factor model could be improved by incorporating an explanatory variate to quantify, and hence enable a correction for, the effect of initial plant size on nal yield. This extension is known as analysis of covariance (ANCOVA), where an explanatory variate is used to account for underlying differences between experimental units. In the second case, we wish to incorporate information on groups into simple (or multiple) linear regression. The groups may arise from the application of different treatments to the experimental units (e.g. different varieties, or levels of water stress) or due to observed differences between experimental units (e.g. males and females of a species, or different soil types). Each group might exhibit a unique pattern of response, so the purpose of analysis is to investigate the differences, which might require separate intercept or slope parameters (or both) for each group. This process is often known as regression with groups or parallel model analysis.