## INTRODUCTION

Previously (Scott *et al*. 2013), we investigated whether two variables might be related; now, we actually want to describe the relationship, to draw inferences about the nature of the relationship and to make predictions of one variable given the value of the other. In this setting, however, the two variables of interest are no longer treated equally: in our discussion of how we can answer this question, we will need to identify the *response variable*, which depends on an *explanatory variable* or *covariate*. We are first going to deal with the simplest case consisting of one response variable and one explanatory variable (simple linear regression), before we deal with one response variable and many explanatory variables (multiple regression). It is worth noting that the linear and multiple regression cases are examples of *linear models*, and this is something that we will return to in a subsequent article.

The first example we will use concerns the weight of a horse's heart. This has been suggested to be a reasonable indicator of its fitness and strength, but is obviously difficult to measure directly. However, it might be possible to estimate the weight indirectly from ultrasound measurements. Forty-six horses destined for euthanasia had echocardiography performed, and their hearts were then weighed post-mortem (O'Callaghan 1985). One of the ultrasound measurements recorded was the thickness, in centimetres, of the outer heart wall during systole (SOW), and direct measurement was made of the post-mortem weight of the heart (in kilograms).

Is it possible to successfully use an ultrasound measurement to predict the weight of the heart? The scatterplot (Fig 1) shows the measurement of heart weight at post-mortem (the response) with the SOW ultrasound measurement (the explanatory) taken on the 46 horses.

A natural question is “Is there a relationship between heart weight and SOW?” If the answer is “yes,” a second question is “Does it appear linear?” If echocardiography is to be a useful guide to determining heart weight in horses, we will be hoping that the answer to both questions is “yes.”

We will shortly illustrate how we might answer these questions. First, however, it will be useful to set the context of this problem a little more formally. The data take the form of a series of n points denoted by (x_{i}, y_{i}), where i=1,2,… n. In this case, x (the explanatory variable) would be systolic outer wall (SOW), y (the response variable) would be heart weight and n (the number of cases) would be 46. With this convention, we can then specify the model that we are going to build.