Chapter 2 Linear models I: multiple linear model

The multiple linear model is a simple but useful statistical model. In short, it allows us to analyze the (assumed) linear relation between a response $Y$ and multiple predictors, $X_1,\ldots,X_p$ in a proper way:

$$\begin{align*} Y=\beta_0+\beta_1X_1+\beta_2X_2+\cdots+\beta_pX_p+\varepsilon \end{align*}$$

The simplest case corresponds to $p=1,$ known as the simple linear model:

$$\begin{align*} Y=\beta_0+\beta_1X+\varepsilon \end{align*}$$

This model would be useful, for example, to predict $Y$ given $X$ from a sample $(X_1,Y_1),\ldots,(X_n,Y_n)$ such that its scatterplot is the one in Figure 2.1.

Figure 2.1: Scatterplot of a sample $(X_1,Y_1),\ldots,(X_n,Y_n)$ showing a linear pattern.