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.