Simple linear Regression And Multiple Linear Regression

A linear regression model with one explanatory variable is referred to as a simple linear regression model.

A linear regression model with more than one explanatory variable is referred to as a multiple regression model.

Notation

Regression models can be expressed in several ways. For example, the quadratic regression model used in the potatoes example with response variable \(y\) and predictor variables \(x\) can be expressed as

\[y_i = \alpha + \beta x_i+ \gamma x_i^2+ \epsilon_i, \quad \quad i=1,\dots,n.\]

with various assumptions regarding \(\epsilon_1, \ldots, \epsilon_n\) that we will discuss later. What we really mean to say is that given a value \(x\), we expect the value of \(y\) to be \(\alpha + \beta x+ \gamma x^2\).

We may write this as \(E(y_i |x_i)=\alpha + \beta x_i+ \gamma x_i^2\).

More generally, for a simple linear regression with response \(y\) and explanatory variable \(x\), we may write

\[\begin{eqnarray*} E(y_i|x_i) &=& \alpha + \beta x_i. \end{eqnarray*}\]

To ease notation, we may expression our model as \[\begin{eqnarray*} E(y_i) = \alpha + \beta x_i\\ \end{eqnarray*}\]

for \(i=1, \ldots, n.\)