9.1 Mathematical Model

The mathematical model for multiple linear regression equates the value of the continuous outcome \(y_{i}\) to a linear combination of multiple predictors \(x_{1} \ldots x_{p}\) each with their own slope coefficient \(\beta_{1} \ldots \beta_{p}\).

\[ y_{i} = \beta_{0} + \beta_{1}x_{1i} + \ldots + \beta_{p}x_{pi} + \epsilon_{i}\]

where \(i\) indexes the observations \(i = 1 \ldots n\), and \(j\) indexes the number of parameters \(j=1 \ldots p\). This linear combination is often written using summation notation: \(\sum_{i=1}^{p}X_{ij}\beta_{j}\).

The assumptions on the residuals \(\epsilon_{i}\) still hold:

  • They have mean zero
  • They are homoscedastic, that is all have the same finite variance: \(Var(\epsilon_{i})=\sigma^{2}<\infty\)
  • Distinct error terms are uncorrelated: (Independent) \(\text{Cov}(\epsilon_{i},\epsilon_{j})=0,\forall i\neq j.\)

In matrix notation the linear combination of \(X\)’s and \(\beta\)’s is written as \(\mathbf{x}_{i}^{'}\mathbf{\beta}\), (the inner product between the vectors \(\mathbf{x}_{i}\) and \(\mathbf{\beta}\)). Then the model is written as:

\[ \textbf{y} = \textbf{X} \mathbf{\beta} + \mathbf{\epsilon} ,\]

and we say the regression model relates \(y\) to a function of \(\textbf{X}\) and \(\mathbf{\beta}\), where \(\textbf{X}\) is a \(nxp\) matrix of \(p\) covariates on \(n\) observations and \(\mathbf{\beta}\) is a length \(p\) vector of regression coefficients.

Note: Knowledge of Matricies or Linear Algebra is not required to conduct or understand multiple regression, but it is foundational and essential for Statistics and Data Science majors to understand the theory behind linear models.

Learners in other domains should attempt to understand matricies at a high level, as some of the places models can fail is due to problems doing math on matricies.