Multiple Linear Regression

Extends the Simple Linear Regression to accommodate multiple predictors.

In general, suppose that we have $p$ distinct predictors. Then the multiple linear regression model takes the form:

Y = β_{0} + β_{1} X_{1} + β_{2} X_{2} + \dots + β_{p} X_{p} + ϵ

where $X_{j}$ represents the $j^{t h}$ predictor and $β_{j}$ quantifies the association between that variable and the response. We interpret $β_{j}$ as the average effect on $Y$ of a one unit increase in $X_{j}$ , holding all other predictors fixed.

Estimating Coefficients

Given estimates $\hat{β_{0}}, \hat{β_{1}}, \dots, \hat{β_{p}}$ , we can make predictions using the formula:

\hat{y} = \hat{β_{0}} + \hat{β_{1}} x_{1} + \hat{β_{2}} x_{2} + \dots + \hat{β_{p}} x_{p}

We choose $\hat{β_{0}}, \hat{β_{1}}, \dots, \hat{β_{p}}$ to minimize:

R S S = \sum_{i = 1}^{n} (y_{i} - \hat{y})^{2} = \sum_{i = 1}^{n} (y_{i} - \hat{β_{0}} - \hat{β_{1}} x_{i 1} - \hat{β_{2}} x_{i 2} - \dots - \hat{β_{p}} x_{i p})^{2}

Multiple Linear Regression

Multiple Linear Regression

Estimating Coefficients

Assess Coefficient Accuracy

Assess Model Accuracy

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