Multiple Linear Regression

Chapter 11: Regression Analysis: Multiple Linear Regression

Multiple Linear Regression

Regression analysis is a statistical procedure for estimating the relationship between variables. The last subchapter introduced Simple Linear Regression, which is used to predict the value of an outcome variable on the basis of a single predictor variable.

Multiple linear regression is an extension of the Simple Linear Regression model to more than one predictor variable.
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Multiple Linear Regression

Multiple Linear Regression is a statistical procedure that is used to predict the value of a continuous outcome (dependent) variable on the basis of two or more predictor (independent) variables.

The regression line of a Multiple Linear Regression with #n# predictor variables is described by the following regression equation:

\[\hat{Y} = b_0 + b_1X_1 + b_2X_2 + \ldots + b_nX_n\]

Where:

#\hat{Y}# is the predicted value of the outcome variable #Y#.
#X_1 \ldots X_n# are the predictor variables.
#b_0# is the intercept of the regression line and is often labelled the constant.
#b_1 \ldots b_n# are the partial regression coefficients.

Example: Multiple Linear Regression Equation

Consider the following regression equation that describes the relationship between an outcome variable #Y# and three predictor variables #X_1, X_2,# and #X_3#:

\[\hat{Y} = 5+2X_1-X_2+4X_3\]

From this regression equation, it follows that:

If #X_1# increases by one, #\hat{Y}# increases by #b_1=2#.
If #X_2# increases by one, #\hat{Y}# decreases by #1#, since #b_2=-1#.
If #X_3# increases by one, #\hat{Y}# increases by #b_3=4#.
If all predictor variables #X_1 \ldots X_3# are zero, then #\hat{Y} = b_0 = 5#.

So for example, if #X_1 = 1#, #X_2 = 2#, and #X_3 = 3#, then the predicted value of #Y# is:

\[\begin{array}{rcl}
\hat{Y} &=& 5+2X_1-X_2+4X_3\\
&=& 5 + 2\cdot 1 - 2 + 4\cdot3\\
&=& 17
\end{array}\]