### 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.

#\phantom{0}#

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}\]

**Pass Your Math**independent of your university. See pricing and more.

Or visit omptest.org if jou are taking an OMPT exam.