Multiple Regression is an extension of simple regression.  Simple regression has only one independent (explanatory) variable.  Multiple Regression fits a model for one dependent (response) variable based on more than one independent (explanatory) variables.  

An Example for Multiple Regression

Typically, in the summer time as the temperature increases people are thirstier.  In the previous simple linear regression example, we considered the two numerical variables, temperature and water consumption.  For that example, we found that temperature and water consumption are positively correlated (i.e. the higher the temperature, the more water a given person consumes) and could be modeled with linear relationship.  Though this linear relationship was strong (r= 0.963), it was not a perfect linear relationship.  Thus we might look for another variable which would be related to the thirst that occurs in the summer while people are outside.  some examples might include the activity that a person was doing while they were outside.  Perhaps they were mowing, playing sports, sitting under a shade tree and etc.   These activities could affect how thirsty a person would be.

In Multiple Linear Regression, as is the case in Simple Linear Regression, we have a linear model for the real (true) relationship between the input variable(s) and the output variable.  The difference is that with multiple regression there will be more than one input variable. The true model contains the random error term and the true coefficients for the input variables.   We also have a prediction (estimation) model which has the estimated coefficients for the input variables and does not contain an error term.  Each of these models are shown below.

General Form for Multiple Regression

Actual (true) Model

Y is the dependent variable		xi = independent (explanatory) variables

Prediction (estimated) Model

		xi = independent (explanatory) variables

Visualization of the Multiple Regression Model

It is more difficult to visualize multiple regression than simple regression.  In the case where you have two independent (explanatory) variables the "line" of best fit becomes a two dimensional surface (i.e. a plane).  

 3-D Visualization

General Goal of Multiple Regression

You will have to determine which independent (explanatory) variables should be included in the model. This is not always a trivial task and can sometimes be quite complicated.  A mix of theory, understanding of the nature of the problem and statistical diagnostics will help you. 

Next we need to examine the Multiple Regression Model Assumptions 

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