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Multiple Regression
Introduction Simple Linear Regression (SLR) fits a “straight line” through data points. Your dependent variable may or may not have had a strong linear relationship with your independent variable. Now consider another type of regression in which you use more than one independent variable to predict a single dependent variable. This regression equation may even have aninteraction term that captures a relationship between two or more independent variables. You should still be able to predict population values by examining a regression equation of sample data. This procedure is known as multiple regression. Suppose we were concerned with fitting an equation to data containing one dependent variable with several different independent variables. While difficult torepresent graphically, (why?) it is still possible to perform multiple regression using Excel or other statistical packages. Consider the following problem: An economist is studying a sample of households to determine the weekly amount saved by each. Three independent variables seem to hold promise as predictors of the amount saved: Weekly income, weekly amount spent on food, and weekly amount spent onentertainment. The multiple regression equation was computed to be:

$ y = 20.00 + 0.50 x1 − 1.20 x 2 − 1.05x 3
where
$ y is the predicted weekly amount saved,

x1 is the weekly income of the household, x2 is the weekly amount spent on food, and x3 is the weekly amount spent on entertainment For any number of independent variables, n, the general multiple regression equation is:

$ y = b0 +b1 x1 + b2 x2 + b3 x3 + ... + bn x n
For each of the independent variables, an interaction term can be replaced in the general regression equation. For example, each of the following are acceptable multiple regression equations:
$ y = 13 + 2 x1 –3x2 $ y = 6 - 23x1 + 14x2 + .01x1x2 $ y = 13 + 2 x1 –3x3 $ y = 2.3 x1 – 3x2 + x3 – 1.7x1x2

Interpretation of Error and Model Evaluation in MultipleRegression As was the case with simple linear regression, multiple regression equations can refer to population data, or most likely, sample data. Error estimates and terminology apply to multiple regression as well, and Excel can calculate the values readily. No matter which type of regression is performed, the following questions should be asked to determine which equation/model is “best”.(Keep in mind, it may be impossible for all of these conditions to be satisfied. These are just general guidelines that can be used to distinguish better models from others). 1. Are all of the independent variables included? Most of the time, a model including all of the independent variables is best. Is R2 significant? This is determined by the hypothesis test using F0. Is Adjusted R2 high? This iscompared relative to other models. If examined alone, a regression equation with an Adjusted R2 value of 0.70 is considered “fairly high.” Is the Rule of Hierarchy upheld? The Rule of Hierarchy states, “...if an interaction term is used in the regression equation, then the lower order terms that were used to make up the model must also be included.” For example, if you wanted to include a “b3x1x2”term in your regression equation, you must also include an appropriate “b1x1” and “b2x2” term, even if they aren’t significant.

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5. Are all coefficients/variables significant? This is determined by a t-hypothesis test of each variable. If a coefficient of a variable is not significant, the variable should not be included in the regression equation. (Treat the "intercept" as alower order term, b0x0. Thus, unless you want the intercept to be zero because of the physical nature of the system being modeled, you would include it in your estimated regression equation, even if it is not significant.) 6. 7. Is the MSE for the model low compared to other models? Is the model simple? (Fewer variables vs R2, fewer variables vs MSE, etc.) For example, a model with two terms and an...