# 14.8: Introduction to Multiple Regression

Skills to Develop

• State the regression equation
• Define "regression coefficient"
• Define "beta weight"
• Explain what $$R$$ is and how it is related to $$r$$
• Explain why a regression weight is called a "partial slope"
• Explain why the sum of squares explained in a multiple regression model is usually less than the sum of the sums of squares in simple regression
• Define $$R^2$$ in terms of proportion explained
• Test $$R^2$$ for significance
• Test the difference between a complete and reduced model for significance
• State the assumptions of multiple regression and specify which aspects of the analysis require assumptions

In simple linear regression, a criterion variable is predicted from one predictor variable. In multiple regression, the criterion is predicted by two or more variables. For example, in the SAT case study, you might want to predict a student's university grade point average on the basis of their High-School GPA (HSGPA) and their total SAT score (verbal + math). The basic idea is to find a linear combination of HSGPA and SAT that best predicts University GPA (UGPA). That is, the problem is to find the values of $$b_1$$ and $$b_2$$ in the equation shown below that give the best predictions of UGPA. As in the case of simple linear regression, we define the best predictions as the predictions that minimize the squared errors of prediction.