Multiple Regression

David M. Lane

Prerequisites

Simple Linear Regression, Paritioning Sums of Squares, Standard Error of the Estimate, Inferential Statistics for b and r

Learning Objectives

In simple linear regression, a crierion variable is predicted from one predictor variable. In multiple regression, the crierion is predicted by two or more variables. For example, in the SAT case study, you might want to predict a student's 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 b1 and b2 in the equation shown below that gives the best predictions of GPA. As in the case of simple linear regression, we define the best predictions as the predictions that minimize the squared errors of prediction.

UGPA' = b1HSGPA + b2SAT + A

where UGPA' is the predicted value of University GPA and A is a constant. For these data, the best prediction equation is shown below.

UGPA' = 0.541 x HSGPA + 0.008 x SAT + 0.540

In other words, to compute the prediction of a student's University GPA, you add up (a) their High School GPA multiplied by 0.541, (b) their SAT multiplied by 0.008, and (c) 0.540. Table 1 shows the data and predictions for the first five students in the dataset.

 HSGPA UGPA SAT UGPA' 3.45 3.52 1232 3.38 2.78 2.91 1070 2.89 2.52 2.4 1086 2.76 3.67 3.47 1287 3.55 3.24 3.47 1130 3.19