Multiple Regression

Prediction

One common application of multiple regression is for predicting values of a particular dependent [Page 845]variable based on knowledge of its association with certain independent variables. In this context, the independent variables are commonly referred to as predictor variables and the dependent variable is characterized as the criterion variable. In applied settings, it is often desirable for one to be able to predict a score on a criterion variable by using information that is available in certain predictor variables. For example, in the life insurance industry, actuarial scientists use complex regression models to predict, on the basis of certain predictor variables, how long a person will live. In scholastic settings, college and university admissions offices will use predictors such as high school grade point average (GPA) and ACT scores to predict an applicant's college GPA, even before he or she has entered the university.

Multiple regression is most commonly used to predict values of a criterion variable based on linear associations with predictor variables. A brief example using simple regression easily illustrates how this works. Assume that a horticulturist developed a new hybrid maple tree that grows exactly 2 feet for every year that it is alive. If the height of the tree was the criterion variable and the age of the tree was the predictor variable, one could accurately describe the relationship between the age and height of the tree with the formula for a straight line, which is also the formula for a simple regression equation:

where Y is the value of the dependent, or criterion, variable; X is the value of the independent, or predictor, variable; b is a regression coefficient that describes the slope of the line; and a is the Y intercept. The Y intercept is the value of Y when X is 0. Returning to the hybrid tree example, the exact relationship between the tree's age and height could be described as follows:

Notice that the Y intercept is 0 in this case because at 0 years of age, the tree has 0 height. At that point, it is just seed in the ground. It is clear how knowledge of the relationship between the tree's age and height could be used to easily predict the height of any given tree by just knowing what its age is. A 5-year-old tree will be 10 feet tall, an 8-year-old tree will be 16 feet tall, and so on.

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