1 Chapter 17: Introduction to Regression

2 Introduction to Linear Regression The Pearson correlation measures the degree to which a set of data points form a straight line relationship. Regression is a statistical procedure that determines the equation for the straight line that best fits a specific set of data.

3 Introduction to Linear Regression (cont.) Any straight line can be represented by an equation of the form Y = bX + a, where b and a are constants. The value of b is called the slope constant and determines the direction and degree to which the line is tilted. The value of a is called the Y-intercept and determines the point where the line crosses the Y-axis.

5 Introduction to Linear Regression (cont.) How well a set of data points fits a straight line can be measured by calculating the distance between the data points and the line. The total error between the data points and the line is obtained by squaring each distance and then summing the squared values. The regression equation is designed to produce the minimum sum of squared errors.

6 Introduction to Linear Regression (cont.) The equation for the regression line is

8 Introduction to Linear Regression (cont.) The ability of the regression equation to accurately predict the Y values is measured by first computing the proportion of the Y-score variability that is predicted by the regression equation and the proportion that is not predicted.

9 Introduction to Linear Regression (cont.) The unpredicted variability can be used to compute the standard error of estimate which is a measure of the average distance between the actual Y values and the predicted Y values.

10 Introduction to Linear Regression (cont.) Finally, the overall significance of the regression equation can be evaluated by computing an F- ratio. A significant F-ratio indicates that the equation predicts a significant portion of the variability in the Y scores (more than would be expected by chance alone). To compute the F-ratio, you first calculate a variance or MS for the predicted variability and for the unpredicted variability:

11 Introduction to Linear Regression (cont.)

13 Introduction to Multiple Regression with Two Predictor Variables In the same way that linear regression produces an equation that uses values of X to predict values of Y, multiple regression produces an equation that uses two different variables (X 1 and X 2 ) to predict values of Y. The equation is determined by a least squared error solution that minimizes the squared distances between the actual Y values and the predicted Y values.

15 Introduction to Multiple Regression with Two Predictor Variables (cont.) For two predictor variables, the general form of the multiple regression equation is: Ŷ= b 1 X 1 + b 2 X 2 + a The ability of the multiple regression equation to accurately predict the Y values is measured by first computing the proportion of the Y-score variability that is predicted by the regression equation and the proportion that is not predicted.

16 Introduction to Multiple Regression with Two Predictor Variables (cont.) As with linear regression, the unpredicted variability (SS and df) can be used to compute a standard error of estimate that measures the standard distance between the actual Y values and the predicted values.

17 Introduction to Multiple Regression with Two Predictor Variables (cont.) In addition, the overall significance of the multiple regression equation can be evaluated with an F-ratio:

18 Partial Correlation A partial correlation measures the relationship between two variables (X and Y) while eliminating the influence of a third variable (Z). Partial correlations are used to reveal the real, underlying relationship between two variables when researchers suspect that the apparent relation may be distorted by a third variable.

19 Partial Correlation (cont.) For example, there probably is no underlying relationship between weight and mathematics skill for elementary school children. However, both of these variables are positively related to age: Older children weigh more and, because they have spent more years in school, have higher mathematics skills.

20 Partial Correlation (cont.) As a result, weight and mathematics skill will show a positive correlation for a sample of children that includes several different ages. A partial correlation between weight and mathematics skill, holding age constant, would eliminate the influence of age and show the true correlation which is near zero.