1. LINEAR REGRESSION: What it Is and How it Works

2. Overview What is Bivariate Linear Regression? The Regression Equation How It’s Based on r Assumptions

3. What is Bivariate Linear Regression? Predict future scores on Y based on measured scores on X Predictions are based on a correlation from a sample where both X and Y were measured

4. Why is it Bivariate? Two variables: X and Y X - independent variable/predictor variable Y - dependent/outcome/criterion variable

5. Why is it Linear? Based on the linear relationship (correlation) between X and Y The relationship can be described by the equation for a straight line

6. The Regression Equation y = b 1 x i + b 0 + e i y = predicted score on criterion variable b 0 = intercept x i = measured score on predictor variable b 1 = slope e i = residual (error score)

7. Least-Squares Solution Minimize squared error in prediction. Error (residual) = difference between predicted y and actual y

8. How It’s Based on r Replace x and y with z X and z Y : z Y = b 1 z X + b o and the y-intercept becomes 0: z Y = b 1 z X and the slope becomes r: z Y = rz X

9. Assumptions for Bivariate Linear Regression Quantitative data (or dichotomous) Independent observations Predict for same population that was sampled

10. Assumptions for Bivariate Linear Regression Linear relationship –Examine scatterplot Homoscedasticity – equal spread of residuals at different values of predictor –Examine ZRESID vs ZPRED plot

11. Assumptions for Bivariate Linear Regression Independent errors –Durbin Watson should be close to 2 Normality of errors –Examine frequency distribution of residuals

12. Influential Cases Influential cases have greater impact on the slope and y-intercept Select casewise diagnostics and look for cases with large residuals

13. Take-Home Point Linear regression is a way of using information about a correlation to make predictions. The validity of linear regression depends on meeting assumptions.