2. LINEAR REGRESSION: Evaluating Regression Models

3. Overview Assumptions for Linear Regression Evaluating a Regression Model

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

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

6. Checking for Homoscedasticity

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

8. Checking for Normality of Errors

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

11. Standard Error of the Estimate Index of how far off predictions are expected to be Larger r means smaller standard error Standard deviation of y scores around predicted y scores

12. Sums of Squares Total SS – total squared differences of Y scores from the mean of Y Model SS – total squared differences of predicted Y scores from the mean of Y Residual SS – total squared differences of Y scores from predicted Y scores

16. Coefficient of Determination r 2 is the proportion of variance in Y explained by X Adjusted r 2 corrects for the fact that the r 2 often overestimates the true relationship. Adjusted r 2 will be lower when there are fewer subjects.

17. Goodness of Fit Dividing the Model SS by the Total SS produces r 2 The ANOVA F-test determines whether the regression equation accounted for a significant proportion of variance in Y F is the Model Mean Square divided by the Residual Mean Square

18. Coefficients The Constant B under “unstandardized” is the y-intercept b 0 The B listed for the X variable is the slope b 1 The t test is the coefficient divided by its standard error The standardized slope is the same as the correlation

19. Example of Reporting a Regression Analysis The linear regression for predicting quiz enjoyment from level of statistics anxiety did not account of a significant portion of variance, F(1, 24) = 1.75, p =.20, r 2 =.07.

20. Take-Home Points The validity of a regression procedure depends on multiple assumptions. A regression model can be evaluated based on whether and how well it predicts an outcome variable.