1. Regression The basic problem Regression and Correlation
Accuracy of prediction in regression
Hypothesis testing
Regression with multiple predictors

2. The Basic Problem How do we predict one variable from another?
How does one variable change as the other changes?
Cause and effect (can only be inferred if it makes theoretical sense)

3. An Example
Effect of Dow Jones Performance on Darts performance (to what degree can Dow Jones predict Dart performance)
Landwehr, J.M. & Watkins, A.E. (1987) Exploring Data: Teacher’s Edition. Palo Alto, CA: Dale Seymour Publications.

4. The Data

5. Relationship can be represented by line of best fit

6. Why use regression? We may want to make a prediction.
More likely, we want to understand the relationship.
How fast does Darts rise with one unit rise in Dow Jones?

7. Regression Line Formula = the predicted value of Y (Darts)
X = Dow value

8. Regression Coefficients
“Coefficients” are a and b
b = slope (also called rate of change)
Change in predicted Y for one unit change in X
a = intercept
value of when X = 0

9. Calculation
Slope
Intercept

10. For Our Data
b = 11.13/5.43 = 2.04
a = *5.95 = 2.37
See SPSS printout on next slide

11. SPSS Printout for one Predictor
R2, Percentage of Variance

12. SPSS printout cont. Is regression Significant? Intercept Slope
Error of prediction
Is regression
Significant?
Intercept
Slope

13. Note: The values we obtained are shown on printout.
The intercept is labeled “constant.”
Slope is labeled by name of predictor variable.

14. Making a Prediction
Suppose that we want to predict Darts score for a new Dow Score of 200
We predict that Darts will be at when Dow is at 25
Check with data: what is real value of Darts when Dow is 25

15. Prediction
Residual

16. Errors of Prediction Residual variance Standard error of estimate
The variability of predicted values
Standard error of estimate
The standard deviation of predicted values

17. Standard Error of Estimate
A common measure of the accuracy of our predictions
We want it to be as small as possible.

18. r 2 as % Predictable Variability
Define Sum of Squares

19. Major Points
Predicting one dependent variable from multiple predictor variables
Example with Product Advisor Data
Multiple correlation
Regression equation
Predictions

20. The Problem
In the product advisor study, we asked participants to rate the system on a number of aspects: e.g, usefulness, ease of use, trust, kind of product information, number of ratings etc.
Lets think of overall usefulness as our dependent variable. Which of the above factors can predict overall usefulness?
What percentage variance do they explain in the usefulness overall?
What factors play the more important role?

21. Product Advisor Data
Kliewer, W., Lepore, S.J., Oskin, D., & Johnson, P.D. (1998) The role of social and cognitive processes in children’s adjustment to community violence. Journal of Consulting and Clinical Psychology, 66,

22. Correlational Matrix

23. Regression Results (using simple linear regression using method “enter”
R2, Percentage of Variance

24. Regression is significant
Importance of each variable
Is contribution significant?

25. Regression Coefficients
Slopes and an intercept.
Each variable adjusted for all others in the model.
Just an extension of slope and intercept in simple regression
SPSS output on next slide

26. Regression Equation A separate coefficient for each variable
These are slopes
An intercept (here called b0 instead of a)