1. Fundamental Statistics for the Behavioral Sciences, 4th edition
Chapter 10
Regression
Fundamental Statistics for the Behavioral Sciences, 4th edition
David C. Howell
©1999 Brooks/Cole Publishing Company/ITP     

2. Major Points The basic problem An example The regression line
Chapter 10 Regression
Major Points
The basic problem
An example
The regression line
Accuracy of prediction
Hypothesis testing
Review questions

3. The Basic Problem How do we predict one variable from another?
Chapter 10 Regression
The Basic Problem
How do we predict one variable from another?
How does one variable change as the other changes?
Cause and effect

4. An Example Cigarettes and CHD Mortality from Chapter 9
Chapter 10 Regression
An Example
Cigarettes and CHD Mortality from Chapter 9
Data repeated on next slide
We want to predict level of CHD mortality in a country averaging 20 cigarettes per day.
Landwehr, J.M. & Watkins, A.E. (1987) Exploring Data: Teacher’s Edition. Palo Alto, CA: Dale Seymour Publications.

5. Chapter 10 Regression
The Data

6. Chapter 10 Regression

7. Why Do We Care? We may want to make a prediction.
Chapter 10 Regression
Why Do We Care?
We may want to make a prediction.
More likely, we want to understand the relationship.
How fast does CHD mortality rise with a one unit increase in smoking?
Note we speak about predicting, but often don’t actually predict.

8. Regression Line Formula  = the predicted value of Y (CHD mortality)
Chapter 10 Regression
Regression Line
Formula
 = the predicted value of Y (CHD mortality)
X = smoking incidence for that country

9. Regression Coefficients
Chapter 10 Regression
Regression Coefficients
“Coefficients” are a and b
b = slope
Change in predicted Y for one unit change in X
a = intercept
value of  when X = 0

10. Chapter 10 Regression
Calculation
Slope
Intercept

11. For Our Data CovXY = 11.13 s2X = 2.332 = 5.43 b = 11.13/5.43 = 2.04
Chapter 10 Regression
For Our Data
CovXY = 11.13
s2X = = 5.43
b = 11.13/5.43 = 2.04
a = *5.95 = 2.37
See SPSS printout on next slide
Answers are not exact due to rounding error and desire to match SPSS.

12. Chapter 10 Regression
SPSS Printout

13. Note: The values we obtained are shown on printout.
Chapter 10 Regression
Note:
The values we obtained are shown on printout.
The intercept is labeled “constant.”
Slope is labeled by name of predictor variable.

14. Chapter 10 Regression
Making a Prediction
Assume that we want CHD mortality when cigarette consumption of 6.
We predict people/10,000 in that country will die of coronary heart disease.

15. Accuracy of Prediction
Chapter 10 Regression
Accuracy of Prediction
Finnish smokers smoke 6 cigarettes/adult/day
We predict deaths/10,000
They actually have 23 deaths/10,000
Our error (“residual”) = = 8.39
a large error

16. Chapter 10 Regression
Residual
Prediction

17. Errors of Prediction Residual variance Standard error of estimate
Chapter 10 Regression
Errors of Prediction
Residual variance
The variability of predicted values
Standard error of estimate
The standard deviation of predicted values

18. Standard Error of Estimate
Chapter 10 Regression
Standard Error of Estimate
A common measure of the accuracy of our predictions
We want it to be as small as possible.

19. r 2 as % Predictable Variability
Chapter 10 Regression
r 2 as % Predictable Variability
Define Sum of Squares
The remaining error divided by the original error

20. Chapter 10 Regression
For Our Data
r = .713
r 2 = =.508
Approximately 50% in variability of incidence of CHD mortality is associated with variability in smoking.
Elaborate on what this means.

21. Hypothesis Testing Null hypotheses
Chapter 10 Regression
Hypothesis Testing
Null hypotheses
b* = 0
a* = 0
Define b* and a*
population correlation () = 0
We saw how to test the last one in Chapter 9.

22. Testing Slope and Intercept
Chapter 10 Regression
Testing Slope and Intercept
These are given in computer printout as a t test.

23. Chapter 10 Regression
Testing
The t values in the second from right column are tests on slope and intercept.
The associated p values are next to them.
The slope is significantly different from zero, but not the intercept.
Why do we care?
Cont.

24. Testing--cont. What does it mean if slope is not significant?
Chapter 10 Regression
Testing--cont.
What does it mean if slope is not significant?
How does that relate to test on r?
What if the intercept is not significant?
Does significant slope mean we predict quite well?

25. Review Questions How does regression differ from correlation?
Chapter 10 Regression
Review Questions
How does regression differ from correlation?
If the slope is negative, what does that tell us about the sign of r?
Why do we call Y-  a residual?
What does it mean to say that r 2 = % variation accounted for?
Cont.

26. Review Questions--cont.
Chapter 10 Regression
Review Questions--cont.
If the slope is significant, what does that tell us about the significance of r?
What does a nonsignificant slope tell us?