1. Chapter 6 Prediction, Residuals, Influence
Some remarks:
Residual = Observed Y – Predicted Y
Residuals are errors.

2. Chapter 6 Prediction, Residuals, Influence
Example:
X: Age in months
Y: Height in inches X: Y:

3. Chapter 6 Prediction, Residuals, Influence
Linear Model: Height = * Age
Examples
Age = 24 months, Observed Height = 31.9
Predicted Height =
Residual = 31.9 – = .196

4. Chapter 6 Prediction, Residuals, Influence
Age = 30 years months
Predicted Height ~ 10 ft!!
Residual = BIG!
Be aware of Extrapolation!

5. Chapter 6 Prediction, Residuals, Influence

6. Chapter 6 Prediction, Residuals, Influence

7. Chapter 6 Prediction, Residuals, Influence

8. Chapter 7 Correlation and Coefficient of Determination
How strong is the linear relationship between two quantitative variables X and Y?

9. Chapter 7 Correlation and Coefficient of Determination
Answer:
Use scatterplots
Compute the correlation coefficient, r.
Compute the coefficient of determination, r^2.

10. Chapter 7 Correlation and Coefficient of Determination
Properties of Correlation coefficient
r is a number between -1 and 1
r = 1 or r = -1 indicates a perfect correlation case where all data points lie on a straight line
r > 0 indicates positive association
r < 0 indicates negative association
r value does not change when units of measurement are changed (correlation has no units!)
Correlation treats X and Y symmetrically. The correlation of X with Y is the same as the correlation of Y with X

11. Chapter 7 Correlation and Coefficient of Determination
r is an indicator of the strength of linear relationship between X and Y
strong linear relationship for r between .8 and 1 and -.8 and -1:
moderate linear relationship for r between .5 and .8 and -.5 and -.8:
weak linear relationship for r between .-.5 and .5
It is possible to have an r value close to 0 and a strong non-linear relationship between X and Y.
r is sensitive to outliers.

12. Chapter 7 Correlation and Coefficient of Determination

13. Chapter 7 Correlation and Coefficient of Determination
How do we compute r?
r = Sxy/(Sqrt(Sxx)*Sqrt(Syy))
Example: X: Y:
Compute:
Sxy = 72, Sxx = 154 and Syy = 46
Hence r = 72/(Sqrt(154)*Sqrt(46)) = .855

14. Chapter 7 Correlation and Coefficient of Determination
r^2: Coefficient of Determination
r^2 is between 0 and 1.
The closer r^2 is to 1, the stronger the linear relationship between X and Y
r^2 does not change when units of measurement are changed
r^2 measures the strength of linear relatioship

15. Chapter 7 Correlation and Coefficient of Determination
Some Remarks
Quantitative variable condition: Do not apply correlation to categorical variables
Correlation can be misleading if the relationship is not linear
Outliers distort correlation dramatically. Report corrlelation with/without outliers.

16. Chapter 7 Correlation and Coefficient of Determination

17. Chapter 7 Correlation and Coefficient of Determination

18. Chapter 7 Correlation and Coefficient of Determination

19. Chapter 7 Correlation and Coefficient of Determination

20. Chapter 7 Correlation and Coefficient of Determination

21. Chapter 7 Correlation and Coefficient of Determination

22. Chapter 7 Correlation and Coefficient of Determination