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Chapter 6 Prediction, Residuals, Influence

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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

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21 Chapter 7 Correlation and Coefficient of Determination

22 Chapter 7 Correlation and Coefficient of Determination


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