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Chapter 6 Prediction, Residuals, Influence. Some remarks: Residual = Observed Y – Predicted Y Residuals are errors. Chapter 6 Prediction, Residuals, Influence. Example: X: Age in months Y: Height in inches X: 18 19 20 21 22 23 24
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Chapter 6 Prediction, Residuals, Influence • Some remarks: • Residual = Observed Y – Predicted Y • Residuals are errors.
Chapter 6 Prediction, Residuals, Influence Example: • X: Age in months • Y: Height in inches • X: 18 19 20 21 22 23 24 • Y: 29.9 30.3 30.7 31 31.38 31.45 31.9
Chapter 6 Prediction, Residuals, Influence • Linear Model: Height = 25.2 +.271 * Age Examples • Age = 24 months, Observed Height = 31.9 • Predicted Height = 31.704 • Residual = 31.9 – 31.704 = .196
Chapter 6 Prediction, Residuals, Influence • Age = 30 years months • Predicted Height ~ 10 ft!! • Residual = BIG! • Be aware of Extrapolation!
Chapter 7 Correlation and Coefficient of Determination How strong is the linear relationship between two quantitative variables X and Y?
Chapter 7 Correlation and Coefficient of Determination • Answer: • Use scatterplots • Compute the correlation coefficient, r. • Compute the coefficient of determination, r^2.
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
Chapter 7 Correlation and Coefficient of Determination • r is an indicator of the strength of linear relationship between X and Y • strong linearrelationship for r between .8 and 1 and -.8 and -1: • moderate linearrelationship for r between .5 and .8 and -.5 and -.8: • weak linearrelationship 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.
Chapter 7 Correlation and Coefficient of Determination • How do we compute r? • r = Sxy/(Sqrt(Sxx)*Sqrt(Syy)) • Example: • X: 6 10 14 19 21 • Y: 5 3 7 8 12 • Compute: • Sxy = 72, Sxx = 154 and Syy = 46 • Hence r = 72/(Sqrt(154)*Sqrt(46)) = .855
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
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.
