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Descriptive measures of the degree of linear association

Descriptive measures of the degree of linear association. R-squared and correlation. Coefficient of determination. R 2 is a number (a proportion!) between 0 and 1. If R 2 = 1: all data points fall perfectly on the regression line predictor X accounts for all of the variation in Y

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Descriptive measures of the degree of linear association

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  1. Descriptive measures of the degree of linearassociation R-squared and correlation

  2. Coefficient of determination • R2 is a number (a proportion!) between 0 and 1. • If R2 = 1: • all data points fall perfectly on the regression line • predictor X accounts for all of the variation in Y • If R2 = 0: • the fitted regression line is perfectly horizontal • predictor X accounts for none of the variation in Y

  3. Interpretations of R2 • R2 ×100 percent of the variation in Y is reduced by taking into account predictor X. • R2 ×100 percent of the variation in Y is “explained by” the variation in predictor X.

  4. R-sq on Minitab fitted line plot

  5. R-sq on Minitab regression output The regression equation is Mort = 389.189 - 5.97764 Lat S = 19.1150 R-Sq = 68.0 % R-Sq(adj) = 67.3 % Analysis of Variance Source DF SS MS F P Regression 1 36464.2 36464.2 99.7968 0.000 Error 47 17173.1 365.4 Total 48 53637.3

  6. Correlation coefficient • r is a number between -1 and 1, inclusive. • Sign of coefficient of correlation • plus sign if slope of fitted regression line is positive • negative sign if slope of fitted regression line is negative.

  7. Correlation coefficient formulas

  8. Interpretation of correlation coefficient • No clear-cut operational interpretation as for R-squared value. • r = -1 is perfect negative linear relationship. • r = 1 is perfect positive linear relationship. • r = 0 is no linear relationship.

  9. R2 = 100% and r = +1

  10. R2 = 2.9% and r = 0.17

  11. R2 = 70.1% and r = - 0.84 Norway Finland U.S. Italy France

  12. R2 = 82.8% and r = 0.91

  13. R2 = 50.4% and r = 0.71

  14. R2 = 0% and r = 0

  15. Cautions about R2 and r • Summary measures of linear association. Possible to get R2 = 0 with a perfect curvilinear relationship. • Large R2 does not necessarily imply that estimated regression line fits the data well. • Both measures can be greatly affected by one (outlying) data point.

  16. Cautions about R2 and r • A “statistically significant R2” does not imply that slope is meaningfully different from 0. • A large R2 does not necessarily mean that useful predictions can be made. Can still get wide intervals.

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