Descriptive measures of the degree of linear associationSlide 2Slide 3Coefficient of determinationInterpretations of R2R-sq on Minitab fitted line plotR-sq on Minitab regression outputCorrelation coefficientCorrelation coefficient formulasInterpretation of correlation coefficientR2 = 100% and r = +1R2 = 2.9% and r = 0.17R2 = 70.1% and r = - 0.84R2 = 82.8% and r = 0.91R2 = 50.4% and r = 0.71R2 = 0% and r = 0Cautions about R2 and rSlide 18Descriptive measures of the degree of linear a s so c ia ti o nR-squared and correlation0 1 2 3 4 5 6 7 8 9 10405060xyy = 54.4758 - 0.764016 xS = 7.81137 R-Sq = 6.5 % R-Sq(adj) = 3.2 %Regression Plot 6.182721niiyySSTO 5.1708ˆ21niiiyySSE 1.119ˆ21niiyySSRyyˆ10 9 8 7 6 5 4 3 2 1 08070605040302010xyS = 7.81137 R-Sq = 79.9 % R-Sq(adj) = 79.2 %y = 75.5458 - 5.76402 xRegression Plot 3.6679ˆ21niiyySSR 5.1708ˆ21niiiyySSE 8.848721niiyySSTOyyˆCoefficient of determinationSSTOSSESSTOSSRrR 122•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 YInterpretations 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.R-sq on Minitab fitted line plot504030200150100Latitude (at center of state)MortalityS = 19.1150 R-Sq = 68.0 % R-Sq(adj) = 67.3 %Mort = 389.189 - 5.97764 LatRegression PlotR-sq on Minitab regression outputThe regression equation is Mort = 389.189 - 5.97764 Lat S = 19.1150 R-Sq = 68.0 % R-Sq(adj) = 67.3 %Analysis of VarianceSource DF SS MS F PRegression 1 36464.2 36464.2 99.7968 0.000Error 47 17173.1 365.4 Total 48 53637.3Correlation coefficient22rRr •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.Correlation coefficient formulas niniiiniiiYYXXYYXXr1 1221 11212bYYXXrniiniiInterpretation 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.R2 = 100% and r = +10 25 50 75 10020120220CelsiusFahrenheitR2 = 2.9% and r = 0.1752 57 622223242526272829303132Head circumference (in cm)Left forearm (in cm)Lengths of left forearms and head circumferencesof Spring 1998 Stat 250 Studentsn=89 studentsR2 = 70.1% and r = - 0.840 1 2 3 4 5 6 7 8 9100200300Liters of wine per person per yearNo. of Heart Disease Deaths (per 100,000)Annual Wine Consumption versus DeathU.S.NorwayFinlandItalyFranceR2 = 82.8% and r = 0.91110 120 130 140 150 160 170 180 190105115125135145155Actual weight (lbs)Ideal weight (lbs)Actual = IdealWeights of FemalesR2 = 50.4% and r = 0.71150 200 250130140150160170180190200Actual weight (lbs)Ideal weight (lbs)Actual = IdealWeights of MalesR2 = 0% and r = 0-5 0 5010203040xyA Perfect Quadratic RelationshipCautions 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.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
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