Statistics as EstimatorsPearson's Sample Correlation CoefficientSpearman's Rank Correlation CoefficientThe Regression FunctionOutlineStatistics as EstimatorsLecture 18Chapter 6: Empirical StatisticsM. George AkritasM. George Akritas Lecture 18 Chapter 6: Empirical StatisticsOutlineStatistics as EstimatorsStatistics as EstimatorsPearson’s Sample Correlation CoefficientSpearman’s Rank Correlation CoefficientThe Regression FunctionM. George Akritas Lecture 18 Chapter 6: Empirical StatisticsOutlineStatistics as EstimatorsPearson’s Sample Correlation CoefficientSpearman’s Rank Correlation CoefficientThe Regression FunctionDefinition (Sample Covariance and Pearson’s Correlation)Let (X1, Y1), . . . , (Xn, Yn) be a sample from a bivariate population.IThe sample covariance, is defined asbσX ,Y=1n − 1nXi =1Xi− XYi− Y=1n − 1"nXi =1XiYi−1n nXi =1Xi! nXi =1Yi!#,IThe Pearson’s sample correlation coefficient is defined asbρX ,Y=bσX ,YSXSY, where SXand SYarethe sample standard deviations of the X - and Y -samples.M. George Akritas Lecture 18 Chapter 6: Empirical StatisticsOutlineStatistics as EstimatorsPearson’s Sample Correlation CoefficientSpearman’s Rank Correlation CoefficientThe Regression FunctionIbρX ,Yhas the same properties as its population counterpart:1. If ac > 0, then bρaX +b ,cY +d= bρX ,Y.2. −1 ≤ bρ(X , Y ) ≤ 1.3. bρXY= 1 or −1 if and only if Yi= aXi+ b, i = 1, . . . , n, forsome constants a, b.M. George Akritas Lecture 18 Chapter 6: Empirical StatisticsOutlineStatistics as EstimatorsPearson’s Sample Correlation CoefficientSpearman’s Rank Correlation CoefficientThe Regression FunctionExampleFind Pearson’s sample correlation coefficient from the n = 10 pairsof (X , Y )-values:X 4.97 2.18 3.48 2.29 1.74 4.07 2.30 3.60 2.27 0.72Y 95.2 0.52 21.5 0.73 0.14 75.9 0.78 28.3 0.66 0.01Solution: Here,PXi= 27.62,PYi= 223.74,PXiYi= 965.105,and so the sample covariance isbσX ,Y=19965.105 −110(27.62)(223.74)= 38.5706.Also, SX= 1.249, SY= 35.1.Thus, Pearson’s sample correlationisρX ,Y=38.5706(1.249)(35.1)= 0.88.M. George Akritas Lecture 18 Chapter 6: Empirical StatisticsOutlineStatistics as EstimatorsPearson’s Sample Correlation CoefficientSpearman’s Rank Correlation CoefficientThe Regression FunctionExampleFind Pearson’s sample correlation coefficient from the n = 10 pairsof (X , Y )-values:X 4.97 2.18 3.48 2.29 1.74 4.07 2.30 3.60 2.27 0.72Y 95.2 0.52 21.5 0.73 0.14 75.9 0.78 28.3 0.66 0.01Solution: Here,PXi= 27.62,PYi= 223.74,PXiYi= 965.105,and so the sample covariance isbσX ,Y=19965.105 −110(27.62)(223.74)= 38.5706.Also, SX= 1.249, SY= 35.1.Thus, Pearson’s sample correlationisρX ,Y=38.5706(1.249)(35.1)= 0.88.M. George Akritas Lecture 18 Chapter 6: Empirical StatisticsOutlineStatistics as EstimatorsPearson’s Sample Correlation CoefficientSpearman’s Rank Correlation CoefficientThe Regression FunctionExampleFind Pearson’s sample correlation coefficient from the n = 10 pairsof (X , Y )-values:X 4.97 2.18 3.48 2.29 1.74 4.07 2.30 3.60 2.27 0.72Y 95.2 0.52 21.5 0.73 0.14 75.9 0.78 28.3 0.66 0.01Solution: Here,PXi= 27.62,PYi= 223.74,PXiYi= 965.105,and so the sample covariance isbσX ,Y=19965.105 −110(27.62)(223.74)= 38.5706.Also, SX= 1.249, SY= 35.1.Thus, Pearson’s sample correlationisρX ,Y=38.5706(1.249)(35.1)= 0.88.M. George Akritas Lecture 18 Chapter 6: Empirical StatisticsOutlineStatistics as EstimatorsPearson’s Sample Correlation CoefficientSpearman’s Rank Correlation CoefficientThe Regression FunctionExampleFind Pearson’s sample correlation coefficient from the n = 10 pairsof (X , Y )-values:X 4.97 2.18 3.48 2.29 1.74 4.07 2.30 3.60 2.27 0.72Y 95.2 0.52 21.5 0.73 0.14 75.9 0.78 28.3 0.66 0.01Solution: Here,PXi= 27.62,PYi= 223.74,PXiYi= 965.105,and so the sample covariance isbσX ,Y=19965.105 −110(27.62)(223.74)= 38.5706.Also, SX= 1.249, SY= 35.1.Thus, Pearson’s sample correlationisρX ,Y=38.5706(1.249)(35.1)= 0.88.M. George Akritas Lecture 18 Chapter 6: Empirical StatisticsOutlineStatistics as EstimatorsPearson’s Sample Correlation CoefficientSpearman’s Rank Correlation CoefficientThe Regression FunctionExampleFind Pearson’s sample correlation coefficient from the n = 10 pairsof (X , Y )-values:X 4.97 2.18 3.48 2.29 1.74 4.07 2.30 3.60 2.27 0.72Y 95.2 0.52 21.5 0.73 0.14 75.9 0.78 28.3 0.66 0.01Solution: Here,PXi= 27.62,PYi= 223.74,PXiYi= 965.105,and so the sample covariance isbσX ,Y=19965.105 −110(27.62)(223.74)= 38.5706.Also, SX= 1.249, SY= 35.1. Thus, Pearson’s sample correlationisρX ,Y=38.5706(1.249)(35.1)= 0.88.M. George Akritas Lecture 18 Chapter 6: Empirical StatisticsOutlineStatistics as EstimatorsPearson’s Sample Correlation CoefficientSpearman’s Rank Correlation CoefficientThe Regression Function• Given a sample X1, . . . , Xnthe rank of Xiis the number ofobservations that are less than or equal to it.• Thus, the smallest observation has rank 1 while the largest hasrank n.DefinitionLet (X1, Y1), . . . , (Xn, Yn) be a sample from a bivariate population.IDenote the ranks of X1, X2, . . . , Xnand of Y1, Y2, . . . , Ynby RX1, RX2, . . . , RXnand RY1, RY2, . . . , RYn, respectively.ISpearman’s rank correlation coefficient is Pearson’s linearcorrelation coefficient computed on the pairs of ranks(RX1, RY1), . . . , (RXn, RYn).M. George Akritas Lecture 18 Chapter 6: Empirical StatisticsOutlineStatistics as EstimatorsPearson’s Sample Correlation CoefficientSpearman’s Rank Correlation CoefficientThe Regression Function• Given a sample X1, . . . , Xnthe rank of Xiis the number ofobservations that are less than or equal to it.• Thus, the smallest observation has rank 1 while the largest hasrank n.DefinitionLet (X1, Y1), . . . , (Xn, Yn) be a sample from a bivariate population.IDenote the ranks of X1, X2, . . . , Xnand of Y1, Y2, . . . , Ynby RX1, RX2, . . . , RXnand RY1, RY2, . . . , RYn, respectively.ISpearman’s rank correlation coefficient is Pearson’s linearcorrelation coefficient computed on the pairs of ranks(RX1, RY1), . . . , (RXn, RYn).M. George Akritas Lecture 18 Chapter 6: Empirical StatisticsOutlineStatistics as EstimatorsPearson’s Sample Correlation CoefficientSpearman’s Rank Correlation CoefficientThe Regression Function• Given a sample X1, . . . , Xnthe rank of Xiis the number ofobservations that are less than or equal to it.• Thus, the smallest observation has rank 1 while the largest hasrank n.DefinitionLet