Outline Statistics as Estimators Lecture 18 Chapter 6 Empirical Statistics M George Akritas M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators Statistics as Estimators Pearson s Sample Correlation Coefficient Spearman s Rank Correlation Coefficient The Regression Function M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators Pearson s Sample Correlation Coefficient Spearman s Rank Correlation Coefficient The Regression Function Definition Sample Covariance and Pearson s Correlation Let X1 Y1 Xn Yn be a sample from a bivariate population I The sample covariance is defined as bX Y n 1 X Xi X Yi Y n 1 i 1 n n n X X 1 1 X Xi Yi Xi Yi n 1 n i 1 I i 1 i 1 The Pearson s sample correlation coefficient is defined as bX Y bX Y where SX and SY are SX SY the sample standard deviations of the X and Y samples M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators I Pearson s Sample Correlation Coefficient Spearman s Rank Correlation Coefficient The Regression Function bX Y has the same properties as its population counterpart 1 If ac 0 then baX b cY d bX Y 2 1 b X Y 1 3 bXY 1 or 1 if and only if Yi aXi b i 1 n for some constants a b M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators Pearson s Sample Correlation Coefficient Spearman s Rank Correlation Coefficient The Regression Function Example Find Pearson s sample correlation coefficient from the n 10 pairs of X Y values X Y 4 97 2 18 3 48 2 29 1 74 4 07 2 30 3 60 2 27 0 72 95 2 0 52 21 5 0 73 0 14 75 9 0 78 28 3 0 66 0 01 P P P Solution Here Xi 27 62 Yi 223 74 Xi Yi 965 105 Also SX 1 249 SY 35 1 M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators Pearson s Sample Correlation Coefficient Spearman s Rank Correlation Coefficient The Regression Function Example Find Pearson s sample correlation coefficient from the n 10 pairs of X Y values X Y 4 97 2 18 3 48 2 29 1 74 4 07 2 30 3 60 2 27 0 72 95 2 0 52 21 5 0 73 0 14 75 9 0 78 28 3 0 66 0 01 P P P Solution Here Xi 27 62 Yi 223 74 Xi Yi 965 105 Also SX 1 249 SY 35 1 M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators Pearson s Sample Correlation Coefficient Spearman s Rank Correlation Coefficient The Regression Function Example Find Pearson s sample correlation coefficient from the n 10 pairs of X Y values X Y 4 97 2 18 3 48 2 29 1 74 4 07 2 30 3 60 2 27 0 72 95 2 0 52 21 5 0 73 0 14 75 9 0 78 28 3 0 66 0 01 P P P Solution Here Xi 27 62 Yi 223 74 Xi Yi 965 105 and so the sample covariance is 1 1 bX Y 965 105 27 62 223 74 38 5706 9 10 Also SX 1 249 SY 35 1 M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators Pearson s Sample Correlation Coefficient Spearman s Rank Correlation Coefficient The Regression Function Example Find Pearson s sample correlation coefficient from the n 10 pairs of X Y values X Y 4 97 2 18 3 48 2 29 1 74 4 07 2 30 3 60 2 27 0 72 95 2 0 52 21 5 0 73 0 14 75 9 0 78 28 3 0 66 0 01 P P P Solution Here Xi 27 62 Yi 223 74 Xi Yi 965 105 and so the sample covariance is 1 1 bX Y 965 105 27 62 223 74 38 5706 9 10 Also SX 1 249 SY 35 1 M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators Pearson s Sample Correlation Coefficient Spearman s Rank Correlation Coefficient The Regression Function Example Find Pearson s sample correlation coefficient from the n 10 pairs of X Y values X Y 4 97 2 18 3 48 2 29 1 74 4 07 2 30 3 60 2 27 0 72 95 2 0 52 21 5 0 73 0 14 75 9 0 78 28 3 0 66 0 01 P P P Solution Here Xi 27 62 Yi 223 74 Xi Yi 965 105 and so the sample covariance is 1 1 bX Y 965 105 27 62 223 74 38 5706 9 10 Also SX 1 249 SY 35 1 Thus Pearson s sample correlation is 38 5706 X Y 0 88 1 249 35 1 M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators Pearson s Sample Correlation Coefficient Spearman s Rank Correlation Coefficient The Regression Function Given a sample X1 Xn the rank of Xi is the number of observations that are less than or equal to it Thus the smallest observation has rank 1 while the largest has rank n Definition Let X1 Y1 Xn Yn be a sample from a bivariate population I Denote the ranks of X1 X2 Xn by I R1X R2X RnX and R1Y R2Y RnY respectively Spearman s rank correlation coefficient is Pearson s linear correlation coefficient computed on the pairs of ranks R1X R1Y RnX RnY M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators Pearson s Sample Correlation Coefficient Spearman s Rank Correlation Coefficient The Regression Function Given a sample X1 Xn the rank of Xi is the number of observations that are less than or equal to it Thus the smallest observation has rank 1 while the largest has rank n Definition Let X1 Y1 Xn Yn be a sample from a bivariate population I Denote the ranks of X1 X2 Xn by I R1X R2X RnX and R1Y R2Y RnY respectively Spearman s rank correlation coefficient is Pearson s linear correlation coefficient computed on the pairs of ranks R1X R1Y RnX RnY M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators Pearson s Sample Correlation Coefficient Spearman s Rank Correlation Coefficient The Regression Function Given a sample X1 Xn the rank of Xi is the number of observations that are less than or equal to it Thus the smallest observation has rank 1 while the largest has rank n Definition Let X1 Y1 Xn Yn be a sample from a bivariate population I Denote the ranks of X1 X2 Xn by I R1X R2X RnX and R1Y R2Y RnY respectively Spearman s rank correlation coefficient is Pearson s linear correlation coefficient computed on the pairs of ranks R1X R1Y RnX RnY M George Akritas Lecture 18 Chapter 6 Empirical Statistics Outline Statistics as Estimators Pearson s Sample Correlation Coefficient Spearman s Rank Correlation …