Lecture 18 Chapter 6: Empirical Statistics

Transcription

1 Lecture 18 Chapter 6: Empirical Statistics M. George Akritas

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3 Definition (Sample Covariance and Pearson s Correlation) Let (X 1, Y 1 ),..., (X n, Y n ) be a sample from a bivariate population. The sample covariance, is defined as σ X,Y = = 1 n 1 1 n 1 n ( Xi X ) ( Y i Y ) i=1 [ n ( n ( n )] X i Y i 1 X i) Y i, n i=1 i=1 i=1 The Pearson s sample correlation coefficient is defined as ρ X,Y = σ X,Y S X S Y, where S X and S Y are the sample standard deviations of the X - and Y -samples.

4 ρ X,Y has the same properties as its population counterpart: 1. If ac > 0, then ρ ax +b,cy +d = ρ X,Y ρ(x, Y ) ρ XY = 1 or 1 if and only if Y i = ax i + b, i = 1,..., n, for some constants a, b.

5 Example Outline Find Pearson s sample correlation coefficient from the n = 10 pairs of (X, Y )-values: X Y Solution: Here, X i = 27.62, Y i = , X i Y i = , Also, S X = 1.249, S Y = 35.1.

6 Example Outline Find Pearson s sample correlation coefficient from the n = 10 pairs of (X, Y )-values: X Y Solution: Here, X i = 27.62, Y i = , X i Y i = , Also, S X = 1.249, S Y = 35.1.

7 Example Outline Find Pearson s sample correlation coefficient from the n = 10 pairs of (X, Y )-values: X Y Solution: Here, X i = 27.62, Y i = , X i Y i = , and so the sample covariance is σ X,Y = 1 [ ] 9 (27.62)(223.74) = Also, S X = 1.249, S Y = 35.1.

8 Example Outline Find Pearson s sample correlation coefficient from the n = 10 pairs of (X, Y )-values: X Y Solution: Here, X i = 27.62, Y i = , X i Y i = , and so the sample covariance is σ X,Y = 1 [ ] 9 (27.62)(223.74) = Also, S X = 1.249, S Y = 35.1.

9 Example Outline Find Pearson s sample correlation coefficient from the n = 10 pairs of (X, Y )-values: X Y Solution: Here, X i = 27.62, Y i = , X i Y i = , and so the sample covariance is σ X,Y = 1 [ ] 9 (27.62)(223.74) = Also, S X = 1.249, S Y = Thus, Pearson s sample correlation is ρ X,Y = (1.249)(35.1) = 0.88.

10 Given a sample X 1,..., X n the rank of X i 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 (X 1, Y 1 ),..., (X n, Y n ) be a sample from a bivariate population. Denote the ranks of X 1, X 2,..., X n by R X 1, R X 2,..., R X n and R Y 1, R Y 2,..., R Y n, respectively. Spearman s rank correlation coefficient is Pearson s linear correlation coefficient computed on the pairs of ranks (R X 1, R Y 1 ),..., (R X n, R Y n ).

11 Given a sample X 1,..., X n the rank of X i 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 (X 1, Y 1 ),..., (X n, Y n ) be a sample from a bivariate population. Denote the ranks of X 1, X 2,..., X n by R X 1, R X 2,..., R X n and R Y 1, R Y 2,..., R Y n, respectively. Spearman s rank correlation coefficient is Pearson s linear correlation coefficient computed on the pairs of ranks (R X 1, R Y 1 ),..., (R X n, R Y n ).

12 Given a sample X 1,..., X n the rank of X i 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 (X 1, Y 1 ),..., (X n, Y n ) be a sample from a bivariate population. Denote the ranks of X 1, X 2,..., X n by R X 1, R X 2,..., R X n and R Y 1, R Y 2,..., R Y n, respectively. Spearman s rank correlation coefficient is Pearson s linear correlation coefficient computed on the pairs of ranks (R X 1, R Y 1 ),..., (R X n, R Y n ).

13 Given a sample X 1,..., X n the rank of X i 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 (X 1, Y 1 ),..., (X n, Y n ) be a sample from a bivariate population. Denote the ranks of X 1, X 2,..., X n by R X 1, R X 2,..., R X n and R Y 1, R Y 2,..., R Y n, respectively. Spearman s rank correlation coefficient is Pearson s linear correlation coefficient computed on the pairs of ranks (R X 1, R Y 1 ),..., (R X n, R Y n ).

14 Given a sample X 1,..., X n the rank of X i 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 (X 1, Y 1 ),..., (X n, Y n ) be a sample from a bivariate population. Denote the ranks of X 1, X 2,..., X n by R X 1, R X 2,..., R X n and R Y 1, R Y 2,..., R Y n, respectively. Spearman s rank correlation coefficient is Pearson s linear correlation coefficient computed on the pairs of ranks (R X 1, R Y 1 ),..., (R X n, R Y n ).

15 Given a sample X 1,..., X n the rank of X i 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 (X 1, Y 1 ),..., (X n, Y n ) be a sample from a bivariate population. Denote the ranks of X 1, X 2,..., X n and of Y 1, Y 2,..., Y n by R X 1, R X 2,..., R X n and R Y 1, R Y 2,..., R Y n, respectively. Spearman s rank correlation coefficient is Pearson s linear correlation coefficient computed on the pairs of ranks (R X 1, R Y 1 ),..., (R X n, R Y n ).

16 Example Find the rank correlation coefficient for the (X, Y )-values given in the previous example: X Y Solution: The pairs of ranks (R X 1, RY 1 ),..., (RX n, R Y n ) are: Ranks of X-values Ranks of Y-values Pearson s correlation coefficient on the ranks, which is Spearman s rank correlation coefficient, is 1.

17 Figure: Scatter Plot of Example Data

18 Let (X, Y ) be a bivariate random variable and suppose that We have seen that E(Y X = x) = α 1 + β 1 x. β 1 = Cov(X, Y ), and α 1 = E(Y ) β 1 E(X ). Var(X ) Thus, given a s.r. sample (X 1, Y 1 ),..., (X n, Y n ), the intercept and slope of the regression line can be estimated by β 1 = Ĉov(X, Y ) SX 2, and α 1 = Y β 1 X µ Y X (x) = α 1 + β 1 x is called the estimated regression line

19 The computational formula for β 1 is β 1 = n X i Y i ( X i )( Y i ) n Xi 2 ( X i ) 2. Example Suppose that n = 10 data points on X =stress applied and Y =time to failure yield X i = 200, Xi 2 = , Yi = 484, X i Y i = Then, β 1 = 10(8407.5) (200)(484) 10(5412.5) (200) 2 = , α 1 = 1 (484) ( ) =

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