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Rank correlation 8.36. Derive Spearman s rank correlation formula (36), page 271.
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Here we are considering nx values (e.g., weights) and n corresponding y values (e.g., heights). Let xj be the rank given to the jth x value, and yj the rank given to the jth y value. The ranks are the integers 1 through n. The mean of the xj is then x # while the variance is s2 x x2 # x2 # 12 n(n n2 12 using the results 1 and 2 of Appendix A. Similarly, the mean y and variance s2 are equal to (n 1)>2 and # y (n2 1)>12, respectively. Now if dj xj yj are the deviations between the ranks, the variance of the deviations, s2, is given in terms d of s2, s2 and the correlation coefficient between ranks by x y s2 d s2 x s2 y 2rranksx sy 1 22 c n n2 1)>6 a a n 2 1 2 1 b
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CHAPTER 8 Curve Fitting, Regression, and Correlation
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Then (1) # Since d (2) 0, s2 d rrank (gd2)>n and (1) becomes (n2 rrank 1)>12 (n2 (n2 1)>12 1)>6 Q a d 2 R >n 1 6 ad 2 n(n2 1) s2 x s2 y 2sx sy s2 d
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8.37. Table 8-17 shows how 10 students were ranked according to their achievements in both the laboratory and lecture portions of a biology course. Find the coefficient of rank correlation. Table 8-17 Laboratory Lecture 8 9 3 5 9 10 2 1 7 8 10 7 4 3 6 4 1 2 5 6
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The difference of ranks d in laboratory and lecture for each student is given in Table 8-18. Also given in the table are d 2 and gd 2. Table 8-18 Difference of Ranks (d ) d2 1 1 2 4 1 1 1 1 1 1 3 9 1 1 2 4 1 1 1 1 gd 2 24
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rrank
6 ad 2 n(n2 1)
6(24) 10(102 1)
indicating that there is a marked relationship between achievements in laboratory and lecture.
8.38. Calculate the coefficient of rank correlation for the data of Problem 8.11, and compare your result with the correlation coefficient obtained by other methods.
Arranged in ascending order of magnitude, the fathers heights are (1) 62, 63, 64, 65, 66, 67, 67, 68, 68, 69, 70, 71
Since the 6th and 7th places in this array represent the same height (67 inches), we assign a mean rank 6.5 to both these places. Similarly, the 8th and 9th places are assigned the rank 8.5. Therefore, the fathers heights are assigned the ranks (2) 1, 2, 3, 4, 5, 6.5, 6.5, 8.5, 8.5, 10, 11, 12 Similarly, the sons heights arranged in ascending order of magnitude are (3) 65, 65, 66, 66, 67, 68, 68, 68, 68, 69, 70, 71
and since the 6th, 7th, 8th, and 9th places represent the same height (68 inches), we assign the mean rank 7.5 (6 7 8 9)>4 to these places. Therefore, the sons heights are assigned the ranks (4) 1.5, 1.5, 3.5, 3.5, 5, 7.5, 7.5, 7.5, 7.5, 10, 11, 12 Using the correspondences (1) and (2), (3) and (4), Table 8-3 becomes Table 8-19. Table 8-19 Rank of Father Rank of Son 4 7.5 2 3.5 6.5 7.5 3 1.5 8.5 10 1 3.5 11 7.5 5 1.5 8.5 12 6.5 5 10 7.5 12 11
CHAPTER 8 Curve Fitting, Regression, and Correlation
The differences in ranks d, and the computations of d2 and gd2 are shown in Table 8-20.
Table 8-20
d d2 3.5 12.25 1.5 2.25 1.0 1.00 1.5 2.25 1.5 2.25 2.5 6.25 3.5 3.5 3.5 1.5 2.5 1.0 gd 2 72.50
12.25 12.25 12.25 2.25 6.25 1.00 6(72.50) 12(122 1)
Then
rrank
6 a d2 n(n2 1)
which agrees well with the value r
0.7027 obtained in Problem 8.26(b).
Probability interpretation of regression and correlation 8.39. Derive (39) from (37).
Assume that the regression equation is y E(Y Z X x) a bx
For the least-squares regression line we must consider E5[Y (a bX)]26 E5[(Y E[(Y s2 Y mY) mY)2] b2s2 X b(X 2bsXY mX) (mY (mY mX)2] bmX bmX 2bE[(X a)2 a)]26 mX)(Y mY)] (mY bmX a)2
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