The hypothesis of independence. Let (X1, Y1), . . . , (X N , YN ) be a sample from a bivariate distribution, and (X(1), Z1), . . . , (X(N), ZN ) be the same sample arranged according to increasing values of the X’s so that the Z’s are a permutation of the Y ’s. Let Ri be the rank of Xi among the X’s, Si the rank of Yi among the Y ’s, and Ti the rank of Zi among the Z’s, and consider the hypothesis of independence of X and Y against the alternatives of positive regression dependence.

(i) Conditionally, given (X(1),..., X(N)), this problem is equivalent to testing the hypothesis of randomness of the Z’s against the alternatives of an upward trend.

(ii) The test (6.73) is equivalent to rejecting when the rank correlation coefficient

(Ri − R¯)(Si − S¯)

(Ri − R¯ 2)

(Si − S¯)2

= 12 N3 − N



Ri − N + 1 2

Si − N + 1 2

is too large.

(iii) An alternative expression for the rank correlation coefficient8 is 1 − 6 N3 − N

(Si − Ri)

2 = 1 − 6 N3 − N

(Ti − i)

2

.

(iv) The test U > C of Problem 6.63(ii) is equivalent to rejecting when Kendall’s t-statistic i< j Vi j /N(N − 1) is too large where Vi j is +1 or −1 as (Yj −

Yi)(X j − Xi) is positive or negative.

(v) The tests (ii) and (iv) are unbiased against the alternatives of positive regression dependence.

Section 6.11