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Last updated on Fri May 21 11:52:15 IST 2010.

Tests of association

Spearman's rank correlation coefficient

Here is another test for association using bivariate data. We have the same set up as for Kendal's τ test. We want to test
H0: X and Y are independent Vs H1: They are not independent.
Let
Ri = rank of Xi
Qi = rank of Yi

Definition Spearman's rank correlation coefficient, rR is defined as the product-moment correlation between the Ri's and the Qi's.

Example: Below we show X's, Y's , R's and Q's for a sample of size 4. 

   Xi : 1.2   2.4   -4     6
   Yi : 2   -10      1.6   5
   Ri : 2     3      1     4
   Qi : 3     1      2     4
Here cov(R,Q) = 0.5, Var(R) = Var(Q) = 1.25. So rR=0.4.

Exercise 3.1: Prove that
rR = 1 - 6 ∑ di2 / n(n2 - 1))

Exercise 3.2: Show that under H0 the vector (S1,...,Sn) is distributed uniformly over all possible permutations of 1,...,n.

Exercise 3.3: Express rR as a function of Si's, the concommitants of Y's wrt X's. (These were defined in the last lecture.)

Define
T = ∑ Ri Qi

Exercise 3.4: Show that under H0
E(T) = n(n+1)2/4
[Hint: First express T in terms of the concommitants.]

Exercise 3.5: Show that under H0
Var(T) = n2(n+1)2(n-1)/144.
[Hint: First express T in terms of the concommitants.]

Exercise 3.6: Use the above to show that E(rR) = 0 under H0.

Exercise 3.7: Show that Var(rR) = 1/(n-1) under H0

Theorem Under H0,
sqrt(n-1) rR ~ AN(0,1).

Proof: Not to be done in this course.

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© Arnab Chakraborty (2010)