10.8. Measures of Association 635Proof: Under independence,XiandYj are independent for alli andj; hence,
in particular,R(Xi) is independent ofR(Yi). Furthermore,R(Xi) is uniformly
distributed on the integers{ 1 , 2 ,...,n}. Therefore,E(R(Xi)) = (n+1)/2, which
leads to the result.
Thus the measure of associationrS can be used to test the null hypothe-
sis of independence similar to Kendall’sK. Under independence, because the
Xis are a random sample, the random vector (R(X 1 ),...,R(Xn)) is equilikely
to assume any permutation of the integers{ 1 , 2 ,...,n}and, likewise, the vector
of the ranks of theYis. Furthermore, under independence, the random vector
[R(X 1 ),...,R(Xn),R(Y 1 ),...,R(Yn)] is equilikely to assume any of the (n!)^2 vec-
tors (i 1 ,i 2 ,...,in,j 1 ,j 2 ,...,jn), where (i 1 ,i 2 ,...,in)and(j 1 ,j 2 ,...,jn) are per-
mutations of the integers{ 1 , 2 ,...,n}. Hence, under independence, the statistic
rSis distribution-free. The distribution is discrete and tables of it can be found,
for instance, in Hollander and Wolfe (1999). Similar to Kendall’s statisticK,the
distribution is symmetric about zero and it has an asymptotic normal distribution
with asymptotic variance 1/(n−1); see Exercise 10.8.7 for a proof of the null vari-
ance ofrs. A large sample levelαtest is to reject independence betweenXandY
if|zS|>zα/ 2 ,wherezS=
√
n− 1 rs. We record these results in a theorem, without
proof.
Theorem 10.8.4.Let(X 1 ,Y 1 ),(X 2 ,Y 2 ),...,(Xn,Yn)be a random sample on the
bivariate random vector(X, Y)with continuous cdfF(x, y). Under the null hypoth-
esis of independence betweenXandY, i.e.,F(x, y)=FX(x)FY(y), for all(x, y)
in the support of(X, Y), the test statisticrSsatisfies the following properties:
rSis distribution-free, symmetrically distributed about 0 (10.8.11)
EH 0 [rS] = 0 (10.8.12)VarH 0 (rS)=1
n− 1(10.8.13)
q rS
VarH 0 (rS)is asymptoticallyN(0,1). (10.8.14)Example 10.8.2(Example 10.8.1, Continued).For the data in Example 10.8.1,
the R code for the analysis based on Spearman’sρis:
cor.test(m1500,marathon,method="spearman")
p-value = 2.021e-06; sample estimates: rho 0.9052632
The result is highly significant. For comparison, the value of the asymptotic test
statistic isZS=0. 905
√
19 = 3.94 with thep-value for a two-sided test is 0.00008;
so, the results are quite similar.
If the samples have a strictly increasing monotone relationship, then it is easy to
see thatrS= 1; while if they have a strictly decreasing monotone relationship, then
rS=−1. Like Kendall’sKstatistic,rSis an estimate of a population parameter,
but, except for whenXandYare independent, it is a more complicated expression
thanτ. It can be shown (see Kendall, 1962) that
E(rS)=
3
n+1[τ+(n−2)(2γ−1)], (10.8.15)