Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.8. Measures of Association 633

null variance ofKis given by expression (10.8.6); see, for instance, page 205 of
Hettmansperger (1984). If all pairs (Xi,Yi),(Xj,Yj) of the sample are concordant
thenK= 1, indicating a strictly increasing monotone relationship. On the other
hand, if all pairs are discordant thenK=−1. Thus the range ofKis contained in
the interval [− 1 ,1]. Also, the summands in expression (10.8.3) are either±1. From
the proof of Theorem 10.8.1, the probability that a summand is 1 is 1/2, which does
not depend on the underlying distribution. Hence the statisticKis distribution-
free underH 0. The null distribution ofKis symmetric about 0. This is easily seen
from the fact that for each concordant pair there is an obvious discordant pair (just
reverse an inequality on theYs) and the fact that concordant and discordant pairs
are equilikely underH 0. Also, it can be shown thatKis asymptotically normal
underH 0. We summarize these results, without proof, in a theorem.

Theorem 10.8.2.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 statisticKsatisfies the following properties:

Kis distribution free with a symmetric pmf (10.8.4)

EH 0 [K] = 0 (10.8.5)

VarH 0 (K)=

2

9

2 n+5

n(n−1)

(10.8.6)

q K

VarH 0 (K)

has an asymptoticN(0,1)distribution. (10.8.7)

Most statistical computing packages compute Kendall’sτ. For instance, the R

functioncor.test(x,y,method=c("kendall"),exact=T)obtainsKand the test

discussed above whenxandyare the vectors of theXandYobservations, respec-

tively. The computation of thep-value is with the exact distribution. We illustrate

this test in the next example.

Based on the asymptotic distribution, a large sample levelαtest for the hy-

potheses (10.8.2) is to rejectH 0 ifZK>zα/ 2 ,where

ZK=

K

√

2(2n+5)/ 9 n(n−1)

. (10.8.8)

Example 10.8.1(Olympic Race Times).Table 10.8.1 displays the winning times
for two races in the Olympics beginning with the 1896 Olympics through the 1980
Olympics. The data were taken from Hettmansperger (1984) and can be found in
the data setolym1500mara.rda. The times in seconds are for the 1500 m and the
marathon. The entries in the table for the marathon race are the actual times minus
2 hours. In Exercise 10.8.2 the reader is asked to create a scatterplot of the times
for the two races. The plot shows a strongly increasing monotone trend with one
obvious outlier (1968 Olympics). The following R code computes Kendall’sτ.We
have summarized the results with the estimate of Kendall’sτand thep-value of the
test of no association. Thisp-value is based on the exact distribution.
cor.test(m1500,marathon,method="kendall",exact=T)