2.4.2 Nonparametric Correlation Coe‰cients
Suppose that the data set consists ofnpairs of observationsfðxi;yiÞg, express-
ing a possible relationship between two continuous variables. We characterize
the strength of such a relationship by calculating the coe‰cient of correlation:
r¼P
ðxxÞðyyÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½P
ðxxÞ^2 ½P
ðyyÞ^2 qcalledPearson’s correlation coe‰cient. Like other common statistics, such as
the meanxand the standard deviations, the correlation coe‰cientris very
sensitive to extreme observations. We may be interested in calculating a mea-
sure of association that is more robust with respect to outlying values. There
are not one but two nonparametric procedures: Spearman’s rho and Kendall’s
tau rank correlations.
Spearman’s Rho Spearman’s rank correlation is a direct nonparametric
counterpart of Pearson’s correlation coe‰cient. To perform this procedure, we
first arrange thexvalues from smallest to largest and assign a rank from 1 ton
for each value; letRibe the rank of valuexi. Similarly, we arrange theyvalues
from smallest to largest and assign a rank from 1 tonfor each value; letSibe
the rank of valueyi. If there are tied observations, we assign an average rank,
averaging the ranks that the tied observations take jointly. For example, if the
second and third measurements are equal, they are both assigned 2.5 as their
TABLE 2.15
xyx^2 y^2 xy
42 130 1,764 16,900 5,460
46 115 2,116 13,225 5,290
42 148 1,764 21,904 6,216
71 100 5,041 10,000 7,100
80 156 6,400 24,336 12,480
74 162 5,476 26,224 11,988
70 151 4,900 22,801 10,570
80 156 6,400 24,336 12,480
85 162 7,225 26,224 13,770
72 158 5,184 24,964 11,376
64 155 4,096 24,025 9,920
81 160 6,561 25,600 12,960
41 125 1,681 15,625 5,125
61 150 3,721 22,500 9,150
75 165 5,625 27,225 12,375984 2,193 67,954 325,889 146,260
88 DESCRIPTIVE METHODS FOR CONTINUOUS DATA