Introductory Biostatistics

common rank. The next step is to replace, in the formula of Pearson’s correla-
tion coe‰cientr,xiby its rankRiandyiby its rankSi. The result isSpear-
man’s rho, a popular rank correlation:

r¼

P

ðRi
RÞðSi
SÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

½

P

ðRi
RÞ^2 Š½

P

ðSi
SÞ^2 Š

q

¼ 1

6

P

ðRi
SiÞ^2

nðn^2 
 1 Þ

The second expression is simpler and easier to use.

Example 2.10 Consider again the birth-weight problem of Example 2.8. We
have the data given in Table 2.16. Substituting the value of

P

ðRi
SiÞ^2 into
the formula for rho (r), we obtain

r¼ 1 

ð 6 Þð 560 : 5 Þ

ð 12 Þð 143 Þ

¼
 0 : 96

which is very close to the value ofr(
0.946) obtained in Example 2.8. This
closeness is true when there are few or no extreme observations.

Kendall’s Tau Unlike Spearman’s rho, the other rank correlation—Kendall’s
taut—is defined and calculated very di¤erently, even though the two correla-

TABLE 2.16

Birth Weight Increase in Weight

x(oz) RankRy(%) RankSR
S ðR
SÞ^2

112 10 63 3 7 49
111 9 66 4 5 25
107 8 72 5.5 2.5 6.25
119 12 52 2 10 100
92 4 75 7
39
80 1 118 11
10 100
81 2 120 12
10 100
84 3 114 10
749
118 11 42 1 10 100
106 7 72 5.5 1.5 2.25
103 6 90 8
24
94 5 91 9
416
560.50

COEFFICIENTS OF CORRELATION 89