Biostatistics

13.10 THE SPEARMAN RANK CORRELATION COEFFICIENT 723
TABLE 13.10.4 Ranks for Data of Example 13.10.2
Subject
Number
Rank
(X)
Rank
(Y) d i d 2 i
Subject
Number
Rank
(X)
Rank
(Y) d i d 2 i
1 32.5 35 2:5 6.25 19 17 9 8 64.00
2 35 27 8 64.00 20 28 25 3 9.00
3 34 23 11 121.00 21 21.5 21 .5 .25
4 25 32 7 49.00 22 23.5 22 1.5 2.25
5 32.5 19 13.5 182.25 23 13.5 24 10:5 110.25
6 19.5 11 8.5 72.25 24 27 34 7 49.00
7 1 14 13 169.00 25 6 3 3 9.00
8 13.5 8 5.5 30.25 26 12 7 5 25.00
9 9 6 3 9.00 27 2 15 13 169.00
10 15 10 5 25.00 28 21.5 31 9:5 90.25
11 26 33 7 49.00 29 29 26 3 9.00
12 10 17 7 49.00 30 11 18 7 49.00
13 4 1 3 9.00 31 7 4 3 9.00
14 17 13 4 16.00 32 30 28 2 4.00
15 23.5 20 3.5 12.25 33 17 12 5 25.00
16 5 2 3 9.00 34 31 29 2 4.00
17 8 5 3 9.00 35 3 16 13 169.00
18 19.5 30 10:5 110.25
P d
2
i ¼ 1788:5
Since 4.37 is greater than z ¼ 3:89; p < 2 ð:0001Þ ¼ :0002, and we
reject H 0 and conclude that the two variables under study are not mutually
independent.
For comparative purposes let us correct for ties using Equation 13.10.3
and then compute r s by Equation 13.10.4.
In the rankings of X we had six groups of ties that were broken by
assigning the values 13.5, 17, 19.5, 21.5, 23.5, and 32.5. In five of the groups
two observations tied, and in one group three observations tied. We,
therefore, compute five values of
and one value of
T x ¼ 23 2
12
¼ 6
12 ¼ :5
T x ¼ 33 3
12
¼ 24
12 ¼ 2
From these computations, we have P T x ¼ 5 ð:5Þþ2 ¼ 4:5, so that
X
x 2 ¼ 352 35
12
4:5 ¼ 3565:5