Statistical Methods in Medical Research 4ed

290 Analysing non-normal data
1 It should lie between 1and‡1, taking the value ‡1 when the individuals
are ranked in exactly the same order by x as by y, and
reversed.
1 when the order is
2 For large samples in which the distribution of x is independent of y (and
conversely), the value should be zero.
A satisfactory rank correlation coefficient can be obtained from Kendall's S
statistic, described in §10.3, with a generalization of the definition used there. The
total number of pairs of individuals is 1
2 n…n 1†. Let P be the number of pairs
which are ranked in the same order by x and by y, and Q the number of pairs in
which the rankings are in the opposite order. Then
S ˆ P Q:
(The previous definition is a particular case of this in which one variable
represents a dichotomy into the two groups, with group X being ranked before
group Y, and the other variable represents the measurement under test, both xi
and yj as previously defined being values of this second variable.)
The rank correlation coefficient, t, is now defined by
t ˆ
1
2
S
: …10:11†
n…n 1†
It is fairly easy to see that for complete agreement of rankings t ˆ 1, and for
complete reversal t ˆ 1. A significance test of the null hypothesis that the x
ranking is independent of the y ranking can conveniently be done on S. The null
expectation of S (as of t) is zero, and, in the absence of ties,
If there are ties, (10.12) is modified to give
var…S† ˆn…n 1†…2n ‡ 5†=18: …10:12†
var…S† ˆ1 P
18 ‰n…n 1†…2n ‡ 5† t…t 1†…2t ‡ 5† P
u…u 1†…2u ‡ 5†Š
t
1
‡
9n…n 1†…n 2† ‰P t…t 1†…t 2†Š‰
t
P
u…u 1†…u 2†Š
u
1
‡
2n…n 1† ‰P t…t 1†Š‰
t
P
u…u 1†Š,
u
where the summations are over groups of ties, t being the number of tied
individuals in a group of x values and u the number of a group of tied y
values.
For further discussion of rank correlation, including tables for the significance
of S, see Kendall and Gibbons (1990).
u