198 NONPARAMETRIC STATISTICS13.4 SPEARMAN’S RANK CORRELATIONSpearman’s rank correlation is simply, as its name implies, a correlation of data that
has been ranked or naturally occurs as ranks. It is called Spearman’s rho (the Greek
letter p). If we rank the data given in Table 12.2 on weight and systolic blood pressure
(SBP), we obtain Table 13.3.
Table 13.3 Ranks for Weight, SBP, and Square Differences in RanksWeight Rank (X) SBP Rank(Y) (Rank X- Rank Y)’
165
243
180
152
163
210
203
195
178
2183
10
5 1 2 8 7 6 4 9134 5
155 10
137 7
124 1
128 3
131 4
143 8
136 6
127 2
146 94
0
4
0
1
16
1
0
4
0The sum of (rank X - sumY)’ is 30, and this is called 0’. The correlation of
the ranks can be computed from the formula given in Section 12.3.1 or by using the
following formula for the sample estimate of Spearman’s rho (T,):
T, = 1 - (6D2)/(n(n2 - 1))
= 1 - (6 x 30)/(10(100 - 1))
z= 1 - (180)/990 = 1 - .182
= .818This formula should not be used if there are ties in the Xi or in the Y,. For example,
in the weight column the first weight is 165 lb. If the fifth weight were also 165 lb.,
the average rank would be used for these two measurements similar to what was done
in the Wilcoxon rank sum example. Both the first and fifth observations would have
a rank of 3.5. In this case, the usual method of computing the correlation coefficient
given in Section 12.3 should be used on the ranks.
The null hypothesis that is tested is that X and Y are independent. The alternative
hypothesis is that X and Y are positively or negatively related. The observations
do not need to be normally distributed or interval data, but they must be capable of
being ranked. Spearman’s rank correlation is less sensitive to outliers than the regular
correlation coefficient.
Obtaining the results for Spearman’s rank correlation with statistical packages is
straightforward. For example, Minitab will give the user ranks of their data by using
the Data and then the Rank commands, and then the usual correlation coefficient