Applied Statistics Using SPSS, STATISTICA, MATLAB and R

70 2 Presenting and Summarising the Data
n
∑ =
2
6 d
i 1 i
rs
= 1−
, 2.21
2
N(
N −1)
When tied ranks occur − i.e., two or more cases receive the same rank on the
same variable −, each of those cases is assigned the average of the ranks that would
have been assigned had no ties occurred. When the proportion of tied ranks is
small, formula 2.21 can still be used. Otherwise, the following correction factor is
computed:
T =
g
∑
i=
1
3
i
( t − t ) ,
i
where g is the number of groupings of different tied ranks and ti is the number of
tied ranks in the ith grouping. The Spearman’s rank correlation with correction for
tied ranks is now written as:
r
s
= 1−
( N
( N
3
3
− N)
− 6
− N)
2
− ( T
n
∑ i=
1
x
d
2
i
y
− ( T
3
x
+ T
+ T )( N − N)
+ T T
y
) / 2
x
y
, 2.22
where Tx and Ty are the correction factors for the variables X and Y, respectively.
Table 2.10. Contingency table obtained with SPSS of the NC, PRTGC variables
(cork stopper dataset).
PRTGC Total
0 1 2 3
NC 0 Count 25 9 4 1 39
% of Total 16.7% 6.0% 2.7% .7% 26.0%
1 Count 12 13 10 1 36
% of Total 8.0% 8.7% 6.7% .7% 24.0%
2 Count 1 13 15 9 38
% of Total .7% 8.7% 10.0% 6.0% 25.3%
3 Count 1 1 9 26 37
% of Total .7% .7% 6.0% 17.3% 24.7%
Total Count 39 36 38 37 150
% of Total 26.0% 24.0% 25.3% 24.7% 100.0%
Example 2.8
Q: Compute the rank correlation for the variables N and PRTG of the Cork
PRTG into 4 categories, according to their value falling into the 1 st , 2 nd , 3 rd or 4 th
Stopper’ dataset, using two new variables, NC and PRTGC, which rank N and
quartile intervals.