The BRANKS function computes the tied ranks and the bivariate ranks for an 
matrix and returns an 
matrix of these ranks. The tied ranks of the first column of matrix are contained in the first column of the result matrix; the tied ranks of the second column of matrix are contained in the second column of the result matrix; and the bivariate ranks of matrix are contained in the third column of the result matrix.
The tied rank of an element 
of a vector is defined as
![\[ \mb {R}_ i = \frac{1}{2} + \sum _ j u(x_ i - x_ j) \]](images/imlug_langref0129.png){kind=small}
where
![\[ u(t) = \left\{ \begin{array}{lcl} 1 & & \mbox{if } t>0 \\ \frac{1}{2} & & \mbox{if } t=0 \\ 0 & & \mbox{if } t<0 \end{array} \right. \]](images/imlug_langref0130.png)
The bivariate rank of a pair 
is defined as
![\[ \mb {Q}_ i = \frac{3}{4} + \sum _ j u(x_ i - x_ j)\, u(y_ i - y_ j) \]](images/imlug_langref0132.png){kind=small}
The results of the BRANKS function can be used to compute rank-based correlation coefficients such as the Spearman rank-order
correlation and Hoeffding’s 
statistic.
The following statements compute the bivariate ranks of two columns of data:
z = { 1 2,
2 1,
3 3,
3 5,
4 4,
5 4,
5 4,
4 5 };
b = branks(z);
print b;
Figure 24.60: Tied Ranks and Bivariate Ranks
| b | ||
|---|---|---|
| 1 | 2 | 1 |
| 2 | 1 | 1 |
| 3.5 | 3 | 3 |
| 3.5 | 7.5 | 3.5 |
| 5.5 | 5 | 4 |
| 7.5 | 5 | 4.75 |
| 7.5 | 5 | 4.75 |
| 5.5 | 7.5 | 5 |