Kendall tau: Kendall Score

Question

Currently, I am working with with Kendall tau distance between two rankings. I get a tau value of 0.849, Z (statistical significance of 3.9) and a Kendall score of 56. What does this Kendall score tell us about? Here under are my results and I would appreciate if anyone could enlighten me with these values of S, D and varS - if they are worthy of adding to a research evaluation. Or is only the kendall tau and Z are enough to tell the story?

kendal$s
    [1] 56
    attr(,"Csingle")
    [1] TRUE
kendal$D
[1] 66.00001
attr(,"Csingle")
[1] TRUE
>
kendal$varS
[1] 212.6667
attr(,"Csingle")
[1] TRUE

Answer 1

It looks like this "Kendall score" is only an intermediate step in the computation of τ (the Kendall correlation) and not of any substantive interest itself. There's no need to report it. Focus on the τ.

Answer 2

Sort the data so that one variable is in order (call it $A$) then take each value of $B$ except the rightmost and count how many values of $B$ to the right of it are larger (call it $p$) and smaller (call it $q$). The add up all the $p$ and call it $P$ and similarly to form $Q$. Then $S = P - Q$. Whether that has any meaning in your scientific question is up to you. It is a way of defining the way in which Kendall decided to measure correlation. $S$ also plays a fundamental role in many other so-called non-parametric tests.