Kendall’s tau-b version calculation steps with tied ranks

Question

I am trying to calculate Kendall’s tau coefficient for example given by Scipy in python. It is for tied ranks with tau-b version. The tau should be -0.47140452079103173 but I get a different result. I use formulas in "Handbook of Parametric and Nonparametric. Statistical Procedures. David J. Sheskin. Chapman & Hall/CRC." page 900.

$$x = [12, 2, 1, 12, 2]$$ $$y = [1, 4, 7, 1, 0]$$

rank the data: $$rank x = [4.5, 2.5, 1, 4.5, 2.5]$$ $$rank y = [2.5, 4, 5, 2.5, 1]$$

dimensions: $n=5, m=2 $ $$rank x+rank y = [7, 6.5, 6, 7, 3.5]$$ $$T=7+6.5+6+7+3.5=30$$ $$U=7^2+6.5^2+6^2+7^2+3.5^2=188.5$$

calculate the tie correlation(${\displaystyle\sum_{i=1}^m}{\displaystyle\sum_{a=1}^s}(t^3-t)$):

in x there are 2 tied groups each with 2 members: $(2^3-2)+(2^3-2)=12$

in y there is 1 tied group with 2 members: $(2^3-2)= 6$

total tie correlations: $12+6=18$

$$W=\frac{12U-3m^2n(n+1)^2}{m^2n(n^2-1)-m{\displaystyle\sum_{i=1}^m}{\displaystyle\sum_{a=1}^s}(t^3-t)}=\frac{12(188.5)-3\times 2^2\times 5(5+1)^2}{2^2\times5(5^2-1)-2(18)}=0.22973$$

Sorry for the low-quality question. there are not many numeric questions for tau-b. I hope this question can be helpful also for others who review python's example.

Answer 1

The formula you are using is calculating Kendall's W, also known as Kendall's coefficient of concordance. Kendall's W not the same as Kendall's tau-b.

To calculate the Kendall tau-b for the given data set, you can use the formula in the Wikipedia page. I count $n_0=10$, $n_1=2$, $n_2=1$, $n_c=2$, $n_d=6$, so that $$ \tau_B=\frac{ 2-6}{\sqrt{(10-2)(10-1)}}=-\frac4{\sqrt{72}}=-.4714045. $$

EDIT: How to calculate $n_1$? We see that $x$ has two groups of ties, namely $\{2,2\}$ and $\{12,12\}$, each with two ties per group, so $t_1=2$ and $t_2=2$ and $$n_1:=\frac12\sum_it_i(t_i-1)=\frac12(2\cdot1)+\frac12(2\cdot1)=2.$$ As for $n_2$, we see that $y$ has one group of ties, namely $\{1,1\}$, so $u_1=2$ and $$n_2:=\frac12\sum_ju_j(u_j-1)=\frac12(2\cdot1)=1.$$