Discrepancy in manually calculated Kendall's tau-b vs SPSS output

Question

stats enthusiasts!

I'm having trouble with a manual calculation of Kendall's tau-b. My result differs from the SPSS output, and I can't figure out where I went wrong. Here's my process:

I'm analyzing the association between exercise levels (0,1,2) and optimism levels (0,1,2) with this data:

Optimism(y): 0,0,0,1,1,1,1,1,2,2,2,2,1,2,1

Exercise(x): 0,0,0,0,0,1,1,1,1,1,2,2,2,2,2

I calculated:

Concordant pairs (C): 49

Discordant pairs (D): 4

Ties in y (Ty): 34

Ties in x (Tx): 30

I used this formula:

τb = (Nc - Nd) / √(Nc + Nd + Tx) * √(Nc + Nd + Ty)

Plugging in the values:

τb = (49 - 4) / √(49 + 4 + 30) * √(49 + 4 + 34) ≈ 0.5295

However, SPSS gives a result of about 0.617.

I've double-checked my calculations for C, D, Tx, and Ty using online calculators(https://statpages.info/ordinal.html), and they seem correct. So, I'm wondering:

Are my parentheses wrong in the formula?

Did I miscalculate concordant or discordant pairs?

Is there an error in my tie calculations?

I'd really appreciate any insights on where I might have gone wrong.

Answer 1

SPSS here gives the same as R's cor(, method="kendall"), which says it uses Kendall's $\tau_b$ in the case of ties. Following Wikipedia

> a
 [1] 0 0 0 1 1 1 1 1 2 2 2 2 1 2 1
> b
 [1] 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2
> ta<-table(a)
> sum(ta*(ta-1)/2)->n2
> tb<-table(b)
> sum(tb*(tb-1)/2)->n1
> n0<-n*(n-1)/2
> (49-4)/sqrt((n0-n1)*(n0-n2))
[1] 0.6166698

is the correct value for $\tau_b$.