I want to quantify correlation between rankings. Rankings in this case have got ties. Also rankings don't rank all the same entries, the rankings might be of different sizes and rank different sets, of course, with non-empty intersection (there are elements that are ranked on both rankings).
I have read through wikipedia articles and found a bunch of correlation coefficients that seem applicable:
- Kendall's $\tau_b$ is adjusted for ties.
- Kendall's $\tau_c$ is adjusted for non-square tables. If I understand correctly, it means that more elements are ranked on one ranking than on the other, just like my case, right?
- Goodman and Kruskal's $\gamma$ disregards the ties altogether.
- Somers' $D$ seems to be somehow normed against the amount of non-ties in one of the rankings.
I am confused among all of this. I am not sure what to take into account when choosing a coefficient. In which cases one coefficient is used, in which cases - a different one?
Somers' $D$ is obviously different from the others. I am not sure if I understand the $\tau_c$ well enough to judge that, but the remaining two are the real confusers.
How do I know if it is better to ignore all the ties and use $\gamma$ or take number of ties into account as in $\tau_b$? Can you name some cases where you would choose each of the cpefficients?