How to estimate Savage scores of tied ranks

Question

I want to calculate the Top-Down Concordance coefficient that quantifies the agreement between two rankings by emphasizing more on the lowest rankings. The original paper is:

R.L. Iman and W.J. Conover, A measure of top-down correlation, Technometrics 29(3) (1987), pp. 351–357.

but it is also described in a more recent one.

My problem is how to calculate the Savage Scores from tied ranks. E.g.

Ranking A:
[  1.,   2.,   3.,   4.,   5.,   7.,   7.,   7.,   9.,  10.,  11., 12.]
Ranking B:
[  1. ,   2. ,   3.5,   3.5,   5. ,   6. ,   7. ,   8. ,   9. , 10. ,  11. , 12. ]

In this case I have taken the average of the scores of the tied observations. As you can see there is a group of 3 tied ranks (7) in Ranking A and a group of two tied ranks (3.5) in Ranking B. How should I calculate the Savage Score for these ranks? For instance, there are several ways I can think to get the Savage Score for Rank 7 in A:

1/7 + 1/9 + 1/10 + 1/11 + 1/12 or 1/7 + 1/7 + 1/9 + 1/10 + 1/11 + 1/12 or 1/7 + 1/7 + 1/7 + 1/9 + 1/10 + 1/11 + 1/12 etc.

Or doesn't Top-Down Concordance support tied ranks at all? In that case is there any other concordance coefficient emphasizing on the lower ranks that you would suggest?

Answer 1

I have found a possible solution in Jerrold Zar's "Biostatistical Analysis", 5th edition.

" If there are tied ranks, then we may use the mean of the Savage scores for the positions of the tied data. For example. if n = 4 and ranks 2 and 3 are tied. then use ((1/2+1/3+1/4)+(1/3+1/4))/2 = (1.083 + 0.583 )/2 = 0.833 for both S2 and S3. "