# Michael S. Rosenberg’s Laboratory

Computational Evolutionary Biology & Bioinformatics

E-mail: msr@asu.edu
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## rational h-index

The rational h-index (Ruane and Tol 2008) (hΔ or hrat) is a continuous version of h which not only measures the standard h-index but includes the fractional progress toward the next higher value of h. It is calculated as

$$h^\Delta = h + 1 -\frac{n}{2h+1},$$where $$n$$ is the number of citations necessary to reach the next value of $$h$$. The divisor, $$2h+1$$, is the maximum number of possible citations needed to move from $$h$$ to $$h+1$$ (one additional citation for each of the $$h$$ publications in the core plus $$h+1$$ citations for a publication outside of the core with no citations). Practically speaking, $$n$$ is the number of publications in the core with exactly $$h$$ citations (thus needing one more to allow a move to $$h+1$$) plus $$h+1-C_{h+1}$$ (the number of citations the $$h+1$$th ranked publication needs to reach $$h+1$$ citations).

### Example

Publications are ordered by number of citations, from highest to lowest.

 Citations (Ci) Rank (i) 42 36 14 11 9 9 3 2 2 2 1 1 1 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 h = 6

The h-index is 6. To reach an h of 7, we would need to add 0 citations to the publications within the core and 4 citations to the 7th publication (for a total of 4). The maximum number of citations that could be required is 13 (if all 6 core publications had exactly 6 citations and the 7th publication had zero), thus the rational h is 6 + 1 - 4/13 = 6.6923.

YearhΔ
19971.3333
19982.8000
19993.7143
20005.8182
20016.6923
20027.8667
20039.8947
200411.9565
200513.8889
200615.9355
200717.9429
200819.9744
200923.8936
201024.9388
201126.8868
201227.9818
201329.9831
201432.9692
201534.9565
201635.8732
201735.8732

## References

• Ruane, F., and R.S.J. Tol (2008) Rational (successive) h-indices: An application to economics in the Republic of Ireland. Scientometrics 75(2):395–405.