rational h-index
The rational h-index (Ruane and Tol 2008) (hΔ or hrat) is a continuous version of the h-index 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) | 47 | 26 | 19 | 15 | 11 | 10 | 4 | 3 | 2 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Rank (i) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
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 3 citations to the 7th publication (for a total of 3). 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 - 3/13 = 6.7692.
History
Year | hΔ |
---|---|
1997 | 1.3333 |
1998 | 2.8000 |
1999 | 3.7143 |
2000 | 5.7273 |
2001 | 6.7692 |
2002 | 8.8235 |
2003 | 11.8696 |
2004 | 12.9600 |
2005 | 15.8710 |
2006 | 17.8857 |
2007 | 19.9487 |
2008 | 21.9535 |
2009 | 25.8824 |
2010 | 26.9245 |
2011 | 29.9322 |
2012 | 32.9538 |
2013 | 33.9254 |
2014 | 34.9710 |
2015 | 35.9577 |
2016 | 35.9859 |
2017 | 37.7867 |
2018 | 37.8133 |
2019 | 37.8667 |
2020 | 38.9481 |
2021 | 40.9506 |
2022 | 41.9639 |
2023 | 42.9529 |
2024 | 42.9529 |
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.