# Michael S. Rosenberg’s Laboratory

Computational Evolutionary Biology & Bioinformatics

E-mail: msr@asu.edu
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## adapted pure h-index (proportional credit)

The adapted pure h-index (Chai et al. 2008) is very similar to the pure h-index, except that it estimates its own core rather than relying on the standard Hirsch core. For a given publication, if one has information on author order and assumes the order directly correlates with effort, one can use a proportional (artihmetic) assignment of credit for each publication as:

$$E_i=\frac{2\left(A_i + 1 - a_i\right)}{A_i\left(A_i + 1\right)},$$

where ai is the position of the target author within the full author list of publication i (i.e., an integer from 1 to Ai). Each publication can then be weighted by the inverse of the author effort, wi = 1/Ei. The effective number of citations for each publication is then calculated as

$$C^{*}_i = \frac{C_i}{\sqrt{w_i}}.$$

Publications are ranked according to these new citation counts and the h-equivalent value, he, is found as the largest rank for which the rank is less than the number of equivalent citations, or

$$h_e = \underset{i}{\max}\left(i \leq C^{*}_i\right).$$The adapted pure h-index is calculated by interpolating between this value and the next largest, as

$$h_{ap.\text{prop}}= \frac{\left(h_e+1\right)C^{*}_{h_e}-h_e C^{*}_{h_e +1}}{C^{*}_{h_e}-C^{*}_{h_e+1}+1}.$$

### Example

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

 Citations (Ci) Authors (Ai) Author Position (ai) Author Effort (Ei) Weight (wi) Adjusted Citations ($$C^*_i$$) Rank (i) 42 36 9 11 14 9 2 2 3 2 1 1 1 0 0 0 3 3 1 2 3 4 1 1 4 4 1 2 4 4 2 1 1 1 1 2 3 3 1 1 3 1 1 1 3 3 1 1 0.50 0.50 1.00 0.33 0.17 0.20 1.00 1.00 0.20 0.40 1.00 0.67 0.20 0.20 0.67 1.00 2.00 2.00 1.00 3.00 6.00 5.00 1.00 1.00 5.00 2.50 1.00 1.50 5.00 5.00 1.50 1.00 29.70 25.46 9.00 6.35 5.72 4.02 2.00 2.00 1.34 1.26 1.00 0.82 0.45 0.00 0.00 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 he = 5

The largest rank where $$i \leq C^*_i$$ is 5. Interpolating between this and the next largest rank yields hap.prop = 5.2659.

Yearhap.prop
19970.0000
19981.9237
19992.6470
20004.0548
20015.2659
20026.6840
20038.1439
20049.9153
200511.5293
200613.4613
200715.3925
200817.5954
200919.7703
201021.1863
201123.0000
201224.0000
201325.7231
201427.7128
201528.2902
201630.2195
201730.3556

## References

• Chai, J.-c., P.-h. Hua, R. Rousseau, and J.-k. Wan (2008) The adapted pure h-index. Fourth International Conference on Webometrics, Informetrics and Scientometrics & Ninth COLLNET Meeting, H. Kretschmer and F. Havemann, eds. Humboldt-Universität zu Berlin: Institute for Library and Information Science.