Of the existing theoretical formulas for the h-index, those recently suggested by Burrell (J Informetr 7:774-783, 2013b) and by Bertoli-Barsotti and Lando (J Informetr 9(4):762-776, 2015) have proved very effective in estimating the actual value of the h-index Hirsch (Proc Natl Acad Sci USA 102:16569-16572, 2005), at least at the level of the individual scientist. These approaches lead (or may lead) to two slightly different formulas, being based, respectively, on a "standard" and a "shifted" version of the geometric distribution. In this paper, we review the genesis of these two formulas-which we shall call the "basic" and "improved" Lambert-W formula for the h-index-and compare their effectiveness with that of a number of instances taken from the well-known Glänzel-Schubert class of models for the h-index (based, instead, on a Paretian model) by means of an empirical study. All the formulas considered in the comparison are "ready-to-use", i.e., functions of simple citation indicators such as: the total number of publications; the total number of citations; the total number of cited paper; the number of citations of the most cited paper. The empirical study is based on citation data obtained from two different sets of journals belonging to two different scientific fields: more specifically, 231 journals from the area of "Statistics and Mathematical Methods" and 100 journals from the area of "Economics, Econometrics and Finance", totaling almost 100,000 and 20,000 publications, respectively. The citation data refer to different publication/citation time windows, different types of "citable" documents, and alternative approaches to the analysis of the citation process ("prospective" and "retrospective"). We conclude that, especially in its improved version, the Lambert-W formula for the h-index provides a quite robust and effective ready-to-use rule that should be preferred to other known formulas if one's goal is (simply) to derive a reliable estimate of the h-index.

Department of Finance VŠB TU Ostrava Sokolskà 33 70121 Ostrava Czech Republic

Department of Management Economics and Quantitative Methods University of Bergamo via dei Caniana 2 24127 Bergamo Italy

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Abbas AM. Bounds and inequalities relating h-index, g-index, e-index and generalized impact factor: An improvement over existing models. PLoSONE. 2012;7:e33699. doi: 10.1371/journal.pone.0033699. PubMed DOI PMC

Alguliev RM, Aliguliyev RM, Fataliyev TK, Hasanova RS. Weighted consensus index for assessment of the scientific performance of researchers. Collnet Journal of Scientometrics and Information Management. 2014;8:371–400. doi: 10.1080/09737766.2014.954864. DOI

Annibaldi A, Truzzi C, Illuminati S, Scarponi G. Scientometric analysis of national university research performance in analytical chemistry on the basis of academic publications: Italy as case study. Analytical and Bioanalytical Chemistry. 2010;398:17–26. doi: 10.1007/s00216-010-3804-7. PubMed DOI

ANVUR Website. www.anvur.org

Arnold BC. Pareto distributions. Fairland, MD: International Cooperative Publishing House; 1983.

Bador P, Lafouge T. Comparative analysis between impact factor and h-index for pharmacology and psychiatry journals. Scientometrics. 2010;84:65–79. doi: 10.1007/s11192-009-0058-2. PubMed DOI

Bar-Ilan J. Ranking of information and library science journals by JIF and by h-type indices. Journal of Informetrics. 2010;4:141–147. doi: 10.1016/j.joi.2009.11.006. DOI

Bar-Ilan J. Journal report card. Scientometrics. 2012;92:249–260. doi: 10.1007/s11192-012-0671-3. DOI

Bertocchi G, Gambardella A, Jappelli T, Nappi CA, Peracchi F. Bibliometric evaluation vs. informed peer review: Evidence from Italy. Research Policy. 2015;44:451–466. doi: 10.1016/j.respol.2014.08.004. DOI

Bertoli-Barsotti L, Lando T. On a formula for the h-index. Journal of Informetrics. 2015;9(4):762–776. doi: 10.1016/j.joi.2015.07.004. DOI

Bletsas A, Sahalos JN. Hirsch index rankings require scaling and higher moment. Journal of the American Society for Information Science and Technology. 2009;60:2577–2586. doi: 10.1002/asi.21197. DOI

Bornmann L, Marx W, Gasparyan AY, Kitas GD. Diversity, value and limitations of the journal impact factor and alternative metrics. Rheumatology International. 2012;32:1861–1867. doi: 10.1007/s00296-011-2276-1. PubMed DOI

Bornmann L, Werner M, Schier H. Hirsch-type index values for organic chemistry journals: A comparison of new metrics with the journal impact factor. European Journal of Organic Chemistry. 2009;10:1471–1476. doi: 10.1002/ejoc.200801243. DOI

Bouabid H, Dalimi M, El Majid Z. Impact evaluation of the voluntary early retirement policy on research and technology outputs of the faculties of science in Morocco. Scientometrics. 2011;86:125–132. doi: 10.1007/s11192-010-0271-z. DOI

Braun T, Glänzel W, Schubert A. A Hirsch-type index for journals. Scientometrics. 2006;69:169–173. doi: 10.1007/s11192-006-0147-4. DOI

Burrell QL. Hirsch’s h-index: A stochastic model. Journal of Informetrics. 2007;1:16–25. doi: 10.1016/j.joi.2006.07.001. DOI

Burrell QL. Formulae for the h-index: A lack of robustness in Lotkaian informetrics? Journal of the American Society for Information Science and Technology. 2013;64:1504–1514. doi: 10.1002/asi.22845. DOI

Burrell QL. The h-index: A case of the tail wagging the dog? Journal of Informetrics. 2013;7:774–783. doi: 10.1016/j.joi.2013.06.004. DOI

Burrell QL. A stochastic approach to the relation between the impact factor and the uncitedness factor. Journal of Informetrics. 2013;7:676–682. doi: 10.1016/j.joi.2013.03.001. DOI

Burrell QL. The individual author’s publication-citation process: Theory and practice. Scientometrics. 2014;98:725–742. doi: 10.1007/s11192-013-1018-4. DOI

Corless RM, Jeffrey DJ. The Lambert W function. In: Higham NJ, Dennis M, Glendinning P, Martin P, Santosa F, Tanner J, editors. The Princeton companion to applied mathematics. Princeton: Princeton University Press; 2015. pp. 151–155.

Csajbók E, Berhidi A, Vasas L, Schubert A. Hirsch-index for countries based on essential science indicators data. Scientometrics. 2007;73:91–117. doi: 10.1007/s11192-007-1859-9. DOI

Egghe L. The functional relation between the impact factor and the uncitedness factor revisited. Journal of Informetrics. 2013;7:183–189. doi: 10.1016/j.joi.2012.10.007. DOI

Egghe L, Liang L, Rousseau R. A relation between h-index and impact factor in the power-law model. Journal of the American Society for Information Science and Technology. 2009;60:2362–2365. doi: 10.1002/asi.21144. DOI

Egghe L, Rousseau R. An informetric model for the Hirsch-index. Scientometrics. 2006;69:121–129. doi: 10.1007/s11192-006-0143-8. DOI

Egghe L, Rousseau R. The Hirsch-index of a shifted Lotka function and applications to the relation with the impact factor. Journal of the American Society for Information Science and Technology. 2012;63:1048–1053. doi: 10.1002/asi.22617. DOI

Elango B, Rajendran P, Bornmann L. Global nanotribology research output (1996–2010): A scientometric analysis. PLoSONE. 2013;8:e81094. doi: 10.1371/journal.pone.0081094. PubMed DOI PMC

Glänzel W. Towards a model of diachronous and synchronous citation analyses. Scientometrics. 2004;60:511–522. doi: 10.1023/B:SCIE.0000034391.06240.2a. DOI

Glänzel W. On the h-index—a mathematical approach to a new measure of publication activity and citation impact. Scientometrics. 2006;67:315–321. doi: 10.1007/s11192-006-0102-4. DOI

Glänzel W. Some new applications of the h-index. ISSI Newsletter. 2007;3:28–31.

Glänzel W. On some new bibliometric applications of statistics related to the h-index. Scientometrics. 2008;77:187–196. doi: 10.1007/s11192-007-1989-0. DOI

Harzing AWK, van der Wal R. A google scholar h-index for journals: An alternative metric to measure journal impact in economics & business? Journal of the American Society for Information Science and Technology. 2009;60:41–46. doi: 10.1002/asi.20953. DOI

Hirsch JE. An index to quantify an individual’s scientific research output. Proceedings of the National Academy of Sciences of the USA. 2005;102:16569–16572. doi: 10.1073/pnas.0507655102. PubMed DOI PMC

Hodge DR, Lacasse JR. Evaluating journal quality: Is the h-index a better measure than impact factors? Research on Social Work Practice. 2010;21:222–230. doi: 10.1177/1049731510369141. DOI

Hsu J-W, Huang D-W. A scaling between impact factor and uncitedness. Physica A. 2012;391:2129–2134. doi: 10.1016/j.physa.2011.11.028. DOI

Iglesias J, Pecharroman C. Scaling the h-index for different scientific ISI fields. Scientometrics. 2007;73:303–320. doi: 10.1007/s11192-007-1805-x. DOI

Ingwersen P. The pragmatics of a diachronic journal impact factor. Scientometrics. 2012;92:319–324. doi: 10.1007/s11192-012-0701-1. DOI

Ingwersen P, Larsen B, Rousseau R, Davis M. The publication-citation matrix and its derived quantities. Chinese Science Bulletin. 2001;46:524–528. doi: 10.1007/BF03187274. DOI

Ionescu G, Chopard B. An agent-based model for the bibliometric h-index. The European Physical Journal B. 2013;86:426. doi: 10.1140/epjb/e2013-40207-0. DOI

Johnson NL, Kemp AW, Kotz S. Univariate discrete distributions. 3. New York: Wiley; 2005.

Johnson NL, Kotz S, Balakrishnan N. Continuous univariate distributions. 2. New York: Wiley; 1994.

Leydesdorff L, Opthof T. Scopus’s source normalized impact per paper (SNIP) versus a journal impact factor based on fractional counting of citations. Journal of the American Society for Information Science and Technology. 2010;61:2365–2369. doi: 10.1002/asi.21371. DOI

Liu YX, Rao IKR, Rousseau R. Empirical series of journal h-indices: the JCR category Horticulture as a case study. Scientometrics. 2009;80:59–74. doi: 10.1007/s11192-007-2026-z. DOI

Lomax KS. Business failures: Another example of the analysis of failure data. Journal of the American Statistical Association. 1954;49(268):847–852. doi: 10.1080/01621459.1954.10501239. DOI

Malesios C. Some variations on the standard theoretical models for the h-index: A comparative analysis. Journal of the Association for Information Science and Technology. 2015;66:2384–2388. doi: 10.1002/asi.23410. DOI

Mingers J, Macri F, Petrovici D. Using the h-index to measure the quality of journals in the field of business and management. Information Processing and Management. 2012;48:234–241. doi: 10.1016/j.ipm.2011.03.009. DOI

Panaretos J, Malesios C. Assessing scientific research performance and impact with single indices. Scientometrics. 2009;81:635–670. doi: 10.1007/s11192-008-2174-9. DOI

Petersen AM, Stanley HE, Succi S. Statistical regularities in the rank-citation profile of scientists. Scientific Reports. 2011;1:181. doi: 10.1038/srep00181. PubMed DOI PMC

Prathap G. Is there a place for a mock h-index? Scientometrics. 2010;84:153–165. doi: 10.1007/s11192-009-0066-2. DOI

Prathap G. The 100 most prolific economists using the p-index. Scientometrics. 2010;84:167–172. doi: 10.1007/s11192-009-0068-0. DOI

R Development Core Team. (2012). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. http://www.R-project.org

Radicchi F, Castellano C. Analysis of bibliometric indicators for individual scholars in a large data set. Scientometrics. 2013;97:627–637. doi: 10.1007/s11192-013-1027-3. DOI

Schreiber M. Restricting the h-index to a citation time window: A case study of a timed Hirsch index. Journal of Informetrics. 2015;9:150–155. doi: 10.1016/j.joi.2014.12.005. DOI

Schreiber M, Malesios CC, Psarakis S. Exploratory factor analysis for the Hirsch index, 17 h-type variants, and some traditional bibliometric indicators. Journal of Informetrics. 2012;6:347–358. doi: 10.1016/j.joi.2012.02.001. DOI

Schubert A. A Hirsch-type index of co-author partnership ability. Scientometrics. 2012;91:303–308. doi: 10.1007/s11192-011-0559-7. DOI

Schubert A, Glänzel W. A systematic analysis of Hirsch-type indices for journals. Journal of Informetrics. 2007;1:179–184. doi: 10.1016/j.joi.2006.12.002. DOI

Schubert A, Korn A, Telcs A. Hirsch-type indices for characterizing networks. Scientometrics. 2009;78:375–382. doi: 10.1007/s11192-008-2218-1. DOI

Seiler C, Wohlrabe K. How robust are journal rankings based on the impact factor? Evidence from the economic sciences. Journal of Informetrics. 2014;8:904–911. doi: 10.1016/j.joi.2014.09.001. DOI

Shalizi, R. C. (2007). Maximum likelihood estimation for q-exponential (Tsallis) distributions. arXiv:math/0701854v2

Stern DI. Uncertainty measures for economics journal impact factors. Journal of Economic Literature. 2013;51:173–189. doi: 10.1257/jel.51.1.173. DOI

Tahira M, Alias RA, Bakri A. Scientometric assessment of engineering in Malaysian universities. Scientometrics. 2013;96:865–879. doi: 10.1007/s11192-013-0961-4. DOI

Tsallis C, de Albuquerque MP. Are citations of scientific papers a case of nonextensivity? European Physical Journal B. 2000;13(4):777–780. doi: 10.1007/s100510050097. DOI

Vanclay JK. On the robustness of the h-index. Journal of the American Society for Information Science and Technology. 2007;58:1547–1550. doi: 10.1002/asi.20616. DOI

Vanclay J. Ranking forestry journals using the h-index. Journal of Informetrics. 2008;2:326–334. doi: 10.1016/j.joi.2008.07.002. DOI

Vinkler P. The π-index: A new indicator for assessing scientific impact. Journal of Information Science. 2009;35:602–612. doi: 10.1177/0165551509103601. DOI

Vinkler P. Quantity and impact through a single indicator. Journal of the American Society for Information Science and Technology. 2013;64:1084–1085. doi: 10.1002/asi.22833. DOI

Waltman L. A review of the literature on citation impact indicators. Journal of Informetrics. 2016;10:365–391. doi: 10.1016/j.joi.2016.02.007. DOI

Wolfram R. Mathematica 10.0. Champaign, IL: Wolfram Research, Inc.; 2014.

Xu F, Liu WB, Mingers J. New journal classification methods based on the global h-index. Information Processing and Management. 2015;51:50–61. doi: 10.1016/j.ipm.2014.10.011. DOI

Ye FY. An investigation on mathematical models of the h-index. Scientometrics. 2009;81:493–498. doi: 10.1007/s11192-008-2169-6. DOI

Ye FY. Academic spectra: A visualization method for research assessment. Cybermetrics: International Journal of Scientometrics, Informetrics and Bibliometrics. 2010;14:1.

Zhao SX, Zhang PL, Li J, Tan AM, Ye FY. Abstracting the core subnet of weighted networks based on link strengths. Journal of the Association for Information Science and Technology. 2014;65:984–994. doi: 10.1002/asi.23030. DOI

The h-index as an almost-exact function of some basic statistics