, Volume 116, Issue 1, pp 645–653 | Cite as

The repeat rate: from Hirschman to Stirling

  • Ronald RousseauEmail author


In this short note we recall the history and definition of the repeat rate, also known as the Hirschman–Herfindahl index or as the Simpson index, and show that its generalization to a measure that includes disparity between items, known as the Rao-Stirling index, or a monotone transformation of it, is an acceptable diversity measure which, however, does not meet the ‘monotonicity of balance’ requirement.


Repeat rate Hirschman index Simpson index Herfindahl index Rao-Stirling index Measuring diversity Interdisciplinarity 



We thank Loet Leydesdorff for helpful observations about an earlier version of this note.


  1. Good, I. J. (1982). Comment [on Patil & Taillie, 1982]. Journal of the American Statistical Association, 77(379), 561–563.Google Scholar
  2. Herdan, G. (1966). The Advanced theory of language as choice and chance. Berlin: Springer-Verlag.CrossRefzbMATHGoogle Scholar
  3. Herfindahl, O. C. (1950). Concentration in the U.S. steel industry. Doctoral dissertation, Columbia University.Google Scholar
  4. Hill, M. (1973). Diversity and evenness: A unifying notation and its consequences. Ecology, 54(2), 427–432.CrossRefGoogle Scholar
  5. Hirschman, A. O. (1945). National power and the structure of foreign trade. Berkeley: University of California Press.Google Scholar
  6. Hirschman, A. O. (1964). The paternity of an index. The American Economic Review, 54(5), 761–762.Google Scholar
  7. Jost, L. (2006). Entropy and diversity. Oikos, 113(2), 363–375.CrossRefGoogle Scholar
  8. Jost, L. (2007). Partitioning diversity into independent alpha and beta components. Ecology, 88(10), 2427–2439.CrossRefGoogle Scholar
  9. Jost, L. (2009). Mismeasuring biological diversity: Response to Hoffmann and Hoffmann (2008). Ecological Economics, 68(4), 925–928.CrossRefGoogle Scholar
  10. Junge, K. (1994). Diversity of ideas about diversity measurement. Scandinavian Journal of Psychology, 35(1), 16–26.CrossRefGoogle Scholar
  11. Leinster, T., & Cobbold, C. A. (2012). Measuring diversity: The importance of species similarity. Ecology, 93(3), 477–489.CrossRefGoogle Scholar
  12. Leydesdorff, L. (2015). Can technology life-cycles be indicated by diversity in patent classifications? The crucial role of variety. Scientometrics, 105(3), 1441–1451.CrossRefGoogle Scholar
  13. Leydesdorff, L., Wagner, C. S., & Bornmann, L. (2018). Betweenness and diversity in journal citation networks as measures of interdisciplinarity—A tribute to Eugene Garfield. Scientometrics, 114(2), 567–592.CrossRefGoogle Scholar
  14. Mugabushaka, A.-M., Kyriakou, A., & Papazoglou, T. (2015). Bibliometric indicators of interdisciplinarity exploring new class of diversity measures. In A. A. Salah, Y. Tonta, A. A. A. Salah, C. Sugimoto, & U. Al (Eds.), Proceedings of ISSI 2015 (pp. 397–402). Istanbul: Boğaziçi University Printhouse.Google Scholar
  15. Nei, M., & Li, W. H. (1979). Mathematical model for studying genetic variation in terms of restriction endonucleases. Proceedings of the National Academy of Science USA, 76(10), 5269–5273.CrossRefzbMATHGoogle Scholar
  16. Nijssen, D., Rousseau, R., & Van Hecke, P. (1998). The Lorenz curve: A graphical representation of evenness. Coenoses, 13(1), 33–38.Google Scholar
  17. Patil, G. P., & Taillie, C. (1982). Diversity as a concept and its measurement. Journal of the American Statistical Association, 77(379), 548–561.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Pavoine, S. (2012). Clarifying and developing analyses of biodiversity: Towards a generalisation of current approaches. Methods in Ecology and Evolution, 3(3), 509–518.CrossRefGoogle Scholar
  19. Rafols, I., Leydesdorff, L., O’Hare, A., Nightingale, P., & Stirling, A. (2012). How journal rankings can suppress interdisciplinary research: A comparison between Innovation Studies and Business & Management. Research Policy, 41(7), 1262–1282.CrossRefGoogle Scholar
  20. Rafols, I., & Meyer, M. (2010). Diversity and network coherence as indicators of interdisciplinarity: Case studies in bionanoscience. Scientometrics, 82(2), 263–287.CrossRefGoogle Scholar
  21. Rao, C. R. (1982). Diversity and dissimilarity coefficients: A unified approach. Theoretical Population Biology, 21(1), 24–43.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Ray, J. L., & Singer, J. D. (1973). Measuring the concentration of power in the international system. Sociological Methods and Research, 1(4), 403–437.CrossRefGoogle Scholar
  23. Ricotta, C., & Szeidl, L. (2006). Towards a unifying approach to diversity measures: Bridging the gap between the Shannon entropy and Rao’s quadratic index. Theoretical Population Biology, 70(3), 237–243.CrossRefzbMATHGoogle Scholar
  24. Rousseau, R., Hu, X. J., & Zhang, L. (2018). Knowledge integration: Its meaning and measurement. In: W. Glänzel, H. Moed, U. Schmoch, & M. Thelwall (Eds.), Springer Handbook of Science and Technology Indicators. (To appear).Google Scholar
  25. Simpson, E. H. (1949). Measurement of diversity. Nature, 163(4148), 688.CrossRefzbMATHGoogle Scholar
  26. Smouse, P. E., Banks, S. C., & Peakall, R. (2017). Converting quadratic entropy to diversity: Both animals and alleles are diverse, but some are more diverse than others. PLoS ONE, 12(10), e0185499.CrossRefGoogle Scholar
  27. Stirling, A. (2007). A general framework for analysing diversity in science, technology and society. Journal of the Royal Society, Interface, 4(15), 707–719.CrossRefGoogle Scholar
  28. Wagner, C. S., Roessner, J. D., Bobb, K., Klein, J. T., Boyack, K. W., Keyton, J., et al. (2011). Approaches to understanding and measuring interdisciplinary scientific research (IDR): A review of the literature. Journal of Informetrics, 5(1), 14–26.CrossRefGoogle Scholar
  29. Zhang, L., Rousseau, R., & Glänzel, W. (2016). Diversity of references as an indicator for interdisciplinarity of journals: Taking similarity between subject fields into account. Journal of the Association for Information Science and Technology, 67(5), 1257–1265.CrossRefGoogle Scholar
  30. Zhou, Q. J., Rousseau, R., Yang, L. Y., Yue, T., & Yang, G. L. (2012). A general framework for describing diversity within systems and similarity between systems with applications in informetrics. Scientometrics, 93(3), 787–812.CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Faculty of Social SciencesUniversity of AntwerpAntwerpBelgium
  2. 2.Facultair Onderzoekscentrum ECOOMKU LeuvenLouvainBelgium

Personalised recommendations