Relative difference in diversity between populations

  • Khursheed Alam
  • Calvin L. Williams


An entropy is conceived as a functional on the space of probability distributions. It is used as a measure of diversity (variability) of a population. Cross entropy leads to a measure of dissimilarity between populations. In this paper, we provide a new approach to the construction of a measure of dissimilarity between two populations, not depending on the choice of an entropy function, measuring diversity. The approach is based on the principle of majorization which provides an intrinsic method of comparing the diversities of two populations. We obtain a general class of measures of dissimilarity and show some interesting properties of the proposed index. In particular, it is shown that the measure provides a metric on a probability space. The proposed measure of dissimilarity is essentially a measure of relative difference in diversity between two populations. It satisfies an invariance property which is not shared by other measures of dissimilarity which are used in ecological studies. A statistical application of the new method is given.

Key words and phrases

Diversity dissimilarity cross entropy majorization Schur-convexity ranking and selection 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Agresti, A. (1990).Categorical Data Analysis, Wiley Series in Probability and Mathematical Statistics, Wiley, New York.Google Scholar
  2. Alam, K., Mitra, A., Rizvi, M. H. and Saxena, K. M. Lal (1986). Selection of the most diverse multinomial population,Amer. J. Math. Management Sci., Special Volume6, 65–86.Google Scholar
  3. Dennis, B., Patil, G. P., Rossi, O. and Taillie, C. (1979). A bibliography of literature on ecological diversity and related methodology,Ecological Diversity in Theory and Practice, 319–354, International Co-operative Publishing House, Jerusalem.Google Scholar
  4. Dudewicz, E. J. and Van der Meulen, E. C. (1981). Selection procedures for the best binomial population with generalized entropy goodness,Tamkang J. Math.,12, 206–208.Google Scholar
  5. Gini, C. (1912). Variabilitá e Mutabilitá, Studi Economico-Giuridici della facoltá di Giurisprodenza dell, Universitá di Cagliari, Anno 3, Part 2, p. 80.Google Scholar
  6. Gower, J. C. (1985). Measures of similarity, dissimilarity, and distance,Encyclopedia of Statistical Sciences (eds. S. Kotz and N. L. Johnson),5, 397–405, Wiley-Interscience, New York.Google Scholar
  7. Grassle, J. F. and Smith, W. K. (1976). A similarity measure sensitive to the contribution of rare species and its use in investigation of variation in marine benthic communities,Oecologia,25, 13–25.Google Scholar
  8. Gupta, S. S. and Huang, D. Y. (1976). On subset selection procedures for the entropy function associated with the binomial populations,Sankhyā Ser. A,38, 153–173.Google Scholar
  9. Gupta, S. S. and Wong, W. Y. (1975). Subset selection procedures for finite schemes in information theory,Colloq. Math. Soc. János Bolyai,16, 279–291.Google Scholar
  10. Haberman, S. J. (1982). Analysis of dispersion of multinomial responses,J. Amer. Statist. Assoc.,77, 568–580.Google Scholar
  11. Kostreva, M. M. (1989). Generalization of Murty's direct algorithm to linear and convex quadratic programming,J. Optim. Theory Appl.,62, 63–76.Google Scholar
  12. Light, R. J. and Margolin, B. H. (1971). An analysis of variance for categorical data,J. Amer. Statist. Assoc.,66, 534–544.Google Scholar
  13. Lorenz, M. O. (1905). Methods of measuring concentration of wealth,J. Amer. Statist. Assoc.,9, 209–212.Google Scholar
  14. Marshall, A. V. and Olkin, I. (1979).Inequalities: Theory of Majorization and Its Applications, Academic Press, San Diego.Google Scholar
  15. Patil, G. P. and Taillie, C. (1982). Diversity as a concept and its measurement,J. Amer. Statist. Assoc.,77, 548–561.Google Scholar
  16. Rao, C. R. (1982a). Diversity and dissimilarity coefficients, a unified approach,Theoret. Population Biol.,21, 24–43.Google Scholar
  17. Rao, C. R. (1982b). Diversity: its measurement, decomposition, appartionment and analysis,Sankhyã Ser. A,44, 1–22.Google Scholar
  18. Rao, C. R. (1984). Convexity properties of entropy functions and analysis of diversity,Inequalities in Statistics and Probability, IMS Lecture Notes - Monograph Series, Vol. 5, 68–77, Hayward, California.Google Scholar
  19. Rao, C. R. and Nayak, T. K. (1985). Cross entropy, dissimilarity measures, and characterizations of quadratic entropy,IEEE Trans. Inform. Theory,31 (5), 589–593.Google Scholar
  20. Rizvi, M. H., Alam, K. and Saxena, K. M. Lal (1987). Selection procedure for multinomial populations with respect to diversity indices,Contribution to the Theory and Application of Statistics (ed. A. E. Gelfard), Academic Press, New York.Google Scholar
  21. Schmidt, P. and Strauss, R. P. (1975). The prediction of occupation using multiple logit models,Internat. Econom. Rev.,16, 471–486.Google Scholar
  22. Schur, I. (1923). Uber eine Klasse von Mittelbildungen mit Anwendungen die Determinanten,Theorie Sitzungsber. Berlin Math. Gesellschaft,22, 9–20 (Issai Collected Works (eds. A. Brauer and H. Rohrbach), Vol. II, 416–427, Springer, Berlin, 1973).Google Scholar
  23. Shannon, C. E. (1948). A mathematical theory of communication,Bell Syst. Tech. J.,27, 379–423 and 626–656.Google Scholar
  24. Simpson, E. H. (1949). Measurement of diversity,Nature,163, 688.Google Scholar
  25. Smith, W. (1989). ANOVA-like similarity analysis using expected species shared,Biometrics,45, 873–881.Google Scholar
  26. Smith, W., Grassle, J. F. and Kravitz, D. (1979). Measures of diversity with unbiased estimators,Ecological Diversity in Theory and Practice, 177–191, International Co-operative Publishing House, Jerusalem.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 1993

Authors and Affiliations

  • Khursheed Alam
    • 1
  • Calvin L. Williams
    • 1
  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA

Personalised recommendations