A Review on Heterogeneity Test: Some Permutation Procedures

  • Stefano Bonnini
  • Eleonora Carrozzo
  • Luigi SalmasoEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 274)


When dealing with categorical data, generally the notion of heterogeneity may be used instead of that of variability. There are many fields where data may be only represented by nominal categorical variables or by ordinal variables, e.g. opinion polls, performance qualitative assessments, psycho aptitude tests and so on. In this paper we provide a review of some nonparametric methods concerning testing for heterogeneity, based on permutation procedures. Examples of real applications in different frameworks are also shown.


Heterogeneity tests Permutation test Nonparametric framework 



The work was also partially supported by University of Ferrara, which funded the FIR (Research Incentive Fund) project “Advanced Statistical Methods for data analysis in complex problems”. The work was also partially supported by University of Padova BIRD185315/18.


  1. 1.
    Agresti, A., Klingenberg, B.: Multivariate tests comparing binomial probabilities, with application to safety studies for drugs. J. R. Stat. Soc. Ser. C (Appl. Stat.) 54, 691–706 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Al-Nasser, A.D.: Customer satisfaction measurment models: generalized maximum entropy approach. Pak. J. Stat. 19(2), 213–226 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ciavolino, E., Carpita, M.: The GME estimator for the regression model with a composite indicator as explanatory variable. Qual. Quant. 49(3), 955–965 (2015)CrossRefGoogle Scholar
  4. 4.
    Ciavolino, E., Al-Nasser, A.D.: Comparing generalized maximum entropy and partial least squares methods for structural equation models. J. Nonparametric Stat. 21(8), 1017–1036 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Doucet, H., Shah, M.K., Cummings, T.L., Kahm, M.J.: Comparison of internal medicine, pediatric and medicine/pediatric applicants and factors influencing career choices. South. Med. J. 92, 296–299 (1999)CrossRefGoogle Scholar
  6. 6.
    Frosini, B.V.: Heterogeneity indeces and distances between distributions. Metron 34, 95–108 (1981)zbMATHGoogle Scholar
  7. 7.
    Golan, A., Judge, G.G., Miller, D.: Maximum Entropy Econometrics: Robust Estimation with Limited Data. Wiley, New York (1996)zbMATHGoogle Scholar
  8. 8.
    Han, K.E., Catalano, P.J., Senchaudhuri, P., Mehta, C.: Exact analysis of dose-response for multiple correlated binary outcomes. Biometrics 4(60), 216–224 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hirotsu, C.: Cumulative chi-squared statistic as a tool for testing goodness-of-fit. Biometrika 73, 165–173 (1986)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620–630 (1957)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kvam, P.H., Vidakovic, B.: Nonparametric Statistics with Applications to Science and Engineering. Wiley, Hoboken (2007)CrossRefGoogle Scholar
  12. 12.
    Leti, G.: Sull’entropia, su un indice del Gini e su altre misure dell’eterogeneità di un collettivo. Metron 24, 332–378 (1965)MathSciNetGoogle Scholar
  13. 13.
    Loughin, T.M.: A Systematic comparison of methods for combining p-values from independent tests. Comput. Stat. Data Anal. 47, 467–485 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Loughin, T.M., Scherer, P.N.: Testing for association in contingency tables with multiple column responses. Biometrics 54, 630–637 (1998)CrossRefGoogle Scholar
  15. 15.
    Lumley, T.: Generalized estimating equations for ordinal data: a note on working correlation structures. Biometrics 52, 354–361 (1996)CrossRefGoogle Scholar
  16. 16.
    Nettleton, D., Banerjee, T.: Testing the equality of distributions of random vectors with categorical components. Comput. Stat. Data Anal. 37, 195–208 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Piccolo, D.: Statistica, 2nd edn. Il Mulino, Bologna (2000)Google Scholar
  18. 18.
    Pesarin, F., Salmaso, L.: Permutation tests for univariate and multivariate ordered categorical data. Aust. J. Stat. 35, 315–324 (2006)Google Scholar
  19. 19.
    Pesarin, F., Salmaso, L.: Permutation Tests for Complex Data. Theory, Applications and Software. Wiley, Chichester (2010)CrossRefGoogle Scholar
  20. 20.
    Shorack, G.R., Wellner, J.A.: Empirical Processes with Applications to Statistics. Wiley, New York (1986)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Stefano Bonnini
    • 1
  • Eleonora Carrozzo
    • 2
  • Luigi Salmaso
    • 2
    Email author
  1. 1.Department of Economics and ManagementUniversity of FerraraFerraraItaly
  2. 2.Department of Management and EngineeringUniversity of PadovaPaduaItaly

Personalised recommendations