Statistical Papers

, Volume 55, Issue 4, pp 1059–1077 | Cite as

Divergence-based tests of homogeneity for spatial data

  • Tomáš HobzaEmail author
  • Domingo Morales
  • Leandro Pardo
Regular Article


The problem of testing homogeneity in contingency tables when the data are spatially correlated is considered. We derive statistics defined as divergences between unrestricted and restricted estimated joint cell probabilities and we show that they are asymptotically distributed as linear combinations of chi-square random variables under the null hypothesis of homogeneity. Monte Carlo simulation experiments are carried out to investigate the behavior of the new divergence test statistics and to make comparisons with the statistics that do not take into account the spatial correlation. We show that some of the introduced divergence test statistics have a significantly better behavior than the classical chi-square test for the problem under consideration when we compare them on the basis of the simulated sizes and powers.


Test of homogeneity Divergence statistics Chi-square statistic Spatial data 



The authors thank the referees for careful reading of the paper and many interesting improvements. Supported by the Grants MTM2012-37077-C02-01, MTM2012-33740 and SGS12/197/OHK4/3T/14.


  1. Ali SM, Silvey SD (1966) A general class of coefficient of divergence of one distribution from another. J R Stat Soc Ser B 286:131–142MathSciNetGoogle Scholar
  2. Basu A, Shioya H, Park C (2011) Statistical inference: the minimum distance approach. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  3. Bhatia R (1997) Matrix analysis. Springer, New YorkCrossRefGoogle Scholar
  4. Cerioli A (1997) Modified tests of independence in \(2\times 2\) tables with spatial data. Biometrics 53:619–628CrossRefzbMATHGoogle Scholar
  5. Cerioli A (2002a) Test of homogeneity for spatial populations. Stat Probab Lett 58:123–130MathSciNetCrossRefzbMATHGoogle Scholar
  6. Cerioli A (2002b) Testing mutual independence between two discrete-valued spatial processes: a correction to Pearson chi-squared. Biometrics 58:888–897MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cressie N (1993) Statistics for spatial data. Wiley, New YorkGoogle Scholar
  8. Csiszár I (1963) Eine Informationstheoretische Ungleidung und ihre Anwendung auf den Bewis del Ergodizitt on Markhoffschen Ketten. Publ Math Inst Hung Acad Sci 8:84–108Google Scholar
  9. Dale MRT, Fortin MJ (2009) Spatial autocorrelation and statistical tests: some solutions. J Agric Biol Environ Stat 14(2):188–206MathSciNetCrossRefGoogle Scholar
  10. Fingleton B (1983) Independence, stationarity, categorical spatial data and the chi-squared test. Environ Plan 15(4):483–499CrossRefGoogle Scholar
  11. Haining R (2003) Spatial data analysis. Theory and practice. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  12. Hobza T, Morales D, Pardo L (2009) Rényi statistics for testing equality of autocorrelation coefficients. Stat Methodol 6:424–436MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kullback S (1959) Information theory and statistics. Wiley, New YorkzbMATHGoogle Scholar
  14. Kullback S, Leibler R (1951) On information and sufficiency. Ann Math Stat 22:79–86MathSciNetCrossRefzbMATHGoogle Scholar
  15. Liese F, Vajda I (1987) Convex statistical distances. Teubner, LeipzigzbMATHGoogle Scholar
  16. Landaburu E, Pardo L (2005) Matching moments for a closer approximation of the weighted \((h,\phi )\)-divergence test statistics in goodness-of-fit for finite samples. J Frankl Inst 342:115–129MathSciNetCrossRefzbMATHGoogle Scholar
  17. Lindley DV (1956) On a measure of the information provided by an experiment. Ann Math Stat 27:986–1005MathSciNetCrossRefzbMATHGoogle Scholar
  18. Menéndez ML, Morales D, Pardo L, Salicrú M (1995a) Asymptotic behaviour and statistical applications of divergence measures in multinomial populations: a unified approach. Stat Pap 36:1–29CrossRefzbMATHGoogle Scholar
  19. Menéndez ML, Morales D, Pardo L, Vajda I (1995b) Divergence based estimation and testing of statistical models of classification. J Multivar Anal 54(2):329–354CrossRefzbMATHGoogle Scholar
  20. Menéndez ML, Morales D, Pardo L, Vajda I (1998) Two approaches to grouping of data and related disparity statistics. Commun Stat (Theory and Methods) 27(3):609–633CrossRefzbMATHGoogle Scholar
  21. Menéndez ML, Pardo JA, Pardo L, Zografos K (2003) On tests of homogeneity based on minimum \(\phi \)-divergence estimator with constraints. Comput Stat Data Anal 43:215–234CrossRefzbMATHGoogle Scholar
  22. Menéndez ML, Pardo JA, Pardo L, Zografos K (2005) On tests of symmetry, marginal homogeneity and quasi-symmetry in two contingency tables based on minimum \(\phi \)-divergence estimator with constraints. J Stat Comput Simul 75(7):555–580MathSciNetCrossRefzbMATHGoogle Scholar
  23. Menéndez ML, Pardo JA, Pardo L, Zografos K (2007) A test for marginal homogeneity based on a \(\phi \)-divergence statistic. Appl Math Lett 20(1):7–12MathSciNetCrossRefzbMATHGoogle Scholar
  24. Morales D, Pardo L, Pardo MC (2001) Likelihood divergence statistics for sending hypotheses about multiple populations. Commun Stat Simul Comput 30:867–884MathSciNetCrossRefzbMATHGoogle Scholar
  25. Pardo L, Pardo MC, Zografos K (1999) Homogeneity for multinomial populations based on \(\phi \)-divergences. J Japan Stat Soc 29(2):213–228MathSciNetCrossRefzbMATHGoogle Scholar
  26. Pardo L (2006) Statistical inference based on divergence measures. Chapman & Hall, Baco RatonzbMATHGoogle Scholar
  27. Parr WC (1981) Minimum distance estimation: a bibliography. Commun Stat (Theory and Methods) 10:1205–1224MathSciNetCrossRefGoogle Scholar
  28. Read TRC, Cressie NAC (1988) Goodness of fit statistics for discrete multivariate data. Springer, New YorkCrossRefzbMATHGoogle Scholar
  29. Rényi A (1961) On measures of entropy and information. In: Fourth Berkeley symposium on mathematics, statistics and probability, vol. 1, pp. 547–561. University California Press, BerkeleyGoogle Scholar
  30. Salicrú M, Morales D, Menéndez ML, Pardo L (1994) On the applications of divergence type measures in testing statistical hypotheses. J Multivar Anal 51:372–391CrossRefzbMATHGoogle Scholar
  31. Shannon CE (1948) The mathematical theory of communicaion. Bell Syst Tech J 27:379–423MathSciNetCrossRefzbMATHGoogle Scholar
  32. Tavaré S (1983) Serial dependence in contingency tables. J R Stat Soc Ser B 45:100–106zbMATHGoogle Scholar
  33. Vajda I (1989) Theory of statistical inference and information. Kluwer, DordrechtzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsCzech Technical University in PraguePrague 2Czech Republic
  2. 2.Operations Research CenterMiguel Hernández University of ElcheElcheSpain
  3. 3.Department of Statistics and Operations ResearchComplutense University of MadridMadridSpain

Personalised recommendations