Journal of Classification

, Volume 2, Issue 1, pp 77–108 | Cite as

On some significance tests in cluster analysis

  • H. H. Bock
Authors Of Articles


We investigate the properties of several significance tests for distinguishing between the hypothesisH of a “homogeneous” population and an alternativeA involving “clustering” or “heterogeneity,” with emphasis on the case of multidimensional observationsx1, ...,x n ε p . Four types of test statistics are considered: the (s-th) largest gap between observations, their mean distance (or similarity), the minimum within-cluster sum of squares resulting from a k-means algorithm, and the resulting maximum F statistic. The asymptotic distributions underH are given forn→∞ and the asymptotic power of the tests is derived for neighboring alternatives.


Significance test Homogeneity Heterogeneity Gap test Minimum within-cluster sum of squares Maximum F statistics Asymptotic normal distribution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BARNETT, V., KAY, R., and SNEATH, P.H.A. (1979), “A Familiar Statistic in an Unfamiliar Guise — A Problem in Clustering,”The Statistican, 28, 185–191.Google Scholar
  2. BAUBKUS, W. (1985), “Minimizing the Variance Criterion in Cluster Analysis: Optimal Configurations in the Multidimensional Normal Case,” Diplomarbeit, Institute of Statistics, Technical University Aachen, 117 p.Google Scholar
  3. BICKEL, P.J., and BREIMAN, L. (1983), “Sums of Functions of Nearest Neighbor Distances, Moment Bounds, Limit Theorems and a Goodness of Fit Test,”Annals of Probability, 11, 185–214.Google Scholar
  4. BINDER, D.A. (1978), “Bayesian Cluster Analysis,”Biometrika, 65, 31–38.Google Scholar
  5. BOCK, H.H. (1972), “Statistische Modelle und Bayes'sche Verfahren zur Bestimmung einer unbekannten Klassifikation normalverteilter zufälliger Vektoren,”Metrika, 18, 120–132.Google Scholar
  6. BOCK, H.H. (1974),Automatische Klassifikation. Theoretische und praktische Methoden zur Gruppierung und Strukturierung von Daten (Clusteranalyse), Göttingen: Vandenhoeck & Ruprecht, 480 p.Google Scholar
  7. BOCK, H.H. (1977), “On Tests Concerning the Existence of a Classification,” inProceedings First International Symposium on Data Analysis and Informatics, Le Chesnay, France, Institut de Recherche en Informatique et en Automatique (IRIA), 449–464.Google Scholar
  8. BOCK, H.H. (1981), “Statistical Testing and Evaluation Methods in Cluster Analysis,” inProceedings on the Golden Jubilee Conference in Statistics: Applications and New Directions, December 1981, Calcutta, Indian Statistical Institute, 1984, 116–146.Google Scholar
  9. BOCK, H.H. (1983), “Statistische Testverfahren im Rahmen der Clusteranalyse,”Proceedings of the 7th Annual Meeting of the Gesellschaft für Klassifikation e.V., inStudien zur Klassifikation, Vol. 13, ed. M. Schader, Frankfurt: Indeks-Verlag, 161–176.Google Scholar
  10. BRYANT, P., and WILLIAMSON, J.A. (1978), “Asymptotic Behavior of Classification Maximum Likelihood Estimates,”Biometrika, 65, 273–281.Google Scholar
  11. COX, D.R. (1957), “Note on Grouping,”Journal of the American Statistical Association, 52, 543–547.Google Scholar
  12. DAVID, H.A. (1981),Order Statistics, New York: Wiley, chap. 9.3, 9.4.Google Scholar
  13. DEGENS, P.O. (1978), “Clusteranalyse auf topologisch-masstheoretischer Grundlage,” Dissertation, Fachbereich Mathematik, Universitaet Muenchen.Google Scholar
  14. DEL PINO, G.E. (1979), “On the Asymptotic Distribution of k-spacings with Applications to Goodness-of-Fit Tests,”Annals of Statistics, 7, 1058–1065.Google Scholar
  15. DUBES, R., and JAIN, A.K. (1979), “Validity Studies in Clustering Methodologies,”Pattern Recognition, 11, 235–254.Google Scholar
  16. EBERL, W., and HAFNER, R. (1971), “Die asymptotische Verteilung von Koinzidenzen,”Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 18, 322–332.Google Scholar
  17. ENGELMAN, L., and HARTIGAN, J.A. (1969), “Percentage Points of a Test for Clusters,”Journal of the American Statistical Association, 64, 1647–1648.Google Scholar
  18. FLEISCHER, P.E. (1964), “Sufficient Conditions for Achieving Minimum Distortion in a Quantizer,”IEEE Int. Conv. Rec., part 1, 104–111.Google Scholar
  19. GHOSH, J.K., and SEN, P.K. (1984), “On the Asymptotic Distribution of the Log Likelihood Ratio Statistic for the Mixture Model and Related Results,” Preprint, Calcutta: Indian Statistical Institute.Google Scholar
  20. GIACOMELLI, F., WIENER, J., KRUSKAL, J.B., v. POMERANZ, J., and LOUD, A.V. (1971), “Subpopulations of Blood Lymphocytes Demonstrated by Quantitative Cytochemistry,”Journal of Histochemistry and Cytochemistry, 19, 426–433.Google Scholar
  21. GRAY, R.M., and KARNIN, E.D. (1982), “Multiple Local Optima in Vector Quantizers,”IEEE Trans. Information Theory, IT-28, 256–261.Google Scholar
  22. HARTIGAN, J.A. (1975),Clustering Algorithms, New York: Wiley.Google Scholar
  23. HARTIGAN, J.A. (1977), “Distribution Problems in Clustering,” inClassification and Clustering, ed. J. van Ryzin, New York: Academic Press, 45–72.Google Scholar
  24. HARTIGAN, J.A. (1978), “Asymptotic Distributions for Clustering Criteria,”Annals of Statistics, 6, 117–131.Google Scholar
  25. HENZE, N. (1981), “An Asymptotic Result on the Maximum Nearest Neighbor Distance Between Independent Random Vectors with an Application for Testing Goodness-of-Fit in ℝp on Spheres,” Dissertation, University of Hannover, published inMetrika, 30, 245–260.Google Scholar
  26. HENZE, N. (1982), “The Limit Distribution for Maxima of Weightedr-th Nearest Neighbor Distances,”Journal of Applied Probability, 19, 334–354.Google Scholar
  27. KIEFFER, J.C. (1983), “Uniqueness of Locally Optimal Quantizer for Log-concave Density and Convex Error Weighting Function,”IEEE Trans. Infromation, IT-29, 42–27.Google Scholar
  28. KUO, M., and RAO, J.S. (1981), “Limit Theory and Efficiences for Tests Based on Higher Order Spacings,” inProceedings on the Golden Jubilee Conference in Statistics: Applications and New Directions, December 1981, Calcutta: Indian Statistical Institute, 1984.Google Scholar
  29. LEE, K.L. (1979), “Multivariate Tests for Clusters,”Journal of the American Statistical Association, 74, 708–714.Google Scholar
  30. LEHMANN, E.L. (1955), “Ordered Families of Distributions,”Annals of Mathematical Statistics, 26, 399–419.Google Scholar
  31. LOEVE, M. (1963),Probability Theory, Princeton, NJ: van Nostrand.Google Scholar
  32. NEWELL, G.F. (1963), “Distribution for the Smallest Distance Between any Pair of the κ-th Nearest Neighbor Random Points on a Line,” inProc. Symp. Time Series Analysis, ed. M. Rosenblatt, New York: Wiley, 89–103.Google Scholar
  33. OGAWA, J. (1951), “Contributions to the Theory of Systematic Statistics I,”Osaka Mathematical Journal, 3, 175–213.Google Scholar
  34. OGAWA, J. (1962), “Determination of Optimum Spacings in the Case of Normal Distribution,” inContributions to Order Statistics, eds. A.E. Sarhan and B.G. Greenberg, New York: Wiley, p. 277 ff.Google Scholar
  35. PERRUCHET, C. (1982), “Les Epreuves de Classifiabilité en Analyse des Données,” Note technique NT/PAA/ATR/MTI/810, Issy-les-Moulineaux, France: Centre National d'Etudes de Télécommunications, September 1982.Google Scholar
  36. PERRUCHET, C. (1983), “Significance Tests for Clusters: Overview and Comments,” inNumerical Taxonomy, ed. J. Felsenstein, Berlin: Springer, 199–208.Google Scholar
  37. POLLARD, D. (1981), “Strong Consistency of k-means Clustering,”Annals of Statistics, 9, 135–140.Google Scholar
  38. POLLARD, D. (1982a), “A Central Limit Theorem for k-means Clustering,”Annals of Probability, 10, 919–926.Google Scholar
  39. POLLARD, D. (1982b), “Quantization and the Method of k-means,”IEEE Trans. Information Theory, IT-28, 119–205.Google Scholar
  40. RANDLES, R.H., and WOLFE, D.A. (1979),Introduction to the Theory of Non-parametric Statistics, New York: Wiley.Google Scholar
  41. SCHILLING, M.F. (1983a), “Goodness of Fit Testing in ℝm Based on the Weighted Empirical Distribution of Certain Nearest Neighbor Statistics,”Annals of Statistics, 11, 1–12.Google Scholar
  42. SCHILLING, M.F. (1983b), “An Infinite-dimensional Approximation for Nearest Neighbor Goodness of Fit,”Annals of Statistics, 11, 13–24.Google Scholar
  43. SILVERMAN, B.W. (1976), “Limit Theorems for Dissociated Random Variables,”Advances in Applied Probability, 8, 806–819.Google Scholar
  44. SNEATH, P.H.A. (1977a), “A Method for Testing the Distinctness of Clusters: A Test of the Disjunction of Two Clusters in Euclidean Space as Measured by their Overlap,”Jour. Int. Assoc. Math. Geol., 9, 123–143.Google Scholar
  45. SNEATH, P.H.A. (1977b), “Cluster Significance Tests and Their Relation to Measures of Overlap,” inProceedings First International Symposium on Data Analysis and Informatics, Versailles, September 1977, Institut de Recherche d'Informatique et d'Automatique (IRIA), Le Chesnay, France, 1, 15–36.Google Scholar
  46. SNEATH, P.H.A. (1979a), “The Sampling Distribution of the W Statistic of Disjunction for the Arbitrary Division of a Random Rectangular Distribution,”Journal. Int. Assoc. Math. Geol., 11, 423–429.Google Scholar
  47. SNEATH, P.H.A. (1979b), “Basic Program for a Significance Test for 2 Clusters in Euclidean Space as Measured by Their Overlap,”Computers and Geosciences, 5, 143–155.Google Scholar
  48. SPAETH, H. (1982),Cluster Analysis Algorithms, Chichester: Horwood.Google Scholar
  49. SPAETH, H. (1983),Cluster-Formation und -Analyse, München-Wien: Oldenbourg.Google Scholar
  50. TRUSHKIN, A.V. (1982), “Sufficient Conditions for Uniqueness of a Locally Optimal Quantizer for a Class of Convex Error Weighting Functions,”IEEE Trans. Information Theory, IT-28, 187–198.Google Scholar
  51. WALLENSTEIN, S.R., and NAUS, J.I. (1973), “Probabilities for ak-th Nearest Neighbor Problem on the Line,”Ann. Probab., 1, 188–190.Google Scholar
  52. WALLENSTEIN, S.R., and NAUS, J.I. (1974), “Probabilities of the Size of Largest Clusters and Smallest Intervals,”Journal of the American Statistical Association, 69, 690–697.Google Scholar
  53. WEISS, L. (1960), “A Test of Fit Based on the Largest Sample Spacing,”SIAM Journal of the Society for Industrial and Applied Mathematics, 8, 295–299.Google Scholar
  54. WITTING, H., and NOELLE, G. (1979),Angewandte Mathematische Statistik, Stuttgart: B.G. Teubner, theorem 2.10.Google Scholar
  55. WOLFE, J.H. (1970), “Pattern Clustering by Multivariate Mixture Analysis,”Multivariate Behavioral Research, 5, 329–350.Google Scholar
  56. WOLFE, J.H. (1981), “A Monte Carlo Study of the Sampling Distribution of the Likelihood Ratio for Mixture of Multinormal Distribution,” Technical Bulletin STB 72-2, San Diego: U.S. Naval Personal and Training Research Laboratory.Google Scholar
  57. WOLFE, S.J. (1975), “On the Unimodality of Spherically Symmetric Stable Distribution Functions,”Journal of Multivariate Analysis, 5, 236–242.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1985

Authors and Affiliations

  • H. H. Bock
    • 1
  1. 1.Institut für Statistik und WirtschaftsmathematikTechnical University AachenAachenWest Germany

Personalised recommendations