Download PDF
Methods Inf Med 1968; 07(03): 187-200
DOI: 10.1055/s-0038-1636297
Original Article
Schattauer GmbH

Einige χ2-Tests zur Hypothesenprüfung bei Kontingenztafeln [*])

SOME χ-TESTS FOR TESTING HYPOTHESES IN CONTINGENCY TABLES
H. J. Jesdinsky
1   Aus dem Institut für Medizinische Statistik und Dokumentation der Universität Freiburg/Br. (Direktor: Prof. Dr. E. Walter)
› Author Affiliations
Further Information

Publication History

Publication Date:
17 February 2018 (online)

    Permissions and Reprints

Die vorliegende TJbersichtsarbeit behandelt Unabhängigkeitstests für Kontingenztafeln. Ausführlich dargestellt werden Aufteilungen der Testgrößen entsprechend bestimmter Vergleiche bei rXs-Tafeln sowie die Auswertung der 2 X 2 X t-Tafel. Die bei multiplen Vergleichen auftauchenden Schwierigkeiten werden berücksichtigt. An einer Reihe von medizinischen Beispielen werden die praktischen Folgerungen aus den Testergebnissen eingehend besprochen.

This review deals with tests of independence for contingency tables. The partition of the test statistic in rXs tables for certain contrasts and the treatment of a 2 X 2 X t table are given in detail. Difficulties arising in multiple comparisons are considered. For some medical examples the practical conclusions from the statistical results are discussed.

* Herrn Prof. Dr. Dr. S. KOLLER zum 60. Geburtstag gewidmet.


 

  • Literaturverzeichnis

  • 1 Ahmitage P. Tests for linear trends in proportions and frequencies. Biometrics 33: 375-386 1955;
    PubMedGoogle Scholar
  • 2 Ahmitage P. The chi-square test for heterogeneity of proportions after adjustment for stratification. J. roy. statist. Soc. B, 28: 150-163 1966;
    PubMedGoogle Scholar
  • 3 Bahnard G. D. Significance for 2X2 tables. Biometrika 34: 123-138 1947;
    PubMedGoogle Scholar
  • 4 Bahtlett M. S. Contingency table interactions. J. roy. statist. Soc. Suppl 2: 248-252 1935;
    CrossrefPubMedGoogle Scholar
  • 5 Bahtlett M. S, and Kendall D. G. The statistical analysis of variance-heterogeneity and the logarithmic transformation. J. roy. statist. Soc. Suppl 8: 128-138 1946;
    CrossrefPubMedGoogle Scholar
  • 6 Bennett B. M, and Nakamuha E. Tables for testing significance in a 2X3 contingency table. Technometrics 5: 501-511 1963;
    CrossrefPubMedGoogle Scholar
  • 7 Bennett B. M, and Nakamuha E. The power function of the exact test for the 2X3 contingency table. Technometrics 6: 439-458 1964;
    PubMedGoogle Scholar
  • 8 Bennett C. A, and Fhanklin N. L. Statistical Analysis. pp. 602 615 ff. Wiley; New York: 1963
    Google Scholar
  • 9 Bennett R. W. Sizes of the χ2 test in the multinomial distribution. Austr. J. Statist 4: 86-88 1962;
    PubMedGoogle Scholar
  • 10 Cochran W. G. The χ2 test of goodness of fit. Ann. math. Statist 23: 315-345 1952;
    CrossrefPubMedGoogle Scholar
  • 11 Cochran W. G. Some methods of strengthening the common χ2 tests. Biometrics 10: 417-451 1954;
    CrossrefPubMedGoogle Scholar
  • 12 Cox D. R. Some statistical methods connected with series of events. J. roy. statist. Soc. B, 17: 129-157 1955;
    PubMedGoogle Scholar
  • 13 Daly C. A simple test for trends in a contingency table. Biometrics 18: 114-119 1962;
    CrossrefPubMedGoogle Scholar
  • 14 Darroch J. N. Interactions in multi-factor contingency tables. J. roy. statist. Soc. B, 24: 251-263 1962;
    PubMedGoogle Scholar
  • 15 Dyke G. V, and Patterson H. D. Analysis of factorial arrangements when the data are proportions. Biometrics 8: 1-12 1952;
    CrossrefPubMedGoogle Scholar
  • 16 Egg K, Rüst H, und van der Waerden B. L. Die Irrtumswahrscheinlichkeit des χ2-Tests im Grenzfall der Pois-sonverteilunrr. Z. Wahrscheinlichkeitstheorie verw. Geb 4: 260-264 1965;
    CrossrefPubMedGoogle Scholar
  • 17 Finney D. J. Probit Analysis. 2. Edit. University Press; Cambridge: 1962
    Google Scholar
  • 18 Fisher R. A, and Yates F. Statistical tables. 6. Edit., pp. 74-77. Oliver & Boyd; Edinburgh: 1963
    Google Scholar
  • 19 Gart J. J. Alternative analyses of contingency tables. J. roy. statist. Soc. B, 28: 164-179 1966;
    PubMedGoogle Scholar
  • 20 Good I. J. Maximum entropy for hypothesis formulation, especially for multidimensional contingency tables. Ann. math. Statist 34: 911-934 1963;
    CrossrefPubMedGoogle Scholar
  • 21 Goodman L. A. On methods for comparing contingency tables. J. roy. statist. Soc. A, 326: 94-108 1963;
    PubMedGoogle Scholar
  • 22 Goodman L. A. On Plackett’s test for continrrencv table interactions. J. roy. statist. Soc. B, 25: 179-188 1963;
    PubMedGoogle Scholar
  • 23 Goodman L. A. Simple methods for analysing three-factor interactions in contingency tables. J. Amer. statist. Ass 59: 319-352 1964;
    CrossrefPubMedGoogle Scholar
  • 24 Goodman L. A. Interactions in multidimensional contingency tables. Ann. math. Statist 35: 632-646 1964;
    CrossrefPubMedGoogle Scholar
  • 25 Goodman L. A. Simultaneous confidence intervals for contrasts among multinomial populations. Ann. math. Statist 35: 716-725 1964;
    CrossrefPubMedGoogle Scholar
  • 26 Goodman L. A, and Kruskal W. H. Measures of association for cross classifications I, II, III. J. Amer. statist. Ass. 49: 732-764 1954; 54: 123-163, 1959; 58: 310-364, 1963.
    PubMedGoogle Scholar
  • 27 Grosse H. Durch »Berkson’s fallacy« vorgetäuschte negative Korrelationen der Arteriosklerose. Z. ges. inn. Med 39: 566-569 1964;
    PubMedGoogle Scholar
  • 28 Irwin J. O. A note on the subdivision of %2 into components. Biometrika 36: 130-134 1949;
    PubMedGoogle Scholar
  • 29 Kastenbaum M. A, and Lamphiear D. E. Calculation of χ2 to test the no three-factor interaction hypothesis. Biometrics 35: 107-115 1959;
    PubMedGoogle Scholar
  • 30 Kastenbaum M. A. A note on the additive partitioning of chi-square in contingency tables. Biometrics 36: 416-422 1960;
    PubMedGoogle Scholar
  • 31 Kimball A. W. Short-cut formulas for the exact partition of χ2 in contingency tables. Biometrics 30: 452-458 1954;
    PubMedGoogle Scholar
  • 32 Koller S. Systematik der statistischen Schlufifehler. Method. Inform. Med 3: 113-117 1964;
    PubMedGoogle Scholar
  • 33 Kullback S, Kuppermann M, and Ku H. H. Tests of contingency tables and Markov chains. Technometrics 4: 573-608 1962;
    PubMedGoogle Scholar
  • 34 Kullback S. Information Theory and Statistics; pp. 155-188. Wiley; New York: 1959
    Google Scholar
  • 35 Ku H. H. A note on contingency tables involving zero frequencies and the 2 i test. Technometrics 5: 398-400 1963;
    PubMedGoogle Scholar
  • 36 Lancaster H. O. The exact partition of χ2 and its application to the problem of pooling of small expectations. Biometrika 37: 267-270 1950;
    PubMedGoogle Scholar
  • 37 Lancaster H.O. Complex contingency tables treated by the partition of χ2 . J. roy. statist. Soc. B, 33: 242-249 1951;
    PubMedGoogle Scholar
  • 38 Lancaster H. O. On tests of independence in several dimensions. J. Aust. math. Soc 3: 241-254 1960;
    PubMedGoogle Scholar
  • 39 Lancaster H. O, and Hamdan M. A. Estimation of the correlation coefficient in contingency tables with possibly non-metrical characters. Psychometrika 29: 383-391 1964;
    CrossrefPubMedGoogle Scholar
  • 40 Lange H. J. Syntropie von Krankheiten. Method. Inform. Med 4: 141-145 1965;
    PubMedGoogle Scholar
  • 41 Lewis B. N. On the analysis of interaction in multidimensional contingency tables. J. roy. statist. Soc. A, 325: 88-117 1962;
    PubMedGoogle Scholar
  • 42 Lewontin R. C, and Felsenstein J. The robustness of homogeneity tests in 2Xn tables. Biometrics 23: 19-33 1965;
    PubMedGoogle Scholar
  • 43 Mainland D. Elementary medical statistics; pp. 117 ff. Saunders; Philadelphia: 1963
    Google Scholar
  • 44 Maxwell A. E. Analysing qualitative data. Methuen; London: 1961
    Google Scholar
  • 45 Neyman J. Contributions to the theory of the χ2 test. Proceedings of the 1st Berkeley Symposium on Mathematical Statistics and Probability. pp. 239-273. University of California Press; Berkeley: 1949
    Google Scholar
  • 46 Norton H. W. Calculation of chi-square for complex contingency tables. J. Amer. statist. Ass 40: 251-258 1945;
    CrossrefPubMedGoogle Scholar
  • 47 Patau K. Eine neue χ2-Tafel. z. Indukt. Abstamm.- u. Vererb.-L 80: 558-564 1942;
    PubMedGoogle Scholar
  • 48 Pearson K. On the criterion that a given system of deviations is such that it can be reasonable supposed to have arisen from random sampling. Philos. Mag. Ser. B 50: 157-172 1900; zit. nach (56).
    PubMedGoogle Scholar
  • 49 Pfanzagl J. Tests und Konfidenzintervalle fur exponentielle Verteilungen und deren Anwendung auf einige diskrete Verteilungen. Metrika 3: 1-25 1960;
    CrossrefPubMedGoogle Scholar
  • 50 Plackett R. L. A note on interactions in contingency tables. J. roy. statist. Soc. B, 24: 162-166 1962;
    PubMedGoogle Scholar
  • 51 Roy S. N. Some aspects of multivariate analysis. pp. 113-134. Eka Press; Calcutta: 1958
    Google Scholar
  • 52 Roy S. N, and Bhafkar V. P. Some nonparametric analogs of ‘normal’ anova, manova, and of studies in ‘normal’ association. In Olkin I. (Edit.) Contributions to Probability and Statistics. pp. 371-387. Stanford University Press; Stanford: 1960
    Google Scholar
  • 53 Roy S. N, and Kastenbaum M. A. On the hypothesis of ‘no interaction’ in a multiway contingency table. Ann. math. Statist 27: 749-757 1956;
    CrossrefPubMedGoogle Scholar
  • 54 Roy S. N, and Mitra S. K. An introduction to some non-parametric generalizations of ANOVA in multivariate analysis. Biometrika 43: 361-376 1956;
    CrossrefPubMedGoogle Scholar
  • 55 Sachs L. Der Vergleich zweier Prozentsätze — Unabhängigkeitsteste für Mehrfeldertafeln. Biometr. Z 7: 55-60 1965;
    CrossrefPubMedGoogle Scholar
  • 56 Schäffer K. A. Der Likelihood-Anpassungstest. Mittbl. math. Statist S: 27-54 1957;
    PubMedGoogle Scholar
  • 57 Scheffe H. A method for judging all contrasts in the analysis of variance. Biometrika 40: 87-104 1953;
    PubMedGoogle Scholar
  • 58 Schneider B. Probitmodell und Logitmodell in ihrer Bedeutung für die experimentelle Prüfung von Arzneimitteln. Antibiot. et Chemother 12: 271-286 1964;
    PubMedGoogle Scholar
  • 59 Simpson E. H. The interpretation of interaction in contingency tables. J. roy. statist. Soc. B, 33: 238-241 1951;
    PubMedGoogle Scholar
  • 60 Snedecor G. W. Chi-squares of Bartlett, Mood, and Lancaster in a 23 contingency table. Biometrics 14: 560-562 1958;
    CrossrefPubMedGoogle Scholar
  • 61 Tallis G. M. The use of models in the analysis of some classes of contingency tables. Biometrics 20: 832-839 1964;
    CrossrefPubMedGoogle Scholar
  • 62 Tukey J. W. Comparing individual means in the analysis of variance. Biometrics 5: 99-114 1949;
    CrossrefPubMedGoogle Scholar
  • 63 Wagner G. Bedeutung, Grenzen und Gefahren der Statistik in der Medizin. Dtsch. med. Wschr 82: 1427-1432 und 1484-1491, 1957;
    Article in Thieme ConnectPubMedGoogle Scholar
  • 64 Wilks S. S. The likelihood test of independence in contingency tables. Ann. math. Statist 6: 190-196 1935;
    CrossrefPubMedGoogle Scholar
  • 65 Wilks S. S. Mathematical Statistics. pp. 290-291. Wiley; New York: 1962
    Google Scholar