Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T09:23:02.770Z Has data issue: false hasContentIssue false
  • English
  • Français

On Tests of independence in several dimensions

Published online by Cambridge University Press:  09 April 2009

H. O. Lancaster
H. O. Lancaster
Affiliation:
Department of Mathematical Statistics, University of Sydney
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Introductory. This paper considers a canonical form, or rather a class of canonical forms, for three dimensional probability distributions subject to a rather mild restriction. These canonical forms are used to develop suitable tests of independence and lead to a consideration of the partition of χ2 in the analysis of complex contingency tables. Where these methods and Bartlett's are both applicable it is shown that they give comparable results; but the partitioning methods are more general.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 1 , Issue 2 , February 1960 , pp. 241 - 254
Copyright
Copyright © Australian Mathematical Society 1960

References

[1]Barnard, G. A., Significance tests for 2 x 2 tables, Biometrika 34, (1947) 123–138.Google ScholarPubMed
[2]Bartlett, M. S., Contingency table interactions, J. roy. statist. Soc. Suppl. 2 (1935), 248–252.Google Scholar
[3]Bartlett, M. S., Properties of sufficiency and statistical tests, Proc. Roy. Soc. A, 160 (1936), 268–282.Google Scholar
[4]Cochran, W. G., The analysis of variance when experimental errors follow the Poisson or binomial laws, Ann. math. Statist. 11 (1940), 335–347.CrossRefGoogle Scholar
[5]Dyke, G. V. and Patterson, H. D., Analysis of factorial arrangements when the data are Proportions, Biometrics, 8 (1952), 1–12.CrossRefGoogle Scholar
[6]Fisher, R. A., The logic of inductive inference, J. roy. statist. Soc., 98 (1935), 39–54.CrossRefGoogle Scholar
[7]Fog, D., Contingency tables and approximate χ2 distributions, Math. Scand. 1 (1953), 93–103.CrossRefGoogle Scholar
[8]Freeman, G. H. and Halton, J. H., Note on the exact treatment of contingency, goodness of fit, and other problems of significance, Biometrika, 38 (1951), 141–149.Google Scholar
[9]Garner, W. R. and McGill, W. J., The relation between information and variance analyses. Psychometrika, 21 (1956), 219–228.CrossRefGoogle Scholar
[10]Irwin, J. O., A note on the subdivision of χ2 into components, Biometrika, 36 (1949), 130–134.Google ScholarPubMed
[11]Kendall, M. G., Proof of relations connected with the tetrachoric series and its generalization, Biometrika, 32 (1941), 196–198.Google Scholar
[12]Kendall, M. G., The Advanced Theory of Statistics, Vol. 1, C. Griffin & Co., London (1943).Google Scholar
[13]Lancaster, H. O., The derivation and partition of χ2 in certain discrete distributions, Biometrika 36 (1949), 117–129.Google ScholarPubMed
[14]Lancaster, H. O., Some properties of the bivariate normal distribution considered in the form of a contingency table. Biometrika, 44 (1957) 289–292.CrossRefGoogle Scholar
[15]Lancaster, H. O., The structure of bivariate distributions. Ann. math. Statist. 29 (1958), 719–736.CrossRefGoogle Scholar
[16]Lancaster, H. O., Complex contingency tables treated by the partition of χ2, J. roy. statist. Soc., Ser. B, 13 (1951), 242–249.Google Scholar
[17]Lancaster, H. O., (unpublished), Analogues of the Pearson χ2.Google Scholar
[18]Lombard, H. L. and Doering, C. R., Treatment of the fourfold table by partial association and partial correlation as it relates to public health problems, Biometrics, 3 (1947), 123–128.CrossRefGoogle ScholarPubMed
[19]McGill, W. J., Multivariate information transmission, Psychometrika, 19 (1954), 97–116.CrossRefGoogle Scholar
[20]Mitra, S. K., Contributions to the statistical analysis of categorical data, Instit. Statistics, Univ. Nth. Carolina (1955), Mimeo series 142.Google Scholar
[21]Mood, A. M., Introduction to the Theory of Statistics, McGraw-Hill Book Co. Inc., New York (1950).Google Scholar
[22]Neyman, J. and Pearson, E. S., On the use and interpretation of certain test criteria, Part II, Biometrika. 20 A (1928) 263–294.Google Scholar
[23]Norton, H. W., Calculation of chi-square for complex contingency tables, J. Amer. statist. Assoc. 40 (1945), 251–258.CrossRefGoogle Scholar
[24]Pearson, E. S., The choice of statistical tests illustrated on the interpretation of data classed in a 2 x 2 table, Biometrika, 34 (1947), 139–167.Google Scholar
[25]Pearson, K., Mathematical contributions to the theory of evolution — XIII. On the theory of contingency and its relation to association and normal correlation. Draper' Co. Res. Memoirs (1904) Biometric Series 1, pp 34.Google Scholar
[26]Pearson, K., On the general theory of multiple contingency with special reference to partial contingency, Biometrika, 11 (1916), 145–158.CrossRefGoogle Scholar
[27]Roy, S. N. and Kastenbaum, M. A., On the hypothesis of ‘no interaction’ in a multi-way contingency table, Ann. math. Statistics, 27 (1956), 749–756.CrossRefGoogle Scholar
[28]Roy, S. N. and Kastenbaum, M. A., A generalization of analysis of variance and multivariate analysis to data based on frequencies in qualitative categories or class intervals, Instit. Statistics, Univ. Nth. Carolina mimeo. No. 131.Google Scholar
[29]Simpson, E. H., The interpretation of interaction in contingency tables, J. roy. statist. Soc., Ser. B. 13 (1951), 238–241.Google Scholar
[30]Wilks, S. S., The likelihood test of independence in contingency tables, Ann. Math. Statistics, 6 (1935), 190–196.CrossRefGoogle Scholar
[31]Wilson, E. B. and Worcester, J., The association of three attributes, Proc. nat. Acad. Sci. U.S.A. 28 (1942), 384–390.CrossRefGoogle ScholarPubMed
[32]Winsor, C. P., Factorial analysis of a multiple dichotomy, Hum. Biol., 20 (1948), 195–204.Google Scholar
[33]Yates, F., Contingency tables involving small numbers and the χ2 test, J. roy. statist. Soc. Suppl. 1 (1934), 217–235.Google Scholar
[34]Yates, F., A note on the application of the combination of probabilities test to a set of 2 x 2 tables, Biometrika, 42, (1955), 404–411.Google Scholar