Abstract
The population measures of association for cross classifications, discussed in the authors’ prior publications, have sample analogues that are approximately normally distributed for large samples. (Some qualifications and restrictions are necessary.) These large sample normal distributions with their associated standard errors, are derived for various measures of association and various methods of sampling. It is explained how the large sample normality may be used to test hypotheses about the measures and about differences between them, and to construct corresponding confidence intervals. Numerical results are given about the adequacy of the large sample normal approximations. In order to facilitate extension of the large sample results to other measures of association, and to other modes of sampling, than those treated here, the basic manipulative tools of large sample theory are explained and illustrated.
This research was supported in part by the Army Research Office, the Office of Naval Research, and the Air Force Office of Scientific Research by Contract No. Nonr-2121(23) ; and in part by Research Grant NSF-G21058 from the Division of Mathematical, Physical, and Engineering Sciences of the National Science Foundation.
Part of Mr. Goodman’s work on this paper was done at the Statistical Laboratory of the University of Cambridge under a Fulbright Award and a Social Science Research Council Fellowship. Part of Mr. Eruskal’s work on this paper was done at the Department of Statistics, University of California, Berkeley.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bulmer, M. G., “Confidence intervals for distance in the analysis of variance,” Biometrika, 45 (1958), 360–9.
Cramér, Harald, Mathematical Methods of Statistics. Princeton, New Jersey: Princeton University Press, 1946.
Daniels, H. E., and Kendall, M. G., “The significance of rank correlations where parental correlation exists,” Biometrika, 34 (1947), 198–208.
Daniels, H. E., “Rank correlations and population models,” Journal of the Royal Statistical Society, Series B, 12 (1950), 171–81.
El-Bradry, M. A., and Stephan, F. F., “On adjusting sample tabulations to census counts,” Journal of the American Statistical Association, 50 (1955), 738–62.
Friedlander, D., “A technique for estimating a contingency table, given the marginal totals and some supplementary data,” Journal of the Royal Statistical Society, Series A, 124 (1961), 412–20.
Goodman, Leo A., “On methods for comparing contingency tables,” Journal of the Royal Statistical Society, Series A, 126 (1963), 94–108.
Goodman, Leo A., and Kruskal, William H., “Measures of association for cross classifications,” Journal of the American Statistical Association, 49 (1954), 732–64.
Goodman, Leo A., and Kruskal, William H., “Measures of association for cross classifications. II: Further discussion and references,” Journal of the American Statistical Association, 54 (1959), 123–63.
Greenwood, Robert E., and Glasgow, Mark O., “Distribution of maximum and minimum frequencies in a sample drawn from a multinomial distribution,” Annals of Mathematical Statistics, 21 (1950), 416–24.
Hoeffding (Höffding), Wassily, “On the distribution of the rank correlation coefficient τ when the variates are not independent,” Biometrika, 34 (1947), 183–96.
Hoeffding, Wassily, “A class of statistics with asymptotically normal distribution,” Annals of Mathematical Statistics, 19 (1948), 293–325.
Hoeffding, Wassily, and Robbins, Herbert, “The central limit theorem for dependent random variables,” Duke Mathematical Journal, 15 (1948), 773–80.
Hoeffding, Wassily, “An upper bound for the variance of Kendall’s ‘tau’ and of related statistics,” pp. 258–64 in Contributions to Probability and Statistics. Essays in Honor of Harold Hotelling, edited by Ingram Olkin and others, Stanford University Press, Stanford, 1960.
Kondo, Tsutomu, “On the standard error of the mean square contingency,” Biometrika, 21 (1929), 377–428.
Kozelka, Robert M., “On some special order statistics from the multinomial distribution,” Unpublished Ph.D. dissertation, Harvard University, 1952.
Kozelka, Robert M., “Approximate upper percentage points for extreme values in multinomial sampling,” Annals of Mathematical Statistics, 27 (1956), 507–12.
Kruskal, William H., “Ordinal measures of association,” Journal of the American Statistical Association, 53 (1958), 814–61.
Mann, H. B., and Wald, A., “On stochastic limit and order relationships,” Annals of Mathematical Statistics, 14 (1943), 217–26. (See also Wald, Abraham, Selected Papers in Statistics and Probability, McGraw-Hill Book Company, New York, 1955.)
Noether, Gottfried E., “Two confidence intervals for the ratio of two probabilities and some measures of effectiveness,” Journal of the American Statistical Association, 52 (1957), 36–45.
Pearson, Karl, “On the probable error of a coefficient of mean square contingency,” Biometrika, 10 (1915), 570–3.
Sneyers, R., “Remarques à propos d’une généralisation de l’indice de similitude de M. Bouët,” Archiv fÜr Meteorologie, Geophysik und Bioklimatologie, Series A: Meteorologie und Geophysik, Band 11, 1. Heft, 1959, 126–35.
Stuart, A., “The estimation and comparison of strengths of association in contingency tables,” Biometrika, 40 (1953), 105–10.
Sundrum, R. M., “Moments of the rank correlation coefficient τ in the general case,” Biometrika, 40 (1953), 409–20.
Weichselberger, K., “Über die Parameterschätzungen bei Kontingenztafeln, deren Randsummen vorgegeben sind,” Metrika, 2 (1959), 100–30 and 198–229.
Young, Andrew W., and Pearson, Karl, “On the probable error of a coefficient of contingency without approximation,” Biometrika, 11 (1915), 215–30.
Yule, G. Udny, “On the association of attributes in statistics with illustrations from the material from the childhood society, &c.,” Philosophical Transactions of the Royal Society of London, Series A, 194 (1900), 257–319.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1979 Springer-Verlag New York
About this chapter
Cite this chapter
Goodman, L.A., Kruskal, W.H. (1979). Measures of Association for Cross Classifications III: Approximate Sampling Theory. In: Measures of Association for Cross Classifications. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-9995-0_3
Download citation
DOI: https://doi.org/10.1007/978-1-4612-9995-0_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-9997-4
Online ISBN: 978-1-4612-9995-0
eBook Packages: Springer Book Archive