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Matching Models in the Analysis of Cross-Classifications

Published online by Cambridge University Press:  01 January 2025

Lawrence J. Hubert
Lawrence J. Hubert*
Affiliation:
The University of California, Santa Barbara
*
Requests for reprints should be sent to Lawrence J. Hubert, Department of Education, The University of California, Santa Barbara, California, 93106.
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Abstract

Inference models motivated by the combinatorial chance literature and the concept of object matching may be used in the analysis of a contingency table if the conditional assumption of fixed row and column totals is imposed. More specifically, by developing a matching reinterpretation for several problems of interest in the prediction analysis of cross-classifications—as defined by Hildebrand, Laing and Rosenthal, appropriate significance tests can be given that may differ from those justified by the more common multinomial models. In the course of the paper the distinction between a degree-1 statistic (based on the relationship between single objects) and a degree-2 statistic (based on the relationship between object pairs) is reviewed in some detail. Also, several specializations are presented to topics of current methodological importance in psychology; for instance, a number of references are made to the measurement of nominal scale response agreement between two raters.

Keywords

distribution-free testscombinatorial chancenominal scale response agreement
Type
Original Paper
Information
Psychometrika , Volume 44 , Issue 1 , March 1979 , pp. 21 - 41
Copyright
Copyright © 1979 The Psychometric Society

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Footnotes

Partial support for this research was provided by the National Science Foundation through GSOC-77-28227.

References

Reference Note

Hubert, L. J.Alternative inference models based on matching for a weighted index of nominal scale response agreement. Unpublished manuscript, 1978.Google Scholar

References

Abe, O. A central limit theorem for the number of edges in the random intersection of two graphs. Annals of Mathematical Statistics, 1969, 40, 144–151.CrossRefGoogle Scholar
Bishop, Y. M. M. Feinberg, S. E. & Holland, P. W. Discrete multivariate analysis: Theory and practice, 1975, Cambridge, Mass.: The M.I.T. Press.Google Scholar
Brennan, R. L. & Light, R. J. Measuring agreement when two observers classify people into categories not defined in advance. The British Journal of Mathematical and Statistical Psychology, 1974, 27, 154–163.CrossRefGoogle Scholar
Brown, M. B. The asymptotic standard errors of some estimates of uncertainty in the two-way contingency table. Psychometrika, 1975, 40, 291–296.CrossRefGoogle Scholar
Camilli, G. & Hopkins, K. D. Applicability of chi-square to 2 × 2 contingency tables with small expected cell frequencies. Psychological Bulletin, 1978, 85, 163–167.CrossRefGoogle Scholar
Cohen, J. A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 1960, 20, 37–46.CrossRefGoogle Scholar
Cohen, J. Weighted kappa: Nominal scale agreement with provision for scaled disagreement or partial credit. Psychological Bulletin, 1968, 70, 213–220.CrossRefGoogle ScholarPubMed
Cohen, J. Weighted chi square: An extension of the kappa method. Educational and Psychological Measurement, 1972, 32, 61–74.CrossRefGoogle Scholar
Daniels, H. E. The relation between measures of correlation in the universe of sample permutations. Biometrika, 1944, 33, 129–135.CrossRefGoogle Scholar
David, F. N. & Barton, D. E. Combinatorial chance, 1962, New York: Hafner.CrossRefGoogle Scholar
Edgington, E. S. Statistical inference: The distribution-free approach, 1969, New York: McGraw-Hill.Google Scholar
Fleiss, J. L., Cohen, J., & Everitt, B. S. Large sample standard errors of kappa and weighted kappa. Psychological Bulletin, 1969, 72, 323–327.CrossRefGoogle Scholar
Goodman, L. A. & Kruskal, W. H. Measures of association for cross classifications, IV: Simplification of asymptotic variances. Journal of the American Statistical Association, 1972, 67, 415–421.CrossRefGoogle Scholar
Graves, G. W. & Whinston, A. B. An algorithm for the quadratic assignment problem. Management Science, 1970, 17, 453–471.CrossRefGoogle Scholar
Hartigan, J. A. Clustering algorithms, 1975, New York: Wiley.Google Scholar
Hawkes, R. K. The multivariate analysis of ordinal measures. American Journal of Sociology, 1971, 76, 908–926.CrossRefGoogle Scholar
Hays, W. L. Statistics for social scientists, 1973, New York: Holt, Rinehart and Winston.Google Scholar
Hildebrand, D. K. Laing, J. D. & Rosenthal, H. Prediction analysis of cross-classifications, 1977, New York: Wiley.Google Scholar
Hoeffding, W. A combinatorial central limit theorem. Annals of Mathematical Statistics, 1951, 22, 558–566.CrossRefGoogle Scholar
Hope, A. C. A. A simplified Monte Carlo significance test procedure. Journal of the Royal Statistical Society, 1968, 30, 582–598.CrossRefGoogle Scholar
Hotelling, H. The selection of variates for use in prediction with some comments on the general problem of nuisance parameters. The Annals of Mathematical Statistics, 1940, 11, 271–283.CrossRefGoogle Scholar
Hubert, L. J. A relationship between the assignment problem and some simple statistical techniques. Quality and Quantity, 1976, 10, 341–348.CrossRefGoogle Scholar
Hubert, L. J. Kappa revisited. Psychological Bulletin, 1977, 84, 289–297. (a)CrossRefGoogle Scholar
Hubert, L. J. Nominal scale response agreement as a generalized correlation. The British Journal of Mathematical and Statistical Psychology, 1977, 30, 98–103. (b)CrossRefGoogle Scholar
Hubert, L. J. A general formula for the variance of Cohen's weighted kappa. Psychological Bulletin, 1978, 85, 183–184.CrossRefGoogle Scholar
Hubert, L. J. & Baker, F. B. Analyzing distinctive features. Journal of Educational Statistics, 1977, 2, 79–98.CrossRefGoogle Scholar
Hubert, L. J. & Baker, F. B. Applications of combinatorial programming to data analysis: The traveling salesman and related problems. Psychometrika, 1978, 43, 81–91. (a)CrossRefGoogle Scholar
Hubert, L. J. & Baker, F. B. Evaluating the conformity of sociometric measurements. Psychometrika, 1978, 43, 31–41. (b)CrossRefGoogle Scholar
Hubert, L. J. & Levin, J. R. Inference models for categorical clustering. Psychological Bulletin, 1977, 84, 878–887.CrossRefGoogle Scholar
Hubert, L. J. & Schultz, J. V. Quadratic assignment as a general data analysis strategy. The British Journal of Mathematical and Statistical Psychology, 1976, 29, 190–241.CrossRefGoogle Scholar
Kendall, M. G. Rank correlation methods, 4th edition, New York: Hafner, 1970.Google Scholar
Klauber, M. R. Two-sample randomization tests for space-time clustering. Biometrics, 1971, 27, 129–142.CrossRefGoogle Scholar
Krippendorff, K. Bivariate agreement coefficients for reliability of data. In Borgatta, E. F.(Eds.), Sociological methodology 1970, 1970, San Francisco: Jossey-Bass.Google Scholar
Lancaster, H. O. The chi-squared distribution, 1969, New York: Wiley.Google Scholar
Lehmann, E. L. Nonparametrics, 1975, San Francisco: Holden-Day.Google Scholar
Light, R. J. & Margolin, B. H. An analysis of variance for categorical data. The Journal of the American Statistical Association, 1971, 66, 534–544.CrossRefGoogle Scholar
Margolin, B. H. & Light, R. J. An analysis of variance for categorical data, II: Small sample comparisons with chi-square and other computations. The Journal of the American Statistical Association, 1974, 69, 755–764.CrossRefGoogle Scholar
Motoo, M. On the Hoeffding's combinatorial central unit theorem. Annals of the Institute of Statistical Mathematics, 1957, 8, 145–154.CrossRefGoogle Scholar
Puri, M. L. & Sen, P. K. Nonparametric methods in multivariate analysis, 1971, New York: Wiley.Google Scholar
Rand, W. M. Objective criteria in the evaluation of clustering methods. The Journal of the American Statistical Association, 1971, 66, 846–850.CrossRefGoogle Scholar
Rao, C. R. Linear statistical inference and its applications, 1973, New York: Wiley.CrossRefGoogle Scholar
Scott, W. A. Reliability of content analysis: The case of nominal scale coding. Public Opinion Quarterly, 1955, 19, 321–325.CrossRefGoogle Scholar