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Statistical Applications of Linear Assignment

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, Graduate School of Education, The University of California, Santa Barbara, California, 93106.
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Abstract

A comprehensive statistical framework is presented which encompasses a wide range of existing nonparametric methods. The basic strategy, referred to as linear assignment (LA), depends on a simple index of correspondence defined between two object sets that have been matched in some a priori manner. In this broad sense, LA can be interpreted as a general correlational technique. A variety of extensions are discussed along with the attendant problems of significance testing and the construction of normalized descriptive indices.

Keywords

linear assignmentpermutation testsnonparametric statisticscorrelationspatial statisticsconcordance
Type
Original Paper
Information
Psychometrika , Volume 49 , Issue 4 , December 1984 , pp. 449 - 473
Copyright
Copyright © 1984 The Psychometric Society

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Footnotes

Given as the Presidential Address to the Psychometric Society's Annual Meeting, June, 1984. I wish to thank M. Jambu, C. Perruchet, and their colleagues at the Centre National d'Études des Télécommunications for assistance during the summer of 1983 when most of this paper was written. Partial support for this research was also supplied by the National Institute of Justice, Grant #82-IJ-CX-0019.

References

Agresti, A., Wackerly, D., & Boyett, J. (1979). Exact conditional tests for cross-classifications: Approximation of obtained significance levels. Psychometrika, 44, 75–83.CrossRefGoogle Scholar
Bachi, R. (1962). Standard distance measures and related methods for spatial analysis. Papers of The Regional Science Association, 10, 83–132.CrossRefGoogle Scholar
Bradley, J. V. (1968). Distribution-free statistical tests, Englewood Cliffs, N.J.: Prentice-Hall.Google Scholar
Buffington, V., Martin, D. C., & Becker, J. (1981). VER similarity between alcoholic probands and their first-degree relatives. Psychophysiology, 18, 529–533.CrossRefGoogle ScholarPubMed
Campbell, D. T., & Fiske, D. W. (1959). Convergent and discriminant validation by the multitrait-multimethod matrix. Psychological Bulletin, 56, 81–105.CrossRefGoogle ScholarPubMed
Cliff, A. D., & Ord, J. K. (1981). Spatial processes: Models and applications, London: Pion.Google Scholar
Cohen, J. (1968). Weighted kappa: Nominal scale agreement with provision for scaled disagreement or partial credit. Psychological Bulletin, 70, 213–220.CrossRefGoogle ScholarPubMed
Cohen, J. (1972). Weighted chi-square: An extension of the kappa method. Educational and Psychological Measurement, 32, 61–74.CrossRefGoogle Scholar
Daniels, H. E. (1944). The relation between measures of correlation in the universe of sample permutations. Biometrika, 33, 129–135.CrossRefGoogle Scholar
David, F. N., & Barton, D. E. (1962). Combinatorial chance, New York: Hafner.CrossRefGoogle Scholar
Diaconis, P., & Graham, R. L. (1977). Spearman's footrule as a measure of disarray. Journal of the Royal Statistical Society, B, 39, 262–268.CrossRefGoogle Scholar
Dietz, E. J. (1983). Permutation tests for association between two distance matrices. Systematic Zoology, 32, 21–26.CrossRefGoogle Scholar
Dwyer, P. S. (1964). The mean and variance of the distribution of group assembly sums. Psychometrika, 29, 397–408.CrossRefGoogle Scholar
Edgington, E. S. (1969). Statistical inference: The distribution-free approach, New York: McGraw-Hill.Google Scholar
Epp, R. J., Tukey, J. W., & Watson, G. S. (1971). Testing unit vectors for correlation. Journal of Geophysical Research, 76, 8480–8483.CrossRefGoogle Scholar
Fraser, D. A. S. (1956). A vector form of the Wald-Wolfowitz-Hoeffding theorem. Annals of Mathematical Statistics, 27, 540–543.CrossRefGoogle Scholar
Hájek, J., & Šidák, Z. (1967). Theory of rank tests, New York: Academic Press.Google Scholar
Harter, H. L. (1969). A new table of percentage points of the Pearson Type III distribution. Technometrics, 11, 177–187.CrossRefGoogle Scholar
Hildebrand, D., Laing, M., & Rosenthal, A. (1977). Prediction analysis of cross-classifications, New York: Wiley.Google Scholar
Hoeffding, W. (1951). A combinatorial central limit theorem. Annals of Mathematical Statistics, 22, 558–566.CrossRefGoogle Scholar
Hubert, L. J. (1977). Kappa revisited. Psychological Bulletin, 84, 289–297.CrossRefGoogle Scholar
Hubert, L. J. (1978). A general formula for the variance of Cohen's weighted kappa. Psychological Bulletin, 85, 183–184.CrossRefGoogle Scholar
Hubert, L. J. (1979). The comparison of sequences. Psychological Bulletin, 86, 1098–1106.CrossRefGoogle Scholar
Hubert, L. J. (1983). Inference procedures for the evaluation and comparison of proximity matrices. In Felsenstein, J. (Eds.), Numerical Taxonomy (pp. 209–228). New York: Springer-Verlag.CrossRefGoogle Scholar
Hubert, L. J., & Golledge, R. G. (1982). Comparing rectangular data matrices. Environment and Planning, A, 14, 1087–1095.CrossRefGoogle Scholar
Hubert, L. J., & Golledge, R. G. (1982). Measuring association between spatially defined variables: Tjøstheim's index and some extensions. Geographical Analysis, 14, 273–278.CrossRefGoogle Scholar
Hubert, L. J., & Golledge, R. G. (1983). Rater agreement for complex assessments. British Journal of Mathematical and Statistical Psychology, 36, 207–216.CrossRefGoogle Scholar
Hubert, L. J., Golledge, R. G., Costanzo, C. M., & Richardson, G. D. (1981). Assessing homogeneity in cross-classified proximity data. Geographical Analysis, 13, 38–50.CrossRefGoogle Scholar
Hubert, L. J., Golledge, R. G., Kenney, T., & Costanzo, C. M. (1981). Aggregation in data tables: Implications for evaluating criminal justice statistics. Environment and Planning, A, 13, 185–199.CrossRefGoogle Scholar
Kendall, M. G. (1970). Rank correlation methods 4th edition,, London: Griffin.Google Scholar
Kendall, M. G., & Stuart, A. (1979). The advanced theory of statistics, Volume 2 4th edition,, New York: Macmillan.Google Scholar
Klauber, M. R. (1971). Two sample randomization tests for space-time clustering. Biometrics, 27, 129–142.CrossRefGoogle Scholar
Lehmann, E. L. (1975). Nonparametrics: Statistical methods based on ranks, New York: Holden-Day.Google Scholar
Lerman, I. C. (1980). Combinatorial analysis in the statistical treatment of behavioral data. Quality and Quantity, 14, 431–469.CrossRefGoogle Scholar
Martin, D. C., Buffington, V., & Becker, J. (1981). Randomization test of paired data: Application to evoked responses. Psychophysiology, 18, 524–528.CrossRefGoogle ScholarPubMed
Mielke, P. W. (1979). On asymptotic non-normality of null distributions of MRPP statistics. Communications in Statistics—Theory and Methods, A8, 1541–1550.CrossRefGoogle Scholar
Mielke, P. W., & Berry, K. J. (1982). An extended class of permutation techniques for matched pairs. Communications in Statistics—Theory and Methods, A11, 1197–1207.CrossRefGoogle Scholar
Mielke, P. W., Berry, K. J., & Brier, G. W. (1981). Application of multi-response permutation procedures for examining seasonal changes in monthly mean sea-level pressure patterns. Monthly Weather Review, 109, 120–126.2.0.CO;2>CrossRefGoogle Scholar
Mielke, P. W., & Iyer, H. K. (1982). Permutation techniques for analyzing multi-response data from randomized block experiments. Communications in Statistics—Theory and Methods, A11, 1427–1437.Google Scholar
Mosteller, F., & Bush, R. R. (1954). Selected quantitative techniques. In Lindzey, G. (Eds.), Handbook of Social Psychology (Volume 1), Cambridge: Addison-Wesley.Google Scholar
Motoo, M. (1957). On the Hoeffding's combinatorial central limit theorem. Annals of the Institute of Statistical Mathematics, 8, 145–154.CrossRefGoogle Scholar
Page, E. B. (1963). Ordered hypothesis for multiple treatments: A significance test for linear ranks. Journal of the American Statistical Association, 58, 216–230.CrossRefGoogle Scholar
Pirie, W. R., & Hollander, M. (1972). A distribution-free normal scores test for ordered alternatives in the randomized block design. Journal of the American Statistical Association, 67, 855–857.CrossRefGoogle Scholar
Puri, M. L., & Sen, P. K. (1971). Nonparametric methods in multivariate analysis, New York: Wiley.Google Scholar
Schucany, W. R., & Frawley, W. H. (1973). A rank test for two-group concordance. Psychometrika, 38, 249–258.CrossRefGoogle Scholar
Sneath, P. H. A., & Sokal, R. R. (1973). Numerical taxonomy, San Francisco: Freeman.Google Scholar
Sokal, R. R. (1979). Testing statistical significance of geographical variation patterns. Systematic Zoology, 28, 227–232.CrossRefGoogle Scholar
Thorndike, R. L. (1950). The problem of classification of personnel. Psychometrika, 15, 215–235.CrossRefGoogle ScholarPubMed
Tjøstheim, D. (1978). A measure of association for spatial variables. Biometrika, 65, 109–114.CrossRefGoogle Scholar
Tsutakawa, R. K., & Yang, S. L. (1974). Permutation tests applied to antibiotic drug resistance. Journal of the American Statistical Association, 69, 87–92.CrossRefGoogle Scholar
Wald, A., & Wolfowitz, J. (1944). Statistical tests based on permutations of the observations. Annals of Mathematical Statistics, 15, 358–372.CrossRefGoogle Scholar
Ward, J. H. (1958). The counseling assignment problem. Psychometrika, 23, 55–65.CrossRefGoogle Scholar