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Combinatorial Data Analysis: Association and Partial Association

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, CA 93106.
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Abstract

A combinatorial data analysis strategy is reviewed that is designed to compare two arbitrary measures of proximity defined between the objects from some set. Based on a particular cross-product definition of correspondence between these two numerically specified notions of proximity (typically represented in the form of matrices), extensions are then pursued to indices of partial association that relate the observed pattern of correspondence between the first two proximity measures to a third. The attendant issues of index normalization and significance testing are discussed; the latter is approached through a simple randomization model implemented either through a Monte Carlo procedure or distributional approximations based on the first three moments. Applications of the original comparison strategy and its extensions to partial association may be developed for a variety of methodological and substantive tasks. Besides rank correlation, we emphasize the topics of spatial autocorrelation for one variable and spatial association between two and mention the connection to the usual randomization approach for one-way analysis-of-variance.

Keywords

partial associationpartial correlationnonparametric statisticsspatial statisticspermutation tests
Type
Original Paper
Information
Psychometrika , Volume 50 , Issue 4 , December 1985 , pp. 449 - 467
Copyright
Copyright © 1985 The Psychometric Society

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References

Campbell, D. T., Kruskal, W. H., Wallace, W. P. (1966). Seating aggregation as an index of attitude. Sociometry, 29, 1–15.CrossRefGoogle Scholar
Cliff, A. D., Ord, J. K. (1981). Spatial processes: Models and applications, London: Pion.Google Scholar
Constanzo, C. M., Hubert, L. J., Golledge, R. G. (1983). A higher moment for spatial statistics. Geographical Analysis, 15, 347–351.CrossRefGoogle Scholar
Daniels, H. E. (1944). The relation between measures of correlation in the universe of sample permutations. Biometrika, 33, 129–135.CrossRefGoogle Scholar
Davis, J. A. (1971). Elementary survey analysis, Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
Dietz, E. J. (1983). Permutation tests for association between two distance matrices. Systematic Zoology, 32, 21–26.CrossRefGoogle Scholar
Douglas, M. E., Endler, J. A. (1982). Quantitative matrix comparisons in ecological and evolutionary investigations. Journal of Theoretical Biology, 99, 777–795.CrossRefGoogle Scholar
Freeman, L. C. (1978). Segregation in social networks. Sociological Methods and Research, 6, 411–429.CrossRefGoogle Scholar
Glick, B. J. (1979). Tests for space-time clustering used in cancer research. Geographical Analysis, 11, 202–208.CrossRefGoogle Scholar
Hubert, L. J. (1983). Inference procedures for the evaluation and comparison of proximity matrices. In Felsenstein, J. (Eds.), Numerical taxonomy, New York: Springer-Verlag.Google Scholar
Hubert, L. J., Golledge, R. G., Costanzo, C. M. (1981). Generalized procedures for evaluating spatial auto-correlation. Geographical Analysis, 13, 224–233.CrossRefGoogle Scholar
Hubert, L. J., Golledge, R. G., Costanzo, C. M. (1982). Analysis of variance procedures based on a proximity measure between subjects. Psychological Bulletin, 91, 424–430.CrossRefGoogle Scholar
Hubert, L. J., Golledge, R. G., Costanzo, C. M., Gale, N. (1985). Measuring association between spatially defined variables: An alternative procedure. Geographical Analysis, 17, 36–46.CrossRefGoogle Scholar
Hubert, L. J., Golledge, R. G., Costanzo, C. M., Gale, N., Halperin, W. C. (1984). Nonparametric tests for directional data. In Bahrenberg, G., Fischer, M., Nijkamp, P. (Eds.), Recent developments in spatial analysis: Methodology, measurement, models (pp. 171–190). Aldershot, UK: Gower.Google Scholar
Hubert, L. J., Schultz, J. V. (1976). Quadratic assignment as a general data analysis strategy. British Journal of Mathematical and Statistical Psychology, 29, 190–241.CrossRefGoogle Scholar
Klauber, M. R. (1975). Space-time clustering for more than two samples. Biometrics, 31, 719–726.CrossRefGoogle ScholarPubMed
Mantel, N. (1967). The detection of disease clustering and a generalized regression approach. Cancer Research, 27, 209–220.Google Scholar
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., Brier, G. W. (1981). Application of multiresponse 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
Pike, M. C., Smith, P. G. (1974). A case-control approach to examine diseases for evidence of contagion, including diseases with long latent periods. Biometrics, 30, 263–279.CrossRefGoogle ScholarPubMed
Puri, M. L., Sen, P. K. (1971). Nonparametric methods in multivariate analysis, New York: Wiley.Google Scholar
Quade, D. (1974). Nonparametric partial correlation. In Blalock, H. M. Jr. (Eds.), Measurement in the social sciences: Theories and strategies (pp. 369–398). Chicago: Aldine.CrossRefGoogle Scholar
Raveh, A. (in press). On measures of monotone association. American Statistician.Google Scholar
Reynolds, H. T. (1977). The analysis of class classifications, New York: Free Press.Google Scholar
Sokal, R. R. (1979). Testing statistical significance of geographical variation patterns. Systematic Zoology, 28, 227–232.CrossRefGoogle Scholar
Somers, R. H. (1959). The rank analogue of product-moment partial correlation and regression, with application to manifold, ordered contingency tables. Biometrika, 46, 241–246.CrossRefGoogle Scholar
Winsborough, H. H., Quarantelli, E. L., Yutzy, D. (1963). The similarity of connected observations. American Sociological Review, 28, 977–983.CrossRefGoogle Scholar