Abstract
The association between constant-sum variables Xi and Xj expressed as percentages can be calculated as a product-moment correlation between Xi and Xj/(100 − Xi ) and a correlation between Xj and Xi/(100 − Xj ). An asymmetric, square matrix may be formed from these coefficients, and multivariate analysis performed by two methods: singular value decomposition and canonical decomposition. Either analysis avoids problems in the interpretation of correlation coefficients determined from closed arrays, and provides information about dependencies among the variables beyond that obtained from the usual correlation coefficient between Xi and Xj.Two examples show the canonical decomposition to have the greater usefulness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, R. Y., 1977, Short term sedimentation response in lakes in western United States as measured by automated sampling: Limnol. Oceanogr., v. 22, no. 3, p. 423–433.
Butler, J. C., 1976, Principal components analysis using the hypothetical closed array: Jour. Math. Geol., v. 8, no. 1, p. 25–36.
Chayes, F., 1962, Numerical correlation and petrographic variation: Jour. Geol., v. 70, no. 4, p. 440–452.
Chayes, F., 1971, Ratio correlation: The University of Chicago Press, Chicago.
Chayes, F. and Kruskal, W., 1966, An approximate statistical test for correlation between proportions: Jour. Geol., v. 74, no. 5, p. 692–702.
Connor, R. J. and Mosimann, J. E., 1969, Concepts of independence for proportions with a generalization of the Dirichlet distribution: Jour. Amer. Statist. Assoc., v. 64, p. 194–206.
Constantine, A. G. and Gower, J. C., 1978, Graphical representation of asymmetric matrices: Appl. Statist., v. 27, no. 3, p. 297–304.
Darroch, J. N., and Ratcliff, D., 1970, Null correlation for proportions II: Jour. Math. Geol., v. 2, no. 3, p. 307–312.
Darroch, J. N., and Ratcliff, D., 1978, No-association of proportions: Jour. Math. Geol., v. 10, no. 4, p. 361–368.
Gabriel, K. R., 1971, The biplot graphic display of matrices with applications to principal components: Biometrika, v. 58, no. 3, p. 453–467.
Gower, J. C., 1977, The analysis of asymmetry and orthogonality,in Recent developments in statistics: North Holland, Amsterdam, p. 109–123.
Kork, J. O., 1977, Examination of the Chayes-Kruskal procedure for testing correlations between proportions: Jour. Math. Geol., v. 9, no. 6, p. 543–562.
Miesch, A. T., 1969, The constant sum problem in geology,in Computer applications in the earth sciences; an international symposium: Plenum Press, New York, p. 161–176.
Nuhfer, E. B., 1979, Temporal and lateral variations in geochemistry and mineralogy in suspended sediments of lakes as studied through automated sampling: unpublished Ph.D. thesis, Univ. New Mexico.
Trochimczyk, J. and Chayes, F., 1978, Some properties of principal component scores: Jour. Math. Geol., v. 10, no. 1, p. 43–52.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hohn, M.E., Nuhfer, E.B. Asymmetric measures of association, closed data, and multivariate analysis. Mathematical Geology 12, 235–246 (1980). https://doi.org/10.1007/BF01091206
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01091206