A new approach to null correlations of proportions

  • J. Aitchison
Article
  • 251 Downloads

Abstract

Much work on the statistical analysis of compositional data has concentrated on the difficulty of interpreting correlations between proportions with an assortment of tests for nullcorrelations, for independence except for the constraint, F-independence of bounded variables, neutrality in the mean and in the median. This paper questions the appropriateness of characterizing the dependence structure of proportions in terms of such concepts, suggests an alternative method of modeling, develops necessary distribution theory and tests, and illustrates the methodology in applications.

Key words

closed and open array compositional data correlations between proportions Dirichlet distributions logistic-normal distributions petrogenesis tests for basis independence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aitchison, J., 1981a, Distributions on the simplex for the analysis of neutrality,in Statistical distributions in scientific work, Taillie, C., Patil, G. P., and Baldessari, B. (eds.): D. Reidel Publishing Company, Dordrecht, Holland, to appear.Google Scholar
  2. Aitchison, J., 1981b, Testing for additive isometry and proportional invariance: Biometrics, v. 37, to appear.Google Scholar
  3. Aitchison, J. and Brown, J. A. C., 1957, The lognormal distribution: Cambridge University Press, New York.Google Scholar
  4. Aitchison, J. and Shen, S. M., 1980, Logistic-normal distributions: some properties and uses: Biometrika, v. 67, p. 261–272.Google Scholar
  5. Bartlett, N. R. and Darroch, J. N., 1978, Regression and correlation of bounded-sum variables: Vistelius Commemoration Volume.Google Scholar
  6. Butler, J. C., 1979, Trends in ternary petrologic variation diagrams—fact or fantasy?: Amer. Mineralogist, v. 64, p. 1115–1121.Google Scholar
  7. Chayes, F., 1960, On correlation between variables of constant sum: Jour. Geophys. Res., v. 65, p. 4185–4193.Google Scholar
  8. Chayes, F., 1962, Numerical correlation and petrographic variation: Jour. Geol., v. 70, p. 440–452.Google Scholar
  9. Chayes, F. and Kruskal, W., 1966, An approximate statistical test for correlations between proportions: Jour. Geol., v. 74, p. 692–702.Google Scholar
  10. Chernoff, H., 1954, On the distribution of the likelihood ratio: Ann. Math. Stat., v. 25, p. 573–578.Google Scholar
  11. Clisby, K. H. and Sears, P. B., 1955, Palynology in southern North America. Part III. Microfossil profiles under Mexico City correlated with sedimentary profiles: Geol. Soc. America Bull., v. 66, no. 5, p. 511–520.Google Scholar
  12. Connor, J. R. 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.Google Scholar
  13. Darroch, J. N., 1969, Null correlations for proportions: Jour. Internat. Assoc. Math. Geol., v. 1, no. 2, p. 221–227.Google Scholar
  14. Darroch, J. N. and James, I. R., 1974, F-independence and null correlations of continuous, bounded-sum, positive variables: Jour. Roy. Stat. Soc., Ser. B, v. 36, no. 3, p. 467–483.Google Scholar
  15. Darroch, J. N. and Ratcliff, D., 1970, Null correlations for proportions. II.: Jour. Internat. Assoc. Math. Geol., v. 2, p. 307–312.Google Scholar
  16. Darroch, J. N. and Ratcliff, D., 1971, A characterization of the Dirichlet distribution: Jour. Amer. Stat. Assoc., v. 66, p. 641–643.Google Scholar
  17. Darroch, J. N. and Ratcliff, D., 1978, No-association of proportions: Jour. Internat. Assoc. Math. Geol., v. 10, p. 361–368.Google Scholar
  18. Feder, P. I., 1968, On the distribution of the log likelihood ratio test statistic when the true parameter is ‘near’ the boundaries of the hypothesis regions: Ann. Math. Stat., v. 39, p. 2044–2055.Google Scholar
  19. Gleser, L. J. and Olkin, I., 1973, Multivariate statistical inference under marginal structure: Brit. Jour. Math. Stat. Psychol., v. 26, p. 98–123.Google Scholar
  20. Krumbein, W. C., 1962, Open and closed number systems stratigraphic mapping: Bull. Amer. Assoc. Petrol. Geologists, v. 46, p. 2229–2245.Google Scholar
  21. Kullback, S. and Leibler, R. A., 1951, On information and sufficiency: Ann. Math. Stat., v. 22, p. 525–540.Google Scholar
  22. McCammon, R. B., 1975, Concepts in geostatistics; John Wiley & Sons, New York.Google Scholar
  23. Miesch, A. T., 1969, The constant sum problem in geochemistry,in Computer applications in the earth sciences, Merriam, D. F. (ed.): Plenum Press, New York, p. 161–177.Google Scholar
  24. Mosimann, J. E., 1962, On the compound multinomial distribution, the multivariateβ-distribution and correlations among proportions: Biometrika, v. 49, p. 65–82.Google Scholar
  25. Mosimann, J. E., 1963, On the compound negative multinomial distribution and correlations among inversely sampled pollen counts: Biometrika, v. 50, p. 47–54.Google Scholar
  26. Mosimann, J. E., 1975a, Statistical problems of size and shape. I. Biological applications and basic theorems,in Statistical distributions in scientific work, v. 2, Patil, G. P., Kotz, S., and Ord, J. K. (eds.): D. Reidel Publishing Company, Dordrecht, Holland, p. 187–217.Google Scholar
  27. Mosimann, J. E., 1975b, Statistical problems of size and shape. II. Characterizations of the lognormal, gamma, and Dirichlet distributions,in Statistical distributions in scientific work, v. 2, Patil, G. P., Kotz, S., and Ord, J. K. (eds.): D. Reidel Publishing Company, Dordrecht, Holland, p. 219–239.Google Scholar
  28. Mukherjee, B. N., 1970, Likelihood ratio tests on statistical hypotheses associated with patterned covariance matrices in psychology: Brit. Jour. Math. Stat. Psychol., v. 23, p. 89–120.Google Scholar
  29. Snow, J. W., 1975, Association of proportions: Jour. Internat. Assoc. Math. Geol., v. 7, no. 1, p. 63–73.Google Scholar
  30. Steiner, A., 1958, Petrographic implications of the 1954 Ngauruhoe lava and its xenoliths: New Zealand Jour. Geol. Geophys., v. 1, p. 325–363.Google Scholar
  31. Thompson, R. N., Esson, J., and Duncan, A. C., 1972, Major element chemical variation in the Eocene lavas of the Isle of Skye, Scotland: Jour. Petrology, v. 13, p. 219–253.Google Scholar
  32. Wald, A., 1943, Tests of statistical hypotheses concerning several parameters when the number of observations is large: Trans. Amer. Math. Soc., v. 54, p. 426–482.Google Scholar

Copyright information

© Plenum Publishing Corporation 1981

Authors and Affiliations

  • J. Aitchison
    • 1
  1. 1.Department of StatisticsUniversity of Hong KongHong Kong

Personalised recommendations