Journal of Classification

, Volume 9, Issue 2, pp 275–290 | Cite as

An investigation of three-matrix permutation tests

  • Neal L. Oden
  • Robert R. Sokal
Article

Abstract

Several methods have recently been introduced for investigating relations between three interpoint proximity matricesA, B, C, each of which furnishes a different type of distance between the same objects. Smouse, Long, and Sokal (1986) investigate the partial correlation betweenA andB conditional onC. Dow and Cheverud (1985) ask whethercorr (A, C), equalscorr (B, C). Manly (1986) investigates regression-like models for predicting one matrix as a function of others.

We have investigated rejection rates of these methods when their null hypotheses are true, but data are spatially autocorrelated (SA). That is,A, andB are distance matrices from independent realizations of the same SA generating process, andC is a matrix of geographic connections.

SA causes all the models to be liberal because the hypothesis of equally likely row/column permutations invoked, by all these methods, is untrue when data are SA. Consequently, we cannot unreservedly recommend the use of any of these methods with SA data. However, if SA is weak, the Smouse-Long-Sokal method, used with a conservative critical value, is unlikely to reject falsely.

Keywords

Spatial autocorrelation Quadratic assignment Partial correlations Permutation tests Proximity matrices 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. CLIFF, A. D., and ORD, J. K. (1981),Spatial Processes, London: Pion.MATHGoogle Scholar
  2. DOW, M. M., and CHEVERUD, J. M. (1985), “Comparison of Distance Matrices in Studies of Population Structure and Genetic Microdifferentiation: Quadratic Assignment,”American Journal of Physical Anthropology 68, 367–373.CrossRefGoogle Scholar
  3. DOW, M. M., CHEVERUD, J. M., and FRIEDLANDER, J. S. (1987), “Partial Correlation of Distance Matrices in Studies of Population Structure,”American Journal of Physical Anthropology, 72, 343–352.CrossRefGoogle Scholar
  4. HUBERT, L. J., (1985), “Combinatorial Data Analysis: Association and Partial Association,”Psychometrika 50, 449–467.MATHCrossRefMathSciNetGoogle Scholar
  5. HUBERT, L. J., and GGOLLEDGE, R. G. (1981), “A Heuristic Method for the Comparison of Related Structures,”Journal of Mathematical Psychology, 23, 214–226.CrossRefGoogle Scholar
  6. MANLY, B. F. J. (1986), “Randomization and Regression Methods for Testing for Associations with Geographical, Environmental, and Biological Distances Between Populations,”Researches in Population Ecology, 28, 201–218.CrossRefGoogle Scholar
  7. MANTEL, N. (1967), “The Detection of Disease Clustering and a Generalized Regression Approach,”Cancer Research, 27, 209–220.Google Scholar
  8. ODEN, N.L. (1992), “Spatial Aurocorrelation Invalidates the Dow-Cheverud Test,”American can Journal of Physical Anthropology (in press).Google Scholar
  9. SLATKIN, M., and ARTER H. (1991), “Methods in Population Genetics,”American Naturalist, 138, 499–517.CrossRefGoogle Scholar
  10. SMOUSE P. E., LONG J. C., and SOKAL, R. R. (1986)., “Multiple Regression and Correlation Extensions of the Mantel Test of Matrix Correspondence,”Systematic Zoology, 35, 627–632.CrossRefGoogle Scholar
  11. SOKAL, R. R., and ODEN, N. L. (1978a), “Spatial Autocorrelation in Biology 1. Methodology,”Biological Journal of the Linnean Society, 10, 199–228.Google Scholar
  12. SOKAL, R. R., and ODEN, N. L. (1978b), “Spatial autocorrelation in Biology 2. Some Biological Implications and Four Applications of Evolutionary and Ecological Interest,”Biological Journal of the Linnean Society, 10, 229–249.Google Scholar
  13. SOKAL, R. R., ODEN, N. L., and WILSON, C. (1991), “Genetic Evidence for the Spread of Agriculture in Europe by Demic Diffusion,”Nature, 351, 143–145.CrossRefGoogle Scholar
  14. SOKAL, R. R., SMOUSE, P. E., and NEEL, J. V. (1986), “The Genetic Structure of a Tribal Population, the Yanomama Indians. XV. Patterns Inferred by Autocorrelation Analysis,”Genetics, 114, 259–281.Google Scholar
  15. SOKAL, R. R. (1988), “Genetic, Geographic, and Linguistic Distances in Europe”,Proceedings of the National Academy of Sciences USA, 85, 1722–1726.CrossRefMathSciNetGoogle Scholar
  16. TILLEY, S. G., VERRELL, P. A., and ARNOLD, S. J. (1990), “Correspondence between Sexual Isolation and Allozyme Differentiation: A Test in the SalamanderDesmognathus ochrophaeus,”Proceedings of the National Academy of Sciences USA, 87, 2715–2719.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Neal L. Oden
    • 1
  • Robert R. Sokal
    • 2
  1. 1.Department of Preventive Medicine, Division of Epidemiology, Health Sciences CenterState Univeresity of New YorkStony BrookUSA
  2. 2.Department of Ecology and EvolutionState University of New YorkStony BrookUSA

Personalised recommendations