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An investigation of three-matrix permutation tests

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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.

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Authors and Affiliations

  1. Department of Preventive Medicine, Division of Epidemiology, Health Sciences Center, State Univeresity of New York, 11794-8036, Stony Brook, New York, USA

    Neal L. Oden

  2. Department of Ecology and Evolution, State University of New York, 11794-5245, Stony Brook, New York, USA

    Robert R. Sokal

Authors
  1. Neal L. Oden
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  2. Robert R. Sokal
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Oden, N.L., Sokal, R.R. An investigation of three-matrix permutation tests. Journal of Classification 9, 275–290 (1992). https://doi.org/10.1007/BF02621410

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