# Clustering and Classification methods for Biologists

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## What is it?

The Mantel test is a very useful, if somewhat obscure test. Basically it is used test if two distance matrices are correlated. For example, suppose that you have two distance matrices.

1. a matrix of genetic distances between populations (i.e. a measure of how similar their gene frequencies are)
2. a matrix of geographical distances (i.e. a measure of how separate the populations are)

It is reasonable to ask if genetic distance is correlated with geographical distance. You can't use a simple correlation coefficient because the cases are not independent (e.g. the distance between case 1 & 3 is not independent of the distance between case 1 & 4 because case 1 is involved in both).

A correlation coefficient, R0, is calculated for the original matrices. Now we permutate the rows and columns within one of the matrices and recalculate the correlation. We do this 100's of times. If the original matrices had been correlated the disruption caused by the permutations should reduce the correlation coefficient. Our measure of significance is the number of times that the original correlation coefficient ( R0) was exceeded by the permutated values. For example, if did 999 permutations and only 1 of the permutated coefficients exceeded R0 this would give use a significance of 0.001. Conversely, if the matrices were uncorrelated there is no reason to assume that the permutations would decrease the correlation coefficient, they may indeed increase it.

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## Examples from the literature.

Eller, E. 1999. Population substructure and isolation by distance in three continental regions. American Journal of Physical Anthropology, 108: 147-159.
Weider, LJ, Hobaek, A, Hebert, PDN, Crease, TJ. 1999. Holarctic phylogeography of an asexual species complex - II. Allozymic variation and clonal structure in Arctic Daphnia. Molecular Ecology, 8: 1-13.
West, DF, Black, WC. 1998. Breeding structure of three snow pool Aedes mosquito species in northern Colorado. Heredity, 81: 371-380.
HattonEllis, TW, Noble, LR, Okamura, B. 1998. Genetic variation in a freshwater bryozoan. I: Populations in the Thames basin, UK. Molecular Ecology, 7: 1575-1585.
Erkkila, HMJB. 1998. Seed banks of grazed and ungrazed Baltic seashore meadows. Journal of Vegetation Science, 9: 395-408.
Parker, MA, Spoerke, JM. 1998. Geographic structure of lineage associations in a plant-bacterial mutualism. Journal of Evolutionary Biology, 11: 549-562.
Miller, KJ. 1998. Short-distance dispersal of black coral larvae: inference from spatial analysis of colony genotypes. Marine Ecology-Progress Series, 163,: 225-233.
Heipel, DA, Bishop, JDD, Brand, AR, Thorpe, JP. 1998. Population genetic differentiation of the great scallop Pecten maximus in western Britain investigated by randomly amplified polymorphic DNA. Marine Ecology-Progress Series, 162: 163-171.
Cesaroni, D, Matarazzo, P, Allegrucci, G, Sbordoni, V. 1997. Comparing patterns of geographic variation in cave crickets by combining geostatistic methods and Mantel tests. Journal of Biogeography, 24: 419-431.
Rossi, JP. 1996. Statistical tool for soil biology .11. Autocorrelogram and Mantel test. European Journal of Soil Biology, 32: 195-203.
BlackSamuelsson, S, Eriksson, G, Gustafsson, L, Gustafsson, P. 1997. RAPD and morphological analysis of the rare plant species Vicia pisiformis (Fabaceae). Biological Journal of The Linnean Society, 61: 325-343.
Kyes, S, Harding, R, Black, G, Craig, A, Peshu, N, Newbold, C, Marsh, K. 1997. Limited spatial clustering of individual Plasmodium falciparum alleles in field isolates from coastal Kenya. American Journal off Tropical Medicine And Hygiene, 57: 205-215.
RuizGarcia, M. 1997. Genetic relationships among some new cat populations sampled in Europe: A spatial autocorrelation analysis. Journal of Genetics, 76: 1-24.
Sokal, RR, Oden, NL, Walker, J, Waddle, DM. 1997. Using distance matrices to choose between competing theories and an application to the origin of modern humans. Journal of Human Evolution, 32: 501-522.
Malhotra, A, Thorpe, RS. 1997. Size and shape variation in a Lesser Antillean anole, Anolis oculatus (Sauria: Iguanidae) in relation to habitat. Biological Journal of the Linnean Society, 60: 53-72.

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## Resources

PopTools is a very versatile Excel addin from CSIRO. In addition to Mantel tests it also incorporates a range of Matrix methods and resampling techniques.

A Mantel test is included in a suite of Excel programs designed for use with population genetics studies.

Manly, B.F.J. 1994. RT: A program for randomisation testing. Version 1.02c, July 1994. Centre for Application of Statistics and Mathematics, University of Otago, Dunedin, New Zealand.

Manly, B. F. J. 1993. A review of computer intensive multivariate methods in ecology. Pages 307-346 in G. P. Patil and C. R. Rao (eds.) Multivariate Environmental Statistics. Elsevier Science, London.

Partial Mantel analysis is described with great clarity (p779-) in Legendre and Legendre's Numerical Ecology (2nd Edition) - published by Elsevier in 1998.

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