## 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.

- a matrix of genetic distances between populations (i.e. a measure of how similar their gene frequencies are)
- 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, R_{0},
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
( R_{0}) was exceeded by the permutated
values. For example, if did 999 permutations and only 1 of the permutated coefficients
exceeded R_{0 }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.

## 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.

## 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.