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| The '''Mantel test''', named after [[Nathan Mantel]], is a [[statistics|statistical]] test of the [[correlation]] between two [[matrix (mathematics)|matrices]]. The matrices must be of the same [[rank of a matrix|rank]]; in most applications, they are matrices of interrelations between the same [[vector space|vector]]s of objects. The test was first published by [[Nathan Mantel]], a biostatistician at the [[National Institutes of Health]], in 1967.<ref name="Mantel1967">{{cite journal | last1 = Mantel | first1 = N. | year = 1967 | title = The detection of disease clustering and a generalized regression approach | url = | journal = Cancer Research | volume = 27 | issue = 2| pages = 209–220 | pmid = 6018555 }}</ref> Accounts of it can be found in advanced statistics books (e.g., Sokal & Rohlf 1995<ref name="Sokal1995">{{cite book |author=Sokal RR, Rohlf FJ |year=1995 |title=Biometry |edition=3rd |location=New York |publisher=Freeman |isbn=0-7167-2411-1 |pages=813-819}}</ref>).
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| ==Usage==
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| The test is commonly used in [[ecology]], where the data are usually estimates of the "distance" between objects such as [[species]] of organisms. For example, one matrix might contain estimates of the [[genetics|genetic]] distances (i.e., the amount of difference between two different genomes) between all possible pairs of species in the study, obtained by the methods of [[molecular systematics]]; while the other might contain estimates of the geographical distance between the ranges of each species and every other species.
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| ==Method==
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| If there are ''n'' objects, and the matrix is [[symmetric matrix|symmetrical]] (so the distance from object ''a'' to object ''b'' is the same as the distance from ''b'' to ''a'') such a matrix contains
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| :<math>{\frac{n(n-1)}{2}}</math>
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| distances. Because distances are not independent of each other – since changing the "position" of one object would change <math>n-1</math> of these distances (the distance from that object to each of the others) – we can not assess the relationship between the two matrices by simply evaluating the [[Pearson product-moment correlation coefficient|correlation coefficient]] between the two sets of distances and testing its [[statistical significance]]. The Mantel test deals with this problem.
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| The procedure adopted is a kind of randomization or [[Resampling (statistics)#Permutation tests|permutation test]]. The correlation between the two sets of <math>n(n-1)/2</math> distances is calculated, and this is both the measure of correlation reported and the [[test statistic]] on which the test is based. In principle, any correlation coefficient could be used, but normally the [[Pearson product-moment correlation coefficient]] is used.
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| In contrast to the ordinary use of the correlation coefficient, to assess significance of any apparent departure from a zero correlation, the rows and columns of one of the matrices are subjected to [[random permutation]]s many times, with the correlation being recalculated after each permutation. The significance of the observed correlation is the proportion of such permutations that lead to a higher correlation coefficient.
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| The reasoning is that if the [[null hypothesis]] of there being no relation between the two matrices is true, then permuting the rows and columns of the matrix should be equally likely to produce a larger or a smaller coefficient. In addition to overcoming the problems arising from the statistical dependence of elements within each of the two matrices, use of the permutation test means that no reliance is being placed on assumptions about the statistical distributions of elements in the matrices.
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| Many [[List of statistical packages|statistical packages]] include routines for carrying out the Mantel test.
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| ==Criticism==
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| The various papers introducing the Mantel test and its extension the partial Mantel test lack a clear statistical framework specifying fully the null and alternative hypotheses. This may convey the wrong idea that these tests are universal. For example, the Mantel and partial Mantel tests can be flawed in presence of spatial auto-correlation and return erroneously low p-values
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| See e.g. Guillot and Rousset, 2013 <ref name="Guillot2013">{{cite arXiv |eprint=1112.0651|author=Guillot G, Rousset F |year=2013 |title=Dismantling the Mantel tests |journal=Methods in Ecology and Evolution |volume=4 |number=4| pages=336-344|doi=10.1111/2041-210x.12018}}</ref>)
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| ==See also==
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| *[[Non-parametric statistics]]
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| ==References==
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| {{Reflist}}
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| == External links ==
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| *[http://www.nceas.ucsb.edu/files/scicomp/doc/SpatialEcologyMantelTest.pdf The Mantel test in ecology]
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| [[Category:Statistical tests]]
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| [[Category:Non-parametric statistics]]
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| [[Category:Permutations]]
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