Method
If there are n objects, and the matrix is symmetrical (so the distance from object a to object b is the same as the distance from b to a) such a matrix contains
distances. Because distances are not independent of each other – since changing the "position" of one object would change 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 correlation coefficient between the two sets of distances and testing its statistical significance. The Mantel test deals with this problem.
The procedure adopted is a kind of randomization or permutation test. The correlation between the two sets of 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.
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 permutations 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.
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.
Many statistical packages include routines for carrying out the Mantel test.
Read more about this topic: Mantel Test
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