Mantel test

The Mantel test is a statistical test of the correlation between two matrices. The matrices must be of the same rank, and in the usual applications, they are matrices of interrelations between the same vectors of objects.

The test is commonly used in ecology, where the data are usually estimates of the "distance" between objects such as species of organism. For example, one matrix might contain estimates of the genetic distances between the species obtained by the methods of molecular systematics. The other might contain estimates of the geographical distance between the ranges of the species.

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


 * n(n &minus; 1)/2

distances. They are not independent of each other: changing the "position" of one object would change n &minus; 1 of these distances - the distance from that object to each of the others. So we can't just assess the relationship between the two matrices by 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 test. The correlation between the two sets of n(n &minus; 1)/2 distances is calculated, and this is the measure of correlation reported. In principle any correlation coefficient could be used, but normally the Pearson product-moment correlation coefficient is used.

To assess significance, however, 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 no correlation 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.

Many statistical packages include routines for carrying out the Mantel test.

The test was first published by Nathan Mantel, a biostatistician at the National Institutes of Health, in 1967. Accounts of it can be found in advanced statistics books, e.g. Sokal & Rohlf (1995), pp. 813-819.