# Mantel test

The Mantel test tests the correlation between two proximity matrices. Available in Excel using the XLSTAT add-on statistical software.

## What is the Mantel test

The **Mantel test **measures and tests the linear correlation between **two proximity matrices** (simple Mantel test) or between two matrices while taking into account the linear correlation with a third matrix (partial Mantel test).

### Simple Mantel test

Mantel proposed a first statistic to measure the correlation between two proximity (similarity or dissimilarity) and symmetric A and B matrices of size n:

Z(AB) = ∑_{(i-n...n-1) } ∑_{(j-i+1...n)} a_{ij} b_{ij}

In the case where the matrices are not symmetric, the computations are possible.

While it is not a problem to compute the correlation coefficient between two sets of proximity coefficients, testing their significance cannot be done using the usual approach that is used to test correlations: to use the latter tests, one needs to assume the independence of the data, which is not the case here. A permutation test has been proposed to determine if the correlation coefficient can be considered as showing a significant correlation between the matrices or not. It can be one- or two-sided.

The Mantel test consists of computing the correlation coefficient that would be obtained after permuting the rows and columns of one of the matrices. The p-value is calculated using the distribution of the r(AB) coefficients obtained from S permutations. In the case where n, the number of rows and columns of the matrices, is lower than 10, all the possible permutations can easily be computed. If n is greater than 10, one needs to randomly generate a set of S permutations in order to estimate the distribution of r(AB).

### Partial Mantel test

A Mantel test for more than two matrices has been proposed: when we have three proximity matrices A, B and C, the partial Mantel statistic r(AB.C) for the A and B matrices knowing the C matrix is computed as a partial correlation coefficient. In order to determine if the coefficient is significantly different from 0, a p-value is computed using random permutations.