This function computes Mantel's permutation test for similarity of two matrices. It permutes the rows and columns of the second matrix randomly and calculates a \(Z\)-statistic.

```
mantel.test(m1, m2, nperm = 999, graph = FALSE,
alternative = "two.sided", ...)
```

m1

a numeric matrix giving a measure of pairwise distances, correlations, or similarities among observations.

m2

a second numeric matrix giving another measure of pairwise distances, correlations, or similarities among observations.

nperm

the number of times to permute the data.

graph

a logical indicating whether to produce a summary graph (by default the graph is not plotted).

alternative

a character string defining the alternative
hypothesis: `"two.sided"`

(default), `"less"`

,
`"greater"`

, or any unambiguous abbreviation of these.

…

further arguments to be passed to `plot()`

(to add a
title, change the axis labels, and so on).

the \(Z\)-statistic (sum of rows*columns of lower triangle) of the data matrices.

\(P\)-value (quantile of the observed \(Z\)-statistic in the permutation distribution).

the alternative hypothesis.

The function calculates a \(Z\)-statistic for the Mantel test, equal to the sum of the pairwise product of the lower triangles of the permuted matrices, for each permutation of rows and columns. It compares the permuted distribution with the \(Z\)-statistic observed for the actual data.

The present implementation can analyse symmetric as well as (since version 5.1 of ape) asymmetric matrices (see Mantel 1967, Sects. 4 and 5). The diagonals of both matrices are ignored.

If `graph = TRUE`

, the functions plots the density estimate of
the permutation distribution along with the observed \(Z\)-statistic
as a vertical line.

The `…`

argument allows the user to give further options to
the `plot`

function: the title main be changed with `main=`

,
the axis labels with `xlab =`

, and `ylab =`

, and so on.

Mantel, N. (1967) The detection of disease clustering and a
generalized regression approach. *Cancer Research*, **27**,
209--220.

Manly, B. F. J. (1986) *Multivariate statistical methods: a primer.*
London: Chapman & Hall.

# NOT RUN { q1 <- matrix(runif(36), nrow = 6) q2 <- matrix(runif(36), nrow = 6) diag(q1) <- diag(q2) <- 0 mantel.test(q1, q2, graph = TRUE, main = "Mantel test: a random example with 6 X 6 matrices representing asymmetric relationships", xlab = "z-statistic", ylab = "Density", sub = "The vertical line shows the observed z-statistic") # }