Mantel and Partial Mantel Tests for Dissimilarity Matrices

Function mantel finds the Mantel statistic as a matrix correlation between two dissimilarity matrices, and function mantel.partial finds the partial Mantel statistic as the partial matrix correlation between three dissimilarity matrices. The significance of the statistic is evaluated by permuting rows and columns of the first dissimilarity matrix.

mantel(xdis, ydis, method="pearson", permutations=999, strata = NULL,
    na.rm = FALSE, parallel = getOption("mc.cores"))
mantel.partial(xdis, ydis, zdis, method = "pearson", permutations = 999,
    strata = NULL, na.rm = FALSE, parallel = getOption("mc.cores"))

Arguments

xdis, ydis, zdis	Dissimilarity matrices ordist objects. The first object xdis will be permuted in permutation tests.
method	Correlation method, as accepted by cor: "pearson", "spearman" or "kendall".
permutations	a list of control values for the permutations as returned by the function how, or the number of permutations required, or a permutation matrix where each row gives the permuted indices.
strata	An integer vector or factor specifying the strata for permutation. If supplied, observations are permuted only within the specified strata.
na.rm	Remove missing values in calculation of Mantel correlation. Use this option with care: Permutation tests can be biased, in particular if two matrices had missing values in matching positions.
parallel	Number of parallel processes or a predefined socket cluster. With parallel = 1 uses ordinary, non-parallel processing. The parallel processing is done with parallel package.

Details

Mantel statistic is simply a correlation between entries of two dissimilarity matrices (some use cross products, but these are linearly related). However, the significance cannot be directly assessed, because there are \(N(N-1)/2\) entries for just \(N\) observations. Mantel developed asymptotic test, but here we use permutations of \(N\) rows and columns of dissimilarity matrix. Only the first matrix (xdist) will be permuted, and the second is kept constant. See permutations for additional details on permutation tests in Vegan.

Partial Mantel statistic uses partial correlation conditioned on the third matrix. Only the first matrix is permuted so that the correlation structure between second and first matrices is kept constant. Although mantel.partial silently accepts other methods than "pearson", partial correlations will probably be wrong with other methods.

The function uses cor, which should accept alternatives pearson for product moment correlations and spearman or kendall for rank correlations.

Value

The function returns a list of class mantel with following components:

Call

Function call.

method

Correlation method used, as returned by cor.test.

statistic

The Mantel statistic.

signif

Empirical significance level from permutations.

perm

A vector of permuted values. The distribution of permuted values can be inspected with permustats function.

permutations

Number of permutations.

control

A list of control values for the permutations as returned by the function how.

References

The test is due to Mantel, of course, but the current implementation is based on Legendre and Legendre.

Legendre, P. and Legendre, L. (2012) Numerical Ecology. 3rd English Edition. Elsevier.

Note

Legendre & Legendre (2012, Box 10.4) warn against using partial Mantel correlations.

See also

cor for correlation coefficients, protest (Procrustes test) for an alternative with ordination diagrams, anosim and mrpp for comparing dissimilarities against classification. For dissimilarity matrices, see vegdist or dist. See bioenv for selecting environmental variables.

Examples

## Is vegetation related to environment?
data(varespec)
data(varechem)
veg.dist <- vegdist(varespec) # Bray-Curtis
env.dist <- vegdist(scale(varechem), "euclid")
mantel(veg.dist, env.dist)
#> 
#> Mantel statistic based on Pearson's product-moment correlation 
#> 
#> Call:
#> mantel(xdis = veg.dist, ydis = env.dist) 
#> 
#> Mantel statistic r: 0.3047 
#>       Significance: 0.002 
#> 
#> Upper quantiles of permutations (null model):
#>   90%   95% 97.5%   99% 
#> 0.107 0.141 0.167 0.202 
#> Permutation: free
#> Number of permutations: 999
#> 
mantel(veg.dist, env.dist, method="spear")
#> 
#> Mantel statistic based on Spearman's rank correlation rho 
#> 
#> Call:
#> mantel(xdis = veg.dist, ydis = env.dist, method = "spear") 
#> 
#> Mantel statistic r: 0.2838 
#>       Significance: 0.001 
#> 
#> Upper quantiles of permutations (null model):
#>   90%   95% 97.5%   99% 
#> 0.118 0.153 0.179 0.213 
#> Permutation: free
#> Number of permutations: 999
#>