Description
The Mantel test (Mantel, 1967) is a significance test of the correlation between two distance matrices. This Python implementation takes two distance matrices (either redundant matrices or condensed vectors) and returns: the veridical correlation, the empirical pvalue, and a standard score (zscore).
Optionally, you can specify: the number of permutations to produce (a larger number gives a more reliable pvalue and zscore but takes longer to run), which type of correlation coefficient to use (Pearson’s r or Spearman’s ρ), and which tail to test in the calculation of the empirical pvalue.
Requirements
 Python 2 or 3
 NumPy
 SciPy
Parameters

X
array_like: First distance matrix (condensed or redundant). 
Y
array_like: Second distance matrix (condensed or redundant), where the order of elements corresponds to the order of elements in X. 
perms
int, optional: The number of permutations to perform (default: 10000). A larger number gives more reliable results but takes longer to run. If the actual number of possible permutations is smaller, the program will enumerate all permutations. Enumeration can be forced by setting this argument to 0. 
method
str, optional: Type of correlation coefficient to use; either 'pearson' or 'spearman' (default: 'pearson'). 
tail
str, optional: Which tail to test in the calculation of the empirical pvalue; either 'upper', 'lower', or 'twotail' (default: 'twotail').
Return values

r
float: Veridical correlation 
p
float: Empirical pvalue 
z
float: Standard score (zscore)
Usage example
First import the module:
import Mantel
Let’s say we have a set of four objects and we want to correlate X (the distances between the four objects using one measure) with Y (the corresponding distances between the four objects using another measure). For example, your “objects” might be species of animal, and your two measures might be genetic distance and geographical distance (the hypothesis being that species that live far away from each other will tend to be more genetically different).
For four objects, there are six pairwise distances. First you should compute the pairwise distances for each measure and store the distances in two lists or arrays (i.e. condensed distance vectors). Alternatively, you can compute the full redundant distance matrices; this program will accept either format. No distance functions are included in this module, since the metrics you use will be specific to your particular data.
Let’s say our data looks like this:
# E.g. species A through D
# A~B A~C A~D B~C B~D C~D
dists1 = [0.2, 0.4, 0.3, 0.6, 0.9, 0.4] # E.g. genetic distances
dists2 = [0.3, 0.3, 0.2, 0.7, 0.8, 0.3] # E.g. geographical distances
We pass the data to the test()
function of the Mantel
module and optionally specify the number of permutations to test against, a correlation method to use (either ‘pearson’ or ‘spearman’), and which tail to test (either ‘upper’, ‘lower’, or ‘twotail’). In this case, we’ll use the Pearson correlation and test the upper tail, since we’re expecting to find a positive correlation.
Mantel.test(dists1, dists2, perms=10000, method='pearson', tail='upper')
This will measure the veridical Pearson correlation between the two sets of pairwise distances. It then repeatedly measures the correlation again and again under permutations of one of the distance matrices to produce a distribution of correlations under the null hypothesis. Finally, it computes the empirical pvalue (the proportion of correlations that were greater than or equal to the veridical correlation) and compares the veridical correlation with the mean and standard deviation of the correlations to generate a zscore.
In this example, the program will return the following:
# r p z
(0.91489361702127669, 0.041666666666666664, 2.0404024922610229)
Since the pvalue is less than 0.05 and the zscore is greater than 1.96, we can conclude that there is a significant correlation between these two sets of distances. This suggests that the species that live closer together tend to be more genetically related, while those that live further apart tend to be less genetically related.
Computation time
To estimate how long it will take to perform a Mantel test on your data, refer to computation_time.pdf
, which shows computation time (on a laptop computer) for 3×3 through 500×500 matrices using 1,000, 10,000, and 100,000 permutations.
Deterministic vs. stochastic Mantel tests
In the example above, we requested 10,000 permutations (the default). However, for four objects there are only 4! = 24 possible permutations of the matrix. If the number of requested permutations is greater than the number of possible permutations (as is the case here), then the program ignores your request and tests the veridical against all possible permutations of the matrix. This gives a deterministic result and can be forced by setting the perms
argument to 0
. Otherwise the program randomly samples the space of possible permutations the requested number of times. This is useful because, in the case of large matrices, it may be intractable to compute all possible permutations. For example, for 13 objects, it would take several days to compute a deterministic result, for 15 objects you’d be looking at multiple years, and 23 objects would take longer than the current age of the universe! However, for small matrices, a deterministic result should be preferred, since it is reproducible.
License
MantelTest is licensed under the terms of the MIT License.
References and links
Mantel, N. (1967). The detection of disease clustering and a generalized regression approach. Cancer Research, 27(2), 209–220.
Mantel Test on Wikipedia: https://en.wikipedia.org/wiki/Mantel_test
A guide to the Mantel test for linguists: https://joncarr.net/s/aguidetothemanteltestforlinguists.html