mantel 2.2.3

Python implementation of the Mantel test, a significance test of the correlation between two distance matrices

mantel

Python implementation of the Mantel test (Mantel, 1967), which is a significance test of the correlation between two distance matrices.

Installation

The mantel package is available on PyPI and can be installed using pip:

$ pip install mantel

Once installed, the package can be imported in the normal way:

import mantel

Usage

The mantel package contains one main function, test() with the following call signature:

def test(X: array_like, Y: array_like, perms: int = 10000, method: str = "pearson",
		tail: str = "two-tail", ignore_nans: bool = False) -> MantelResult:

• 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 consistent results but takes longer to run. If the number of possible permutations of the distance matrix is smaller than perms, all possible permutations will be enumerated leading to a deterministic result (this can be forced by setting perms 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 p-value; either upper, lower, or two-tail (default: two-tail).
• ignore_nans bool, optional: Ignore np.nan values in the Y matrix (default: False). This can be useful if you have missing values in one of the matrices.

The function returns a MantelResult object with the following properties:

• MantelResult.r float: Veridical correlation
• MantelResult.p float: Empirical p-value
• MantelResult.z float: Standard score (z-score)
• MantelResult.stochastic_test bool: True if the test was performed by randomly sampling possible permutations; False if the test was performed by enumerating all possible permutations.
• MantelResult.correlations array: Correlation coefficients produced by the test
• MantelResult.mean float: Mean of correlations
• MantelResult.std float: Sample standard deviation of correlations (if stochastic_test==True) or population standard deviation of correlations (if stochastic_test==False)

Worked Example

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 condensed or redundant distance matrices (the test() function will accept either format). No distance functions are included in this package, 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 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 ‘two-tail’). In this case, we’ll use the Pearson correlation and test the upper tail, since we’re expecting to find a positive correlation.

result = 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. 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. In such a scenario, where the number of requested permutations is greater than the number of possible permutations, mantel.test() simply tests against all possible permutations of the matrix, yielding a deterministic result.

Printing the result shows the veridical correlation, empirical p-value, and z-score:

print(result)
# MantelResult(0.9148936170212766, 0.041666666666666664, 2.040402492261023)

Individual results can be accessed by the relevant property name. For example, to check if the veridical correlation is significant, we could do:

print(result.p < 0.05)
# True

Since the p-value is less than 0.05, we can conclude that there is a significant correlation between the 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.

Plotting

The mantel package also provides a plot() function, which plots the distribution of sample correlations against the veridical. For example, here we generate two random distance matrices and plot the results:

dists1 = np.random.random(351)
dists2 = np.random.random(351)

result = mantel.test(dists1, dists2)

fig, axis = mantel.plot(result)
fig.savefig('example.svg')

The plot() function has several other arguments for customization:

• alpha float: Significance level for rejecting the null hypothesis (default: 5%)
• hist_colorstr: Color used for the histogram bars (default: 'lightgray').
• gaussian_color str: Color used for the normal distribution curve and the confidence interval limits (default: 'black').
• acceptance_color str: Color used for drawing the vertical line and the label of the veridical correlation if the null hypothesis is rejected according to the significance level value (default: 'black').
• rejection_color str: Color used for drawing the vertical line and the label of the veridical correlation if the null hypothesis cannot be rejected according to the significance level value (default: 'black').

Contributing

This project is in the stable/maintenance phase and is not under active development. I am only aiming to fix bugs and keep it compatible with current versions of Python so that it remains useful to the community. With that in mind, contributions are very welcome, but please read the CONTRIBUTING file before submitting a pull request.

License

This package 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/a-guide-to-the-mantel-test-for-linguists.html