Mantel Correlation Test

A Mantel Correlation Test is a non-parametric correlational hypothesis test between two matrices.

• AKA: Mantel-Haenszel Test, Mantel Test.
• Example(s)
  • Partial Mantel Test.
  • Mantel Correlogram.
  • Mantel Test on Geographic Distance.
  • Mantel Test on a Predictor Matrix.
  • Partial Mantel Test on Three Distance Matrices.
  • Partial Mantel Test on Multiple Predictor Variables.
  • …
• Counter-Example(s)
  • Pearson's Correlation Test.
• See: Correlational Hypothesis Test, National Institutes of Health, Nathan Mantel, Statistics, Correlation, Matrix (Mathematics), Rank of a Matrix, Vector Space.

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References

2017a

• (ITL-SED, 2017) ⇒ Retrieved 2017-01-08 from NIST (National Intitute of Standards and Technology, US) website http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/mantel.htm
  • The Mantel-Haenszel test can be used to estimate the common odds ratio and to test whether the overall degree of association is significant. It is a consistent estimator in the following two cases:

1. When the number of tables is fixed, and possibly small, but each table has large marginal frequencies.
2. The number of tables is large. The marginal frequencies can be small in the individual tables.

(...)

The MANTEL-HAENSZEL TEST generates the following output:

1. A summary table of various statistics (odds ratio, log(odds ratio), standard error of log(odds ratio)) for each group.
2. The estimates of the common log(odds ratio) and the standard error of the common log(odds ratio).
3. A table for the Mantel-Haenszel chi-square test for the overall degree of association.
4. A large sample confidence interval for the log(odds ratio).

2017b

• (QIIME Tutorials, 2017) ⇒ Retrieved on 2017-01-15 from http://qiime.org/tutorials/distance_matrix_comparison.html
  • The Mantel test tests the correlation between two distance matrices. It is non-parametric and computes the significance of the correlation through permutations of the rows and columns of one of the input distance matrices. The test statistic is the Pearson product-moment correlation coefficient r. r falls in the range of -1 to +1, where being close to -1 indicates strong negative correlation and +1 indicates strong positive correlation. An r value of 0 indicates no correlation.

2017c

• (ST:733, 2017) ⇒ Retrieved on 2017-01-15 from http://www.stat.ncsu.edu/people/fuentes/courses/madrid/lectures/mantel
  • Mantel's (1967) test is an approach that overcomes some of the problems inherent in explaining species-environment relationships (Legendre and Fortin 1989). Mantel's test is a regression in which the variables are themselves distance or dissimilarity matrices summarizing pairwise similarities among sample locations. For example, instead of "abundance of species X on plot i" the dependent variable might be "similarity of basal area of species X on plots i and j." Similarly, the predictor variable might be "similarity of soil type" between samples instead of "soil type" for a single sample. The operative question is, "Do samples that are similar in terms of the predictor (environmental) variables also tend to be similar in terms of the dependent (species) variable?" One important case that Mantel's test considers explicitly is the case where the predictor variable is space itself, measured as geographic location (e.g., as (x,y) coordinates). In this case, the question is "Are samples that are close together also compositionally similar?" Reciprocally, "Are samples that are spatially removed (or environmentally dissimilar) from each other also compositionally dissimilar?"

2016

• (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Mantel_test Retrieved:2016-12-17.
  • The Mantel test, named after Nathan Mantel, is a statistical test of the correlation between two matrices. The matrices must be of the same rank; in most applications, they are matrices of interrelations between the same vectors of objects. The test was first published by Nathan Mantel, a biostatistician at the National Institutes of Health, in 1967. Accounts of it can be found in advanced statistics books (e.g., Sokal & Rohlf 1995).

2013

• (Diniz-Filho et al., 2013) ⇒ Diniz-Filho, José Alexandre F., et al. “Mantel test in population genetics." Genetics and Molecular Biology 36.4 (2013): 475-485 doi:10.1590/S1415-47572013000400002.
  • ABSTRACT: The comparison of genetic divergence or genetic distances, estimated by pairwise FST and related statistics, with geographical distances by Mantel test is one of the most popular approaches to evaluate spatial processes driving population structure. There have been, however, recent criticisms and discussions on the statistical performance of the Mantel test. Simultaneously, alternative frameworks for data analyses are being proposed. Here, we review the Mantel test and its variations, including Mantel correlograms and partial correlations and regressions. For illustrative purposes, we studied spatial genetic divergence among 25 populations of Dipteryx alata (“Baru”), a tree species endemic to the Cerrado, the Brazilian savannas, based on 8 microsatellite loci. We also applied alternative methods to analyze spatial patterns in this dataset, especially a multivariate generalization of Spatial Eigenfunction Analysis based on redundancy analysis. The different approaches resulted in similar estimates of the magnitude of spatial structure in the genetic data. Furthermore, the results were expected based on previous knowledge of the ecological and evolutionary processes underlying genetic variation in this species. Our review shows that a careful application and interpretation of Mantel tests, especially Mantel correlograms, can overcome some potential statistical problems and provide a simple and useful tool for multivariate analysis of spatial patterns of genetic divergence.

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