Mantel test
Encyclopedia
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 vector
s of objects.
The test is commonly used in ecology
, where the data are usually estimates of the "distance" between objects such as species
of organisms. For example, one matrix might contain estimates of the genetic
distances (i.e., the amount of difference between two different genomes) between all possible pairs of species in the study, obtained by the methods of molecular systematics; while the other might contain estimates of the geographical distance between the ranges of each species and every other species.
If there are n objects, and the matrix is symmetrical (so the distance from object a to object b is the same as the distance from b to a) such a matrix contains
distances. Because distances are not independent of each other – since changing the "position" of one object would change n − 1 of these distances (the distance from that object to each of the others) – we can't assess the relationship between the two matrices by simply evaluating the correlation coefficient
between the two sets of distances and testing its statistical significance
. The Mantel test deals with this problem.
The procedure adopted is a kind of randomization or permutation test. The correlation between the two sets of n(n − 1)/2 distances is calculated, and this is both the measure of correlation reported and the test statistic
on which the test is based. In principle any correlation coefficient could be used, but normally the Pearson product-moment correlation coefficient
is used.
In contrast to the ordinary use of the correlation coefficient, to assess significance of any apparent departure from a zero correlation, the rows and columns of one of the matrices are subjected to random permutation
s many times, with the correlation being recalculated after each permutation. The significance of the observed correlation is the proportion of such permutations that lead to a higher correlation coefficient.
The reasoning is that if the null hypothesis
of there being no relation between the two matrices is true, then permuting the rows and columns of the matrix should be equally likely to produce a larger or a smaller coefficient. In addition to overcoming the problems arising from the statistical dependence of elements within each of the two matrices, use of the permutation test means that no reliance is being placed on assumptions about the statistical distributions of elements in the matrices.
Many statistical packages include routines for carrying out the Mantel test.
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), pp. 813–819.
Nathan Mantel
Nathan Mantel was a biostatistician best known for his work with William Haenszel which led to the Mantel–Haenszel test and its associated estimate, the Mantel–Haenszel odds ratio...
, is a statistical
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
test of the correlation
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
between two matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. The matrices must be of the same rank, in most applications they are matrices of interrelations between the same vector
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s of objects.
The test is commonly used in ecology
Ecology
Ecology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...
, where the data are usually estimates of the "distance" between objects such as species
Species
In biology, a species is one of the basic units of biological classification and a taxonomic rank. A species is often defined as a group of organisms capable of interbreeding and producing fertile offspring. While in many cases this definition is adequate, more precise or differing measures are...
of organisms. For example, one matrix might contain estimates of the genetic
Genetics
Genetics , a discipline of biology, is the science of genes, heredity, and variation in living organisms....
distances (i.e., the amount of difference between two different genomes) between all possible pairs of species in the study, obtained by the methods of molecular systematics; while the other might contain estimates of the geographical distance between the ranges of each species and every other species.
If there are n objects, and the matrix is symmetrical (so the distance from object a to object b is the same as the distance from b to a) such a matrix contains
- n(n − 1)/2
distances. Because distances are not independent of each other – since changing the "position" of one object would change n − 1 of these distances (the distance from that object to each of the others) – we can't assess the relationship between the two matrices by simply evaluating the correlation coefficient
Pearson product-moment correlation coefficient
In statistics, the Pearson product-moment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...
between the two sets of distances and testing its statistical significance
Statistical significance
In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. The phrase test of significance was coined by Ronald Fisher....
. The Mantel test deals with this problem.
The procedure adopted is a kind of randomization or permutation test. The correlation between the two sets of n(n − 1)/2 distances is calculated, and this is both the measure of correlation reported and the test statistic
Test statistic
In statistical hypothesis testing, a hypothesis test is typically specified in terms of a test statistic, which is a function of the sample; it is considered as a numerical summary of a set of data that...
on which the test is based. In principle any correlation coefficient could be used, but normally the Pearson product-moment correlation coefficient
Pearson product-moment correlation coefficient
In statistics, the Pearson product-moment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...
is used.
In contrast to the ordinary use of the correlation coefficient, to assess significance of any apparent departure from a zero correlation, the rows and columns of one of the matrices are subjected to random permutation
Random permutation
A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and simulation...
s many times, with the correlation being recalculated after each permutation. The significance of the observed correlation is the proportion of such permutations that lead to a higher correlation coefficient.
The reasoning is that if the null hypothesis
Null hypothesis
The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...
of there being no relation between the two matrices is true, then permuting the rows and columns of the matrix should be equally likely to produce a larger or a smaller coefficient. In addition to overcoming the problems arising from the statistical dependence of elements within each of the two matrices, use of the permutation test means that no reliance is being placed on assumptions about the statistical distributions of elements in the matrices.
Many statistical packages include routines for carrying out the Mantel test.
The test was first published by Nathan Mantel
Nathan Mantel
Nathan Mantel was a biostatistician best known for his work with William Haenszel which led to the Mantel–Haenszel test and its associated estimate, the Mantel–Haenszel odds ratio...
, a biostatistician at the National Institutes of Health
National Institutes of Health
The National Institutes of Health are an agency of the United States Department of Health and Human Services and are the primary agency of the United States government responsible for biomedical and health-related research. Its science and engineering counterpart is the National Science Foundation...
, in 1967. Accounts of it can be found in advanced statistics books, e.g. Sokal & Rohlf (1995), pp. 813–819.