The Mantel test (Mantel 1967;
Mantel and Valand 1970)
is an extremely versatile statistical test that has many uses, including
spatial analysis. The Mantel test examines the relationship between two
square matrices (often distance matrices) **X**
and **Y**. The values within each
matrix (*X**ij*
or *Y**ij*)
represent a relationship between points *i*
and *j.* The relationship represented
by a matrix could be geographic distance, a data distance, an angle, a
binary matrix, or almost any other conceivable data. Often one matrix
is a binary matrix representing a hypothesis of relationships among the
points or some other relationship (*e.g.*,
*X**ij*
may equal 1 if points *i* and *j* are from the same country and
0 if they are not). By definition, the diagonals of both matrices are
always filled with zeros.

The basic Mantel statistic is simply the sum of the products of the corresponding elements of the matrices

,

where is the double sum
over all *i* and all *j
*where* i* ≠ *j*.
Because *Z* is can take on any
value depending on the exact nature of **X**
and **Y**, one usually uses a normalized
Mantel coefficient, calculated as the correlation between the pair wise
elements of **X** and **Y**.
Like any product-moment coefficient, it ranges from –1 to 1.

While more or less interpretable as a correlation coefficient, the significance
of the normalized Mantel coefficient cannot be found from standard statistical
tests because the observations within the matrices are not independent.
In general, the significance is tested through a randomization test by
randomly permuting the order of the elements within one matrix (rows and
columns are permuted in tandem). When the number of points is large (*n* > 40), it is possible to transform the Mantel statistic into a *t* statistic. The significance
of *t* is obtained from an asymptotic
approximation of the *t*-test,
which is more reliable for larger *n*.
See Fortin and Gurevitch
(1993) and Dutilleul
*et al.* (2000) for more information
on the Mantel test.

The basic Mantel test allows for the comparison of two matrices. A number
of extensions to include additional matrices have been developed (Dow
and Cheverud 1985; Hubert 1987;
Manly 1968, 1997; Smouse
*et al.* 1986). One extension
is to perform a partial Mantel test, where a third (or more) matrix is
held constant while the relationship of the first two is determined (Smouse *et
al.* 1986). This test is done by regressing the elements of
**X** and **Y**
onto the additional matrix (multiple regression for more than one additional
matrix), and using the residuals from the regressions as the input for
the standard Mantel test.

The traditional approach to estimating significance of a partial Mantel
test is to permute the rows and columns of the residual matrices (*e.g.*, this was the method used
in *PASSaGE* 1). While computationally
quick (relative to other approaches), this method has been found to have
poor statistical properties (Legendre
2000). In this version of *PASSaGE,*
one of the original matrices is permuted prior to the regression (or multiple
regression), the regression for that matrix is repeated and then the partial
Mantel correlation determined. While slower than the original approach,
this should lead to more accurate estimates of the significance or partial
Mantel statistics. See Oden
and Sokal (1992) and Legendre
(2000) for more information on partial Mantel tests.

The Mantel permutation tests are traditionally performed by comparing
the magnitudes of *Z,* which is
fine for one-tailed tests, but can give misleading results for two-tailed
tests. For a two-tailed permutation test, it is important to transform
each *Z* to an *r,*
which may be more computationally intensive, particularly for partial
Mantel tests (for simple pairwise tests it should make little difference).

Menu: | Analysis→Basic→Mantel Test |

Button: | |

Batch: | Mantel |

Mantel Test window.

The basic input for a Mantel test is a pair of square matrices: these
can be distance matrices, angle matrices, or binary matrices. All matrices
must be the same size. Optionally, additional matrices can be specified
to hold constant. You may also choose to perform permutation tests (recommended)
for the determination of significance. You have the option of one-tailed
or two-tailed permutation tests (the two-tailed tests may take a lot longer).
The output of a Mantel test includes both the observed *Z*
and the Mantel correlation. It lists the results of the asymptotic *t*-test and significance.

Mantel Test

Matrix 1: Geo Distance Matrix

Matrix 2: Data Distance Matrix

Matrices are 355 x 355

Observed Z = 3894672599.85108

Correlation = 0.31206

t = 17.81190

Left-tailed p = 1.00000

Right-tailed p = 0.00000

Two-tailed p = 0.00000

Permutation Results (999 permutations)

# of Permutations < Observed = 999

# of Permutations > Observed = 0

# of Permutations = Observed = 1

Left-tailed p = 1.00000

Right-tailed p = 0.00100

Example of Mantel Test output.

If permutation tests were performed, results follow the asymptotic test.
They include the number of times replicates (including the observed value)
are less than, greater to, or equal to the observed value, and *P*-values
for one-tailed tests (both left and right) and, if requested, for the
two-tailed test.