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# MANTEL procedure

Assesses the association between similarity matrices (J.W. McNicol, E.I. Duff & D.A. Elston).

### Options

`PRINT` = string token Controls printed output (`test`); default `*` i.e. none The type of metric by which to compare the distance matrices (`correlation`, `rankcorrelation`, `mantel`); default `corr` The number of permutations of the units in the second distance matrix `X` on which the significance of the correlation between `Y` and `X` is to be based; default 100

### Parameters

`Y` = symmetric matrices The first distance or similarity matrix: the order of the units of this matrix is held fixed The second distance or similarity matrix: the rows of `X` are permuted to allow the significance of the correlation between `Y` and `X` to be assessed Random number seed for the permutations; default set by `RANDOMIZE` Association between `Y` and `X` Associations between `Y` and the permuted `X`‘s The proportion of `MPERMUTED` values greater than or equal to `M` Variate to save the off-diagonal elements of the distance/similarity matrix `Y` Variate to save the off-diagonal elements of the distance/similarity matrix `X`

### Description

The extent to which two similarity/distance matrices describe the same relationships among the units can be measured by comparing their off-diagonal elements. The metrics to be used can be selected using the `METHOD` option: product-moment correlation (`correlation`), rank correlation (`rankcorrelation`) and `SUM(X*Y)` (`Mantel`). The last of these is the metric originally proposed by Mantel (1967). If the metric `rankcorrelation` is selected, the data are restricted to non-missing units and Spearman’s rank correlation is used.

The significance of the association is assessed by a permutation test. The rows/columns of the second matrix are permuted at random and the association is recalculated for each permutation. Significance is estimated by the percentage of the permutations with association less/more than or equal to that of the original association.

If the number of random permutations, specified by the `NPERMUTATIONS` option, is set to a number greater than or equal to the total number of distinct permutations d!, where d is the dimension of the symmetric matrices, the full randomization test is implemented. Otherwise the rows/columns of the second matrix are permuted at random without regard to the duplication of specific permutations. By default, 100 permutations are done. The `SEED` parameter can supply a seed for the random numbers used to generate the random permutations. By default `SEED`=0, so the random numbers will continue any existing sequence, used earlier in the Genstat program, or be initialised by the `RANDOMIZE` directive.

The two matrices to be compared are specified by the `Y` and `X` parameters. The `M` parameter allows the value of the statistic for the original matrices to be saved, the `MPERMUTED` parameter saves the values from the permuted matrices, and the `CUPROB` parameter saves the proportion of the permuted associations that are greater than the association between the original matrices. The off-diagonal elements of the matrices, on which the calculations are based, can be saved as variates using the `XOFFDIAGONAL` and `YOFFDIAGONAL` parameters.

The `PRINT` option can be set to `test` to print the values of `M` and `CUPROB`; by default there is no output.

Options: `PRINT`, `METHOD`, `NPERMUTATIONS`.

Parameters: `Y`, `X`, `SEED`, `M`, `MPERMUTED`, `CLPROB`, `YOFFDIAGONAL`, `XOFFDIAGONAL`.

### Method

The off-diagonal elements of the symmetric matrices are transferred to variates by `EQUATE`, and the association is derived by `CALCULATE` for methods `correlation` and `Mantel`, and by `SPEARMAN` for `rankcorr`. If the full randomization test is used, all possible permutations of the rows of the second matrix are generated by `PERMUTE`. Otherwise a random set of permutations is generated by permuting an index to the rows of the matrix using `RANDOMIZE`. The permutations are then performed using `CALCULATE`, with the permuted indices as a qualified identifier.

Mantel, N. (1967). The detection of disease clustering and a generalized regression approach. Cancer Research, 27, 209-220.

Manly, B.F.J. (1991). Randomization and Monte Carlo Methods in Biology. Chapman & Hall, London.

### See also

Procedure: `ECANOSIM`.

Commands for Multivariate and cluster analysis.

### Example

```CAPTION   'MANTEL example',\
!t('Data are from Tables 1.1, 1.2 and 1.3 of Manly B.F.J.',\
'(1991) Randomization and Monte Carlo Methods in Biology.');\
STYLE=meta,plain
SYMMETRIC [ROWS=8] Assoc,Dist1,Dist2
READ      Assoc
1
.30   1
.14  .50   1
.23  .50  .54    1
.30  .40  .50  .61    1
-.04  .04  .11  .03  .15   1
.02  .09  .14 -.16  .11  .14   1
-.09 -.06  .05 -.16  .03 -.06  .36  1 :
READ      Dist1
0
1 0
2 1 0
1 2 3 0
2 3 4 1 0
3 4 5 2 1 0
2 3 4 3 4 5 0
1 2 3 2 3 4 1 0 :
READ      Dist2
0
1 0
2 1 0
1 1 1 0
2 1 1 1 0
3 2 2 2 1 0
2 1 2 2 2 3 0
1 2 3 2 3 4 1 0 :
PRINT     [SERIAL=yes] Assoc,Dist1,Dist2; FIELD=7; DECIMALS=2
MANTEL    [PRINT=test; NPERMUTATIONS=25] Y=Assoc; X=Dist1; SEED=615023
MANTEL    [PRINT=test; NPERMUTATIONS=25] Y=Assoc; X=Dist2; SEED=712378
```