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 |
|---|---|
METHOD = string token |
The type of metric by which to compare the distance matrices (correlation, rankcorrelation, mantel); default corr |
NPERMUTATIONS = scalar |
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 |
|---|---|
X = symmetric matrices |
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 |
SEED = scalars |
Random number seed for the permutations; default set by RANDOMIZE |
M = scalars |
Association between Y and X |
MPERMUTED = variates |
Associations between Y and the permuted X‘s |
CUPROB = scalars |
The proportion of MPERMUTED values greater than or equal to M |
YOFFDIAGONAL = variates |
Variate to save the off-diagonal elements of the distance/similarity matrix Y |
XOFFDIAGONAL = variates |
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.
References
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