using eigen-decomposition of the transformed dissimilarity matrix, whereas in
NMDS the solution is found by an iterative approximation algorithm. The literature
shows that some researchers favour the former and others the latter, generally
without explicit justification. Users may wonder: are the two methods equally
suitable in all situations? Do they produce identical or fairly similar solutions? If
not, are there situations where one method is preferable over the other? How do they
compare in their principles and applications? Some of their properties can be
advocated in favour of the one or the other, depending on the application. Let us
enumerate them.
NMDS
1. The iterative algorithm may find different solutions depending on the starting
point of the calculation, which, in most instances, is a randomly chosen
configuration.
2. The dissimilarities are distorted (stretched or squeezed) during NMDS calculation. That is an acknowledged property of the method. The distances in the
ordination solution do not exactly correspond to the starting dissimilarities.
3. The first NMDS axis is not bound to maximize the variance of the observations.
However, most if not all NMDS programs compute a PCA rotation of the NMDS
result, so that the first axis maximizes the variance of the NMDS solution.
4. Different criteria are available in different NMDS functions to minimize the
stress. The solution may differ depending on the criterion that one (or the R
function) chooses to optimise.
5. R functions may contain several other arguments that may influence the solution.
Except for highly experienced users, it is difficult to determine which combination of options is best for a particular data set.
6. Users must set k, the number of dimensions of the ordination solution. All
ordination axes can be produced, i.e. min[p, nÀ1], but then the solution is not
exact since the distances are distorted. It is recommended to set k < (n – 1)/2.
7. The stress statistic does not indicate the proportion of the variance of the data
represented in the ordination solution. Instead, it indicates the amount of deformation of the original dissimilarities, which is very different.
Application to ecological analysis – When it is necessary to squeeze in two
dimensions a PCA or PCoA solution that requires 3 or 4 dimensions, for instance
in a figure drawn for publication, NMDS is useful to represent well the main
dissimilarity relationships among the sites in 2-D. In such a case, it is preferable to
use the PCA or PCoA ordination axes as input into NMDS to make sure that the
NMDS solution will not diverge markedly from the metric ordination.
5.6 Nonmetric Multidimensional Scaling (NMDS)
197
NMDS the solution is found by an iterative approximation algorithm. The literature
shows that some researchers favour the former and others the latter, generally
without explicit justification. Users may wonder: are the two methods equally
suitable in all situations? Do they produce identical or fairly similar solutions? If
not, are there situations where one method is preferable over the other? How do they
compare in their principles and applications? Some of their properties can be
advocated in favour of the one or the other, depending on the application. Let us
enumerate them.
NMDS
1. The iterative algorithm may find different solutions depending on the starting
point of the calculation, which, in most instances, is a randomly chosen
configuration.
2. The dissimilarities are distorted (stretched or squeezed) during NMDS calculation. That is an acknowledged property of the method. The distances in the
ordination solution do not exactly correspond to the starting dissimilarities.
3. The first NMDS axis is not bound to maximize the variance of the observations.
However, most if not all NMDS programs compute a PCA rotation of the NMDS
result, so that the first axis maximizes the variance of the NMDS solution.
4. Different criteria are available in different NMDS functions to minimize the
stress. The solution may differ depending on the criterion that one (or the R
function) chooses to optimise.
5. R functions may contain several other arguments that may influence the solution.
Except for highly experienced users, it is difficult to determine which combination of options is best for a particular data set.
6. Users must set k, the number of dimensions of the ordination solution. All
ordination axes can be produced, i.e. min[p, nÀ1], but then the solution is not
exact since the distances are distorted. It is recommended to set k < (n – 1)/2.
7. The stress statistic does not indicate the proportion of the variance of the data
represented in the ordination solution. Instead, it indicates the amount of deformation of the original dissimilarities, which is very different.
Application to ecological analysis – When it is necessary to squeeze in two
dimensions a PCA or PCoA solution that requires 3 or 4 dimensions, for instance
in a figure drawn for publication, NMDS is useful to represent well the main
dissimilarity relationships among the sites in 2-D. In such a case, it is preferable to
use the PCA or PCoA ordination axes as input into NMDS to make sure that the
NMDS solution will not diverge markedly from the metric ordination.
5.6 Nonmetric Multidimensional Scaling (NMDS)
197