Function metaMDS
performs Nonmetric
Multidimensional Scaling (NMDS), and tries to find a stable solution
using several random starts. In addition, it standardizes the
scaling in the result, so that the configurations are easier to
interpret, and adds species scores to the site ordination. The
metaMDS
function does not provide actual NMDS, but it calls
another function for the purpose. Currently monoMDS
is
the default choice, and it is also possible to call the
isoMDS
(MASS package).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  metaMDS(comm, distance = "bray", k = 2, try = 20, trymax = 20,
engine = c("monoMDS", "isoMDS"), autotransform =TRUE,
noshare = (engine == "isoMDS"), wascores = TRUE, expand = TRUE,
trace = 1, plot = FALSE, previous.best, ...)
## S3 method for class 'metaMDS'
plot(x, display = c("sites", "species"), choices = c(1, 2),
type = "p", shrink = FALSE, ...)
## S3 method for class 'metaMDS'
points(x, display = c("sites", "species"),
choices = c(1,2), shrink = FALSE, select, ...)
## S3 method for class 'metaMDS'
text(x, display = c("sites", "species"), labels,
choices = c(1,2), shrink = FALSE, select, ...)
## S3 method for class 'metaMDS'
scores(x, display = c("sites", "species"), shrink = FALSE,
choices, ...)
metaMDSdist(comm, distance = "bray", autotransform = TRUE,
noshare = TRUE, trace = 1, commname, zerodist = "ignore",
distfun = vegdist, ...)
metaMDSiter(dist, k = 2, try = 20, trymax = 20, trace = 1, plot = FALSE,
previous.best, engine = "monoMDS", maxit = 200,
parallel = getOption("mc.cores"), ...)
initMDS(x, k=2)
postMDS(X, dist, pc=TRUE, center=TRUE, halfchange, threshold=0.8,
nthreshold=10, plot=FALSE, ...)
metaMDSredist(object, ...)

comm 
Community data. Alternatively, dissimilarities either as
a 
distance 
Dissimilarity index used in 
k 
Number of dimensions. NB., the number of points n should be n > 2*k + 1, and preferably higher in nonmetric MDS. 
try, trymax 
Minimum and maximum numbers of random starts in
search of stable solution. After 
engine 
The function used for MDS. The default is to use the

autotransform 
Use simple heuristics for possible data
transformation of typical community data (see below). If you do
not have community data, you should probably set

noshare 
Triggering of calculation stepacross or extended
dissimilarities with function 
wascores 
Calculate species scores using function

expand 
Expand weighted averages of species in

trace 
Trace the function; 
plot 
Graphical tracing: plot interim results. You may want to set

previous.best 
Start searches from a previous solution. 
x 

choices 
Axes shown. 
type 
Plot type: 
display 
Display 
shrink 
Shrink back species scores if they were expanded originally. 
labels 
Optional test to be used instead of row names. 
select 
Items to be displayed. This can either be a logical
vector which is 
X 
Configuration from multidimensional scaling. 
commname 
The name of 
zerodist 
Handling of zero dissimilarities: either

distfun 
Dissimilarity function. Any function returning a

maxit 
Maximum number of iterations in the single NMDS run;
passed to the 
parallel 
Number of parallel processes or a predefined socket
cluster. If you use predefined socket clusters (say,

dist 
Dissimilarity matrix used in multidimensional scaling. 
pc 
Rotate to principal components. 
center 
Centre the configuration. 
halfchange 
Scale axes to halfchange units. This defaults

threshold 
Largest dissimilarity used in halfchange scaling. 
nthreshold 
Minimum number of points in halfchange scaling. 
object 
A result object from 
... 
Other parameters passed to functions. Function

Nonmetric Multidimensional Scaling (NMDS) is commonly
regarded as the most robust unconstrained ordination method in
community ecology (Minchin 1987). Function metaMDS
is a
wrapper function that calls several other functions to combine
Minchin's (1987) recommendations into one command. The complete
steps in metaMDS
are:
Transformation: If the data values are larger than common
abundance class scales, the function performs a Wisconsin double
standardization (wisconsin
). If the values look
very large, the function also performs sqrt
transformation. Both of these standardizations are generally found
to improve the results. However, the limits are completely
arbitrary (at present, data maximum 50 triggers sqrt
and >9 triggers wisconsin
). If you want to
have a full control of the analysis, you should set
autotransform = FALSE
and standardize and transform data
independently. The autotransform
is intended for community
data, and for other data types, you should set autotransform
= FALSE
. This step is perfomed using metaMDSdist
.
Choice of dissimilarity: For a good result, you should use
dissimilarity indices that have a good rank order relation to
ordering sites along gradients (Faith et al. 1987). The default
is BrayCurtis dissimilarity, because it often is the test
winner. However, any other dissimilarity index in
vegdist
can be used. Function
rankindex
can be used for finding the test winner
for you data and gradients. The default choice may be bad if you
analyse other than community data, and you should probably select
an appropriate index using argument distance
. This step is
performed using metaMDSdist
.
Stepacross dissimilarities: Ordination may be very difficult
if a large proportion of sites have no shared species. In this
case, the results may be improved with stepacross
dissimilarities, or flexible shortest paths among all sites. The
default NMDS engine
is monoMDS
which is able
to break tied values at the maximum dissimilarity, and this often
is sufficient to handle cases with no shared species, and
therefore the default is not to use stepacross
with
monoMDS
. Function isoMDS
does
not handle tied values adequately, and therefore the default is to
use stepacross
always when there are sites with no
shared species with engine = "isoMDS"
. The
stepacross
is triggered by option noshare
. If
you do not like manipulation of original distances, you should set
noshare = FALSE
. This step is skipped if input data were
dissimilarities instead of community data. This step is performed
using metaMDSdist
.
NMDS with random starts: NMDS easily gets trapped into local
optima, and you must start NMDS several times from random starts
to be confident that you have found the global solution. The
strategy in metaMDS
is to first run NMDS starting with the
metric scaling (cmdscale
which usually finds a good
solution but often close to a local optimum), or use the
previous.best
solution if supplied, and take its solution
as the standard (Run 0
). Then metaMDS
starts NMDS
from several random starts (minimum number is given by try
and maximum number by trymax
). These random starts are
generated by initMDS
. If a solution is better (has a lower
stress) than the previous standard, it is taken as the new
standard. If the solution is better or close to a standard,
metaMDS
compares two solutions using Procrustes analysis
(function procrustes
with option
symmetric = TRUE
). If the solutions are very similar in their
Procrustes rmse
and the largest residual is very small, the
solutions are regarded as convergent and the better one is taken as
the new standard. The conditions are stringent, and you may have
found good and relatively stable solutions although the function is
not yet satisfied. Setting trace = TRUE
will monitor the final
stresses, and plot = TRUE
will display Procrustes overlay
plots from each comparison. This step is performed using
metaMDSiter
. This is the only step performed if input data
(comm
) were dissimilarities.
Scaling of the results: metaMDS
will run postMDS
for the final result. Function postMDS
provides the
following ways of “fixing” the indeterminacy of scaling and
orientation of axes in NMDS: Centring moves the origin to the
average of the axes; Principal components rotate the configuration
so that the variance of points is maximized on first dimension
(with function MDSrotate
you can alternatively rotate
the configuration so that the first axis is parallel to an
environmental variable); Halfchange scaling scales the
configuration so that one unit means halving of community
similarity from replicate similarity. Halfchange scaling is
based on closer dissimilarities where the relation between
ordination distance and community dissimilarity is rather linear
(the limit is set by argument threshold
). If there are
enough points below this threshold (controlled by the parameter
nthreshold
), dissimilarities are regressed on distances.
The intercept of this regression is taken as the replicate
dissimilarity, and halfchange is the distance where similarity
halves according to linear regression. Obviously the method is
applicable only for dissimilarity indices scaled to 0 …
1, such as Kulczynski, BrayCurtis and Canberra indices. If
halfchange scaling is not used, the ordination is scaled to the
same range as the original dissimilarities.
Species scores: Function adds the species scores to the final
solution as weighted averages using function
wascores
with given value of parameter
expand
. The expansion of weighted averages can be undone
with shrink = TRUE
in plot
or scores
functions, and the calculation of species scores can be suppressed
with wascores = FALSE
.
Function metaMDS
returns an object of class
metaMDS
. The final site ordination is stored in the item
points
, and species ordination in the item species
,
and the stress in item stress
(NB, the scaling of the stress
depends on the engine
: isoMDS
uses
percents, and monoMDS
proportions in the range 0
… 1). The other items store the information on the steps taken
and the items returned by the engine
function. The object has
print
, plot
, points
and text
methods.
Functions metaMDSdist
and metaMDSredist
return
vegdist
objects. Function initMDS
returns a
random configuration which is intended to be used within
isoMDS
only. Functions metaMDSiter
and
postMDS
returns the result of NMDS with updated
configuration.
The function tries hard to find two convergent solutions, but it
may fail. With default engine = "monoMDS"
the function will
tabulate the stopping criteria used, so that you can see which
criterion should be made more stringent. The criteria can be given
as arguments to metaMDS
and their current values are
described in monoMDS
. In particular, if you reach
the maximum number of iterations, you should increase the value of
maxit
. You may ask for a larger number of random starts
without losing the old ones giving the previous solution in
argument previous.best
.
In addition to too slack convergence criteria and too low number
of random starts, wrong number of dimensions (argument k
)
is the most common reason for not finding convergent
solutions. NMDS is usually run with a low number dimensions
(k=2
or k=3
), and for complex data increasing
k
by one may help. If you run NMDS with much higher number
of dimensions (say, k=10
or more), you should reconsider
what you are doing and drastically reduce k
. For very
heterogeneous data sets with partial disjunctions, it may help to
set stepacross
, but for most data sets the default
weakties = TRUE
is sufficient.
Please note that you can give all arguments of other
metaMDS*
functions and NMDS engine (default
monoMDS
) in your metaMDS
command,and you
should check documentation of these functions for details.
metaMDS
uses monoMDS
as its
NMDS engine
from vegan version 2.00, when it replaced
the isoMDS
function. You can set argument
engine
to select the old engine.
Function metaMDS
is a simple wrapper for an NMDS engine
(either monoMDS
or isoMDS
) and
some support functions (metaMDSdist
,
stepacross
, metaMDSiter
, initMDS
,
postMDS
, wascores
). You can call these support
functions separately for better control of results. Data
transformation, dissimilarities and possible
stepacross
are made in function metaMDSdist
which returns a dissimilarity result. Iterative search (with
starting values from initMDS
with monoMDS
) is
made in metaMDSiter
. Processing of result configuration is
done in postMDS
, and species scores added by
wascores
. If you want to be more certain of reaching
a global solution, you can compare results from several independent
runs. You can also continue analysis from previous results or from
your own configuration. Function may not save the used
dissimilarity matrix (monoMDS
does), but
metaMDSredist
tries to reconstruct the used dissimilarities
with original data transformation and possible
stepacross
.
The metaMDS
function was designed to be used with community
data. If you have other type of data, you should probably set some
arguments to nondefault values: probably at least wascores
,
autotransform
and noshare
should be FALSE
. If
you have negative data entries, metaMDS
will set the previous
to FALSE
with a warning.
Jari Oksanen
Faith, D. P, Minchin, P. R. and Belbin, L. (1987). Compositional dissimilarity as a robust measure of ecological distance. Vegetatio 69, 57–68.
Minchin, P.R. (1987) An evaluation of relative robustness of techniques for ecological ordinations. Vegetatio 69, 89–107.
monoMDS
(and isoMDS
),
decostand
,
wisconsin
,
vegdist
, rankindex
, stepacross
,
procrustes
, wascores
, MDSrotate
,
ordiplot
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  ## The recommended way of running NMDS (Minchin 1987)
##
data(dune)
# Global NMDS using monoMDS
sol < metaMDS(dune)
sol
plot(sol, type="t")
## Start from previous best solution
sol < metaMDS(dune, previous.best = sol)
## Local NMDS and stress 2 of monoMDS
sol2 < metaMDS(dune, model = "local", stress=2)
sol2
## Use Arrhenius exponent 'z' as a binary dissimilarity measure
sol < metaMDS(dune, distfun = betadiver, distance = "z")
sol

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