R: Nonmetric Multidimensional Scaling with Stable Solution from...

metaMDS {vegan} | R Documentation

## Nonmetric Multidimensional Scaling with Stable Solution from Random Starts, Axis Scaling and Species Scores

### Description

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).

### Usage

    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, ...)

### Arguments

`comm` | Community data. Alternatively, dissimilarities either as a `dist` structure or as a symmetric square matrix. In the latter case all other stages are skipped except random starts and centring and pc rotation of axes.
`distance` | Dissimilarity index used in `vegdist`.
`k` | Number of dimensions. NB., the number of points n should be n > 2*k + 1, and preferably higher in non-metric MDS.
`try, trymax` | Minimum and maximum numbers of random starts in search of stable solution. After `try` has been reached, the iteration will stop when two convergent solutions were found or `trymax` was reached.
`engine` | The function used for MDS. The default is to use the `monoMDS` function in vegan, but for backward compatibility it is also possible to use `isoMDS` of MASS.
`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 `autotransform = FALSE`.
`noshare` | Triggering of calculation step-across or extended dissimilarities with function `stepacross`. The argument can be logical or a numerical value greater than zero and less than one. If `TRUE`, extended dissimilarities are used always when there are no shared species between some sites, if `FALSE`, they are never used. If `noshare` is a numerical value, `stepacross` is used when the proportion of site pairs with no shared species exceeds `noshare`. The number of pairs with no shared species is found with `no.shared` function, and `noshare` has no effect if input data were dissimilarities instead of community data.
`wascores` | Calculate species scores using function `wascores`.
`expand` | Expand weighted averages of species in `wascores`.
`trace` | Trace the function; `trace = 2` or higher will be more voluminous.
`plot` | Graphical tracing: plot interim results. You may want to set `par(ask = TRUE)` with this option.
`previous.best` | Start searches from a previous solution.
`x` | `metaMDS` result (or a dissimilarity structure for `initMDS`.
`choices` | Axes shown.
`type` | Plot type: `"p"` for points, `"t"` for text, and `"n"` for axes only.
`display` | Display `"sites"` or `"species"`.
`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 `TRUE` for displayed items or a vector of indices of displayed items.
`X` | Configuration from multidimensional scaling.
`commname` | The name of `comm`: should not be given if the function is called directly.
`zerodist` | Handling of zero dissimilarities: either `"fail"` or `"add"` a small positive value, or `"ignore"`. `monoMDS` accepts zero dissimilarities and the default is `zerodist = "ignore"`, but with `isoMDS` you may need to set `zerodist = "add"`.
`distfun` | Dissimilarity function. Any function returning a `dist` object and accepting argument `method` can be used (but some extra arguments may cause name conflicts).
`maxit` | Maximum number of iterations in the single NMDS run; passed to the `engine` function `monoMDS` or `isoMDS`.
`parallel` | Number of parallel processes or a predefined socket cluster. If you use pre-defined socket clusters (say, `clus`), you must issue `clusterEvalQ(clus, library(vegan))` to make available internal vegan functions. With `parallel = 1` uses ordinary, non-parallel processing. The parallel processing is done with parallel package.
`dist` | Dissimilarity matrix used in multidimensional scaling.
`pc` | Rotate to principal components.
`center` | Centre the configuration.
`halfchange` | Scale axes to half-change units. This defaults `TRUE` when dissimilarities were evaluated within `metaMDS` and the dissimilarity index has an upper limit of 1. If `FALSE`, the ordination dissimilarities are scaled to the same range as the input dissimilarities.
`threshold` | Largest dissimilarity used in half-change scaling.
`nthreshold` | Minimum number of points in half-change scaling.
`object` | A result object from `metaMDS`.
`...` | Other parameters passed to functions. Function `metaMDS` passes all arguments to its component functions `metaMDSdist`, `metaMDSiter`, `postMDS`, and to `distfun` and `engine`.

### Details

Non-metric 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:

1. 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`.

2. 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 Bray-Curtis 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`.

3. Step-across 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`.

4. 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.

5. 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); Half-change scaling scales the configuration so that one unit means halving of community similarity from replicate similarity. Half-change 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 half-change 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, Bray-Curtis and Canberra indices. If half-change scaling is not used, the ordination is scaled to the same range as the original dissimilarities.

6. 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`.

### Value

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.

### Convergence Problems

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.

### Warning

`metaMDS` uses `monoMDS` as its NMDS `engine` from vegan version 2.0-0, when it replaced the `isoMDS` function. You can set argument `engine` to select the old engine.

### Note

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 non-default 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.

### Author(s)

Jari Oksanen

### References

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.

### See Also

`monoMDS` (and `isoMDS`), `decostand`, `wisconsin`, `vegdist`, `rankindex`, `stepacross`, `procrustes`, `wascores`, `MDSrotate`, `ordiplot`.

### Examples

    ## 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

[Package vegan version 2.4-3 Index]