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

Arguments

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, and the step is skipped if input were dissimilarities.

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, and the step is skipped if input were dissimilarities.

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, and the step is skipped always when input were dissimilarities.

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 first step performed if input data (comm) were dissimilarities. Random starts can be run with parallel processing (argument parallel).

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. Half-change scaling is skipped by default if input were dissimilarities, but can be turned on with argument halfchange = TRUE. NB., The PC rotation only changes the directions of reference axes, and it does not influence the configuration or solution in general.

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. This step is skipped if input were dissimilarities and community data were unavailable. However, the species scores can be added or replaced with sppscores.

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.

Common Wrong Claims

NMDS is often misunderstood and wrong claims of its properties are common on the Web and even in publications. It is often claimed that the NMDS configuration is non-metric which means that you cannot fit environmental variables or species onto that space. This is a false statement. In fact, the result configuration of NMDS is metric, and it can be used like any other ordination result. In NMDS the rank orders of Euclidean distances among points in ordination have a non-metric monotone relationship to any observed dissimilarities. The transfer function from observed dissimilarities to ordination distances is non-metric (Kruskal 1964a, 1964b), but the ordination result configuration is metric and observed dissimilarities can be of any kind (metric or non-metric).

The ordination configuration is usually rotated to principal components in metaMDS. The rotation is performed after finding the result, and it only changes the direction of the reference axes. The only invariant feature in the NMDS solution is the rank order of Euclidean ordination distances, and this does not change in rotation. Similarly, the rank order of distances does not change in uniform scaling or centring of configuration of points.

Function stressplot displays the method graphically: it plots the observed dissimilarities against distances in ordination space, and also shows the non-metric monotone regression.

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.

Kruskal, J.B. (1964a). Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika 29, 1–28.

Kruskal, J.B. (1964b). Nonmetric multidimensional scaling: a numerical method. Psychometrika 29, 115–129.

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, sppscores, MDSrotate, ordiplot, stressplot.

Examples