### Introduction

### Theory, Examples & Exercises

en:pcoa_nmds

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This method is also known as MDS (Metric Multidimensional Scaling). While PCA preserves Euclidean distances among samples and CA chi-square distances, PCoA provides Euclidean representation of a set of objects whose relationship is measured by any similarity or distance measure chosen by the user. As well as PCA and CA, PCoA returns a set of orthogonal axes whose importance is measured by eigenvalues. This means that calculating PCoA on Euclidean distances among samples yields the same results as PCA calculated on covariance matrix of the same dataset (if scaling 1 is used).

(library`cmdscale`

`vegan`

) - calculates PCoA on matrix of distances among samples (this could be calculated e.g. by function`vegdist`

from library`vegan`

). Use function`ordiplot`

to project the ordination diagram.(library`pcoa`

`ape`

) - another way how to achieve PCoA analysis. Use`biplot.pcoa`

function to project ordination diagram.

Non-metric alternative to PCoA analysis - it can use any distance measure among samples, and the main focus is on projecting the relative position of sample points into low dimensional ordination space (two or three axes). The method is distance based, not eigenvalue based - it means that it does not attempt to maximize the variance preserved by particular ordination axes and resulting projection could therefore be rotated in any direction.

The algorithm goes like this (simplified):

- Specify the number of dimensions
*m*you want to use (into which you want to*scale*down the distribution of samples in multidimensional space - that's why it's scaling). - Construct initial configuration of all samples in
*m*dimensions as a starting point of iterative process. The result of the whole iteration procedure may depend on this step, so it's somehow crucial - the initial configuration could be generated by random, but better way is to help it a bit, e.g. by using PCoA ordination as a starting position. - An iterative procedure tries to reshuffle the objects in given number of dimension in such a way that the real distances among objects reflects best their compositional dissimilarity. Fit between these two parameters is expressed as so called
*stress value*- the lower stress value the better. - Algorithm stops when new iteration cannot lower the stress value - the solution has been reached.
- After the algorithm is finished, the final solution is rotated using PCA to ease its interpretation (that's why final ordination diagram has ordination axes, even if original algorithm doesn't produce any).

(library`metaMDS`

`vegan`

) - rather advanced function, composed of many subroutine steps. See example below for details.(library`stressplot`

`vegan`

) - draws Shepards stress plot, which is the relationship between real distances between samples in resulting*m*dimensional ordination solution, and their particular compositional dissimilarities expressed by selected dissimilarity measure.(library`goodness`

`vegan`

) - returns goodness-of-fit of particular samples. See example how can be this result visualized (inspired by Borcard et al. 2011).

vltava.spe <- read.delim ('http://www.davidzeleny.net/anadat-r/data-download/vltava-spe.txt', row.names = 1) NMDS <- metaMDS (vltava.spe)

Square root transformation Wisconsin double standardization Run 0 stress 0.2022791 Run 1 stress 0.2193042 Run 2 stress 0.2130607 Run 3 stress 0.208742 Run 4 stress 0.2022791 ... procrustes: rmse 9.278716e-06 max resid 3.31574e-05 *** Solution reached

NMDS

Call: metaMDS(comm = vltava.spe) global Multidimensional Scaling using monoMDS Data: wisconsin(sqrt(vltava.spe)) Distance: bray Dimensions: 2 Stress: 0.2022791 Stress type 1, weak ties Two convergent solutions found after 4 tries Scaling: centring, PC rotation, halfchange scaling Species: expanded scores based on ‘wisconsin(sqrt(vltava.spe))’

If the default setting of metaMDS function is used, the data are automatically (if necessary) transformed (in this case, combination of wisconsin and sqrt transformation was used). In this case, stress value is 20.2.

To draw the result, use the function `ordiplot`

. In this case, using `type = 't`

' will add text labels (default setting adds only points):

ordiplot (NMDS, type = 't')

par (mfrow = c(1,2)) # this function divides plotting window into two columns stressplot (NMDS) plot (NMDS, display = 'sites', type = 't', main = 'Goodness of fit') # this function draws NMDS ordination diagram with sites points (NMDS, display = 'sites', cex = goodness (NMDS)*200) # and this adds the points with size reflecting goodness of fit (bigger = worse fit)

Use data from the variable `eurodist`

, which is available in R (you don't need to install any library, just type `eurodist`

). This variable contains real geographical distances among big European cities (in km).

- Using this distance matrix, calculate PCoA analysis and draw the PCoA ordination diagram - result will look somehow like a map of Europe.
- Draw also screeplot of eigenvalues for individual PCoA axes.

For hints click here ☛

For hints click here ☛

`cmdscale`

,`ordiplot`

with argument`type = 't`

'. To make the illusion perfect, you will perhaps need to rotate scores on the second ordination axis (the vertical one) to put Stockholm at the north and Rome at the south (you need to multiply these scores by -1).`barplot`

. You need to calculate these eigenvalues - in`cmdscale`

, use the argument`eig = TRUE`

, and extract the resulting eigenvalues in the resulting object (list) as $eig.

Use data about confusion of different Morse codes, originating from Rothkopf's experiment with Morse codes. This is a classical data set, used by Shepard (1962)^{1)} to demonstrate the use of NMDS analysis.

- After importing the dataset into R, the column names contain letters and row names contain Morse codes - this needs to be unified, so as column names also contain Morse codes. In R, you will need to copy row names into column names.
- Use the distance matrix between the Morse codes (so called confusion matrix, each number represents the number of cases, when respondents consider given pair of codes as being different) to calculate NMDS analysis.
- What is the stress value of the resulting analysis?
- Draw ordination diagram and Shepard diagram.

For hints click here ☛

For hints click here ☛

`colnames`

or`names`

,`rownames`

`metaMDS`

from`vegan`

- check the results of
`metaMDS`

`ordiplot`

,`stressplot`

Check the example Betadiversity of coral reefs after disturbance to apply NMDS analysis on community data from coral reefs.

Shepard, R. N. (1962): The Analysis of Proximities: Multidimensional Scaling with an Unknown Distance Function, I and II. *Psychometrika*, 27: 125-139 and 219-246.

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