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Section: Ordination analysis

PCoA & NMDS (distance-based unconstrained ordination)

Example 1 - PCoA on the matrix of distances between European cities

We use data from the variable eurodist, which is available in R (you don't need to install any library, just type euro disc). This distance matrix contains real geographical distances among big European cities (driving distance, in km). We will use this matrix to calculate PCoA and draw the PCoA ordination diagram, and also a screeplot of eigenvalues for individual PCoA axes.

To calculate PCoA, use the base R function cmdscale (note that vegan contains the function wcmdscale, which in default setting is doing the same):

pcoa <- cmdscale (eurodist, eig = TRUE)

Note that I set up the argument eig = TRUE - in this way, the cmdscale function returns also the eigenvalues for individual axes (in the default setting this argument is set to FALSE and the function returns only the data frame of sample scores on individual PCoA axes).

Now we can draw the ordination diagram of the cities:

library (vegan)
ordiplot (pcoa, display = 'sites', type = 'text')

You can see that the distances between cities make intuitive sense (Athens are far from Stockholm, for example), and it almost looks like the map of Europe, except that Athens are at north and Stockholm at south. Let's flip the y-axis (the second axis of PCoA) and draw the ordination diagram again. The sample scores in PCoA ordination are in the object pcoa, in the component points1) (use str (pcoa) if you wish to see the structure of pcoa object).

pcoa$points[,2] <- -pcoa$points[,2]
ordiplot (pcoa, display = 'sites', type = 't')

Now the distribution of cities make better geographical sense!

Finally, let's draw the screeplot with eigenvalues for individual axes; these eigenvalues are stored in the component $eig of the object pcoa:

barplot (pcoa$eig, names = paste ('PCoA', 1:21), las = 3, ylab = 'eigenvalues')

Note that names argument adds the names to tickmarks on horizontal axis, las argument influences rotation of labels on both x and y axis (see ?par for explanation) and ylab adds the label to the vertical axis.

Example 2: NMDS on the Morse code confusion matrix

Use data about the confusion of different Morse codes, originating from Rothkopf's experiment with Morse codes. This is a classical data set, used by Shepard (1962)2) to demonstrate the use of NMDS analysis.

library (vegan)
morse.dist <- read.delim ('', row.names = 1, head = T)
names (morse.dist) <- rownames (morse.dist)
NMDS <- metaMDS (morse.dist)

The stress values are in the following output:

metaMDS(comm = morse.dist) 

global Multidimensional Scaling using monoMDS

Data:     morse.dist 
Distance: user supplied 

Dimensions: 2 
Stress:     0.1800255 
Stress type 1, weak ties
Two convergent solutions found after 1 tries
Scaling: centring, PC rotation 
Species: scores missing

The ordination diagram and Shepard diagram could be drawn in the following way:

par (mfrow = c(1,2))
ordiplot (NMDS, cex = 1.5, type = 't')
stressplot (NMDS)

From left to right there is a gradient of increasing code length, from bottom-up increases the proportion of long beeps within the code. This relationship can be also directly displayed by projecting these two attributes as supplementary variables onto the NMDS ordination diagram:

morse.attr <- read.delim ('', row.names = 1, head = T)
ef <- envfit (NMDS, morse.attr)
ordiplot (NMDS, cex = 1.5, type = 't')
plot (ef)

Example 3: NMDS of river valley dataset

vltava.spe <- read.delim ('', 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
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)

If we have calculated cmdscale with eig = FALSE, the structure of pcoa object would be simpler, it would be just a data frame with sample scores; with eig = TRUE the object became a list with the score data frame nested inside within the component $points.
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.
en/pcoa_nmds_examples.txt · Last modified: 2021/08/04 23:53 by David Zelený

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