Chapter-9

Joey and Sean

Plan for today

NMDS
Correspondence analysis
Big thanks to Tim Bowles for sharing his code on vegan!!
I will use a lot of it today!

Load packages and functions

# (vegan must be loaded after ade4 to avoid some conflicts)
library(ade4)
library(vegan)
library(gclus)
library(ape)

Load additional functions

# (files must be in the working directory)
source("evplot.R")
source("cleanplot.pca.R")
source("PCA.R")
source("CA.R")

NMDS

the objective is to plot dissimilar objects far apart in the ordination space and similar objects close to one another.
can use any distance matrix
can deal with missing data
NMDS is rank based approach which attempts to represent pairwise dissimilarity between objects in low dimensional space

NMDS applied to the varespec data: vegetation lichen pastures

data(varespec) ## 44 species, lots of zeros, so  not amenable to PCA
str(varespec)

## 'data.frame':	24 obs. of  44 variables:
##  $ Callvulg: num  0.55 0.67 0.1 0 0 ...
##  $ Empenigr: num  11.13 0.17 1.55 15.13 12.68 ...
##  $ Rhodtome: num  0 0 0 2.42 0 0 1.55 0 0.35 0.07 ...
##  $ Vaccmyrt: num  0 0.35 0 5.92 0 ...
##  $ Vaccviti: num  17.8 12.1 13.5 16 23.7 ...
##  $ Pinusylv: num  0.07 0.12 0.25 0 0.03 0.12 0.1 0.1 0.05 0.12 ...
##  $ Descflex: num  0 0 0 3.7 0 0.02 0.78 0 0.4 0 ...
##  $ Betupube: num  0 0 0 0 0 0 0.02 0 0 0 ...
##  $ Vacculig: num  1.6 0 0 1.12 0 0 2 0 0.2 0 ...
##  $ Diphcomp: num  2.07 0 0 0 0 0 0 0 0 0.07 ...
##  $ Dicrsp  : num  0 0.33 23.43 0 0 ...
##  $ Dicrfusc: num  1.62 10.92 0 3.63 3.42 ...
##  $ Dicrpoly: num  0 0.02 1.68 0 0.02 0.02 0 0.23 0.2 0 ...
##  $ Hylosple: num  0 0 0 6.7 0 0 0 0 9.97 0 ...
##  $ Pleuschr: num  4.67 37.75 32.92 58.07 19.42 ...
##  $ Polypili: num  0.02 0.02 0 0 0.02 0.02 0 0 0 0 ...
##  $ Polyjuni: num  0.13 0.23 0.23 0 2.12 1.58 0 0.02 0.08 0.02 ...
##  $ Polycomm: num  0 0 0 0.13 0 0.18 0 0 0 0 ...
##  $ Pohlnuta: num  0.13 0.03 0.32 0.02 0.17 0.07 0.1 0.13 0.07 0.03 ...
##  $ Ptilcili: num  0.12 0.02 0.03 0.08 1.8 0.27 0.03 0.1 0.03 0.25 ...
##  $ Barbhatc: num  0 0 0 0.08 0.02 0.02 0 0 0 0.07 ...
##  $ Cladarbu: num  21.73 12.05 3.58 1.42 9.08 ...
##  $ Cladrang: num  21.47 8.13 5.52 7.63 9.22 ...
##  $ Cladstel: num  3.5 0.18 0.07 2.55 0.05 ...
##  $ Cladunci: num  0.3 2.65 8.93 0.15 0.73 0.25 2.38 0.82 0.05 0.95 ...
##  $ Cladcocc: num  0.18 0.13 0 0 0.08 0.1 0.17 0.15 0.02 0.17 ...
##  $ Cladcorn: num  0.23 0.18 0.2 0.38 1.42 0.25 0.13 0.05 0.03 0.05 ...
##  $ Cladgrac: num  0.25 0.23 0.48 0.12 0.5 0.18 0.18 0.22 0.07 0.23 ...
##  $ Cladfimb: num  0.25 0.25 0 0.1 0.17 0.1 0.2 0.22 0.1 0.18 ...
##  $ Cladcris: num  0.23 1.23 0.07 0.03 1.78 0.12 0.2 0.17 0.02 0.57 ...
##  $ Cladchlo: num  0 0 0.1 0 0.05 0.05 0.02 0 0 0.02 ...
##  $ Cladbotr: num  0 0 0.02 0.02 0.05 0.02 0 0 0.02 0.07 ...
##  $ Cladamau: num  0.08 0 0 0 0 0 0 0 0 0 ...
##  $ Cladsp  : num  0.02 0 0 0.02 0 0 0.02 0.02 0 0.07 ...
##  $ Cetreric: num  0.02 0.15 0.78 0 0 0 0.02 0.18 0 0.18 ...
##  $ Cetrisla: num  0 0.03 0.12 0 0 0 0 0.08 0.02 0.02 ...
##  $ Flavniva: num  0.12 0 0 0 0.02 0.02 0 0 0 0 ...
##  $ Nepharct: num  0.02 0 0 0 0 0 0 0 0 0 ...
##  $ Stersp  : num  0.62 0.85 0.03 0 1.58 0.28 0 0.03 0.02 0.03 ...
##  $ Peltapht: num  0.02 0 0 0.07 0.33 0 0 0 0 0.02 ...
##  $ Icmaeric: num  0 0 0 0 0 0 0 0.07 0 0 ...
##  $ Cladcerv: num  0 0 0 0 0 0 0 0 0 0 ...
##  $ Claddefo: num  0.25 1 0.33 0.15 1.97 0.37 0.15 0.67 0.08 0.47 ...
##  $ Cladphyl: num  0 0 0 0 0 0 0 0 0 0 ...

metaMDS - a wrapper function

Wraps several recommended procedures into one command:

takes raw data and performs 'Wisconsin double standardization' (so abundance isn't influencing similarity)
calculates specified dissimilarity matrix
runs vegan function monoMDS many times with random starts, until it finds two similar configurations with minimized stress value
rotates solution so largest variation of site score is on first axis
other details in vegan tutor

Running NMDS

varespec.nmds.bray <- metaMDS(varespec, distance="bray", trace=FALSE, trymax=100) 
## trace = FALSE, won't show all the outputs, trymax= number of starts
varespec.nmds.bray

## 
## Call:
## metaMDS(comm = varespec, distance = "bray", trymax = 100, trace = FALSE) 
## 
## global Multidimensional Scaling using monoMDS
## 
## Data:     wisconsin(sqrt(varespec)) 
## Distance: bray 
## 
## Dimensions: 2 
## Stress:     0.1825658 
## Stress type 1, weak ties
## Two convergent solutions found after 14 tries
## Scaling: centring, PC rotation, halfchange scaling 
## Species: expanded scores based on 'wisconsin(sqrt(varespec))'

plotting NMDS

plot(varespec.nmds.bray, type = "t")

With many variables/sites, ordination plots can quickly become overwhelming

some built-in function to help include:

make.cepnames (shortens latin names to 6 char variables)
orditorp (reduces overlapping points and variable names)
ordilabel

evaluating NMDS mapping

## plots the observed disimilarity values vs. their ordination distance. 
## If NMDS is a good representation of actual values, you'll see a good fit
stressplot(varespec.nmds.bray)

goodness of fit

gof <- goodness(varespec.nmds.bray) # gof for each site. 
plot(varespec.nmds.bray, type="t", main="goodness of fit") 
# larger circles represent plots that don't have a strong fit with original disimilarity matrix.
points(varespec.nmds.bray, display="sites", cex=gof*100)

Comparing ordinations

Comparing different ordinations can be difficult because of slightly different orientation and scaling. Procrustes rotation using procrustes allows comparison

varespec.nmds.eu <- metaMDS(varespec, distance="eu", trace=FALSE, trymax=100) 
# use euclidean distance - probably not a good choice for most community analyses
pro <- procrustes(varespec.nmds.bray, varespec.nmds.eu)

Comparing ordinations

plot(pro, cex=1.5)

Comparing ordinations

plot(pro, kind=2) # shows the shift in sites between two ordinations.

dissimilarity indices

metaMDS automatically standardizes and then calculates specified dissimilarity index
vegdist will take a matrix of sites (rows) and variables/species (columns) and calculate specied dissimilarity index, outputs class dist

Overlaying environmental vectors onto ordination

What if you want to see how environmental variables are related to your ordination??
Let's look at environmental data, paired with varespec data

data(varechem)
str(varechem)

## 'data.frame':	24 obs. of  14 variables:
##  $ N       : num  19.8 13.4 20.2 20.6 23.8 22.8 26.6 24.2 29.8 28.1 ...
##  $ P       : num  42.1 39.1 67.7 60.8 54.5 40.9 36.7 31 73.5 40.5 ...
##  $ K       : num  140 167 207 234 181 ...
##  $ Ca      : num  519 357 973 834 777 ...
##  $ Mg      : num  90 70.7 209.1 127.2 125.8 ...
##  $ S       : num  32.3 35.2 58.1 40.7 39.5 40.8 33.8 27.1 42.5 60.2 ...
##  $ Al      : num  39 88.1 138 15.4 24.2 ...
##  $ Fe      : num  40.9 39 35.4 4.4 3 ...
##  $ Mn      : num  58.1 52.4 32.1 132 50.1 ...
##  $ Zn      : num  4.5 5.4 16.8 10.7 6.6 9.1 7.4 5.2 9.3 9.1 ...
##  $ Mo      : num  0.3 0.3 0.8 0.2 0.3 0.4 0.3 0.3 0.3 0.5 ...
##  $ Baresoil: num  43.9 23.6 21.2 18.7 46 40.5 23 29.8 17.6 29.9 ...
##  $ Humdepth: num  2.2 2.2 2 2.9 3 3.8 2.8 2 3 2.2 ...
##  $ pH      : num  2.7 2.8 3 2.8 2.7 2.7 2.8 2.8 2.8 2.8 ...

Overlaying vectors with envfit

# Will base the significance of the fit based on permutations
# first two columns are direction cosines of the vectors, and `r2` correlation coefficient between the env variable and the two axes. 
# when plotted, vectors should be scaled by square root of `r2`. `plot` does this automatically (see next slide)
# significances (`Pr>r`) are based on  random permutations of the data
# if you often get as good or better R2 with randomly permuted data, your values are insignificant.
fit <- envfit(varespec.nmds.bray, varechem, permu=999)
fit

## 
## ***VECTORS
## 
##             NMDS1    NMDS2     r2 Pr(>r)    
## N        -0.05735 -0.99835 0.2536  0.046 *  
## P         0.61976  0.78479 0.1938  0.103    
## K         0.76650  0.64225 0.1809  0.113    
## Ca        0.68523  0.72832 0.4118  0.004 ** 
## Mg        0.63256  0.77451 0.4270  0.004 ** 
## S         0.19141  0.98151 0.1752  0.134    
## Al       -0.87158  0.49026 0.5269  0.001 ***
## Fe       -0.93598  0.35204 0.4451  0.002 ** 
## Mn        0.79868 -0.60175 0.5231  0.001 ***
## Zn        0.61759  0.78650 0.1879  0.109    
## Mo       -0.90308  0.42947 0.0609  0.476    
## Baresoil  0.92485 -0.38032 0.2508  0.047 *  
## Humdepth  0.93280 -0.36038 0.5200  0.001 ***
## pH       -0.64793  0.76170 0.2308  0.062 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Permutation: free
## Number of permutations: 999

Plotting envfit output

The arrow points to the direction of most rapid change in the env. variable (this is called the direction of the gradient)
The length of the arrow is proportional to the correlation between ordination and environmental variable. Often this is called the strength of the gradient.

plot(varespec.nmds.bray, display="sites")
plot(fit, p.max=0.05) #only display variables that are significant

Other options

envfit also works with factors
vector fitting implies a linear relationship between ordination and environment
function ordisurf fits surfaces of environmental variables to ordinations based on generalized additive models in function gam of package mgcv.
see vegan tutor or help files for more details.

Import data

# (files must be in the working directory)
spe <- read.csv("DoubsSpe.csv", row.names=1)
env <- read.csv("DoubsEnv.csv", row.names=1)
spa <- read.csv("DoubsSpa.csv", row.names=1)
# Remove empty site 8
spe <- spe[-8,]
env <- env[-8,]
spa <- spa[-8,]

Intro to CA

CA is well suited to the analysis of species abundance data without pre-transformation.
data submitted to CA must be frequencies or frequency-like, dimensionally homogeneous and non-negative (as is the case of species counts or presence–absence data)
raw data are transformed into a matrix Q, of cell-by-cell contributions to the pearson χ 2 statistic (which is calculated on relative counts, and not on the original ones, and it standardizes by the mean and not by the variance)
get an ordination, where the χ 2 distance is preserved among sites instead of the Euclidean distance. The χ 2 distance is not influenced by double zeros

How CA differs from PCA

In both PCA and CA the weights are derived by eigenanalysis
In PCA, the matrix of species abundances is transformed into a matrix of covariances or correlations, each abundance value being replaced by a measure of its correlation (or covariance) with other abundances in other quadrats
In CA, the abundance data is transformed to a chi-square statistic, which is used to depict the degree associations among sites depart from independence
the chi-square metric in CA preserves ecological distance by modeling differences in associations rather than abundances of single species.
the system of weights used to score sites or quadrats is derived from a metric of species associations, and the more these associations depart from independence, the further separated final scores will be

CA of the raw species dataset (original species abundances)

# Compute CA
## things to note: the first axis has a very large eigenvalue. 
## In CA, a value over 0.6 indicate a very strong gradient in the data. 
## Note that the eigenvalues are the same in both scalings. 
## The scaling affects the eigenvectors, not the eigenvalues. 
spe.ca <- cca(spe)
summary(spe.ca)		# default scaling 2

## 
## Call:
## cca(X = spe) 
## 
## Partitioning of mean squared contingency coefficient:
##               Inertia Proportion
## Total           1.167          1
## Unconstrained   1.167          1
## 
## Eigenvalues, and their contribution to the mean squared contingency coefficient 
## 
## Importance of components:
##                         CA1    CA2     CA3     CA4     CA5     CA6     CA7
## Eigenvalue            0.601 0.1444 0.10729 0.08337 0.05158 0.04185 0.03389
## Proportion Explained  0.515 0.1237 0.09195 0.07145 0.04420 0.03586 0.02904
## Cumulative Proportion 0.515 0.6388 0.73069 0.80214 0.84634 0.88220 0.91124
##                           CA8     CA9    CA10    CA11     CA12     CA13
## Eigenvalue            0.02883 0.01684 0.01083 0.01014 0.007886 0.006123
## Proportion Explained  0.02470 0.01443 0.00928 0.00869 0.006760 0.005250
## Cumulative Proportion 0.93594 0.95038 0.95965 0.96835 0.975100 0.980350
##                           CA14     CA15     CA16     CA17     CA18
## Eigenvalue            0.004867 0.004606 0.003844 0.003067 0.001823
## Proportion Explained  0.004170 0.003950 0.003290 0.002630 0.001560
## Cumulative Proportion 0.984520 0.988470 0.991760 0.994390 0.995950
##                           CA19     CA20      CA21      CA22      CA23
## Eigenvalue            0.001642 0.001295 0.0008775 0.0004217 0.0002149
## Proportion Explained  0.001410 0.001110 0.0007500 0.0003600 0.0001800
## Cumulative Proportion 0.997360 0.998470 0.9992200 0.9995900 0.9997700
##                            CA24      CA25      CA26
## Eigenvalue            0.0001528 8.949e-05 2.695e-05
## Proportion Explained  0.0001300 8.000e-05 2.000e-05
## Cumulative Proportion 0.9999000 1.000e+00 1.000e+00
## 
## Scaling 2 for species and site scores
## * Species are scaled proportional to eigenvalues
## * Sites are unscaled: weighted dispersion equal on all dimensions
## 
## 
## Species scores
## 
##          CA1       CA2      CA3       CA4       CA5       CA6
## CHA  1.50075 -1.410293  0.26049 -0.307203  0.271777 -0.003465
## TRU  1.66167  0.444129  0.57571  0.166073 -0.261870 -0.326590
## VAI  1.28545  0.285328 -0.04768  0.018126  0.043847  0.200732
## LOC  0.98662  0.360900 -0.35265 -0.009021 -0.012231  0.253429
## OMB  1.55554 -1.389752  0.80505 -0.468471  0.471301  0.225409
## BLA  0.99709 -1.479902 -0.48035  0.079397 -0.105715 -0.332445
## HOT -0.54916 -0.051534  0.01123 -0.096004 -0.382763  0.134807
## TOX -0.18478 -0.437710 -0.57438  0.424267 -0.587150  0.091866
## VAN  0.01337 -0.095342 -0.57672  0.212017  0.126668 -0.389103
## CHE  0.01078  0.140577 -0.34811 -0.538268  0.185286  0.167087
## BAR -0.33363 -0.300682 -0.04929  0.170961 -0.157203  0.103068
## SPI -0.38357 -0.255310 -0.20136  0.374057 -0.385866  0.239001
## GOU -0.32152 -0.034382 -0.07423 -0.031236  0.014417 -0.156351
## BRO -0.26165  0.187282  0.00617  0.183771  0.295142 -0.262808
## PER -0.28913  0.121044 -0.18919  0.367615  0.218087 -0.163675
## BOU -0.60298 -0.057369  0.20341  0.214299 -0.050977  0.211926
## PSO -0.58669 -0.082467  0.21198  0.050175 -0.120456  0.108724
## ROT -0.61815  0.124733  0.13339  0.147190  0.317736 -0.340380
## CAR -0.57951 -0.110732  0.20173  0.308547  0.006854  0.153224
## TAN -0.37880  0.138023 -0.07825  0.095793  0.256285 -0.029245
## BCO -0.70235  0.011155  0.40242  0.211582  0.138186  0.132297
## PCH -0.73238 -0.009098  0.55678  0.321852  0.281812  0.172271
## GRE -0.69300  0.038971  0.37688 -0.183965 -0.051945 -0.011126
## GAR -0.44181  0.176915 -0.23691 -0.345104  0.129676 -0.043802
## BBO -0.70928  0.032317  0.40924  0.030224  0.049050  0.114560
## ABL -0.63114  0.053594  0.15204 -0.661381 -0.414796 -0.206611
## ANG -0.63578 -0.041894  0.30093  0.224044  0.030444  0.203160
## 
## 
## Site scores (weighted averages of species scores)
## 
##         CA1       CA2        CA3      CA4      CA5      CA6
## 1   2.76488  3.076306  5.3657489  1.99192 -5.07714 -7.80447
## 2   2.27540  2.565531  1.2659130  0.87538 -1.89139 -0.13887
## 3   2.01823  2.441224  0.5144079  0.79436 -1.03741  0.56015
## 4   1.28485  1.935664 -0.2509482  0.76470  0.54752  0.10579
## 5   0.08875  1.015182 -1.4555434  0.47672  2.69249 -2.92498
## 6   1.03188  1.712163 -0.9544059  0.01584  0.91932  0.39856
## 7   1.91427  2.256208 -0.0001407  0.39844 -1.07017  0.32127
## 9   0.25591  1.443008 -2.5777721 -3.41400  2.36613  2.71741
## 10  1.24517  1.526391 -1.9635663 -0.41230  0.69647  1.51859
## 11  2.14501  0.110278  1.6108693 -0.82023  0.53918  1.01153
## 12  2.17418 -0.251649  1.5845397 -0.81483  0.52623  1.05501
## 13  2.30944 -2.034439  1.9181448 -0.60481  0.64435 -0.14844
## 14  1.87141 -2.262503  1.1066796 -0.80840  1.09542  0.11038
## 15  1.34659 -1.805967 -0.6441505 -0.52803  0.76871 -0.67165
## 16  0.70214 -1.501167 -1.9735888  0.98502 -0.93585 -1.27168
## 17  0.28775 -0.836803 -1.2259108  0.73302 -1.57036  0.57315
## 18  0.05299 -0.647950 -0.9234228  0.35770 -0.95401  0.77738
## 19 -0.20584 -0.007252 -1.0154343  0.07041 -1.03450  0.51442
## 20 -0.57879  0.042849 -0.3660551 -0.15019 -0.61357  0.10115
## 21 -0.67320  0.038875  0.1194956  0.17256 -0.14686 -0.12018
## 22 -0.71933  0.014694  0.2204186  0.13598  0.09459 -0.02068
## 23 -0.70438  0.735398 -0.6546250 -6.61523 -2.49441 -1.73215
## 24 -0.83976  0.390120  0.5605295 -4.38864 -2.56916 -0.96702
## 25 -0.68476  0.418842 -0.2860819 -2.80336 -0.37540 -3.93791
## 26 -0.75808  0.210204  0.5894091 -0.70004 -0.01880 -0.10779
## 27 -0.75046  0.100869  0.5531191 -0.12946  0.29164  0.11280
## 28 -0.77878  0.088976  0.7379012  0.05204  0.40940  0.43236
## 29 -0.60815 -0.203235  0.5522726  0.43621  0.15010  0.25618
## 30 -0.80860 -0.019592  0.6686542  0.88136  0.52744  0.16456

summary(spe.ca, scaling=1)

## 
## Call:
## cca(X = spe) 
## 
## Partitioning of mean squared contingency coefficient:
##               Inertia Proportion
## Total           1.167          1
## Unconstrained   1.167          1
## 
## Eigenvalues, and their contribution to the mean squared contingency coefficient 
## 
## Importance of components:
##                         CA1    CA2     CA3     CA4     CA5     CA6     CA7
## Eigenvalue            0.601 0.1444 0.10729 0.08337 0.05158 0.04185 0.03389
## Proportion Explained  0.515 0.1237 0.09195 0.07145 0.04420 0.03586 0.02904
## Cumulative Proportion 0.515 0.6388 0.73069 0.80214 0.84634 0.88220 0.91124
##                           CA8     CA9    CA10    CA11     CA12     CA13
## Eigenvalue            0.02883 0.01684 0.01083 0.01014 0.007886 0.006123
## Proportion Explained  0.02470 0.01443 0.00928 0.00869 0.006760 0.005250
## Cumulative Proportion 0.93594 0.95038 0.95965 0.96835 0.975100 0.980350
##                           CA14     CA15     CA16     CA17     CA18
## Eigenvalue            0.004867 0.004606 0.003844 0.003067 0.001823
## Proportion Explained  0.004170 0.003950 0.003290 0.002630 0.001560
## Cumulative Proportion 0.984520 0.988470 0.991760 0.994390 0.995950
##                           CA19     CA20      CA21      CA22      CA23
## Eigenvalue            0.001642 0.001295 0.0008775 0.0004217 0.0002149
## Proportion Explained  0.001410 0.001110 0.0007500 0.0003600 0.0001800
## Cumulative Proportion 0.997360 0.998470 0.9992200 0.9995900 0.9997700
##                            CA24      CA25      CA26
## Eigenvalue            0.0001528 8.949e-05 2.695e-05
## Proportion Explained  0.0001300 8.000e-05 2.000e-05
## Cumulative Proportion 0.9999000 1.000e+00 1.000e+00
## 
## Scaling 1 for species and site scores
## * Sites are scaled proportional to eigenvalues
## * Species are unscaled: weighted dispersion equal on all dimensions
## 
## 
## Species scores
## 
##          CA1      CA2      CA3      CA4      CA5      CA6
## CHA  1.93586 -3.71167  0.79524 -1.06393  1.19669 -0.01694
## TRU  2.14343  1.16888  1.75759  0.57516 -1.15306 -1.59651
## VAI  1.65814  0.75094 -0.14555  0.06277  0.19306  0.98127
## LOC  1.27267  0.94983 -1.07661 -0.03124 -0.05385  1.23887
## OMB  2.00654 -3.65761  2.45774 -1.62244  2.07523  1.10190
## BLA  1.28617 -3.89487 -1.46646  0.27497 -0.46548 -1.62514
## HOT -0.70838 -0.13563  0.03428 -0.33249 -1.68537  0.65900
## TOX -0.23836 -1.15198 -1.75354  1.46935 -2.58533  0.44908
## VAN  0.01724 -0.25092 -1.76067  0.73427  0.55774 -1.90211
## CHE  0.01391  0.36998 -1.06276 -1.86417  0.81585  0.81679
## BAR -0.43036 -0.79135 -0.15048  0.59208 -0.69219  0.50384
## SPI -0.49478 -0.67194 -0.61472  1.29546 -1.69904  1.16834
## GOU -0.41473 -0.09049 -0.22662 -0.10818  0.06348 -0.76431
## BRO -0.33751  0.49290  0.01884  0.63645  1.29957 -1.28472
## PER -0.37296  0.31857 -0.57758  1.27315  0.96028 -0.80011
## BOU -0.77780 -0.15099  0.62098  0.74218 -0.22446  1.03599
## PSO -0.75678 -0.21704  0.64715  0.17377 -0.53039  0.53149
## ROT -0.79737  0.32828  0.40724  0.50976  1.39905 -1.66393
## CAR -0.74752 -0.29143  0.61585  1.06858  0.03018  0.74903
## TAN -0.48862  0.36325 -0.23888  0.33176  1.12847 -0.14296
## BCO -0.90598  0.02936  1.22855  0.73277  0.60846  0.64673
## PCH -0.94471 -0.02395  1.69979  1.11466  1.24087  0.84214
## GRE -0.89392  0.10256  1.15059 -0.63712 -0.22872 -0.05439
## GAR -0.56990  0.46561 -0.72328 -1.19519  0.57099 -0.21413
## BBO -0.91492  0.08505  1.24936  0.10467  0.21598  0.56002
## ABL -0.81412  0.14105  0.46416 -2.29054 -1.82642 -1.01000
## ANG -0.82011 -0.11026  0.91871  0.77593  0.13405  0.99313
## 
## 
## Site scores (weighted averages of species scores)
## 
##         CA1       CA2        CA3       CA4       CA5      CA6
## 1   2.14343  1.168878  1.7575907  0.575155 -1.153061 -1.59651
## 2   1.76398  0.974804  0.4146591  0.252762 -0.429551 -0.02841
## 3   1.56461  0.927572  0.1684981  0.229368 -0.235605  0.11459
## 4   0.99607  0.735478 -0.0821999  0.220804  0.124347  0.02164
## 5   0.06880  0.385730 -0.4767740  0.137649  0.611487 -0.59835
## 6   0.79995  0.650556 -0.3126227  0.004573  0.208785  0.08153
## 7   1.48401  0.857273 -0.0000461  0.115048 -0.243045  0.06572
## 9   0.19839  0.548288 -0.8443683 -0.985772  0.537369  0.55588
## 10  0.96530  0.579970 -0.6431806 -0.119049  0.158175  0.31065
## 11  1.66289  0.041901  0.5276521 -0.236838  0.122453  0.20692
## 12  1.68551 -0.095617  0.5190277 -0.235278  0.119512  0.21582
## 13  1.79036 -0.773009  0.6283025 -0.174635  0.146337 -0.03037
## 14  1.45079 -0.859665  0.3625011 -0.233420  0.248779  0.02258
## 15  1.04393 -0.686198 -0.2109962 -0.152467  0.174580 -0.13740
## 16  0.54432 -0.570386 -0.6464636  0.284420 -0.212539 -0.26014
## 17  0.22308 -0.317953 -0.4015561  0.211655 -0.356642  0.11725
## 18  0.04108 -0.246196 -0.3024740  0.103285 -0.216664  0.15902
## 19 -0.15958 -0.002755 -0.3326130  0.020330 -0.234943  0.10523
## 20 -0.44870  0.016281 -0.1199041 -0.043366 -0.139347  0.02069
## 21 -0.52189  0.014771  0.0391417  0.049826 -0.033352 -0.02458
## 22 -0.55765  0.005583  0.0721997  0.039264  0.021482 -0.00423
## 23 -0.54606  0.279423 -0.2144273 -1.910111 -0.566501 -0.35434
## 24 -0.65102  0.148231  0.1836056 -1.267193 -0.583479 -0.19782
## 25 -0.53085  0.159144 -0.0937082 -0.809453 -0.085257 -0.80556
## 26 -0.58769  0.079869  0.1930653 -0.202133 -0.004271 -0.02205
## 27 -0.58178  0.038326  0.1811782 -0.037380  0.066234  0.02307
## 28 -0.60374  0.033808  0.2417050  0.015027  0.092978  0.08844
## 29 -0.47146 -0.077222  0.1809010  0.125954  0.034090  0.05241
## 30 -0.62686 -0.007444  0.2190226  0.254487  0.119787  0.03366

Scaling options

when scaling = 1 the distances among objects approximate their x2 distances (i.e. object points close together are similar in their species frequencies). Any object near the point representing a species is likely to contain a high contribution of that species.
when scaling = 2, ordination of species. Species points close to one another are likely to have similar relative frequencies among objects.

Plot eigenvalues and % of variance for each axis

ev2 <- spe.ca$CA$eig

plot

## things to note: the first axis is extremely dominant. 
evplot(ev2)

CA biplots

par(mfrow=c(1,2))
# Scaling 1: sites are centroids of species
plot(spe.ca, scaling=1, main="CA fish abundances - biplot scaling 1")
# Scaling 2 (default): species are centroids of sites
plot(spe.ca, main="CA fish abundances - biplot scaling 2")

A posteriori projection of environmental variables in a CA

-envfit finds vectors or factor averages of environmental variables. [...] The projections of points onto vectors have maximum correlation with corresponding environmental variables, and the factors show the averages of factor levels”

CA biplot (scaling 2) of the Doubs fish abundance data with a posteriori projection of environmental variables

# The last plot produced (CA scaling 2) must be active
plot(spe.ca, main="CA fish abundances - biplot scaling 2")
(spe.ca.env <- envfit(spe.ca, env))

## 
## ***VECTORS
## 
##          CA1      CA2     r2 Pr(>r)    
## das -0.94801 -0.31825 0.6889  0.001 ***
## alt  0.81141  0.58448 0.8080  0.001 ***
## pen  0.73753  0.67531 0.2976  0.002 ** 
## deb -0.92837 -0.37166 0.4440  0.001 ***
## pH   0.50723 -0.86181 0.0908  0.227    
## dur -0.71728 -0.69678 0.4722  0.002 ** 
## pho -0.99897  0.04533 0.1757  0.071 .  
## nit -0.94906 -0.31511 0.4510  0.003 ** 
## amm -0.97495  0.22241 0.1762  0.064 .  
## oxy  0.93352 -0.35854 0.6263  0.001 ***
## dbo -0.94094  0.33857 0.2237  0.036 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Permutation: free
## Number of permutations: 999

plot(spe.ca.env)
# Plot significant variables with a different colour
plot(spe.ca.env, p.max=0.05, col=3)

Species data table ordered after the CA result

vegemite(spe, spe.ca)

##                                   
##      2322222222211  11 1 11  11 1 
##      40867235190985976604547312231
##  PCH .53122..14...................
##  BBO 155244..3421.................
##  BCO .53234..3411.................
##  GRE 255454.135211................
##  ANG .54232..24211..1.............
##  ABL 5555552355532..2.............
##  ROT .52112.12221.2...............
##  BOU .54233..35322..1.............
##  PSO 134233..25211..11............
##  CAR .54123..23111..11............
##  HOT 111113..22221..1.............
##  GAR 255455115555254211...........
##  SPI .51111..23323..41............
##  TAN .54354..4342131112.11........
##  BAR .33245..45423..32...21.......
##  GOU 154345.2554422.1211121.......
##  PER .52114..342134.211.2.........
##  BRO .43243.1352114.111.2.1.1.....
##  TOX .21.12..22233..44............
##  CHE 23423411243132522221311.11...
##  VAN .32123.1232225.3512.3.1......
##  LOC ..1111..11253234554554551432.
##  BLA .........1.11..25...43.....2.
##  VAI ........11133314344545454445.
##  CHA ............1..12...33..12.2.
##  OMB .........1..1..1....24..12.3.
##  TRU .........1..12.23314455535553
##   sites species 
##      29      27

CA using CA() function

# Ordering of the data table following the first CA axis
# The table is transposed, as in vegemite() output
spe.CA.PL <- CA(spe)
biplot(spe.CA.PL, cex=1)

useful links

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Chap-9-presentation.md

Chap-9-presentation.md

Chapter-9

Plan for today

Load packages and functions

Load additional functions

NMDS

NMDS applied to the varespec data: vegetation lichen pastures

metaMDS - a wrapper function

Running NMDS

plotting NMDS

With many variables/sites, ordination plots can quickly become overwhelming

evaluating NMDS mapping

goodness of fit

Comparing ordinations

Comparing ordinations

Comparing ordinations

dissimilarity indices

Overlaying environmental vectors onto ordination

Overlaying vectors with envfit

Plotting envfit output

Other options

Import data

Intro to CA

How CA differs from PCA

CA of the raw species dataset (original species abundances)

Scaling options

Plot eigenvalues and % of variance for each axis

plot

CA biplots

A posteriori projection of environmental variables in a CA

CA biplot (scaling 2) of the Doubs fish abundance data with a posteriori projection of environmental variables

Species data table ordered after the CA result

CA using CA() function

useful links

Files

Chap-9-presentation.md

Latest commit

History

Chap-9-presentation.md

File metadata and controls

Chapter-9

Plan for today

Load packages and functions

Load additional functions

NMDS

NMDS applied to the varespec data: vegetation lichen pastures

metaMDS - a wrapper function

Running NMDS

plotting NMDS

With many variables/sites, ordination plots can quickly become overwhelming

evaluating NMDS mapping

goodness of fit

Comparing ordinations

Comparing ordinations

Comparing ordinations

dissimilarity indices

Overlaying environmental vectors onto ordination

Overlaying vectors with envfit

Plotting envfit output

Other options

Import data

Intro to CA

How CA differs from PCA

CA of the raw species dataset (original species abundances)

Scaling options

Plot eigenvalues and % of variance for each axis

plot

CA biplots

A posteriori projection of environmental variables in a CA

CA biplot (scaling 2) of the Doubs fish abundance data with a posteriori projection of environmental variables

Species data table ordered after the CA result

CA using CA() function

useful links