Workshop 15: Q-mode MVA Murray Logan 06 Aug 2016 R-mode analyses - - PowerPoint PPT Presentation

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Workshop 15: Q-mode MVA Murray Logan 06 Aug 2016 R-mode analyses - - PowerPoint PPT Presentation

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Workshop 15: Q-mode MVA Murray Logan 06 Aug 2016 R-mode analyses preserve euclidean (PCA/RDA) or 2 (CA/CCA) distances based on either correlation or covariance of variables data restricted by assumptions Q-mode analyses

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SLIDE 1

Workshop 15: Q-mode MVA

Murray Logan 06 Aug 2016

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SLIDE 2

R-mode analyses

  • preserve euclidean (PCA/RDA) or χ2

(CA/CCA) distances

  • based on either correlation or covariance
  • f variables
  • data restricted by assumptions
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SLIDE 3

Q-mode analyses

  • based on object similarity
  • unrestricted choice of similarity/distance

matrix

  • filter data to comply
  • resulting scores not independent
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SLIDE 4

Dissimilarity

Sp1 Sp2 Sp3 Sp4 Site1 2 5 Site2 13 7 10 5 Site3 9 5 55 93 Site4 10 6 76 81 Site5 2 6

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SLIDE 5

Distance measures

  • Euclidean distance

djk =

√∑ (yji − yki)2

> library(vegan) > vegdist(Y,method="euclidean") Site1 Site2 Site3 Site4 Site2 16.431677 Site3 104.129727 98.939375 Site4 107.944430 100.707497 24.228083 Site5 8.306624 15.329710 105.546198 107.596468

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SLIDE 6

Distance measures

  • Bray-Curtis distance

djk =

∑ | yji − yki | ∑

yji + yki

= 1 − 2 ∑

min(yji, yki)

∑

yji + yki

> library(vegan) > vegdist(Y,method="bray") Site1 Site2 Site3 Site4 Site2 0.6666667 Site3 0.9171598 0.7055838 Site4 0.9222222 0.7019231 0.1044776 Site5 1.0000000 0.6279070 0.9058824 0.9116022

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SLIDE 7

Distance measures

  • Hellinger distance

djk =

√ ∑ (√ yji ∑

yj

− √ yki ∑

yk

)2

> library(vegan) > dist(decostand(Y,method="hellinger")) Site1 Site2 Site3 Site4 Site2 0.8423744 Site3 0.6836053 0.5999300 Site4 0.7657483 0.5608590 0.1092925 Site5 1.4142136 0.7918120 0.9028295 0.8159425

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SLIDE 8

Multidimensional scaling

  • re-project objects (sites) in reduced

dimensional space

  • must nominate the number of dimensions

up-front

  • optimized patterns for the nominated

dimensions

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SLIDE 9

Q-mode analyses

.

Site 1

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Site 10

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Site 9

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Site 2

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Site 3

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Site 8

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Site 4

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Site 5

.

Site 6

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Site 7

Sites Sp1 Sp2 Sp3 Sp4 Sp5 Sp6 Sp7 Sp8 Sp9 Sp10 Site1 Site1 5 65 5 Site2 Site2 25 39 6 23 Site3 Site3 6 42 6 31 Site4 Site4 40 14 Site5 Site5 6 34 18 12 Site6 Site6 29 12 22 Site7 Site7 21 5 20 Site8 Site8 13 6 37 Site9 Site9 60 47 4 Site10 Site10 72 34

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SLIDE 10

Multidimensional scaling

  • 1. generate distance matrix

> data.dist <- vegdist(data[,-1], "bray") > data.dist Site1 Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site2 0.6428571 Site3 0.8625000 0.1685393 Site4 1.0000000 0.6870748 0.5539568 Site5 1.0000000 0.7177914 0.6000000 0.2580645 Site6 1.0000000 1.0000000 1.0000000 1.0000000 0.6390977 Site7 1.0000000 1.0000000 1.0000000 1.0000000 0.5862069 0.4128440 Site8 0.9236641 0.4362416 0.2907801 0.3272727 0.4603175 1.0000000 1.0000000 Site9 0.3010753 0.3333333 0.4693878 1.0000000 1.0000000 1.0000000 1.0000000 0.7964072 Site10 0.2265193 0.4070352 0.5811518 1.0000000 1.0000000 1.0000000 1.0000000 0.8395062 0.1336406

Data Distances

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SLIDE 11

Multidimensional scaling

  • 2. choose # dimensions (k=2)
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SLIDE 12

Multidimensional scaling

  • 3. random configuration
  • Site1 Site2

Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

Dimension (axis) 1 Dimension (axis) 2

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SLIDE 13

Multidimensional scaling

  • 4. measure Kruskal฀s stress
  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

Dimension (axis) 1 Dimension (axis) 2

Data Distances Ordination Distances

  • ●
  • Data distance

Ordination distance

Iterations = 0 Stress = 0.368

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SLIDE 14

Multidimensional scaling

  • 5. iterate - gradient descent
  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

Dimension (axis) 1 Dimension (axis) 2

Data Distances Ordination Distances

  • Data distance

Ordination distance

Iterations = 1 Stress = 0.329

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SLIDE 15

Multidimensional scaling

  • 6. continual to iterate
  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

Dimension (axis) 1 Dimension (axis) 2

Data Distances Ordination Distances

  • ●
  • Data distance

Ordination distance

Iterations = 10 Stress = 0.155

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SLIDE 16

Multidimensional scaling

  • 7. continual (stopping criteria)
  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

Dimension (axis) 1 Dimension (axis) 2

Data Distances Ordination Distances

  • ●
  • Data distance

Ordination distance

Iterations = 34 Stress = 0.055

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SLIDE 17

Multidimensional scaling

S t

  • p

p i n g c r i t e r i a

  • convergence tolerance (stress below a

threshold)

  • the stress ratio (improvement in stress)
  • maximum iterations

Ideally stress ฀ 0.2

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SLIDE 18

Multidimensional scaling

F i n a l c

  • n

f i g u r a t i

  • n
  • axes have no real meaning
  • operate together to create ordination

space

  • orientation of points is arbitrary
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SLIDE 19

Multidimensional scaling

S t a r t i n g c

  • n

t i g u r a t i

  • n
  • completely random
  • based on eigen-analysis
  • repeated random starts
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SLIDE 20

Multidimensional scaling

P r

  • c

r u s t e s r

  • t

a t i

  • n
  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

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SLIDE 21

Multidimensional scaling

P r

  • c

r u s t e s r

  • t

a t i

  • n
  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

Root mean square error (rmse)

  • rmse ฀ 0.01, and no one ฀ 0.005

stopping criteria

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SLIDE 22

Multidimensional scaling

metaMDS()

  • transform and scale
  • generates dissimilarity
  • PCoA for starting config
  • up to 20 random starts
  • procrustes used to determine final config
  • final scores scaled
  • PCA-like axes rotations
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SLIDE 23

Multidimensional scaling

> library(vegan) > data.nmds <- metaMDS(data[,-1]) Square root transformation Wisconsin double standardization Run 0 stress 9.575935e-05 Run 1 stress 9.419141e-05 ... New best solution ... procrustes: rmse 0.02551179 max resid 0.04216078 Run 2 stress 0.0005652599 ... procrustes: rmse 0.1885613 max resid 0.2962192 Run 3 stress 0.0003846328 ... procrustes: rmse 0.128145 max resid 0.2569595 Run 4 stress 0.1071568 Run 5 stress 0.08642 Run 6 stress 0.0002599177 ... procrustes: rmse 0.1766213 max resid 0.2933088 Run 7 stress 9.39752e-05 ... New best solution ... procrustes: rmse 0.1050724 max resid 0.2432038 Run 8 stress 9.692363e-05 ... procrustes: rmse 0.101578 max resid 0.2286374 Run 9 stress 9.636878e-05 ... procrustes: rmse 0.03799946 max resid 0.06399347 Run 10 stress 9.756922e-05 ... procrustes: rmse 0.09787159 max resid 0.2172152 Run 11 stress 9.903768e-05 ... procrustes: rmse 0.1377402 max resid 0.2070494 Run 12 stress 0.07877608 Run 13 stress 9.67808e-05 ... procrustes: rmse 0.105466 max resid 0.2688797 Run 14 stress 0.00122647 Run 15 stress 0.002519856 Run 16 stress 9.401245e-05 ... procrustes: rmse 0.01935187 max resid 0.04030324 Run 17 stress 9.207175e-05 ... New best solution ... procrustes: rmse 0.1056994 max resid 0.2401655 Run 18 stress 0.00139044 Run 19 stress 9.781088e-05 ... procrustes: rmse 0.1089896 max resid 0.25471 Run 20 stress 9.529673e-05 ... procrustes: rmse 0.09327002 max resid 0.1897497

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SLIDE 24

Multidimensional scaling

> data.nmds Call: metaMDS(comm = data[, -1]) global Multidimensional Scaling using monoMDS Data: wisconsin(sqrt(data[, -1])) Distance: bray Dimensions: 2 Stress: 9.207175e-05 Stress type 1, weak ties No convergent solutions - best solution after 20 tries Scaling: centring, PC rotation, halfchange scaling Species: expanded scores based on 'wisconsin(sqrt(data[, -1]))'

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SLIDE 25

Multidimensional scaling

S h e p a r d d i a g r a m

> stressplot(data.nmds)

  • ● ●
  • ●
  • 0.2

0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 Observed Dissimilarity Ordination Distance Non−metric fit, R2 = 1 Linear fit, R2 = 1

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SLIDE 26

Multidimensional scaling

O r d i n a t i

  • n

p l

  • t

> ordiplot(data.nmds, type="text")

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 −1.0 −0.5 0.0 0.5 1.0 1.5 NMDS2

Site1 Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10 Sp1 Sp2 Sp3 Sp4 Sp5 Sp6 Sp7 Sp8 Sp9 Sp10
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SLIDE 27

Multidimensional scaling

E n v i r

  • n

m e n t a l f i t

  • rotate axes to planes of maximum

variability

  • observations are not independent
  • permutation tests - e.g based on R2
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SLIDE 28

Multidimensional scaling

E n v i r

  • n

m e n t a l f i t

> ordiplot(data.nmds, type="text") > data.envfit <- envfit(data.nmds,enviro[,-1]) > plot(data.envfit)

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 −1.0 −0.5 0.0 0.5 1.0 1.5 NMDS2

Site1 Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10 Sp1 Sp2 Sp3 Sp4 Sp5 Sp6 Sp7 Sp8 Sp9 Sp10

pH Slope Pressure Altitude SubstrateQuartz SubstrateShale

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SLIDE 29

Multidimensional scaling

E n v i r

  • n

m e n t a l f i t

> data.envfit ***VECTORS NMDS1 NMDS2 r2 Pr(>r) pH 0.94416 -0.32948 0.1605 0.551 Slope

  • 0.18315 -0.98308 0.3033

0.264 Pressure 0.98333 -0.18182 0.8361 0.005 ** Altitude 0.82690 -0.56235 0.8107 0.002 **

  • Signif. codes:

0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Permutation: free Number of permutations: 999 ***FACTORS: Centroids: NMDS1 NMDS2 SubstrateQuartz -0.0849 0.5622 SubstrateShale 0.0849 -0.5622 Goodness of fit: r2 Pr(>r) Substrate 0.2178 0.144 Permutation: free Number of permutations: 999

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SLIDE 30

Section 1 Q-mode Inference testing

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SLIDE 31

Environmental correlates

. .

Site 1

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Site 10

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Site 9

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Site 2

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Site 3

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Site 8

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Site 4

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Site 5

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Site 6

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Site 7

Sites Sp1 Sp2 Sp3 Sp4 Sp5 Sp6 Sp7 Sp8 Sp9 Sp10 Site1 Site1 5 65 5 Site2 Site2 25 39 6 23 Site3 Site3 6 42 6 31 Site4 Site4 40 14 Site5 Site5 6 34 18 12 Site6 Site6 29 12 22 Site7 Site7 21 5 20 Site8 Site8 13 6 37 Site9 Site9 60 47 4 Site10 Site10 72 34

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SLIDE 32

Environmental correlates

. .

Site 1

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Site 10

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Site 9

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Site 2

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Site 3

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Site 8

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Site 4

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Site 5

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Site 6

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Site 7

Site pH Slope Pressure Altitude Substrate Site1 6 4 101325 2 Quartz Site2 7 9 101352 510 Shale Site3 7 9 101356 546 Shale Site4 7 7 101372 758 Shale Site5 7 6 101384 813 Shale Site6 8 8 101395 856 Quartz Site7 8 101396 854 Quartz Site8 7 12 101370 734 Shale Site9 8 8 101347 360 Quartz Site10 6 2 101345 356 Quartz

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SLIDE 33

Environmental correlates

  • PCA/CA
  • relate envir to first few axes
  • MDS with envfit
  • relate envir to reduced optimized axes

All just relate envir to SOME patterns of community data

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SLIDE 34

Permutation tests

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SLIDE 35

Mantel test

  • correlation of two distance matrices
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SLIDE 36

Mantel test

C

  • m

m u n i t y d i s t a n c e s

> library(vegan) > data.dist <- vegdist(wisconsin(sqrt(data[,-1])),"bray") > data.dist Site1 Site2 Site3 Site4 Site5 Site6 Site2 0.67587556 Site3 0.76402116 0.08814559 Site4 1.00000000 0.76728722 0.71732569 Site5 1.00000000 0.76728722 0.71950051 0.43782491 Site6 1.00000000 1.00000000 1.00000000 1.00000000 0.56217509 Site7 1.00000000 1.00000000 1.00000000 1.00000000 0.56217509 0.40287938 Site8 0.85671371 0.24898448 0.18481903 0.61338910 0.71950051 1.00000000 Site9 0.52225127 0.24045299 0.30461844 1.00000000 1.00000000 1.00000000 Site10 0.43930743 0.53960530 0.60377075 1.00000000 1.00000000 1.00000000 Site7 Site8 Site9 Site2 Site3 Site4 Site5 Site6 Site7 Site8 1.00000000 Site9 1.00000000 0.48943747 Site10 1.00000000 0.78858978 0.29915230

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SLIDE 37

Mantel test

E n v i r

  • n

m e n t a l d i s t a n c e s

> library(vegan) > enviro1 <- within(enviro, + Substrate<-as.numeric(Substrate)) > env.dist <- vegdist(decostand(enviro1[,-1], + "standardize"),"euclid") > env.dist 1 2 3 4 5 6 7 2 3.3229931 3 3.4352275 0.3112630 4 4.2003679 1.4095427 1.1199511 5 4.6243094 2.1315293 1.8280978 0.7722349 6 4.9177360 3.1669299 2.9561936 2.3097932 2.1404251 7 4.9080429 4.0132037 3.7487386 3.0169364 2.5115880 2.2216568 8 4.5387150 1.4066733 1.3097554 1.2431118 1.8378136 2.5328911 3.9603544 9 3.9400812 3.2272044 3.1427473 3.3444512 3.5238311 3.0372913 3.7500941 10 1.7177533 3.0391765 3.0117638 3.3462020 3.5745559 3.9177255 3.4387521 8 9 2 3 4 5 6 7 8 9 3.4268548 10 4.0495542 3.7782159

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SLIDE 38

Mantel test

> plot(data.dist, env.dist)

  • 0.2
0.4 0.6 0.8 1.0 1 2 3 4 5 data.dist env.dist
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SLIDE 39

Mantel test

> data.mantel <- mantel(data.dist, env.dist, perm=1000) > data.mantel Mantel statistic based on Pearson's product-moment correlation Call: mantel(xdis = data.dist, ydis = env.dist, permutations = 1000) Mantel statistic r: 0.5593 Significance: 0.003996 Upper quantiles of permutations (null model): 90% 95% 97.5% 99% 0.209 0.297 0.357 0.428 Permutation: free Number of permutations: 1000

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SLIDE 40

Mantel test

> hist(data.mantel$perm) > abline(v=data.mantel$statistic)

Histogram of data.mantel$perm

data.mantel$perm Frequency −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 50 100 150 200 250

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SLIDE 41

Best environmental subsets

  • looks at combinations of environmental

predictors

  • analogous to model selection
  • not an inference test
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SLIDE 42

Best environmental subsets

> # convert the Substrate factor to a numeric via dummy coding > # function definition at the start of this Tutorial > dummy <- function(x) { + nms <- colnames(x) + ff <- eval(parse(text=paste("~",paste(nms,collapse="+")))) + mm <- model.matrix(ff,x) + nms <- colnames(mm) + mm <- as.matrix(mm[,-1]) + colnames(mm) <- nms[-1] + mm + } > enviro1 <- dummy(enviro[,-1]) > enviro1 pH Slope Pressure Altitude SubstrateShale 1 6.1 4.2 101325 2 2 6.7 9.2 101352 510 1 3 6.8 8.6 101356 546 1 4 7.0 7.4 101372 758 1 5 7.2 5.8 101384 813 1 6 7.5 8.4 101395 856 7 7.5 0.5 101396 854 8 7.0 11.8 101370 734 1 9 8.4 8.2 101347 360 10 6.2 1.5 101345 356

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SLIDE 43

Best environmental subsets

> library(car) > vif(lm(1:nrow(enviro1) ~ pH+Slope+Pressure+Altitude+SubstrateShale, data=data.frame(enviro1))) pH Slope Pressure 1.976424 2.187796 45.804821 Altitude SubstrateShale 52.754368 5.118418 > vif(lm(1:nrow(enviro1) ~ pH+Slope+Altitude+SubstrateShale, data=data.frame(enviro1))) pH Slope Altitude 1.968157 2.116732 1.830014 SubstrateShale 2.636302

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SLIDE 44

Best environmental subsets

> # Pressure is in column three > data.bioenv <- bioenv(wisconsin(sqrt(data[,-1])), decostand(enviro1[,-3],"standardize")) > data.bioenv Call: bioenv(comm = wisconsin(sqrt(data[, -1])), env = decostand(enviro1[,

  • 3], "standardize"))

Subset of environmental variables with best correlation to community data. Correlations: spearman Dissimilarities: bray Metric: euclidean Best model has 1 parameters (max. 4 allowed): Altitude with correlation 0.5514973

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SLIDE 45

Permutation Multivariate Analysis of Variance

a d

  • n

i s

  • partitions the total sums of squares
  • contribution of each predictor
  • permutation test of inference
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SLIDE 46

Permutation Multivariate Analysis of Variance

a d

  • n

i s

> data.adonis <- adonis(data.dist ~ pH + Slope + Altitude + + Substrate, data=enviro) > data.adonis Call: adonis(formula = data.dist ~ pH + Slope + Altitude + Substrate, data = enviro) Permutation: free Number of permutations: 999 Terms added sequentially (first to last) Df SumsOfSqs MeanSqs F.Model R2 Pr(>F) pH 1 0.21899 0.21899 1.6290 0.07768 0.209 Slope 1 0.55960 0.55960 4.1628 0.19850 0.015 * Altitude 1 0.98433 0.98433 7.3223 0.34916 0.002 ** Substrate 1 0.38404 0.38404 2.8568 0.13623 0.057 . Residuals 5 0.67215 0.13443 0.23842 Total 9 2.81911 1.00000

  • Signif. codes:

0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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SLIDE 47

Section 2 Worked Examples

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SLIDE 48

Worked Examples

> data <- read.csv('../data/data.csv', strip.white=TRUE) > head(data) Sites Sp1 Sp2 Sp3 Sp4 Sp5 Sp6 Sp7 Sp8 Sp9 Sp10 1 Site1 5 65 5 2 Site2 25 39 6 23 3 Site3 6 42 6 31 4 Site4 40 14 5 Site5 6 34 18 12 6 Site6 29 12 22 > enviro <- read.csv('../data/enviro.csv', strip.white=TRUE) > head(enviro) Site pH Slope Pressure Altitude Substrate 1 Site1 6.1 4.2 101325 2 Quartz 2 Site2 6.7 9.2 101352 510 Shale 3 Site3 6.8 8.6 101356 546 Shale 4 Site4 7.0 7.4 101372 758 Shale 5 Site5 7.2 5.8 101384 813 Shale 6 Site6 7.5 8.4 101395 856 Quartz > data.bc <- vegdist(data[,-1], 'bray') > > data.cpscale<-capscale(data.bc~ pH+Slope+Altitude+Substrate, data=enviro) > plot(data.cpscale) > str(data.cpscale) List of 13 $ call : language capscale(formula = data.bc ~ pH + Slope + Altitude + Substrate, data = enviro) $ grand.total: logi NA $ rowsum : logi NA $ colsum : logi NA $ tot.chi : num 2.76 $ pCCA : NULL $ CCA :List of 12 ..$ eig : Named num [1:4] 1.266 0.978 0.073 0.03 .. ..- attr(*, "names")= chr [1:4] "CAP1" "CAP2" "CAP3" "CAP4" ..$ u : num [1:10, 1:4] -0.66766 -0.04799 -0.00444 0.24882 0.31907 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:10] "1" "2" "3" "4" ... .. .. ..$ : chr [1:4] "CAP1" "CAP2" "CAP3" "CAP4" ..$ v : num [1:6, 1:4] NA NA NA NA NA NA NA NA NA NA ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:6] "Dim1" "Dim2" "Dim3" "Dim4" ... .. .. ..$ : chr [1:4] "CAP1" "CAP2" "CAP3" "CAP4" ..$ wa : num [1:10, 1:4] -0.4575 -0.1783 -0.0215 0.3049 0.4377 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:10] "1" "2" "3" "4" ... .. .. ..$ : chr [1:4] "CAP1" "CAP2" "CAP3" "CAP4" ..$ biplot : num [1:4, 1:4] 0.312 0.138 0.987 0.442 0.256 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:4] "pH" "Slope" "Altitude" "SubstrateShale" .. .. ..$ : chr [1:4] "CAP1" "CAP2" "CAP3" "CAP4" ..$ rank : int 4 ..$ qrank : int 4 ..$ tot.chi : num 2.35 ..$ QR :List of 4 .. ..$ qr : num [1:10, 1:4] 2.0159 0.1687 0.1191 0.0198 -0.0794 ... .. .. ..- attr(*, "dimnames")=List of 2 .. .. .. ..$ : chr [1:10] "1" "2" "3" "4" ... .. .. .. ..$ : chr [1:4] "pH" "Slope" "Altitude" "SubstrateShale" .. ..$ rank : int 4 .. ..$ qraux: num [1:4] 1.47 1.31 1.04 1.19 .. ..$ pivot: int [1:4] 1 2 3 4 .. ..- attr(*, "class")= chr "qr" ..$ envcentre: Named num [1:4] 7.04 6.56 578.9 0.5 .. ..- attr(*, "names")= chr [1:4] "pH" "Slope" "Altitude" "SubstrateShale" ..$ Xbar : num [1:10, 1:6] -1.261 -0.808 -0.401 0.787 1.424 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:10] "1" "2" "3" "4" ... .. .. ..$ : chr [1:6] "Dim1" "Dim2" "Dim3" "Dim4" ... .. ..- attr(*, "scaled:center")= Named num [1:6] 1.80e-16 1.11e-16 -2.76e-16 7.79e-17 1.06e-15 ... .. .. ..- attr(*, "names")= chr [1:6] "Dim1" "Dim2" "Dim3" "Dim4" ... ..$ centroids: num [1:2, 1:4] -0.14 0.14 0.283 -0.283 0.015 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:2] "SubstrateQuartz" "SubstrateShale" .. .. ..$ : chr [1:4] "CAP1" "CAP2" "CAP3" "CAP4" $ CA :List of 9 ..$ eig : Named num [1:5] 0.35573 0.1688 0.04051 0.02265 0.00193 .. ..- attr(*, "names")= chr [1:5] "MDS1" "MDS2" "MDS3" "MDS4" ... ..$ u : num [1:10, 1:5] -0.567 0.262 0.28 -0.269 -0.342 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:10] "1" "2" "3" "4" ... .. .. ..$ : chr [1:5] "MDS1" "MDS2" "MDS3" "MDS4" ... ..$ v : num [1:6, 1:5] NA NA NA NA NA NA NA NA NA NA ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:6] "Dim1" "Dim2" "Dim3" "Dim4" ... .. .. ..$ : chr [1:5] "MDS1" "MDS2" "MDS3" "MDS4" ... ..$ rank : int 5 ..$ tot.chi : num 0.59 ..$ Xbar : num [1:10, 1:6] 0.853 -0.373 -0.123 0.145 0.528 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:10] "1" "2" "3" "4" ... .. .. ..$ : chr [1:6] "Dim1" "Dim2" "Dim3" "Dim4" ... .. ..- attr(*, "scaled:center")= Named num [1:6] 1.80e-16 1.11e-16 -2.76e-16 7.79e-17 1.06e-15 ... .. .. ..- attr(*, "names")= chr [1:6] "Dim1" "Dim2" "Dim3" "Dim4" ... ..$ imaginary.chi : num -0.176 ..$ imaginary.rank : int 3 ..$ imaginary.u.eig: num [1:10, 1:3] -0.00467 -0.04869 0.01533 -0.00407 -0.01243 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:10] "1" "2" "3" "4" ... .. .. ..$ : NULL $ method : chr "capscale" $ inertia : chr "squared Bray distance" $ terms :Classes 'terms', 'formula' length 3 data.bc ~ pH + Slope + Altitude + Substrate .. ..- attr(*, "variables")= language list(data.bc, pH, Slope, Altitude, Substrate) .. ..- attr(*, "factors")= int [1:5, 1:4] 0 1 0 0 0 0 0 1 0 0 ... .. .. ..- attr(*, "dimnames")=List of 2 .. .. .. ..$ : chr [1:5] "data.bc" "pH" "Slope" "Altitude" ... .. .. .. ..$ : chr [1:4] "pH" "Slope" "Altitude" "Substrate" .. ..- attr(*, "term.labels")= chr [1:4] "pH" "Slope" "Altitude" "Substrate" .. ..- attr(*, "specials")=Dotted pair list of 1 .. .. ..$ Condition: NULL .. ..- attr(*, "order")= int [1:4] 1 1 1 1 .. ..- attr(*, "intercept")= int 1 .. ..- attr(*, "response")= int 1 .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv> $ terminfo :List of 3 ..$ terms :Classes 'terms', 'formula' length 2 ~pH + Slope + Altitude + Substrate .. .. ..- attr(*, "variables")= language list(pH, Slope, Altitude, Substrate) .. .. ..- attr(*, "factors")= int [1:4, 1:4] 1 0 0 0 0 1 0 0 0 0 ... .. .. .. ..- attr(*, "dimnames")=List of 2 .. .. .. .. ..$ : chr [1:4] "pH" "Slope" "Altitude" "Substrate" .. .. .. .. ..$ : chr [1:4] "pH" "Slope" "Altitude" "Substrate" .. .. ..- attr(*, "term.labels")= chr [1:4] "pH" "Slope" "Altitude" "Substrate" .. .. ..- attr(*, "order")= int [1:4] 1 1 1 1 .. .. ..- attr(*, "intercept")= int 1 .. .. ..- attr(*, "response")= num 0 .. .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv> ..$ xlev :List of 1 .. ..$ Substrate: chr [1:2] "Quartz" "Shale" ..$ ordered: Named logi [1:4] FALSE FALSE FALSE FALSE .. ..- attr(*, "names")= chr [1:4] "pH" "Slope" "Altitude" "Substrate" $ adjust : num 3

  • attr(*, "class")= chr [1:3] "capscale" "rda" "cca"

> data.cmdscale <- cmdscale(data.bc, k=2) > plot(data.cmdscale) > data.scores <- scores(data.cmdscale)