Workshop 15: Q-mode MVA Murray Logan August 6, 2016 Table of - - PDF document

▶
workshop 15 q mode mva
SMART_READER_LITE
LIVE PREVIEW

Workshop 15: Q-mode MVA Murray Logan August 6, 2016 Table of - - PDF document

Nov 06, 2023 499 likes •753 views

-1- Workshop 15: Q-mode MVA Murray Logan August 6, 2016 Table of contents 1 Q-mode Inference testing 13 2 Worked Examples 19 0.1. R-mode analyses preserve euclidean (PCA/RDA) or 2 (CA/CCA) distances based on either correlation

slide-1
SLIDE 1
  • 1-

Workshop 15: Q-mode MVA

Murray Logan

August 6, 2016

Table of contents

1 Q-mode Inference testing 13 2 Worked Examples 19

0.1. R-mode analyses

  • preserve euclidean (PCA/RDA) or χ2 (CA/CCA) distances
  • based on either correlation or covariance of variables

– data restricted by assumptions

0.2. Q-mode analyses

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

– filter data to comply

  • resulting scores not independent

0.3. 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

0.4. 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

slide-2
SLIDE 2
  • 2-

0.5. 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

0.6. 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

0.7. Multidimensional scaling

  • re-project objects (sites) in reduced dimensional space
  • must nominate the number of dimensions up-front
  • optimized patterns for the nominated dimensions
slide-3
SLIDE 3
  • 3-

0.8. Q-mode analyses

.

S i t e 1

.

Site 10

.

Site 9

.

Site 2

.

Site 3

.

Site 8

.

Site 4

.

Site 5

.

Site 6

.

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

0.9. 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

slide-4
SLIDE 4
  • 4-

Data Distances 0.10. Multidimensional scaling

  • 2. choose # dimensions (k=2)

0.11. Multidimensional scaling

  • 3. random configuration
  • Site1 Site2

Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

Dimension (axis) 1 Dimension (axis) 2 0.12. Multidimensional scaling

  • 4. measure Kruskal’s stress
slide-5
SLIDE 5
  • 5-
  • 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

0.13. 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

0.14. Multidimensional scaling

  • 6. continual to iterate
slide-6
SLIDE 6
  • 6-
  • 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

0.15. 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

0.16. Multidimensional scaling

slide-7
SLIDE 7
  • 7-

0.16.1. Stopping criteria

  • convergence tolerance (stress below a threshold)
  • the stress ratio (improvement in stress)
  • maximum iterations

Ideally stress < 0.2

0.17. Multidimensional scaling

0.17.1. Final configuration

  • axes have no real meaning
  • operate together to create ordination space
  • orientation of points is arbitrary

0.18. Multidimensional scaling

0.18.1. Starting contiguration

  • completely random
  • based on eigen-analysis
  • repeated random starts

0.19. Multidimensional scaling

0.19.1. Procrustes rotation

  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

  • Site1

Site2 Site3 Site4 Site5 Site6 Site7 Site8 Site9 Site10

0.20. Multidimensional scaling

slide-8
SLIDE 8
  • 8-

0.20.1. Procrustes rotation

  • 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

0.21. 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

0.22. 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

slide-9
SLIDE 9
  • 9-

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

0.23. 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]))'

0.24. Multidimensional scaling

0.24.1. Shepard diagram

> stressplot(data.nmds)

slide-10
SLIDE 10
  • 10-
  • ● ●
  • ●
  • 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

0.25. Multidimensional scaling

0.25.1. Ordination plot

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

slide-11
SLIDE 11
  • 11-

−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 NMDS1 NMDS2

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

0.26. Multidimensional scaling

0.26.1. Environmental fit

  • rotate axes to planes of maximum variability
  • observations are not independent
  • permutation tests - e.g based on R2

0.27. Multidimensional scaling

0.27.1. Environmental fit

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

slide-12
SLIDE 12
  • 12-

−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 NMDS1 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

0.28. Multidimensional scaling

0.28.1. Environmental fit

> 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

slide-13
SLIDE 13
  • 13-

Permutation: free Number of permutations: 999

  • 1. Q-mode Inference testing

1.1. Environmental correlates

. .

S i t e 1

.

Site 10

.

Site 9

.

Site 2

.

Site 3

.

Site 8

.

Site 4

.

Site 5

.

Site 6

.

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

slide-14
SLIDE 14
  • 14-

1.2. Environmental correlates

.

S i t e 1

.

Site 10

.

Site 9

.

Site 2

.

Site 3

.

Site 8

.

Site 4

.

Site 5

.

Site 6

.

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

1.3. 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

1.4. Permutation tests 1.5. Mantel test

  • correlation of two distance matrices

1.6. Mantel test

1.6.1. Community distances

> library(vegan) > data.dist <- vegdist(wisconsin(sqrt(data[,-1])),"bray") > data.dist

slide-15
SLIDE 15
  • 15-

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

1.7. Mantel test

1.7.1. Environmental distances

> 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

1.8. Mantel test

> plot(data.dist, env.dist)

slide-16
SLIDE 16
  • 16-
  • 0.2

0.4 0.6 0.8 1.0 1 2 3 4 5 data.dist env.dist

1.9. 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

1.10. Mantel test

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

slide-17
SLIDE 17
  • 17-

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

1.11. Best environmental subsets

  • looks at combinations of environmental predictors
  • analogous to model selection
  • not an inference test

1.12. 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

slide-18
SLIDE 18
  • 18-

1.13. 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

1.14. 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

1.15. Permutation Multivariate Analysis of Variance

1.15.1. adonis

  • partitions the total sums of squares

– contribution of each predictor

  • permutation test of inference

1.16. Permutation Multivariate Analysis of Variance

1.16.1. adonis

> data.adonis <- adonis(data.dist ~ pH + Slope + Altitude + + Substrate, data=enviro) > data.adonis

slide-19
SLIDE 19
  • 19-

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

  • 2. Worked Examples

2.1. 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)

slide-20
SLIDE 20
  • 20-

−1.0 −0.5 0.0 0.5 1.0 −0.5 0.0 0.5 1.0 CAP1 CAP2

1 2 3 4 5 67 8 9 10

pH Slope Altitude −1 1 SubstrateQuartz SubstrateShale

> 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

slide-21
SLIDE 21
  • 21-

.. ..- 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

slide-22
SLIDE 22
  • 22-

.. .. ..$ : 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)

slide-23
SLIDE 23
  • 23-
  • −0.4

−0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2 0.4 data.cmdscale[,1] data.cmdscale[,2]

> data.scores <- scores(data.cmdscale)