Changes to the schedule

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Changes to the schedule
Jan 16 - Introduction to R, data management Jan
23 - (Jake) Analyses of diversity, diversity and
similarity indices Jan 30 - (Sheri) Analyses of
similarity matrices (ANOSIM, MRPP, Analysis of
Variance using distance matrix) Feb 6 - (Jaclyn)
Matrix comparisons (Mantel test) Feb 13 -
(Justin) Clustering Feb 20 - (Allison M.)
Ordination I - Principal coordinates analysis
Feb 27 - (Brianna) Ordination III - Principal
components analysis March 5 - (Rong Su)
Ordination II - Non-metric multidimensional
scaling March 19 - (Mike) Ordination IV -
Correspondence analysis (CA, Reciprocal
averaging) and Detrended correspondence analysis
(DCA) March 26 - No class April 2 - (Jay)
Constrained Ordination I - Canonical
correspondence analysis (CCA) and Redundancy
analysis (RA) April 9 - Constrained Ordination
II - CCA and RA continued, AIC model selection in
CCA/RA, partial CCA/RA April 16 - (Zoli)
Indicator species analysis April 23 - (Joseph
H.) Regression trees, AIC model selection in
multiple regression
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NMDS Stress values as a function of iterations
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NMDS Stress values as a function of axes
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initial value 27.498853 iter 5 value
21.634000 iter 10 value 21.350476 iter 10 value
21.350198 iter 10 value 21.332007 final value
21.332007 converged gt nmds points
,1 ,2 Beaver1_5_2005
-0.4093229008 -0.280313523 Beaver1_6_2005
0.2608121921 0.445974399 Big1_2_2006
-0.0659405439 -0.331529653 Big2_1_2006
-0.0363998707 -0.198943562 Big2_1_2006b
0.0243210327 -0.207362460 Big2_10_2005
-0.0703308114 -0.392953003
isoMDS Results NMDS Plot (axis I and II)
Stress Plot
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metaMDS Results
Square root transformation Wisconsin double
standardization Run 0 stress 24.05053 Run 1
stress 25.32086 Run 2 stress 23.97737 Waiting
to confirm page change... ... New best
solution ... procrustes rmse 0.02192356 max
resid 0.1328728 Run 3 stress 23.94375 Waiting
to confirm page change... ... New best
solution ... procrustes rmse 0.03284727 max
resid 0.253941 Run 4 stress 28.0158 Run 5
stress 24.76548 Run 6 stress 25.93025 Run 7
stress 25.09786 Run 8 stress 23.82973 Waiting
to confirm page change... ... New best
solution ... procrustes rmse 0.02378274 max
resid 0.1851065 Run 9 stress 23.88804 Run 10
stress 24.42519 Run 11 stress 24.27981 Run 12
stress 24.73001 Run 13 stress 23.97843 Run 14
stress 24.64128 Run 15 stress 23.88268 Run 16
stress 23.82276 Waiting to confirm page
change... ... New best solution ... procrustes
rmse 0.006354104 max resid 0.03321683 Run 17
stress 25.93384 Run 18 stress 27.99029 Run 19
stress 24.35741 Run 20 stress 23.9441 Error in
cov.wt(wa, wa.w) 'x' must contain finite values
only
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metaMDS with no rare species
Run 0 stress 21.33201 Run 1 stress 21.34638
Waiting to confirm page change... ...
procrustes rmse 0.04004226 max resid 0.3419202
Run 2 stress 21.73745 Run 3 stress 21.35399
Waiting to confirm page change... ...
procrustes rmse 0.03991896 max resid 0.3410837
Run 4 stress 21.26580 Waiting to confirm page
change... ... New best solution ... procrustes
rmse 0.003445796 max resid 0.01551004 Run 5
stress 21.32380 Run 6 stress 21.28194 Waiting
to confirm page change... ... procrustes rmse
0.006401349 max resid 0.04568366 Run 7 stress
21.32376 Run 8 stress 21.32376 Run 9 stress
21.68528 Run 10 stress 21.32379 Run 11 stress
21.28209 Waiting to confirm page change... ...
procrustes rmse 0.006377127 max resid
0.04522789 Run 12 stress 24.75756 Run 13 stress
21.39340 Run 14 stress 21.26576 Waiting to
confirm page change... ... New best solution ...
procrustes rmse 0.0006280686 max resid
0.003683763 Solution reached
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Note that species here are weighted averages
computed after the ordination is
complete. Points can be moved and rotated in
ordination space so long as the rank similarity
does not change. Axis scores are largely
meaningless by themselves.
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Correspondence Analysis

• Recall some of the weaknesses with PCA
• Assumes species are linearly related with each
  other and/or gradients
• Samples are ordinated in species space
• Results in horseshoe effect where ends of
  ordination axes are distorted
• Correspondence analysis allows for non-linear
  unimodal relationships
• Ordinates both samples and species

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Correspondence Analysis

• Not iterative, only one solution a dataset
• Eigenvalue approach, not distance based
• Ordinates both samples and species
• Steps
• 1. Sample scores weighted average of all
  species abundances
• 2. Species scores weighted average of all
  sample abundances
• 3. Standardize species and sample scores
• Repeat steps until a converged solution is
  reached.

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Correspondence Analysis

• Output species and sample scores that are
  maximally correlated.
• axes n-1 for whichever dimension of the data
  matrix is lower (samples or species)
• Eigenvalues correlation coefficient between
  axis scores and species scores
• Eigenvalues do not translate directly into
  variance explained as in a PCA

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CA Code

• Correspondence Analysis is run as an
  unconstrained cannonical correspondence analysis
  (CCA)
• CCA function with no environmental matrix
• Code
• calt-cca(community)
• Options
• All options for this function apply to CCA
• Transformations are often used
• Log or eliminating rare species
• Function downweight(community, fractionx)

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CA Example
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CA Example
cca(X community) Inertia
Rank Total 1.78 Unconstrained
1.78 9 Inertia is mean squared contingency
coefficient Eigenvalues for unconstrained
axes CA1 CA2 CA3 CA4
CA5 CA6 CA7 CA8 0.849792 0.529798
0.252153 0.095300 0.030295 0.010971 0.006376
0.003073 CA9 0.001793
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CA Example
Sample 17

• Eigenvalue for axis 1 weighted correlation
  between species scores and sample scores.

Sp. 3
Point size abundance
Sp. 2
Sp. 1
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Species scores CA1 CA2 CA3
CA4 CA5 CA6 sp1 -1.3246 1.07471
-0.72818 -0.389798 -0.15597 -0.04794 sp2 -1.1472
0.52343 0.06469 0.327220 0.28065 0.14819 sp3
-0.8980 -0.07284 0.55031 0.357635 0.01755
-0.10184 sp4 -0.5765 -0.60229 0.60013 0.004217
-0.18891 -0.09976 sp5 -0.1995 -0.92301 0.25165
-0.299273 -0.11131 0.10135 sp6 0.1995 -0.92301
-0.25165 -0.299273 0.11131 0.10135 sp7 0.5765
-0.60229 -0.60013 0.004217 0.18891 -0.09976 sp8
0.8980 -0.07284 -0.55031 0.357635 -0.01755
-0.10184 sp9 1.1472 0.52343 -0.06469 0.327220
-0.28065 0.14819 sp10 1.3246 1.07471 0.72818
-0.389798 0.15597 -0.04794 CA1 CA2
CA3 CA4 CA5 CA6 sample2
-1.559e00 2.02852 -2.888e00 -4.0902 -5.148e00
-4.36957 sample3 -1.559e00 2.02852 -2.888e00
-4.0902 -5.148e00 -4.36957 sample4 -1.503e00
1.74971 -2.045e00 -2.0742 -1.287e00
0.42059 sample5 -1.482e00 1.64766 -1.737e00
-1.3363 1.269e-01 2.17391 sample6 -1.415e00
1.35747 -1.058e00 -0.2856 8.942e-01
1.51561 sample7 -1.376e00 1.18993 -6.654e-01
0.3211 1.337e00 1.13554 sample8 -1.288e00
0.87042 -1.320e-01 0.7008 9.562e-01
-0.03145 sample9 -1.223e00 0.63652 2.586e-01
0.9788 6.773e-01 -0.88574 sample10 -1.101e00
0.27771 6.256e-01 0.9065 1.406e-01
-0.84557 sample11 -9.941e-01 -0.03302 9.434e-01
0.8439 -3.241e-01 -0.81078 sample12 -8.125e-01
-0.43743 1.110e00 0.5186 -4.899e-01
-0.10515 sample13 -6.309e-01 -0.84184 1.276e00
0.1933 -6.557e-01 0.60049 sample14 -4.237e-01
-1.07373 9.190e-01 -0.4290 -6.608e-01
0.32383 sample15 -2.165e-01 -1.30562 5.616e-01
-1.0512 -6.659e-01 0.04717
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PCA
NMDS
CA
Review of unconstrained ordinations of our sample
dataset. Recall that the sample dataset had one
strong gradient.
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CA Example

• The arch effect here is unwanted.
• The ends of the axes are also compressed
• Detrending (detrended correspondence analysis,
  DCA) deals with the arch by
• 5 segment smoothing of 1st axis. Divide into
  segments (weights of 1,2,3,2,1), center each at
  0.
• Rescaling of axis into standard deviation units
  of species turnover.
• Assumptions
• Same as for CA
• DCA is not really an analysis. It is a post hoc
  modification of a CA.
• vague bag of tricks

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DCA

• DCA (Decorana) detrending and rescaling of CA
• Only first 4 axes are adjusted
• Units on axes are in SD of species turnover, beta
  diversity in samples measured in length of 1st
  axis
• Code
• Decorana(community)
• Options
• Downweight rare species
• number of rescaling iterations
• number of detrending segments to use
• Whether or not to detrend at all (avoid
  detrending regular CA)

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DCA Example

• Arch effect removed
• First axis a good representation of the original
  gradient
• Species evenly distributed along first axis
• First axis length 6, indicates complete
  turnover in species
• Second axis length 2

Beta diversity length of 1st axis
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CA and DCA

• First axis is usually fine (not very different
  from CA axis 1), second is often problematic.
• Problems adjustments made during detrending are
  arbitrary.

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PCA
NMDS
DCA
CA
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History of Ordinations in Ecology

• From McCune, B. and J. B. Grace. 2002. Analysis
  of Ecological Communities. MJM Software Design.