Chapter 7 from: Borcard, D., Gillet, F. and Legendre, P. 2011. Numerical Ecology with R. Springer. link

### Key Points

Two ordination-based approaches to modeling spatial structure:

1. Do an ordination of the spatial relationships among sampling locations and use these “spatial eigenvectors” to explore variation in a response variable (e.g. PCNM, MEM).
2. Do an ordination of the response (e.g. community data) and use the variogram of the resulting eigenvectors to explore spatial dependence of the primary axes of variation (e.g. MSO).

#### Approach 1: Moran’s Eigenvector Maps (MEM) and Principal Coordinates of Neighbor Matrices (PCNM)

1. Construct spatial variables (eigenvectors) derived from the adjacency of sampling sites.
• PCNM: uses a distance matrix
1. Create a matrix of Euclidean distances among sites and set sites that are far apart to 0. (All sites must be connected.)
2. Compute a PCoA of the distance matrix to create spatial eigenvectors representing the broadest to smallest potential spatial structures that can be detected by your sampling design.
• MEM: uses similarity matrix
1. Define a spatial weights matrix in which weights ($$w_{i,j}$$) are larger between sites that are more similar (e.g. closer together).

$W[i,j] = \begin{cases} 0 & \text{if i and j are not connected} \\ w_{i,j} & \text{if i and j are connected} \end{cases}$

Note: How $$W$$ is defined can have a strong effect on results, so if you don’t have a strong biologically motivated way to construct $$W$$ be sure to do some sensitivity analyses and determine which way of defining $$W$$ is best. The spdep packages has many ways to define connectivity and neighbor matrices (some examples: Delaunay triangulation, $$k$$-nearest neighbors, minimum spanning tree, neighbors within $$d$$ distance)

1. Compute a PCoA without the square root standardization to create spatial eigenvectors from the most positively autocorrelated to the most negatively autocorrelated.
1. Test spatial eigenvectors for significant autocorrelation using Moran’s I (see below).
2. Model $$Y \sim \text{spatial eigenvectors}$$ using linear models or canonical ordination to determine the scales at which $$Y$$ has spatial structure. Some propose using spatial eigenvectors to “control” for spatial autocorrelation (what do you think?), but different authors find that this may work well or poorly.

#### Approach 2: Multiscale Ordination (MSO)

1. Conduct a canonical ordination (RDA) that partitions $$Y$$ into fitted values explained by the covariates and residual values not explained by covariates.
2. Calculate a variogram matrix of the fitted values that shows how the fitted values covary across sites for each distance class. Plot the empirical variogram of this matrix to see the spatial structure of fitted values (e.g. spatial structure related to covariates).
3. Calculate a variogram matrix of the residual values that shows how residuals covary across sites for each distance class. Plot the empirical variogram of this matrix to see the spatial structure of residuals (e.g. spatial structure not related to covariates).

#### Correlograms

• Correlograms are used to assess spatial autocorrelation at different spatial scales.
• Correlograms plot the correlation between observations as a function of the distance between them.
• Several statistics are used in correlograms: Moran’s I, Geary’s c, and the Mantel statistic.
• We can calculate whether data are significantly autocorrelated at a given spatial scale by comparing the observed value of a statistic to its expected value. It is usually better to use permutations to test in case normality is violated.
• If testing multiple distance classes, p-values should be adjusted for multiple tests (e.g. Holm’s or Bonferroni correction).
• We can only test for significance if data are stationary: there is not trend in the mean or the spatial covariance across the spatial extent of the data. If there is a trend, we can try “detrending” by modeling the data as a function of the site-coordinates and then testing the residuals.

#### Other resources

A list of papers that might be useful:

• Legendre and Gauthier 2014. Statistical methods for temporal and space–time analysis of community composition. Proc. R. Soc. B. DOI: 10.1098/rspb.2013.2728

• Wagner 2013. Rethinking the linear regression model for spatial ecological data. Ecology. DOI: 10.1890/12-1899.1 Describes a method called “spatial component regression”.

### Analysis Example

First, let’s get the data:

library(vegan)
## Loading required package: permute
## Loading required package: lattice
## This is vegan 2.4-4
library(adespatial)
## Warning: namespace 'DBI' is not available and has been replaced
## by .GlobalEnv when processing object 'call.'

## Warning: namespace 'DBI' is not available and has been replaced
## by .GlobalEnv when processing object 'call.'

## Warning: namespace 'DBI' is not available and has been replaced
## by .GlobalEnv when processing object 'call.'

## Warning: namespace 'DBI' is not available and has been replaced
## by .GlobalEnv when processing object 'call.'

## Warning: namespace 'DBI' is not available and has been replaced
## by .GlobalEnv when processing object 'call.'

## Warning: namespace 'DBI' is not available and has been replaced
## by .GlobalEnv when processing object 'call.'

## Warning: namespace 'DBI' is not available and has been replaced
## by .GlobalEnv when processing object 'call.'

## Warning: namespace 'DBI' is not available and has been replaced
## by .GlobalEnv when processing object 'call.'

## Warning: namespace 'DBI' is not available and has been replaced
## by .GlobalEnv when processing object 'call.'

## Warning: namespace 'DBI' is not available and has been replaced
## by .GlobalEnv when processing object 'call.'

## Warning: namespace 'DBI' is not available and has been replaced
## by .GlobalEnv when processing object 'call.'
library(ade4)
##
## Attaching package: 'ade4'
## The following object is masked from 'package:adespatial':
##
##     multispati
source("sr.value.R") # from https://raw.githubusercontent.com/JoeyBernhardt/NumericalEcology/master/sr.value.R

# Data converted to semi-quantitative
data <- cbind(c(1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 100, 30, 100, 100, 100, 100, 100, 100, 76, 3, 76, 1, 0, 75, 76, 0, 1, 0, 0, 1, 0, 0, 0, 0), c(75, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 75, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 75, 1, 1, 2, 2, 0, 0, 0, 0, 100, 75, 1, 2, 0, 0, 0, 0, 0), c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 75, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), c(0, 75, 0, 0, 75, 1, 0, 77, 75, 75, 75, 2, 100, 2, 100, 100, 100, 30, 30, 77, 100, 100, 100, 100, 100, 100, 77, 100, 75, 77, 31, 0, 30, 76, 30, 1, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 100, 77, 2, 76, 2, 0, 100, 30, 1), c(75, 75, 100, 75, 0, 100, 100, 0, 0, 0, 0, 0, 30, 75, 0, 0, 0, 30, 30, 100, 75, 30, 75, 30, 0, 0, 75, 0, 0, 0, 30, 0, 0, 0, 75, 0, 0, 76, 100, 100, 100, 100, 100, 100, 100, 100, 77, 100, 100, 100, 100, 100, 0, 0, 0))
colnames(data) <- c("pt", "sp", "co", "ly", "fe") # clades
# vector indicating distance from river
riv.dist <- c(seq(35,1),seq(1,20))
# vector indicating point along transect
trans.dist <- seq(1,nrow(data),1)

#### Two naive examples, using spatial data but no autocorrelation

Now, we can do an RDA using distance from the river, or distance along the transect, as a possible explanatory variable. First, distance from the river:

riv.rda <- rda(data,riv.dist)
summary(riv.rda)
##
## Call:
## rda(X = data, Y = riv.dist)
##
## Partitioning of variance:
##               Inertia Proportion
## Total          5939.8     1.0000
## Constrained     606.8     0.1022
## Unconstrained  5333.0     0.8978
##
## Eigenvalues, and their contribution to the variance
##
## Importance of components:
##                           RDA1       PC1       PC2       PC3       PC4
## Eigenvalue            606.8149 2893.2107 1309.1388 566.22028 459.45026
## Proportion Explained    0.1022    0.4871    0.2204   0.09533   0.07735
## Cumulative Proportion   0.1022    0.5893    0.8096   0.90498   0.98233
##                             PC5
## Eigenvalue            104.94684
## Proportion Explained    0.01767
## Cumulative Proportion   1.00000
##
## Accumulated constrained eigenvalues
## Importance of components:
##                        RDA1
## Eigenvalue            606.8
## Proportion Explained    1.0
## Cumulative Proportion   1.0
##
## Scaling 2 for species and site scores
## * Species are scaled proportional to eigenvalues
## * Sites are unscaled: weighted dispersion equal on all dimensions
## * General scaling constant of scores:  23.79803
##
##
## Species scores
##
##         RDA1      PC1     PC2     PC3     PC4      PC5
## pt  7.461522  -3.8399  6.2365 -4.7412  3.0844  0.11915
## sp -0.602733  -1.6925 -1.7844 -4.7018 -4.9289 -0.02402
## co  0.737645   0.2319  0.5707  0.3657 -0.4354  3.14807
## ly -1.129948  11.3183 -5.4668 -2.9741  2.3755  0.18828
## fe  0.009609 -11.4059 -7.2481 -0.6500  2.0414  0.21429
##
##
## Site scores (weighted sums of species scores)
##
##         RDA1     PC1      PC2       PC3     PC4      PC5
## sit1  -2.393 -4.0862  1.33792 -5.523906 -6.5964  0.73325
## sit2  -3.076 -0.4994 -1.02133 -2.942616  5.7314  2.22260
## sit3  -1.736 -4.5469  0.61506  1.157042  2.5815  1.23294
## sit4  -2.053 -3.6478  1.49117 -0.973518 -2.1364  0.52364
## sit5  -3.218  2.7639  2.84361 -1.408341  1.6333  0.29883
## sit6  -1.627 -4.3713  0.18115  1.557633  2.1734  0.95794
## sit7  -1.736 -4.3489  0.02830  1.881708  1.8712  0.82531
## sit8  -3.257  2.9944  2.31594 -0.975018  1.2091  0.03074
## sit9  -3.218  2.9618  2.25684 -0.683674  0.9230 -0.10879
## sit10 -3.218  3.0113  2.11015 -0.502507  0.7455 -0.21070
## sit11 -3.218  3.0608  1.96346 -0.321341  0.5679 -0.31261
## sit12 -1.792  0.1152  5.01388  3.881304 -3.5681 -1.78811
## sit13 -3.701  2.9451 -1.16681 -1.697408  2.9664  0.59644
## sit14 -2.561 -3.3470 -0.70661 -3.191097 -8.8668 -0.56581
## sit15 -3.706  4.2845  0.28180 -0.973893  1.2133 -0.24983
## sit16 -3.706  4.3339  0.13511 -0.792726  1.0357 -0.35174
## sit17 -3.706  4.3834 -0.01158 -0.611560  0.8581 -0.45364
## sit18 -2.076  0.2926  1.26539  2.888998 -1.5764 -1.20642
## sit19 -2.321  0.3708  1.02334  3.252578 -1.9047 -1.01754
## sit20 -3.240 -0.5464 -5.25098 -0.004959  3.7382  0.94938
## sit21 -3.565  1.4665 -4.90335 -0.877668  3.7132  0.75677
## sit22 -3.701  3.3904 -2.48702 -0.066908  1.3683 -0.32072
## sit23 -3.694  1.5793 -5.24670 -0.427515  3.2877  0.54105
## sit24 -3.701  3.4894 -2.78040  0.295425  1.0131 -0.52454
## sit25 -3.706  4.7793 -1.18510  0.837774 -0.5624 -1.26890
## sit26 -3.706  4.8288 -1.33179  1.018940 -0.7400 -1.37080
## sit27 -3.244  0.8336 -4.82615  1.564193  1.3302 -0.29935
## sit28 -3.706  4.9277 -1.62518  1.381274 -1.0951 -1.57462
## sit29 -3.218  3.9515 -0.67697  2.939659 -2.6283 -2.14693
## sit30 -3.257  4.0830 -0.91125  3.010648 -2.6975 -2.21121
## sit31 10.925 -0.3620  4.34809 -3.214045  2.7712  8.09263
## sit32 11.143 -0.3692  7.16391 -1.167158 -0.1871 -2.67313
## sit33  2.486  1.9486  2.54886  4.016760 -4.4119 20.55116
## sit34  9.659  2.8479  3.54203 -4.991569  3.5789 -1.44691
## sit35 10.570 -2.0909  1.05499 -3.179275  4.4019 -0.80823
## sit36 11.124 -0.1797  6.68005 -0.678747 -0.6656 -2.96004
## sit37 11.143 -0.2703  6.87053 -0.804824 -0.5423 -2.87695
## sit38 10.355 -3.8812  1.48825 -8.487854 -5.2074 -1.30870
## sit39 11.149 -4.5100  1.34299 -2.458189  4.3603 -0.53402
## sit40  8.054 -4.2254  0.29058 -0.531682  2.8481 -0.71789
## sit41 -1.409 -3.1828 -3.31188  5.500724 -2.1182 -1.44999
## sit42  8.057 -4.3297  0.57423 -0.974332  3.0807 -0.20187
## sit43 -1.607 -3.3237 -3.00224  5.598388 -1.7873 -1.30283
## sit44 -1.736 -3.3593 -2.90551  5.505041 -1.6802 -1.21283
## sit45  7.936 -4.4527  0.98836 -1.262605  3.7780 -0.21786
## sit46  8.065 -4.5162  1.18501 -1.531591  4.0260 -0.10405
## sit47 -4.735  0.9326 -6.93903 -8.979466 -8.0480  0.24193
## sit48 -3.893 -0.8721 -6.71322 -6.081058 -5.1626  0.47553
## sit49 -1.786 -3.5308 -2.27395  4.401940 -0.7964 -0.66806
## sit50 -3.241 -0.5502 -5.38246  0.057116  3.2813  0.82385
## sit51 -1.647 -3.6375 -1.91631  4.038878 -0.2583 -0.44994
## sit52 -1.736 -3.7551 -1.73199  4.055708 -0.2596 -0.39757
## sit53 -3.706  4.4329 -0.15827 -0.430393  0.6806 -0.55555
## sit54 -2.339  1.5114  3.05414  3.244653 -2.9376 -1.77078
## sit55 -1.773  0.2721  4.47092  4.661060 -4.3326 -2.21455
##
##
## Site constraints (linear combinations of constraining variables)
##
##           RDA1     PC1      PC2       PC3     PC4      PC5
## con1  -6.67215 -4.0862  1.33792 -5.523906 -6.5964  0.73325
## con2  -6.33393 -0.4994 -1.02133 -2.942616  5.7314  2.22260
## con3  -5.99571 -4.5469  0.61506  1.157042  2.5815  1.23294
## con4  -5.65749 -3.6478  1.49117 -0.973518 -2.1364  0.52364
## con5  -5.31927  2.7639  2.84361 -1.408341  1.6333  0.29883
## con6  -4.98105 -4.3713  0.18115  1.557633  2.1734  0.95794
## con7  -4.64283 -4.3489  0.02830  1.881708  1.8712  0.82531
## con8  -4.30461  2.9944  2.31594 -0.975018  1.2091  0.03074
## con9  -3.96639  2.9618  2.25684 -0.683674  0.9230 -0.10879
## con10 -3.62817  3.0113  2.11015 -0.502507  0.7455 -0.21070
## con11 -3.28995  3.0608  1.96346 -0.321341  0.5679 -0.31261
## con12 -2.95173  0.1152  5.01388  3.881304 -3.5681 -1.78811
## con13 -2.61351  2.9451 -1.16681 -1.697408  2.9664  0.59644
## con14 -2.27529 -3.3470 -0.70661 -3.191097 -8.8668 -0.56581
## con15 -1.93708  4.2845  0.28180 -0.973893  1.2133 -0.24983
## con16 -1.59886  4.3339  0.13511 -0.792726  1.0357 -0.35174
## con17 -1.26064  4.3834 -0.01158 -0.611560  0.8581 -0.45364
## con18 -0.92242  0.2926  1.26539  2.888998 -1.5764 -1.20642
## con19 -0.58420  0.3708  1.02334  3.252578 -1.9047 -1.01754
## con20 -0.24598 -0.5464 -5.25098 -0.004959  3.7382  0.94938
## con21  0.09224  1.4665 -4.90335 -0.877668  3.7132  0.75677
## con22  0.43046  3.3904 -2.48702 -0.066908  1.3683 -0.32072
## con23  0.76868  1.5793 -5.24670 -0.427515  3.2877  0.54105
## con24  1.10690  3.4894 -2.78040  0.295425  1.0131 -0.52454
## con25  1.44512  4.7793 -1.18510  0.837774 -0.5624 -1.26890
## con26  1.78334  4.8288 -1.33179  1.018940 -0.7400 -1.37080
## con27  2.12156  0.8336 -4.82615  1.564193  1.3302 -0.29935
## con28  2.45978  4.9277 -1.62518  1.381274 -1.0951 -1.57462
## con29  2.79800  3.9515 -0.67697  2.939659 -2.6283 -2.14693
## con30  3.13622  4.0830 -0.91125  3.010648 -2.6975 -2.21121
## con31  3.47444 -0.3620  4.34809 -3.214045  2.7712  8.09263
## con32  3.81266 -0.3692  7.16391 -1.167158 -0.1871 -2.67313
## con33  4.15088  1.9486  2.54886  4.016760 -4.4119 20.55116
## con34  4.48909  2.8479  3.54203 -4.991569  3.5789 -1.44691
## con35  4.82731 -2.0909  1.05499 -3.179275  4.4019 -0.80823
## con36  4.82731 -0.1797  6.68005 -0.678747 -0.6656 -2.96004
## con37  4.48909 -0.2703  6.87053 -0.804824 -0.5423 -2.87695
## con38  4.15088 -3.8812  1.48825 -8.487854 -5.2074 -1.30870
## con39  3.81266 -4.5100  1.34299 -2.458189  4.3603 -0.53402
## con40  3.47444 -4.2254  0.29058 -0.531682  2.8481 -0.71789
## con41  3.13622 -3.1828 -3.31188  5.500724 -2.1182 -1.44999
## con42  2.79800 -4.3297  0.57423 -0.974332  3.0807 -0.20187
## con43  2.45978 -3.3237 -3.00224  5.598388 -1.7873 -1.30283
## con44  2.12156 -3.3593 -2.90551  5.505041 -1.6802 -1.21283
## con45  1.78334 -4.4527  0.98836 -1.262605  3.7780 -0.21786
## con46  1.44512 -4.5162  1.18501 -1.531591  4.0260 -0.10405
## con47  1.10690  0.9326 -6.93903 -8.979466 -8.0480  0.24193
## con48  0.76868 -0.8721 -6.71322 -6.081058 -5.1626  0.47553
## con49  0.43046 -3.5308 -2.27395  4.401940 -0.7964 -0.66806
## con50  0.09224 -0.5502 -5.38246  0.057116  3.2813  0.82385
## con51 -0.24598 -3.6375 -1.91631  4.038878 -0.2583 -0.44994
## con52 -0.58420 -3.7551 -1.73199  4.055708 -0.2596 -0.39757
## con53 -0.92242  4.4329 -0.15827 -0.430393  0.6806 -0.55555
## con54 -1.26064  1.5114  3.05414  3.244653 -2.9376 -1.77078
## con55 -1.59886  0.2721  4.47092  4.661060 -4.3326 -2.21455
##
##
## Biplot scores for constraining variables
##
##      RDA1 PC1 PC2 PC3 PC4 PC5
## bip1   -1   0   0   0   0   0

The RDA explains very little of the variance. Now, we can try distance along the transect:

trans.rda <- rda(data,trans.dist)
summary(trans.rda)
##
## Call:
## rda(X = data, Y = trans.dist)
##
## Partitioning of variance:
##               Inertia Proportion
## Total          5939.8    1.00000
## Constrained     352.9    0.05941
## Unconstrained  5586.9    0.94059
##
## Eigenvalues, and their contribution to the variance
##
## Importance of components:
##                            RDA1       PC1       PC2      PC3       PC4
## Eigenvalue            352.90119 2679.1055 1593.7619 711.5639 492.82768
## Proportion Explained    0.05941    0.4510    0.2683   0.1198   0.08297
## Cumulative Proportion   0.05941    0.5105    0.7788   0.8986   0.98154
##                             PC5
## Eigenvalue            109.62157
## Proportion Explained    0.01846
## Cumulative Proportion   1.00000
##
## Accumulated constrained eigenvalues
## Importance of components:
##                        RDA1
## Eigenvalue            352.9
## Proportion Explained    1.0
## Cumulative Proportion   1.0
##
## Scaling 2 for species and site scores
## * Species are scaled proportional to eigenvalues
## * Sites are unscaled: weighted dispersion equal on all dimensions
## * General scaling constant of scores:  23.79803
##
##
## Species scores
##
##       RDA1      PC1     PC2     PC3     PC4      PC5
## pt  3.7706  -4.1971  9.1655  5.0135  0.5828 -0.07387
## sp  0.1774  -1.5634 -1.8156  2.2245 -6.4879  0.01356
## co  0.1678   0.1591  0.7869 -0.2799 -0.1113  3.22393
## ly -3.1127  11.0362 -3.6563  5.2684  1.0794  0.11785
## fe  3.1117 -10.6566 -7.1183  3.1509  1.8385  0.19729
##
##
## Site scores (weighted sums of species scores)
##
##             RDA1     PC1      PC2     PC3        PC4      PC5
## sit1    4.806093 -4.9531 -1.34340 -0.4852 -9.076e+00 -0.10256
## sit2   -2.527394 -1.1876 -2.28327  2.8419  2.987e+00  0.63585
## sit3    6.610543 -5.3882 -1.74148 -1.9326  2.216e+00  0.25929
## sit4    4.456799 -4.4628 -0.96024 -2.1477 -2.919e+00 -0.18358
## sit5   -9.575158  2.1561  1.07543 -0.8284 -3.073e-03 -0.80247
## sit6    6.630093 -5.1630 -1.79941 -1.8941  2.207e+00  0.23380
## sit7    6.610543 -5.1237 -1.86706 -2.0834  2.157e+00  0.21970
## sit8   -9.760174  2.4409  0.93313 -0.7862 -8.666e-04 -0.80961
## sit9   -9.575158  2.4206  0.94986 -0.9792 -6.139e-02 -0.84206
## sit10  -9.575158  2.4868  0.91846 -1.0169 -7.596e-02 -0.85195
## sit11  -9.575158  2.5529  0.88707 -1.0546 -9.054e-02 -0.86185
## sit12  -2.822081 -0.5348  2.61212 -6.7609 -1.782e+00 -1.69483
## sit13  -9.113573  2.5138 -1.18252  2.2045  1.628e+00 -0.03349
## sit14   4.509020 -3.9903 -1.85996 -0.8941 -9.232e+00 -0.20161
## sit15 -11.887855  3.8976  0.15997  0.7358  4.254e-01 -0.61957
## sit16 -11.887855  3.9638  0.12858  0.6981  4.108e-01 -0.62946
## sit17 -11.887855  4.0299  0.09719  0.6604  3.962e-01 -0.63936
## sit18  -2.413905 -0.2126  0.46540 -3.2719 -2.770e-02 -0.88637
## sit19  -2.633034 -0.1130  0.31856 -3.4615 -6.945e-02 -0.57369
## sit20  -0.512565 -0.9371 -4.12787  3.4055  3.737e+00  0.95921
## sit21  -4.840092  1.1493 -3.48128  4.0666  3.285e+00  0.72967
## sit22  -9.113573  3.1091 -1.46507  1.8651  1.497e+00 -0.12257
## sit23  -4.952149  1.2980 -3.60438  3.9173  3.243e+00  0.71694
## sit24  -9.113573  3.2414 -1.52785  1.7897  1.468e+00 -0.14237
## sit25 -11.887855  4.5591 -0.15397  0.3587  2.796e-01 -0.71854
## sit26 -11.887855  4.6252 -0.18536  0.3210  2.650e-01 -0.72844
## sit27  -2.824467  0.5689 -3.17656  1.9805  2.657e+00  0.41803
## sit28 -11.887855  4.7575 -0.24815  0.2456  2.359e-01 -0.74824
## sit29  -9.575158  3.7436  0.32198 -1.7334 -3.529e-01 -1.04001
## sit30  -9.760174  3.8961  0.24246 -1.6158 -3.216e-01 -1.02736
## sit31   8.624789 -0.9010  6.09941  3.4335  9.501e-01  7.55698
## sit32   8.568686 -0.9413  8.06383 -0.2809 -8.792e-01 -2.62209
## sit33  -1.676663  1.6178  3.47697 -3.4711 -1.251e+00 21.33440
## sit34   1.538085  2.4745  6.17242  5.5452  8.374e-01 -1.78498
## sit35  12.729155 -2.5756  3.73462  5.4187  2.701e+00 -0.89784
## sit36   8.476178 -0.6335  7.91419 -0.3541 -9.146e-01 -2.65040
## sit37   8.568686 -0.6106  7.90686 -0.4695 -9.521e-01 -2.67158
## sit38  15.899754 -4.1308  3.39528  5.5592 -8.325e+00 -1.13831
## sit39  17.821566 -4.6561  3.14785  4.1322  2.793e+00 -0.80249
## sit40  15.132186 -4.1957  1.66891  2.3210  2.481e+00 -0.64276
## sit41   6.772244 -2.8498 -2.82553 -2.9230  1.469e+00 -0.11288
## sit42  15.142443 -4.0689  1.59935  2.2742  2.311e+00 -0.35282
## sit43   6.722601 -2.7588 -2.93693 -3.3671  1.645e+00 -0.14369
## sit44   6.610543 -2.6762 -3.02864 -3.4787  1.618e+00 -0.14652
## sit45  15.014856 -3.8424  1.46357  2.0257  2.534e+00 -0.68648
## sit46  15.126914 -3.7927  1.49249  2.0619  2.532e+00 -0.70345
## sit47  -4.240005  2.1900 -5.64630  6.3839 -1.084e+01  0.64692
## sit48  -0.005114  0.4396 -5.84267  4.8826 -7.014e+00  0.77233
## sit49   6.430799 -2.2652 -3.24568 -3.4792  1.453e+00 -0.17216
## sit50  -0.409514  0.9919 -5.06952  2.2621  3.000e+00  0.65359
## sit51   6.537585 -2.1432 -3.23620 -3.5135  1.574e+00 -0.20032
## sit52   6.610543 -2.1471 -3.27979 -3.7804  1.501e+00 -0.22570
## sit53 -11.887855  6.4112 -1.03300 -0.6972 -1.286e-01 -0.99568
## sit54  -5.412302  3.4531  0.61987 -6.1705 -1.751e+00 -1.79484
## sit55  -2.729573  2.2663  1.28624 -8.4601 -2.432e+00 -2.13171
##
##
## Site constraints (linear combinations of constraining variables)
##
##          RDA1     PC1      PC2     PC3        PC4      PC5
## con1  -5.4579 -4.9531 -1.34340 -0.4852 -9.076e+00 -0.10256
## con2  -5.2557 -1.1876 -2.28327  2.8419  2.987e+00  0.63585
## con3  -5.0536 -5.3882 -1.74148 -1.9326  2.216e+00  0.25929
## con4  -4.8514 -4.4628 -0.96024 -2.1477 -2.919e+00 -0.18358
## con5  -4.6493  2.1561  1.07543 -0.8284 -3.073e-03 -0.80247
## con6  -4.4472 -5.1630 -1.79941 -1.8941  2.207e+00  0.23380
## con7  -4.2450 -5.1237 -1.86706 -2.0834  2.157e+00  0.21970
## con8  -4.0429  2.4409  0.93313 -0.7862 -8.666e-04 -0.80961
## con9  -3.8407  2.4206  0.94986 -0.9792 -6.139e-02 -0.84206
## con10 -3.6386  2.4868  0.91846 -1.0169 -7.596e-02 -0.85195
## con11 -3.4364  2.5529  0.88707 -1.0546 -9.054e-02 -0.86185
## con12 -3.2343 -0.5348  2.61212 -6.7609 -1.782e+00 -1.69483
## con13 -3.0322  2.5138 -1.18252  2.2045  1.628e+00 -0.03349
## con14 -2.8300 -3.9903 -1.85996 -0.8941 -9.232e+00 -0.20161
## con15 -2.6279  3.8976  0.15997  0.7358  4.254e-01 -0.61957
## con16 -2.4257  3.9638  0.12858  0.6981  4.108e-01 -0.62946
## con17 -2.2236  4.0299  0.09719  0.6604  3.962e-01 -0.63936
## con18 -2.0214 -0.2126  0.46540 -3.2719 -2.770e-02 -0.88637
## con19 -1.8193 -0.1130  0.31856 -3.4615 -6.945e-02 -0.57369
## con20 -1.6171 -0.9371 -4.12787  3.4055  3.737e+00  0.95921
## con21 -1.4150  1.1493 -3.48128  4.0666  3.285e+00  0.72967
## con22 -1.2129  3.1091 -1.46507  1.8651  1.497e+00 -0.12257
## con23 -1.0107  1.2980 -3.60438  3.9173  3.243e+00  0.71694
## con24 -0.8086  3.2414 -1.52785  1.7897  1.468e+00 -0.14237
## con25 -0.6064  4.5591 -0.15397  0.3587  2.796e-01 -0.71854
## con26 -0.4043  4.6252 -0.18536  0.3210  2.650e-01 -0.72844
## con27 -0.2021  0.5689 -3.17656  1.9805  2.657e+00  0.41803
## con28  0.0000  4.7575 -0.24815  0.2456  2.359e-01 -0.74824
## con29  0.2021  3.7436  0.32198 -1.7334 -3.529e-01 -1.04001
## con30  0.4043  3.8961  0.24246 -1.6158 -3.216e-01 -1.02736
## con31  0.6064 -0.9010  6.09941  3.4335  9.501e-01  7.55698
## con32  0.8086 -0.9413  8.06383 -0.2809 -8.792e-01 -2.62209
## con33  1.0107  1.6178  3.47697 -3.4711 -1.251e+00 21.33440
## con34  1.2129  2.4745  6.17242  5.5452  8.374e-01 -1.78498
## con35  1.4150 -2.5756  3.73462  5.4187  2.701e+00 -0.89784
## con36  1.6171 -0.6335  7.91419 -0.3541 -9.146e-01 -2.65040
## con37  1.8193 -0.6106  7.90686 -0.4695 -9.521e-01 -2.67158
## con38  2.0214 -4.1308  3.39528  5.5592 -8.325e+00 -1.13831
## con39  2.2236 -4.6561  3.14785  4.1322  2.793e+00 -0.80249
## con40  2.4257 -4.1957  1.66891  2.3210  2.481e+00 -0.64276
## con41  2.6279 -2.8498 -2.82553 -2.9230  1.469e+00 -0.11288
## con42  2.8300 -4.0689  1.59935  2.2742  2.311e+00 -0.35282
## con43  3.0322 -2.7588 -2.93693 -3.3671  1.645e+00 -0.14369
## con44  3.2343 -2.6762 -3.02864 -3.4787  1.618e+00 -0.14652
## con45  3.4364 -3.8424  1.46357  2.0257  2.534e+00 -0.68648
## con46  3.6386 -3.7927  1.49249  2.0619  2.532e+00 -0.70345
## con47  3.8407  2.1900 -5.64630  6.3839 -1.084e+01  0.64692
## con48  4.0429  0.4396 -5.84267  4.8826 -7.014e+00  0.77233
## con49  4.2450 -2.2652 -3.24568 -3.4792  1.453e+00 -0.17216
## con50  4.4472  0.9919 -5.06952  2.2621  3.000e+00  0.65359
## con51  4.6493 -2.1432 -3.23620 -3.5135  1.574e+00 -0.20032
## con52  4.8514 -2.1471 -3.27979 -3.7804  1.501e+00 -0.22570
## con53  5.0536  6.4112 -1.03300 -0.6972 -1.286e-01 -0.99568
## con54  5.2557  3.4531  0.61987 -6.1705 -1.751e+00 -1.79484
## con55  5.4579  2.2663  1.28624 -8.4601 -2.432e+00 -2.13171
##
##
## Biplot scores for constraining variables
##
##      RDA1 PC1 PC2 PC3 PC4 PC5
## bip1    1   0   0   0   0   0

This is even worse.

Now, for examples from this week’s reading:

#### PCNM/MEM

We will begin by looking at the locations only and calculating a PCoA based only on the distances between the locations. So we’re not looking at the actual occurrence data yet, just the locations where the data were taken from.

trans.dm <- dist(trans.dist) # create distance matrix
thresh <- 1 # truncation distance set to 1
trans.dm[trans.dm > thresh] <- 4 * thresh # truncation to threshold

# Make the PCoA
trans.pcoa <- cmdscale(trans.dm, eig=TRUE, k=54) # this is the highest possible value of k in this case, see the textbook for other possibilities
## Warning in cmdscale(trans.dm, eig = TRUE, k = 54): only 38 of the first 54
## eigenvalues are > 0
# Count the positive eigenvalues
nb.ev <- length(which(trans.pcoa$eig > 0.0000000000001)) # Matrix of PCNM variables trans.PCNM <- as.data.frame(trans.pcoa$points[,1:nb.ev])

# Plot some of these
par(mfrow=c(3,2))
somePCNM <- c(1,2,4,8,16,37)
for(i in 1:length(somePCNM)) {
plot(trans.PCNM[,somePCNM[i]],type="l",ylab=c("PCNM",somePCNM[i]))
}

So, these are some of the possible autocorrleation patterns that we might expect to see in the data. We can now test these against the actual data.

# Detrend the data
trans.dist.x <- as.data.frame(trans.dist) # need to make this a data frame in order to detrend
data.D <- dist(data)
data.det <- resid(lm(as.matrix(data.D) ~ ., data=trans.dist.x))

# Run PCNM
PCNM <- rda(data.det,trans.PCNM)
anova.cca(PCNM)
## Permutation test for rda under reduced model
## Permutation: free
## Number of permutations: 999
##
## Model: rda(X = data.det, Y = trans.PCNM)
##          Df Variance      F Pr(>F)
## Model    37    82321 2.5016  0.001 ***
## Residual 17    15120
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Compute adj R2, run forward selection of variables
R2a <- RsquareAdj(PCNM)$adj.r.squared PCNM.fwd <- forward.sel(data.det, as.matrix(trans.PCNM),adjR2thresh=R2a) ## Testing variable 1 ## Testing variable 2 ## Testing variable 3 ## Testing variable 4 ## Testing variable 5 ## Testing variable 6 ## Testing variable 7 ## Testing variable 8 ## Testing variable 9 ## Procedure stopped (alpha criteria): pvalue for variable 9 is 0.053000 (> 0.050000) # We can see the variables by viewing the object PCNM.fwd ## variables order R2 R2Cum AdjR2Cum F pval ## 1 V3 3 0.10449149 0.1044915 0.0875951 6.184250 0.003 ## 2 V2 2 0.09930561 0.2037971 0.1731739 6.485648 0.003 ## 3 V1 1 0.08528084 0.2890779 0.2472590 6.117862 0.006 ## 4 V5 5 0.06753658 0.3566145 0.3051437 5.248532 0.014 ## 5 V17 17 0.04971784 0.4063324 0.3457540 4.103600 0.018 ## 6 V4 4 0.03853049 0.4448629 0.3754707 3.331544 0.031 ## 7 V14 14 0.03294297 0.4778058 0.4000322 2.965027 0.046 ## 8 V7 7 0.03254084 0.5103467 0.4251896 3.057017 0.033 # Make object with significant PCNMs PCNM.sign <- sort(PCNM.fwd[,2]) # Now we can write the significant variables to a new object PCNM.red <- trans.PCNM[,c(PCNM.sign)] # Now re-run RDA with only significant PCNMs PCNM2 <- rda(data.det ~ ., data=PCNM.red) # Get new adj R2 R2a <- RsquareAdj(PCNM2)$adj.r.squared

anova.cca(PCNM2)
## Permutation test for rda under reduced model
## Permutation: free
## Number of permutations: 999
##
## Model: rda(formula = data.det ~ V1 + V2 + V3 + V4 + V5 + V7 + V14 + V17, data = PCNM.red)
##          Df Variance     F Pr(>F)
## Model     8    49728 5.993  0.001 ***
## Residual 46    47712
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Number of significant axes
(axes.test <- anova.cca(PCNM2, by="axis")) # only the first 2 are significant
## Permutation test for rda under reduced model
## Forward tests for axes
## Permutation: free
## Number of permutations: 999
##
## Model: rda(formula = data.det ~ V1 + V2 + V3 + V4 + V5 + V7 + V14 + V17, data = PCNM.red)
##          Df Variance       F Pr(>F)
## RDA1      1    33311 32.1159  0.001 ***
## RDA2      1    14191 13.6817  0.002 **
## RDA3      1     1422  1.3708  0.981
## RDA4      1      436  0.4206  1.000
## RDA5      1      248  0.2388  1.000
## RDA6      1       65  0.0631  1.000
## RDA7      1       42  0.0404  1.000
## RDA8      1       13  0.0127  1.000
## Residual 46    47712
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Now we can plot the 2 significant axes
PCNM.axes <- scores(PCNM2, choices=c(1,2), display="lc",scaling=1) # had to use "scores" instead of "scores.cca"
par(mfrow=c(2,1))
plot(trans.dist, PCNM.axes[,2])

Finally, let’s look at the summary output.

summary(PCNM2)
##
## Call:
## rda(formula = data.det ~ V1 + V2 + V3 + V4 + V5 + V7 + V14 +      V17, data = PCNM.red)
##
## Partitioning of variance:
##               Inertia Proportion
## Total           97441     1.0000
## Constrained     49728     0.5103
## Unconstrained   47712     0.4897
##
## Eigenvalues, and their contribution to the variance
##
## Importance of components:
##                            RDA1      RDA2      RDA3      RDA4      RDA5
## Eigenvalue            3.331e+04 1.419e+04 1.422e+03 436.28707 247.65282
## Proportion Explained  3.419e-01 1.456e-01 1.459e-02   0.00448   0.00254
## Cumulative Proportion 3.419e-01 4.875e-01 5.021e-01   0.50657   0.50911
##                           RDA6     RDA7     RDA8       PC1       PC2
## Eigenvalue            65.44374 41.87905 13.16987 2.757e+04 7.946e+03
## Proportion Explained   0.00067  0.00043  0.00014 2.829e-01 8.155e-02
## Cumulative Proportion  0.50978  0.51021  0.51035 7.933e-01 8.748e-01
##                             PC3       PC4       PC5       PC6       PC7
## Eigenvalue            5.048e+03 3147.6078 1.425e+03 1.125e+03 389.20050
## Proportion Explained  5.181e-02    0.0323 1.462e-02 1.155e-02   0.00399
## Cumulative Proportion 9.266e-01    0.9589 9.735e-01 9.851e-01   0.98909
##                             PC8       PC9      PC10      PC11      PC12
## Eigenvalue            213.22477 182.00068 157.15288 122.86219 104.68137
## Proportion Explained    0.00219   0.00187   0.00161   0.00126   0.00107
## Cumulative Proportion   0.99127   0.99314   0.99475   0.99601   0.99709
##                           PC13     PC14     PC15    PC16     PC17     PC18
## Eigenvalue            93.10408 42.98121 41.28744 29.3111 21.71777 17.70461
## Proportion Explained   0.00096  0.00044  0.00042  0.0003  0.00022  0.00018
## Cumulative Proportion  0.99804  0.99849  0.99891  0.9992  0.99943  0.99961
##                           PC19     PC20    PC21    PC22   PC23   PC24
## Eigenvalue            14.00945 12.80920 8.65995 0.93431 0.4566 0.1193
## Proportion Explained   0.00014  0.00013 0.00009 0.00001 0.0000 0.0000
## Cumulative Proportion  0.99976  0.99989 0.99998 0.99999 1.0000 1.0000
##                         PC25    PC26    PC27    PC28    PC29    PC30
## Eigenvalue            0.1038 0.09116 0.08183 0.06728 0.05327 0.05162
## Proportion Explained  0.0000 0.00000 0.00000 0.00000 0.00000 0.00000
## Cumulative Proportion 1.0000 1.00000 1.00000 1.00000 1.00000 1.00000
##                          PC31    PC32    PC33    PC34    PC35     PC36
## Eigenvalue            0.02628 0.02067 0.01651 0.01274 0.00877 0.004766
## Proportion Explained  0.00000 0.00000 0.00000 0.00000 0.00000 0.000000
## Cumulative Proportion 1.00000 1.00000 1.00000 1.00000 1.00000 1.000000
##                           PC37
## Eigenvalue            0.004018
## Proportion Explained  0.000000
## Cumulative Proportion 1.000000
##
## Accumulated constrained eigenvalues
## Importance of components:
##                            RDA1      RDA2      RDA3      RDA4      RDA5
## Eigenvalue            3.331e+04 1.419e+04 1.422e+03 436.28707 247.65282
## Proportion Explained  6.699e-01 2.854e-01 2.859e-02   0.00877   0.00498
## Cumulative Proportion 6.699e-01 9.552e-01 9.838e-01   0.99260   0.99758
##                           RDA6     RDA7     RDA8
## Eigenvalue            65.44374 41.87905 13.16987
## Proportion Explained   0.00132  0.00084  0.00026
## Cumulative Proportion  0.99889  0.99974  1.00000
##
## Scaling 2 for species and site scores
## * Species are scaled proportional to eigenvalues
## * Sites are unscaled: weighted dispersion equal on all dimensions
## * General scaling constant of scores:  47.89425
##
##
## Species scores
##
##        RDA1     RDA2     RDA3       RDA4     RDA5      RDA6
## 1   1.95817 -2.00736 -0.51078 -0.7557549  0.14319 -0.127576
## 2  -1.43754 -3.23113  0.99803  0.4124588  0.23676  0.001500
## 3   3.54096 -3.77896 -0.81971  0.3858870  0.06879  0.075892
## 4   2.21936 -2.62331 -0.84134 -0.0765282  0.02692 -0.131879
## 5  -5.00861 -0.49262 -0.87741  0.1848134 -0.24565  0.182378
## 6   3.52649 -3.76017 -0.80169  0.3971604  0.06389  0.069623
## 7   3.54096 -3.77896 -0.81971  0.3858870  0.06879  0.075892
## 8  -5.08362 -0.50874 -0.79984  0.1787048 -0.22747  0.181139
## 9  -5.00861 -0.49262 -0.87741  0.1848134 -0.24565  0.182378
## 10 -5.00861 -0.49262 -0.87741  0.1848134 -0.24565  0.182378
## 11 -5.00861 -0.49262 -0.87741  0.1848134 -0.24565  0.182378
## 12 -1.73135  0.03379 -1.58902  0.0383099 -0.28483 -0.352434
## 13 -4.78386 -1.86013  0.71410  0.3776534 -0.21408 -0.062255
## 14  1.87732 -2.05894 -0.50445 -0.7581547  0.15415 -0.143686
## 15 -5.89112 -0.77981  0.13636  0.0596541 -0.15109 -0.023991
## 16 -5.89112 -0.77981  0.13636  0.0596541 -0.15109 -0.023991
## 17 -5.89112 -0.77981  0.13636  0.0596541 -0.15109 -0.023991
## 18 -1.66825 -1.19825 -0.67077  0.0267199 -0.13829 -0.548706
## 19 -1.70201 -1.26711 -0.67633  0.0205122 -0.13077 -0.545032
## 20 -0.38571 -3.89616  1.10426  0.4209090  0.34983 -0.135223
## 21 -2.52082 -3.26792  1.45356  0.5347808  0.06384 -0.110352
## 22 -4.78386 -1.86013  0.71410  0.3776534 -0.21408 -0.062255
## 23 -2.54126 -3.29400  1.44687  0.5304250  0.05874 -0.107368
## 24 -4.78386 -1.86013  0.71410  0.3776534 -0.21408 -0.062255
## 25 -5.89112 -0.77981  0.13636  0.0596541 -0.15109 -0.023991
## 26 -5.89112 -0.77981  0.13636  0.0596541 -0.15109 -0.023991
## 27 -1.55895 -3.27206  1.04417  0.4356049  0.23774  0.002796
## 28 -5.89112 -0.77981  0.13636  0.0596541 -0.15109 -0.023991
## 29 -5.00861 -0.49262 -0.87741  0.1848134 -0.24565  0.182378
## 30 -5.08362 -0.50874 -0.79984  0.1787048 -0.22747  0.181139
## 31  0.69767  3.23236  0.05536  0.6184596  0.43741 -0.085362
## 32  0.72377  4.20435 -0.53948  0.8896915  0.24898 -0.147735
## 33 -1.54363  1.51472 -0.53953  0.0726253  0.62908 -0.038876
## 34 -1.61961  3.05903  0.09749  0.7374051  0.50734 -0.122737
## 35  2.62663  1.76358  0.51325  0.6906857 -0.21412 -0.139971
## 36  0.69500  4.19871 -0.53664  0.8954000  0.25120 -0.151155
## 37  0.72377  4.20435 -0.53948  0.8896915  0.24898 -0.147735
## 38  3.29036  1.63539  0.32370 -0.0480905 -0.49369  0.100801
## 39  4.51366  1.13842  0.68441  0.3525279 -0.78551  0.034467
## 40  4.67027  0.04181  0.53922  0.2164533 -0.78997 -0.080313
## 41  3.51883 -3.66407 -0.74627  0.3773316  0.02738  0.051745
## 42  4.66540  0.04747  0.53146  0.2221906 -0.78639 -0.084319
## 43  3.56327 -3.74338 -0.80920  0.3950476  0.05586  0.067077
## 44  3.54096 -3.77896 -0.81971  0.3858870  0.06879  0.075892
## 45  4.65596 -0.01396  0.52765  0.2150589 -0.76975 -0.083593
## 46  4.66385  0.03532  0.53917  0.2154517 -0.77953 -0.080955
## 47 -1.19736 -2.57354  0.33769 -1.0492424  0.08108 -0.202801
## 48 -0.01554 -3.20722  0.34115 -0.7809334  0.12400 -0.199830
## 49  3.45321 -3.79552 -0.79475  0.3762984  0.07506  0.064896
## 50 -0.33493 -3.90483  1.06146  0.3940497  0.35517 -0.126324
## 51  3.47836 -3.76688 -0.78392  0.3932278  0.07000  0.068144
## 52  3.54096 -3.77896 -0.81971  0.3858870  0.06879  0.075892
## 53 -5.89112 -0.77981  0.13636  0.0596541 -0.15109 -0.023991
## 54 -2.97229 -0.17554 -1.29545  0.0007564 -0.27338 -0.249279
## 55 -1.68883  0.04073 -1.59331  0.0383069 -0.28580 -0.350556
##
##
## Site scores (weighted sums of species scores)
##
##        RDA1      RDA2    RDA3     RDA4    RDA5    RDA6
## 1  -13.1234  -1.04039   1.763  51.8830 -24.164  12.749
## 2   -3.2640   6.30742 -17.426 -13.8593 -10.308  24.902
## 3  -15.3875   7.33564  10.613  -8.2939 -10.756  -7.097
## 4  -12.0269   4.21136  11.285   6.9401  -6.523  35.013
## 5    6.7130  -2.33409   7.208  -5.3673  16.623 -18.268
## 6  -14.7148   7.65258  10.686  -9.3606  -9.869  -5.927
## 7  -14.5750   7.71990  11.073  -8.6536 -10.467  -7.142
## 8    7.4898  -2.07622   6.561  -4.9907  16.405 -19.958
## 9    7.5254  -1.94984   7.668  -5.7269  16.912 -18.314
## 10   7.7285  -1.85378   7.783  -5.8168  16.985 -18.326
## 11   7.9317  -1.75771   7.898  -5.9067  17.057 -18.337
## 12  -2.1639  -5.03615  23.201   4.4428   1.019  60.134
## 13   6.9608   1.76292 -13.141  -5.1241   5.149  -6.572
## 14 -10.2763   0.43138   3.066  50.5774 -23.336  13.211
## 15  10.1817  -2.29330  -5.393   5.8263   8.264 -32.325
## 16  10.3848  -2.19723  -5.278   5.7364   8.337 -32.337
## 17  10.5879  -2.10117  -5.163   5.6465   8.409 -32.348
## 18   0.3901   2.59534  16.687 -17.3897  12.401  79.314
## 19   0.6543   2.81733  16.896 -16.4945  11.819  78.216
## 20  -2.8247   7.92330 -21.490  -2.5025 -21.280   6.333
## 21   2.3837   6.00382 -25.103  -2.8549 -15.825   1.496
## 22   8.7888   2.62749 -12.106  -5.9333   5.799  -6.676
## 23   2.8146   6.24550 -24.833  -2.6459 -15.941   1.035
## 24   9.1950   2.81961 -11.876  -6.1131   5.943  -6.699
## 25  12.2128  -1.33266  -4.243   4.9273   8.987 -32.440
## 26  12.4159  -1.23660  -4.128   4.8373   9.059 -32.452
## 27   2.0262   8.66151 -15.425 -15.0063  -9.399  22.572
## 28  12.8221  -1.04447  -3.898   4.6575   9.203 -32.475
## 29  11.5877  -0.02857   9.969  -7.5250  18.357 -18.544
## 30  11.9583   0.03717   9.091  -6.9687  17.993 -20.211
## 31  -3.9131 -13.78425  -5.591 -14.3824 -11.919  15.169
## 32  -4.3436 -18.73510   3.258 -10.3630 -22.100  15.436
## 33   1.7628  -8.97822   6.572  14.4437 -28.605  -5.157
## 34   2.2498 -14.95125 -10.724  -0.1577 -23.893 -16.857
## 35  -7.1036  -8.78730 -13.611 -14.4326  15.772  11.807
## 36  -3.4431 -18.28295   3.638 -11.0092 -21.644  15.649
## 37  -3.3280 -18.25478   3.834 -10.8125 -21.739  15.379
## 38  -9.3230 -11.83348 -13.299  40.2306  -6.784 -29.963
## 39 -11.0065  -8.82384 -15.065   2.1653  25.531 -14.796
## 40 -10.3279  -3.62834  -9.350  -8.9777  37.193   7.700
## 41  -7.5333  10.86599  14.048 -12.6026  -6.018  -3.251
## 42  -9.9151  -3.42786  -9.072  -8.8976  36.964   7.816
## 43  -7.2917  11.10264  15.093 -12.2724  -7.295  -6.916
## 44  -7.0599  11.27424  15.329 -11.9801  -7.795  -7.567
## 45  -9.2617  -2.95043  -8.475  -9.9258  37.344   7.933
## 46  -9.0992  -3.06842  -8.660  -9.7257  37.547   7.370
## 47   2.7544  -0.01225 -22.847  82.5936 -44.084 -42.279
## 48   0.8111   5.06949 -18.109  55.9195 -32.156 -12.436
## 49  -5.8343  11.90239  15.439 -12.6840  -7.184  -5.773
## 50   3.1695  10.91872 -17.623  -5.0061 -18.868   7.329
## 51  -5.4634  12.03085  15.606 -13.6538  -6.484  -5.704
## 52  -5.4350  12.04275  16.249 -12.6994  -7.217  -7.659
## 53  17.8999   1.35711  -1.022   2.4099  11.009 -32.761
## 54  10.2013   1.07995  24.279  -8.1280  11.832  51.856
## 55   6.4371  -0.99572  28.156   1.0071   3.742  59.145
##
##
## Site constraints (linear combinations of constraining variables)
##
##        RDA1      RDA2      RDA3      RDA4     RDA5     RDA6
## 1   -3.9713  -0.39559  -2.60357   5.62084  -0.3656  -4.1152
## 2   -5.9427   0.17291  -1.43956   6.54218   0.1071  -4.9160
## 3   -5.7214   1.99062   3.84388   1.74993   1.2161  -2.0673
## 4   -4.9902   4.44117   9.84609  -5.55837   1.8038   1.5848
## 5   -5.4246   6.29125  12.47862 -10.43042   1.1568   2.4517
## 6   -6.7942   6.39568  10.59326  -9.81752   0.1419  -0.7303
## 7   -7.0163   4.44323   6.88699  -4.62913   0.6813  -5.8412
## 8   -4.2437   1.26280   5.32443   1.24285   3.8124  -9.1543
## 9    0.9868  -1.58494   7.49950   3.96768   8.1666  -8.3218
## 10   5.7307  -2.75290  11.26748   2.46097  10.5728  -4.0007
## 11   6.9899  -1.99339  12.82297  -1.19627   8.6029   1.1066
## 12   4.3968  -0.34134  10.21555  -3.61715   2.9162   4.7093
## 13   0.7094   0.61009   5.03655  -2.60824  -2.8395   6.6098
## 14  -0.4948  -0.11322   0.79570   1.51762  -4.7779   8.2547
## 15   2.0410  -2.08030  -0.38442   6.46709  -1.9625  10.8484
## 16   6.1128  -3.68118   0.38568   9.59304   2.7574  13.9609
## 17   8.0212  -3.26222   0.05872   9.36176   5.1261  15.9071
## 18   5.9433  -0.37000  -3.23452   5.89700   3.1279  15.3781
## 19   1.6902   3.82013  -8.27281   0.65424  -1.2389  12.6301
## 20  -0.7740   7.23622 -11.86041  -4.32100  -3.6510   9.0867
## 21   1.2191   8.28859 -11.87434  -7.21976  -1.2112   5.8593
## 22   6.6719   6.89643  -9.29107  -7.24630   4.9444   2.7792
## 23  11.7238   4.42763  -7.22051  -5.11935  10.4156  -1.0260
## 24  12.9012   2.62693  -7.85530  -2.87458  11.0158  -5.6163
## 25   9.9410   2.34235 -10.27465  -2.73042   6.0088  -9.4519
## 26   5.7602   2.97122 -11.26728  -5.48461  -1.5450 -10.3720
## 27   3.7309   2.98033  -8.46448  -9.58384  -7.4355  -7.6035
## 28   4.7792   1.06634  -2.82045 -11.85915  -9.5452  -2.8761
## 29   6.8528  -2.89633   1.98815  -9.75835  -8.9849   0.6338
## 30   7.0586  -7.75334   2.79946  -3.54958  -8.4415   0.8232
## 31   4.3678 -11.81013  -0.13125   3.34743  -9.4913  -1.5514
## 32   0.4098 -13.85049  -3.31384   6.50081 -11.3172  -3.5290
## 33  -2.2297 -13.76146  -3.21543   3.69505 -11.7558  -2.6053
## 34  -2.4606 -12.43354   0.43590  -3.21862  -9.4281   0.9125
## 35  -1.5566 -11.09511   4.30833  -9.45328  -4.8925   4.0201
## 36  -1.7284 -10.54339   4.46926 -10.72089   0.0794   3.6193
## 37  -3.9035 -10.71481   0.02807  -6.39721   4.1345  -0.6254
## 38  -6.9056 -10.78942  -5.91902   0.05903   7.1620  -5.7255
## 39  -8.7749  -9.72932  -8.93993   3.80285   9.6111  -7.6879
## 40  -8.8375  -6.94888  -7.12934   2.33211  11.2608  -4.7764
## 41  -8.3957  -2.75486  -2.69051  -2.64550  10.9741   1.0701
## 42  -9.2981   1.70812   0.16973  -6.61104   7.8781   5.8487
## 43 -11.7938   5.03172  -0.91984  -5.73048   2.7779   6.7057
## 44 -13.8775   6.35436  -4.42250   0.29263  -1.8555   4.0057
## 45 -12.9498   5.91834  -6.42413   8.11743  -3.6826   0.6029
## 46  -8.3386   4.96061  -4.28930  13.30664  -2.5797  -0.8536
## 47  -2.2575   4.91754   0.93742  13.42920  -0.9625  -0.1237
## 48   1.8744   6.41807   5.53589   9.43892  -1.7535   0.7617
## 49   2.3543   8.76836   6.49895   4.57480  -5.6278  -0.3583
## 50   0.6845  10.35964   4.12796   1.91557 -10.0580  -3.7033
## 51   0.1596   9.78656   1.48728   2.48058 -11.2471  -7.3872
## 52   2.7317   6.90345   1.34885   5.01770  -7.4516  -9.1292
## 53   6.9954   3.02373   3.57240   7.21743  -0.7902  -8.1275
## 54   9.2200   0.05871   5.37069   7.23155   4.2055  -5.4005
## 55   6.6228  -0.81701   4.12471   4.54613   4.2347  -2.4934
##
##
## Biplot scores for constraining variables
##
##        RDA1     RDA2      RDA3     RDA4     RDA5     RDA6
## V1  -0.4802 -0.19901 -0.163856  0.10723 -0.18155 -0.30333
## V2  -0.4435  0.42443  0.578201  0.41837  0.23916  0.03902
## V3   0.3355  0.65486 -0.457695  0.31591 -0.07258  0.20224
## V4   0.2242  0.36235  0.316985 -0.06579 -0.40118 -0.63251
## V5   0.3720 -0.34441  0.325772  0.36066 -0.54469  0.13762
## V7   0.3036  0.01143 -0.002357 -0.21932  0.42871 -0.52486
## V14  0.2134 -0.31522 -0.125176  0.70207  0.38763 -0.27604
## V17  0.3658 -0.04479  0.455048 -0.19561  0.33600  0.30804

RDA1 is our most informative axis. PC1 does explain noticeably more than RDA2, which is maybe not ideal, but there clearly is value in running the RDA.

#### Multiscale Ordination (MSO)

First, we can run a MSO using undetrended data.

# Initial RDA with distance from river as explanatory variable
river.rda <- rda(data,riv.dist)

#Multiscale ordination
river.mso <- mso(river.rda,trans.dist,  perm=999)

#Plot
msoplot(river.mso, ylim=c(0,15000), xlim=c(-20,40))

## Error variance of regression model underestimated by 4.2 percent

Now we can run a MSO using detrended data.

# Initial RDA with distance from river as explanatory variable
river.rda2 <- rda(data.det,riv.dist)

#Multiscale ordination
river.mso2 <- mso(river.rda2,trans.dist,  perm=999)

#Plot
msoplot(river.mso2)

## Error variance of regression model underestimated by 3.8 percent

### Discussion Questions

1. When is MEM/PCNM the most appropriate method to use, and when is MSO the most appropriate method to use?

2. Are there circumstances in which it would be inappropriate to use either of these methods?

3. In what circumstances would we find significant PCNM axes at both the broad and fine scale?

4. Can you think of any examples in which you might apply either of these methods to your data?

5. When would you want to run a MSO using undetrended data, and when would you want to use detrended data?

6. When would it be most appropriate to use MEM and when would it be most appropriate to use PCNM?