Fit Co-Correspondence Analysis Ordination Models

coca is used to fit Co-Correspondence Analysis (CoCA) models. It can fit predictive or symmetric models to two community data matrices containing species abundance data.

coca(y, ...)

# S3 method for default
coca(y, x, method = c("predictive", "symmetric"),
     reg.method = c("simpls", "eigen"), weights = NULL,
     n.axes = NULL, symmetric = FALSE, quiet = FALSE, ...)

# S3 method for formula
coca(formula, data, method = c("predictive", "symmetric"),
     reg.method = c("simpls", "eigen"), weights = NULL,
     n.axes = NULL, symmetric = FALSE, quiet = FALSE, ...)

Arguments

y

a data frame containing the response community data matrix.
x

a data frame containing the predictor community data matrix.
formula

a symbolic description of the model to be fit. The details of model specification are given below.
data

an optional data frame containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which coca is called.
method

a character string indicating which co-correspondence analysis method to use. One of "predictive"(default), or "symmetric", can be abbreviated.
reg.method

One of "simpls" (default) or "eigen". If method is "predictive" then reg.method controls whether the co-correspondence analysis should be fitted using the SIMPLS algorithm or via an eigen analysis.
weights

a vector of length nrow(y) of user supplied weights for \(R_0\). If weights = NULL (default) then the weights are determined from y (default) or x and y (symmetric = TRUE only).
n.axes

the number of CoCA axes to extract. If missing (default) the n.axes is $$min(ncol(y), ncol(x), nrow(y), nrow(x)) - 1$$.
symmetric

if method is "symmetric" then symmetric determines whether weights for \(R_0\) are symmetric and taken as the average of the row sums of x and y (symmetric = TRUE). If symmetric = FALSE (default) then the weights \(R_0\) are taken as the row sums of y unless user defined weights are provided via argument weights. Ignored if method is "predictive".
quiet

logical; suppress messages due to removal of species with no data.
...

additional arguments to be passed to lower level methods.

Details

coca is the main user-callable function.

A typical model has the form response ~ terms where response is the (numeric) response data frame and terms is a series of terms which specifies a linear predictor for response. A typical form for terms is ., which is shorthand for "all variables" in data. If . is used, data must also be provided. If specific species (variables) are required then terms should take the form spp1 + spp2 + spp3.

The default is to fit a predictive CoCA model using SIMPLS via a modified version of simpls.fit from package pls. Alternatively, reg.method = "eigen" fits the model using an older, slower eigen analysis version of the SIMPLS algorithm. reg.method = "eigen" is about 100% slower than reg.method = "simpls".

Value

coca returns a list with method and reg.method determining the actual components returned.

nam.dat

list with components namY and namX containing the names of the response and the predictor(s) respectively.

call

the matched call.

method

the CoCA method used, one of "predictive" or "symmetric".

scores

the species and site scores of the fitted model.

loadings

the site loadings of the fitted model for the response and the predictor. (Predictive CoCA via SIMPLS only.)

fitted

the fitted values for the response. A list with 2 components Yhat (the fitted values on the original scale) and Yhat1 (the fitted values on the chi-square transformed scale). (Predictive CoCA via SIMPLS only.)

varianceExp

list with components Yblock and Xblock containing the variances in the response and the predictor respectively, explained by each fitted PLS axis. (Predictive CoCA via SIMPLS only.)

totalVar

list with components Yblock and Xblock containing the total variance in the response and the predictor respectively. (Predictive CoCA via SIMPLS only.)

lambda

the Eigenvalues of the analysis.

n.axes

the number of fitted axes

Ychi

a list containing the mean-centered chi-square matrices for the response (Ychi1) and the predictor (Ychi2). (Predictive CoCA only.)

R0

the (possibly user-supplied) row weights used in the analysis.

X

X-Matrix (symmetric CoCA only).

residuals

Residuals of a symmetric model (symmetric CoCA only).

inertia

list with components total and residual containing the total and residual inertia for the response and the predictor (symmetric CoCA only).

rowsum

a list with the row sums for the response (rsum1) and the preditor (rsum2) (symmetric CoCA only).

colsum

a list with the column sums for the response (csum1)and the preditor (csum2) (symmetric CoCA only).

References

ter Braak, C.J.F and Schaffers, A.P. (2004) Co-Correspondence Analysis: a new ordination method to relate two community compositions. Ecology 85(3), 834--846

Author

Original Matlab code by C.J.F. ter Braak and A.P. Schaffers. R port by Gavin L. Simpson. Formula method for coca uses a modified version of ordiParseFormula by Jari Oksanen to handle formulea.

See also

crossval for cross-validation and permutest.coca for permutation test to determine the number of PLS axes to retain in for predictive CoCA.

summary.predcoca and summary.symcoca for summary methods.

Examples

# \dontshow{
od <- options(digits = 4)
# }
## symmetric CoCA
data(beetles)
## log transform the bettle data
beetles <- log(beetles + 1)
data(plants)
## fit the model
bp.sym <- coca(beetles ~ ., data = plants, method = "symmetric")
#> 
#> Removed some species that contained no data in: beetles, plants 
bp.sym
#> 
#> Symmetric Co-Correspondence Analysis
#> 
#> Call: symcoca(y = y, x = x, n.axes = n.axes, R0 = weights, symmetric =
#> symmetric, nam.dat = nam.dat)
#> 
#> Inertia:
#>          Total Explained Residual
#> beetles:  5.70      5.60     0.09
#> plants:   6.16      6.07     0.09
#> 
summary(bp.sym)
#> 
#> Symmetric Co-Correspondence Analysis
#> 
#> Call: symcoca(y = y, x = x, n.axes = n.axes, R0 = weights, symmetric =
#> symmetric, nam.dat = nam.dat)
#> 
#> Inertia:
#>          Total Explained Residual
#> beetles:  5.70      5.60     0.09
#> plants:   6.16      6.07     0.09
#> 
#> Eigenvalues:
#>   COCA 1    COCA 2    COCA 3    COCA 4    COCA 5    COCA 6    COCA 7    COCA 8  
#> 5.91e-01  3.16e-01  2.03e-01  1.04e-01  6.69e-02  6.11e-02  5.26e-02  4.45e-02  
#>   COCA 9   COCA 10   COCA 11   COCA 12   COCA 13   COCA 14   COCA 15   COCA 16  
#> 3.51e-02  2.31e-02  2.18e-02  1.43e-02  1.34e-02  1.10e-02  1.02e-02  8.39e-03  
#>  COCA 17   COCA 18   COCA 19   COCA 20   COCA 21   COCA 22   COCA 23   COCA 24  
#> 7.44e-03  4.85e-03  4.15e-03  3.45e-03  2.82e-03  2.42e-03  2.26e-03  1.78e-03  
#>  COCA 25   COCA 26   COCA 27   COCA 28   COCA 29  
#> 1.30e-03  1.06e-03  8.53e-04  2.93e-04  5.56e-05  
biplot(bp.sym)                          # produces a Benzecri biplot


## extract eigenvalues of the analysis
eigenvals(bp.sym)
#>  COCA 1  COCA 2  COCA 3  COCA 4  COCA 5  COCA 6  COCA 7  COCA 8  COCA 9 COCA 10 
#>  0.5906  0.3159  0.2028  0.1042  0.0669  0.0611  0.0526  0.0445  0.0351  0.0231 
#> COCA 11 COCA 12 COCA 13 COCA 14 COCA 15 COCA 16 COCA 17 COCA 18 COCA 19 COCA 20 
#>  0.0218  0.0143  0.0134  0.0110  0.0102  0.0084  0.0074  0.0049  0.0041  0.0034 
#> COCA 21 COCA 22 COCA 23 COCA 24 COCA 25 COCA 26 COCA 27 COCA 28 COCA 29 
#>  0.0028  0.0024  0.0023  0.0018  0.0013  0.0011  0.0009  0.0003  0.0001 

## correlations between beetle and plant score scores on Co-CA axes
corAxis(bp.sym)
#>  COCA 1  COCA 2  COCA 3  COCA 4  COCA 5  COCA 6  COCA 7  COCA 8  COCA 9 COCA 10 
#>  0.8795  0.8830  0.8154  0.8490  0.8880  0.7738  0.7264  0.8193  0.7979  0.7758 
#> COCA 11 COCA 12 COCA 13 COCA 14 COCA 15 COCA 16 COCA 17 COCA 18 COCA 19 COCA 20 
#>  0.8908  0.8277  0.8916  0.9240  0.7915  0.7046  0.6205  0.7882  0.7451  0.7148 
#> COCA 21 COCA 22 COCA 23 COCA 24 COCA 25 COCA 26 COCA 27 COCA 28 COCA 29 
#>  0.9076  0.3968  0.8034  0.6177  0.7608  0.9179  0.5622  0.3823  0.9937 

## predictive CoCA using SIMPLS and formula interface
bp.pred <- coca(beetles ~ ., data = plants)
#> 
#> Removed some species that contained no data in: beetles, plants 
## should retain only the useful PLS components for a parsimonious model

# \donttest{
## Leave-one-out crossvalidation - this takes a while
crossval(beetles, plants)
#> some species contain no data and were removed from data matrix y
#> some species contain no data and were removed from data matrix x
#> LOO - Site: 1 - Complete
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#> 
#> Cross-validation for Predictive Co-Correspondence Analysis
#> 
#> Call: crossval(y = beetles, x = plants)
#> 
#> Cross-validatory %fit of beetles to plants:
#> 
#>  COCA1   COCA2   COCA3   COCA4   COCA5   COCA6   COCA7   COCA8   COCA9  COCA10  
#>    5.1     7.8     7.0     6.9     7.7     8.5     8.7     6.8     6.5     6.7  
#> COCA11  COCA12  COCA13  COCA14  COCA15  COCA16  COCA17  COCA18  COCA19  COCA20  
#>    5.8     3.5     2.7     2.0     2.5    -0.1    -2.1    -2.9    -2.9    -3.5  
#> COCA21  COCA22  COCA23  COCA24  COCA25  COCA26  COCA27  COCA28  COCA29  
#>   -4.2    -4.3    -4.8    -5.7    -8.1   -24.6   -29.2   -37.9   -41.4  
## so 2 axes are sufficient
## permutation test to assess significant PLS components - takes a while
bp.perm <- permutest(bp.pred, permutations = 99)
#> Permutations for axis: 1 - completed
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bp.perm
#> 
#> Permutation test for predictive co-correspondence analysis:
#> 
#> Call: permutest.coca(x = bp.pred, permutations = 99)
#> 
#> Permutation test results:
#> 
#>          Stat. Inertia    Fit  % fit P-value
#> COCA 1  0.0890  5.6964 0.4658 8.1763    0.02
#> COCA 2  0.0761  5.2306 0.3701 6.4973    0.08
#> COCA 3  0.0620  4.8605 0.2839 4.9832    0.23
#> COCA 4  0.0935  4.5766 0.3912 6.8683    0.11
#> COCA 5  0.0569  4.1854 0.2252 3.9530    0.77
#> COCA 6  0.0878  3.9602 0.3196 5.6115    0.40
#> COCA 7  0.0715  3.6406 0.2430 4.2650    0.63
#> COCA 8  0.0627  3.3976 0.2005 3.5195    0.91
#> COCA 9  0.0792  3.1971 0.2347 4.1209    0.85
#> COCA 10 0.0716  2.9624 0.1978 3.4730    0.83
#> COCA 11 0.1044  2.7646 0.2613 4.5880    0.61
#> COCA 12 0.0486  2.5032 0.1160 2.0368    1.00
#> COCA 13 0.0805  2.3872 0.1778 3.1208    0.98
#> COCA 14 0.0717  2.2094 0.1478 2.5938    0.99
#> COCA 15 0.0926  2.0617 0.1747 3.0670    0.99
#> COCA 16 0.1201  1.8870 0.2023 3.5520    0.89
#> COCA 17 0.0606  1.6846 0.0963 1.6903    1.00
#> COCA 18 0.1157  1.5883 0.1647 2.8919    0.91
#> COCA 19 0.0793  1.4236 0.1045 1.8353    1.00
#> COCA 20 0.0921  1.3191 0.1112 1.9524    0.99
#> COCA 21 0.0864  1.2078 0.0961 1.6867    1.00
#> COCA 22 0.1297  1.1118 0.1276 2.2403    0.94
#> COCA 23 0.0793  0.9841 0.0723 1.2692    0.99
#> COCA 24 0.1198  0.9118 0.0976 1.7130    0.97
#> COCA 25 0.1142  0.8143 0.0835 1.4651    0.88
#> COCA 26 0.2439  0.7308 0.1433 2.5157    0.75
#> COCA 27 0.2024  0.5875 0.0989 1.7364    0.79
#> COCA 28 0.6375  0.4886 0.1902 3.3391    0.51
#> COCA 29 1.1924  0.2984 0.1623 2.8489    0.12
#> 
# }

## agrees with the Leave-one-out cross-validation
## refit the model with only 2 PLS components
bp.pred <- coca(beetles ~ ., data = plants, n.axes = 2)
#> 
#> Removed some species that contained no data in: beetles, plants 
bp.pred
#> 
#> Predictive Co-Correspondence Analysis
#> 
#> Call: predcoca.simpls(y = y, x = x, R0 = weights, n.axes = n.axes,
#> nam.dat = nam.dat)
#> 
#> Co-CA Method: simpls
#> 
#>              Role Variance
#> plants  Predictor     6.16
#> beetles  Response     5.70
summary(bp.pred)
#> 
#> Predictive Co-Correspondence Analysis
#> 
#> Call: predcoca.simpls(y = y, x = x, R0 = weights, n.axes = n.axes,
#> nam.dat = nam.dat)
#> 
#> Percentage Variance Explained:
#> 
#> Y-block: variance explained in beetles (response) 
#>              Comp 1  Comp 2
#> Individual:   8.18    6.38 
#> Cumulative:   8.18   14.56 
#> 
#> X-block: variance explained in plants (predictor) 
#>              Comp 1  Comp 2
#> Individual:  21.7    16.2  
#> Cumulative:  21.7    37.9  
biplot(bp.pred)                     # plots correct scores or loadings


## predictive CoCA using Eigen-analysis
data(bryophyte)
data(vascular)
carp.pred <- coca(y = bryophyte, x = vascular, reg.method = "eigen")
carp.pred
#> 
#> Predictive Co-Correspondence Analysis
#> 
#> Call: predcoca.eigen(y = y, x = x, R0 = weights, n.axes = n.axes,
#> nam.dat = nam.dat)
#> 
#> Co-CA Method: eigen
#> 
#> 
#> Eigenvalues:
#> 
#> 
#>  COCA 1  COCA 2  COCA 3  COCA 4  COCA 5  COCA 6  COCA 7  COCA 8  COCA 9 COCA 10 
#>   0.301   0.072   0.022   0.017   0.015   0.010   0.007   0.006   0.005   0.005 
#> COCA 11 COCA 12 COCA 13 COCA 14 COCA 15 COCA 16 COCA 17 COCA 18 COCA 19 COCA 20 
#>   0.004   0.004   0.003   0.003   0.003   0.003   0.002   0.002   0.002   0.002 
#> COCA 21 COCA 22 COCA 23 COCA 24 COCA 25 COCA 26 COCA 27 COCA 28 COCA 29 
#>   0.001   0.001   0.001   0.001   0.001   0.001   0.001   0.001   0.001 

# \donttest{
## determine important PLS components - takes a while
crossval(bryophyte, vascular)
#> LOO - Site: 1 - Complete
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#> 
#> Cross-validation for Predictive Co-Correspondence Analysis
#> 
#> Call: crossval(y = bryophyte, x = vascular)
#> 
#> Cross-validatory %fit of bryophyte to vascular:
#> 
#>  COCA1   COCA2   COCA3   COCA4   COCA5   COCA6   COCA7   COCA8   COCA9  COCA10  
#>   17.9    24.8    26.2    26.6    28.3    28.2    27.1    25.5    24.6    24.6  
#> COCA11  COCA12  COCA13  COCA14  COCA15  COCA16  COCA17  COCA18  COCA19  COCA20  
#>   24.0    23.1    22.9    21.2    20.4    19.8    21.0    19.5    18.0    17.4  
#> COCA21  COCA22  COCA23  COCA24  COCA25  COCA26  COCA27  COCA28  COCA29  
#>   17.2    13.7    13.0    13.5    12.5    11.1     9.7     8.4     8.4  
(carp.perm <- permutest(carp.pred, permutations = 99))
#> Permutations for axis: 1 - completed
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#> 
#> Permutation test for predictive co-correspondence analysis:
#> 
#> Call: permutest.coca(x = carp.pred, permutations = 99)
#> 
#> Permutation test results:
#> 
#>           Stat. Inertia     Fit   % fit P-value
#> COCA 1   0.2617  3.4713  0.7199 20.7388    0.01
#> COCA 2   0.1385  2.7514  0.3346  9.6394    0.01
#> COCA 3   0.0687  2.4168  0.1554  4.4755    0.18
#> COCA 4   0.0669  2.2614  0.1418  4.0854    0.48
#> COCA 5   0.0776  2.1196  0.1526  4.3953    0.33
#> COCA 6   0.0676  1.9670  0.1245  3.5872    0.80
#> COCA 7   0.0671  1.8425  0.1159  3.3385    0.46
#> COCA 8   0.0586  1.7266  0.0956  2.7536    0.78
#> COCA 9   0.0508  1.6310  0.0789  2.2718    0.96
#> COCA 10  0.0521  1.5522  0.0768  2.2136    0.97
#> COCA 11  0.0576  1.4753  0.0804  2.3153    0.91
#> COCA 12  0.0631  1.3950  0.0827  2.3835    0.84
#> COCA 13  0.0692  1.3122  0.0850  2.4481    0.65
#> COCA 14  0.0546  1.2272  0.0636  1.8311    0.94
#> COCA 15  0.0556  1.1637  0.0613  1.7656    0.98
#> COCA 16  0.0559  1.1024  0.0583  1.6802    0.97
#> COCA 17  0.0679  1.0441  0.0664  1.9125    0.59
#> COCA 18  0.0641  0.9777  0.0589  1.6970    0.98
#> COCA 19  0.0562  0.9188  0.0489  1.4084    1.00
#> COCA 20  0.0445  0.8699  0.0371  1.0686    1.00
#> COCA 21  0.0468  0.8328  0.0373  1.0732    1.00
#> COCA 22  0.0464  0.7955  0.0353  1.0162    1.00
#> COCA 23  0.0559  0.7603  0.0403  1.1595    1.00
#> COCA 24  0.0703  0.7200  0.0473  1.3629    0.99
#> COCA 25  0.0456  0.6727  0.0293  0.8450    1.00
#> COCA 26  0.0656  0.6434  0.0396  1.1408    1.00
#> COCA 27  0.0599  0.6038  0.0341  0.9835    0.99
#> COCA 28  0.0630  0.5696  0.0338  0.9725    1.00
#> COCA 29  0.0620  0.5359  0.0313  0.9007    1.00
#> 
# }

## 2 components again, refit
carp.pred <- coca(y = bryophyte, x = vascular,
                  reg.method = "eigen", n.axes = 2)
carp.pred
#> 
#> Predictive Co-Correspondence Analysis
#> 
#> Call: predcoca.eigen(y = y, x = x, R0 = weights, n.axes = n.axes,
#> nam.dat = nam.dat)
#> 
#> Co-CA Method: eigen
#> 
#> 
#> Eigenvalues:
#> 
#> 
#> COCA 1 COCA 2 
#>  0.301  0.072 
## drawn biplot
biplot(carp.pred)

options(od)