Diagnostic Tools for [Constrained] Ordination (CCA, RDA, DCA, CA, PCA)

Functions goodness and inertcomp can be used to assess the goodness of fit for individual sites or species. Function vif.cca and alias.cca can be used to analyse linear dependencies among constraints and conditions. In addition, there are some other diagnostic tools (see 'Details').

Usage

# S3 method for cca
goodness(object, choices, display = c("species", "sites"),
    model = c("CCA", "CA"), summarize = FALSE, addprevious = FALSE, ...)
inertcomp(object, display = c("species", "sites"),
    unity = FALSE, proportional = FALSE)
spenvcor(object)
intersetcor(object)
vif.cca(object)
# S3 method for cca
alias(object, names.only = FALSE, ...)

Arguments

object: A result object from cca, rda, dbrda or capscale.
display: Display "species" or "sites". Species are not available in dbrda and capscale.
choices: Axes shown. Default is to show all axes of the "model".
model: Show constrained ("CCA") or unconstrained ("CA") results.
summarize: Show only the accumulated total.
addprevious: Add the variation explained by previous components when statistic="explained". For model = "CCA" add conditioned (partialled out) variation, and for model = "CA" add both conditioned and constrained variation. This will give cumulative explanation with previous components.
unity: Scale inertia components to unit sum (sum of all items is 1).
proportional: Give the inertia components as proportional for the corresponding total of the item (sum of each row is 1). This option takes precedence over unity.
names.only: Return only names of aliased variable(s) instead of defining equations.
...: Other parameters to the functions.

Details

Function goodness gives cumulative proportion of inertia accounted by species up to chosen axes. The proportions can be assessed either by species or by sites depending on the argument display, but species are not available in distance-based dbrda. The function is not implemented for capscale.

Function inertcomp decomposes the inertia into partial, constrained and unconstrained components for each site or species. Legendre & De Cáceres (2012) called these inertia components as local contributions to beta-diversity (LCBD) and species contributions to beta-diversity (SCBD), and they give these as relative contributions summing up to unity (argument unity = TRUE). For this interpretation, appropriate dissimilarity measures should be used in dbrda or appropriate standardization in rda (Legendre & De Cáceres 2012). The function is not implemented for capscale.

Function spenvcor finds the so-called “species -- environment correlation” or (weighted) correlation of weighted average scores and linear combination scores. This is a bad measure of goodness of ordination, because it is sensitive to extreme scores (like correlations are), and very sensitive to overfitting or using too many constraints. Better models often have poorer correlations. Function ordispider can show the same graphically.

Function intersetcor finds the so-called “interset correlation” or (weighted) correlation of weighted averages scores and constraints. The defined contrasts are used for factor variables. This is a bad measure since it is a correlation. Further, it focuses on correlations between single contrasts and single axes instead of looking at the multivariate relationship. Fitted vectors (envfit) provide a better alternative. Biplot scores (see scores.cca) are a multivariate alternative for (weighted) correlation between linear combination scores and constraints.

Function vif.cca gives the variance inflation factors for each constraint or contrast in factor constraints. In partial ordination, conditioning variables are analysed together with constraints. Variance inflation is a diagnostic tool to identify useless constraints. A common rule is that values over 10 indicate redundant constraints. If later constraints are complete linear combinations of conditions or previous constraints, they will be completely removed from the estimation, and no biplot scores or centroids are calculated for these aliased constraints. A note will be printed with default output if there are aliased constraints. Function alias will give the linear coefficients defining the aliased constraints, or only their names with argument names.only = TRUE.

Value

The functions return matrices or vectors as is appropriate.

References

Greenacre, M. J. (1984). Theory and applications of correspondence analysis. Academic Press, London.

Gross, J. (2003). Variance inflation factors. R News 3(1), 13--15.

Legendre, P. & De Cáceres, M. (2012). Beta diversity as the variance of community data: dissimilarity coefficients and partitioning. Ecology Letters 16, 951--963. doi:10.1111/ele.12141

Author

Jari Oksanen. The vif.cca relies heavily on the code by W. N. Venables. alias.cca is a simplified version of alias.lm.

Examples

data(dune)
data(dune.env)
mod <- cca(dune ~ A1 + Management + Condition(Moisture), data=dune.env)
goodness(mod, addprevious = TRUE)
#>                CCA1      CCA2      CCA3      CCA4
#> Achimill 0.36630013 0.3822685 0.3838616 0.4934158
#> Agrostol 0.67247051 0.6724758 0.6779597 0.7773267
#> Airaprae 0.36213737 0.3698100 0.3816619 0.3908018
#> Alopgeni 0.61547145 0.6966105 0.7042650 0.7212918
#> Anthodor 0.24619147 0.2795001 0.3509172 0.3609709
#> Bellpere 0.41185412 0.4179432 0.4847618 0.4849622
#> Bromhord 0.33487622 0.3397416 0.3870032 0.5505037
#> Chenalbu 0.23594716 0.2684323 0.2828928 0.2885321
#> Cirsarve 0.29041563 0.3013655 0.3080671 0.3591280
#> Comapalu 0.16338257 0.6836790 0.7390659 0.7963425
#> Eleopalu 0.55132024 0.6099415 0.6193301 0.6259818
#> Elymrepe 0.25239595 0.2710266 0.2761491 0.2882666
#> Empenigr 0.27089495 0.3132399 0.3153052 0.3154203
#> Hyporadi 0.31349648 0.3371809 0.3387669 0.3388716
#> Juncarti 0.43923609 0.4492937 0.4871043 0.5224072
#> Juncbufo 0.70439967 0.7226263 0.7228786 0.7257471
#> Lolipere 0.48141171 0.5720410 0.5727299 0.6034007
#> Planlanc 0.54969676 0.6084389 0.6802195 0.6826265
#> Poaprat  0.40267189 0.4944813 0.5014516 0.5326546
#> Poatriv  0.49694972 0.5409439 0.5468830 0.5594817
#> Ranuflam 0.68677962 0.6983001 0.7020461 0.7064850
#> Rumeacet 0.44788204 0.5211145 0.7673956 0.7691199
#> Sagiproc 0.27039747 0.3497634 0.3553109 0.3613746
#> Salirepe 0.64788354 0.7264891 0.7276110 0.7639711
#> Scorautu 0.54312496 0.5510319 0.6078931 0.6140593
#> Trifprat 0.37328840 0.4101104 0.6624199 0.6625703
#> Trifrepe 0.03048149 0.2115857 0.3300132 0.4207437
#> Vicilath 0.17824132 0.1784611 0.3762406 0.4279428
#> Bracruta 0.15585567 0.1641095 0.1672797 0.2449864
#> Callcusp 0.30771429 0.3143582 0.3308502 0.3518027
goodness(mod, addprevious = TRUE, summ = TRUE)
#>  Achimill  Agrostol  Airaprae  Alopgeni  Anthodor  Bellpere  Bromhord  Chenalbu 
#> 0.4934158 0.7773267 0.3908018 0.7212918 0.3609709 0.4849622 0.5505037 0.2885321 
#>  Cirsarve  Comapalu  Eleopalu  Elymrepe  Empenigr  Hyporadi  Juncarti  Juncbufo 
#> 0.3591280 0.7963425 0.6259818 0.2882666 0.3154203 0.3388716 0.5224072 0.7257471 
#>  Lolipere  Planlanc   Poaprat   Poatriv  Ranuflam  Rumeacet  Sagiproc  Salirepe 
#> 0.6034007 0.6826265 0.5326546 0.5594817 0.7064850 0.7691199 0.3613746 0.7639711 
#>  Scorautu  Trifprat  Trifrepe  Vicilath  Bracruta  Callcusp 
#> 0.6140593 0.6625703 0.4207437 0.4279428 0.2449864 0.3518027 
# Inertia components
inertcomp(mod, prop = TRUE)
#>                pCCA        CCA        CA
#> Achimill 0.34271900 0.15069678 0.5065842
#> Agrostol 0.55602406 0.22130269 0.2226733
#> Airaprae 0.06404726 0.32675457 0.6091982
#> Alopgeni 0.34238968 0.37890210 0.2787082
#> Anthodor 0.10259139 0.25837947 0.6390291
#> Bellpere 0.40972447 0.07523776 0.5150378
#> Bromhord 0.33046684 0.22003683 0.4494963
#> Chenalbu 0.11064346 0.17788865 0.7114679
#> Cirsarve 0.26649913 0.09262886 0.6408720
#> Comapalu 0.16096277 0.63537969 0.2036575
#> Eleopalu 0.53954819 0.08643366 0.3740182
#> Elymrepe 0.22234322 0.06592337 0.7117334
#> Empenigr 0.10361994 0.21180040 0.6845797
#> Hyporadi 0.03889627 0.29997533 0.6611284
#> Juncarti 0.43439190 0.08801527 0.4775928
#> Juncbufo 0.66622672 0.05952038 0.2742529
#> Lolipere 0.46273045 0.14067027 0.3965993
#> Planlanc 0.51993753 0.16268893 0.3173735
#> Poaprat  0.39408053 0.13857406 0.4673454
#> Poatriv  0.05598349 0.50349824 0.4405183
#> Ranuflam 0.68509904 0.02138594 0.2935150
#> Rumeacet 0.40125987 0.36786003 0.2308801
#> Sagiproc 0.26050435 0.10087025 0.6386254
#> Salirepe 0.12527838 0.63869277 0.2360289
#> Scorautu 0.10895437 0.50510492 0.3859407
#> Trifprat 0.34544815 0.31712212 0.3374297
#> Trifrepe 0.02132183 0.39942191 0.5792563
#> Vicilath 0.12125433 0.30668844 0.5720572
#> Bracruta 0.07222706 0.17275938 0.7550136
#> Callcusp 0.29447422 0.05732850 0.6481973
inertcomp(mod)
#>                  pCCA         CCA         CA
#> Achimill 0.0173766015 0.007640656 0.02568493
#> Agrostol 0.0456558521 0.018171449 0.01828399
#> Airaprae 0.0066672285 0.034014687 0.06341666
#> Alopgeni 0.0325977567 0.036073980 0.02653486
#> Anthodor 0.0096274015 0.024246897 0.05996790
#> Bellpere 0.0154640710 0.002839669 0.01943887
#> Bromhord 0.0180126793 0.011993496 0.02450059
#> Chenalbu 0.0031913088 0.005130874 0.02052099
#> Cirsarve 0.0110663060 0.003846389 0.02661204
#> Comapalu 0.0127652351 0.050389111 0.01615116
#> Eleopalu 0.0797827194 0.012780901 0.05530588
#> Elymrepe 0.0193932154 0.005749967 0.06207879
#> Empenigr 0.0063826176 0.013046147 0.04216766
#> Hyporadi 0.0046669914 0.035992710 0.07932587
#> Juncarti 0.0359126341 0.007276518 0.03948420
#> Juncbufo 0.0494087668 0.004414156 0.02033917
#> Lolipere 0.0368344271 0.011197683 0.03157023
#> Planlanc 0.0366139947 0.011456552 0.02234944
#> Poaprat  0.0142991623 0.005028142 0.01695757
#> Poatriv  0.0028845344 0.025942611 0.02269759
#> Ranuflam 0.0446783229 0.001394671 0.01914141
#> Rumeacet 0.0288221948 0.026423110 0.01658394
#> Sagiproc 0.0151161507 0.005853146 0.03705718
#> Salirepe 0.0142756439 0.072779924 0.02689581
#> Scorautu 0.0030643984 0.014206339 0.01085478
#> Trifprat 0.0228613139 0.020986733 0.02233067
#> Trifrepe 0.0008339368 0.015622139 0.02265580
#> Vicilath 0.0049088357 0.012415912 0.02315905
#> Bracruta 0.0032317812 0.007730074 0.03378289
#> Callcusp 0.0319130878 0.006212868 0.07024716
# vif.cca
vif.cca(mod)
#>   Moisture.L   Moisture.Q   Moisture.C           A1 ManagementHF ManagementNM 
#>     1.504327     1.284489     1.347660     1.367328     2.238653     2.570972 
#> ManagementSF 
#>     2.424444 
# Aliased constraints
mod <- cca(dune ~ ., dune.env)
mod
#> Call: cca(formula = dune ~ A1 + Moisture + Management + Use + Manure,
#> data = dune.env)
#> 
#>               Inertia Proportion Rank
#> Total          2.1153     1.0000     
#> Constrained    1.5032     0.7106   12
#> Unconstrained  0.6121     0.2894    7
#> Inertia is scaled Chi-square 
#> Some constraints or conditions were aliased because they were redundant
#> 
#> Eigenvalues for constrained axes:
#>   CCA1   CCA2   CCA3   CCA4   CCA5   CCA6   CCA7   CCA8   CCA9  CCA10  CCA11 
#> 0.4671 0.3410 0.1761 0.1532 0.0953 0.0703 0.0589 0.0499 0.0318 0.0260 0.0228 
#>  CCA12 
#> 0.0108 
#> 
#> Eigenvalues for unconstrained axes:
#>     CA1     CA2     CA3     CA4     CA5     CA6     CA7 
#> 0.27237 0.10876 0.08975 0.06305 0.03489 0.02529 0.01798 
#> 
vif.cca(mod)
#>           A1   Moisture.L   Moisture.Q   Moisture.C ManagementHF ManagementNM 
#>     2.208249     2.858927     3.072715     3.587087     6.608315   142.359372 
#> ManagementSF        Use.L        Use.Q     Manure.L     Manure.Q     Manure.C 
#>    12.862713     2.642718     3.007238    80.828330    49.294455    21.433337 
#>     Manure^4 
#>           NA 
alias(mod)
#> Model :
#> dune ~ A1 + Moisture + Management + Use + Manure
#> 
#> Complete :
#>          A1 Moisture.L Moisture.Q Moisture.C ManagementHF ManagementNM
#> Manure^4                                                   8.366600   
#>          ManagementSF Use.L Use.Q Manure.L  Manure.Q  Manure.C 
#> Manure^4                           5.291503 -4.472136  2.645751
#> 
with(dune.env, table(Management, Manure))
#>           Manure
#> Management 0 1 2 3 4
#>         BF 0 2 1 0 0
#>         HF 0 1 2 2 0
#>         NM 6 0 0 0 0
#>         SF 0 0 1 2 3
# The standard correlations (not recommended)
## IGNORE_RDIFF_BEGIN
spenvcor(mod)
#>      CCA1      CCA2      CCA3      CCA4      CCA5      CCA6      CCA7      CCA8 
#> 0.9636709 0.9487249 0.9330741 0.8734876 0.9373716 0.8362687 0.9748793 0.8392720 
#>      CCA9     CCA10     CCA11     CCA12 
#> 0.8748741 0.6087512 0.6633248 0.7581210 
intersetcor(mod)
#>                    CCA1        CCA2        CCA3         CCA4        CCA5
#> A1           -0.5332506  0.13691202 -0.47996401 -0.259859587 -0.09894964
#> Moisture.L   -0.8785505  0.17867589  0.03714134  0.181952935 -0.09826534
#> Moisture.Q   -0.1956664 -0.33044917 -0.27321286 -0.180333890  0.26609291
#> Moisture.C   -0.2023782 -0.09698397  0.28596824 -0.261712720 -0.49103002
#> ManagementHF  0.3473460  0.01680324 -0.51205769  0.194144965  0.30752664
#> ManagementNM -0.5699549 -0.61111645  0.14751127 -0.013777789  0.04571982
#> ManagementSF -0.1197499  0.64084416  0.19780650  0.134892908 -0.09679992
#> Use.L        -0.1871999  0.32990444 -0.30941161 -0.372747011  0.09586963
#> Use.Q        -0.1820298 -0.48874152 -0.01997442 -0.009812946  0.04812588
#> Manure.L      0.3175126  0.65945634  0.03724864 -0.025383543 -0.04077470
#> Manure.Q     -0.4075615 -0.21149073  0.49297244 -0.176686201  0.11973190
#> Manure.C      0.4676279  0.11376054  0.29132473 -0.173382982  0.14219924
#> Manure^4      0.2222349 -0.12789494 -0.12921227  0.108367170 -0.02559567
#>                     CCA6        CCA7        CCA8        CCA9         CCA10
#> A1           -0.15225816  0.25788462  0.19247720 -0.27694466 -0.1158449480
#> Moisture.L   -0.02923342  0.07858647 -0.10772510  0.07101300  0.0952517164
#> Moisture.Q   -0.11211675  0.05062810 -0.48302647  0.06138704 -0.2053304965
#> Moisture.C   -0.23581275 -0.38693407 -0.10144580 -0.21907160  0.1875632770
#> ManagementHF -0.24278705  0.16364055 -0.14053438  0.31066725  0.1310215145
#> ManagementNM -0.06430101  0.23917584  0.14375754 -0.27103732  0.0002768613
#> ManagementSF -0.01611984 -0.49726250  0.08073472 -0.30235728 -0.1381281272
#> Use.L         0.19127262 -0.44624831 -0.18450714  0.12950951  0.0452826749
#> Use.Q         0.13485545  0.10367354 -0.11020112  0.41245485 -0.0766932005
#> Manure.L     -0.22265819 -0.49627772 -0.16971786 -0.03943343 -0.0045229147
#> Manure.Q     -0.19402211 -0.11937394  0.17611673 -0.44002593  0.0903998202
#> Manure.C      0.14760330  0.07842345  0.37774417  0.10181374  0.1055057288
#> Manure^4     -0.36683782  0.05953330  0.40927409 -0.06054381 -0.1500198368
#>                    CCA11       CCA12
#> A1           -0.03550223 -0.08881387
#> Moisture.L    0.06404776 -0.08587882
#> Moisture.Q   -0.21810558  0.16917878
#> Moisture.C    0.13701079 -0.14260914
#> ManagementHF  0.17283125  0.13296499
#> ManagementNM -0.01358436  0.09533598
#> ManagementSF -0.01468592 -0.06614834
#> Use.L        -0.08584883  0.32559307
#> Use.Q         0.41893616  0.04881247
#> Manure.L      0.02396993  0.13049087
#> Manure.Q      0.12987366  0.07137031
#> Manure.C      0.05176927 -0.41550238
#> Manure^4     -0.41603287  0.01661279
## IGNORE_RDIFF_END