Abstract
Canonical ordination associates two or more data sets in the ordination process itself. Consequently, if one wishes to extract structures of a data set that are related to (or can be interpreted by) another data set, and/or formally test statistical hypotheses about the significance of these relationships, canonical ordination is the way to go. in this chapter, you will learn how to choose among various canonical ordination techniques: asymmetric (RDA, db-RDA, CCA, LDA, PRC and CoCA) and symmetric (CCorA, CoIA and MFA); explore methods devoted to the study of the relationships between species traits and environment; compute them using the correct options and properly interpret the results; apply these techniques to the Doubs River and other data sets; explore particular applications of some canonical ordination methods, for instance variation partitioning and multivariate analysis of variance by RDA; and write your own RDA function.
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Notes
- 1.
For convenience, some programs and functions standardize the X variables at the beginning of the calculation. This does not change the RDA results because standardizing X, or not, does not change the matrix Ŷ of fitted values of the multivariate regression, upon which the RDA statistics, tests of significance, and PCA are computed.
- 2.
Obtained with RsquareAdj( spe.rda).
- 3.
Mathematically, to partial out the effect of a matrix W from a canonical ordination of Y by X, one computes the residuals of a multivariate multiple regression of X on W and uses these residuals as the explanatory variables .
- 4.
Two important conditions are that the species must have been sampled along their whole ecological range and that they display unimodal responses towards their main ecological constraints. These conditions are difficult to test formally, but graphs of species abundances in sites arranged along their scores on the first few CA ordination axes may help visualize their distributions along the main ecological gradients.
- 5.
Note that ade4 has been developed around a general mathematical framework involving entities that will not be described here, called duality diagrams (Escoufier 1987); hence the dudi part of the function names. Readers are invited to consult the original publication to learn more about this framework.
Bibliography
Abdi, H., Williams, L.J., Valentin, D.: Multiple factor analysis: principal component analysis for multitable and multiblock data sets. WIREs Comput Stat. 5, 149–179 (2013)
Anderson, M.J.: Distinguishing direct from indirect effects of grazers in intertidal estuarine assemblages. J. Exp. Mar. Biol. Ecol. 234, 199–218 (1999)
Anderson, M.J.: Distance-based tests for homogeneity of multivariate dispersions. Biometrics. 62, 245–253 (2006)
Beamud, S.G., Diaz, M.M., Baccala, N.B., Pedrozo, F.L.: Analysis of patterns of vertical and temporal distribution of phytoplankton using multifactorial analysis: acidic Lake Caviahue, Patagonia, Argentina. Limnologica. 40, 140–147 (2010)
Benjamini, Y., Hochberg, Y.: Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc B. 57, 289–300 (1995)
Bernier, N., Gillet, F.: Structural relationships among vegetation, soil fauna and humus form in a subalpine forest ecosystem: a Hierarchical Multiple Factor Analysis (HMFA). Pedobiologia. 55, 321–334 (2012)
Blanchet, F.G., Legendre, P., Borcard, D.: Forward selection of explanatory variables. Ecology. 89, 2623–2632 (2008a)
Borcard, D., Legendre, P., Drapeau, P.: Partialling out the spatial component of ecological variation. Ecology. 73, 1045–1055 (1992)
Carlson, M.L., Flagstad, L.A., Gillet, F., Mitchell, E.A.D.: Community development along a proglacial chronosequence: are above-ground and below-ground community structure controlled more by biotic than abiotic factors? J. Ecol. 98, 1084–1095 (2010)
Choler, P.: Consistent shifts in Alpine plant traits along a mesotopographical gradient. Arct. Antarct. Alp. Res. 37, 444–453 (2005)
Dolédec, S., Chessel, D.: Co-inertia analysis: an alternative method to study species-environment relationships. Freshw. Biol. 31, 277–294 (1994)
Dolédec, S., Chessel, D., ter Braak, C.J.F., Champely, S.: Matching species traits to environmental variables: a new three-table ordination method. Environ. Ecol. Stat. 3, 143–166 (1996)
Dray, S., Legendre, P.: Testing the species traits-environment relationships: the fourth-corner problem revisited. Ecology. 89, 3400–3412 (2008)
Dray, S., Pettorelli, N., Chessel, D.: Matching data sets from two different spatial samplings. J. Veg. Sci. 13, 867–874 (2002)
Dray, S., Chessel, D., Thioulouse, J.: Co-inertia analysis and the linking of ecological data tables. Ecology. 84, 3078–3089 (2003)
Dray, S., Choler, P., Doledec, S., Peres-Neto, P.R., Thuiller, W., Pavoine, S., ter Braak, C.J.F.: Combining the fourth-corner and the RLQ methods for assessing trait responses to environmental variation. Ecology. 95, 14–21 (2014)
Escofier, B., Pagès, J.: Multiple factor analysis (AFMULT package). Comput Stat Data Anal. 18, 121–140 (1994)
Escoufier, Y.: The duality diagram: a means of better practical applications. In: Legendre, P., Legendre, L. (eds.) Developments in Numerical Ecology, NATO ASI Series Series, Series G: Ecological Sciences, vol. 14, pp. 139–156. Springer, Berlin (1987)
Ezekiel, M.: Methods of Correlational Analysis. Wiley, New York (1930)
Faith, D.P., Minchin, P.R., Belbin, L.: Compositional dissimilarity as a robust measure of ecological distance. Vegetatio. 69, 57–68 (1987)
Geffen, E., Anderson, M.J., Wayne, R.K.: Climate and habitat barriers to dispersal in the highly mobile gray wolf. Mol. Ecol. 13, 2481–2490 (2004)
Hill, M.O., Smith, A.J.E.: Principal component analysis of taxonomic data with multi-state discrete characters. Taxon. 25, 249–255 (1976)
Holm, S.: A simple sequentially rejective multiple test procedure. Scand. J. Stat. 6, 65–70 (1979)
Josse, J., Pagès, J., Husson, F.: Testing the significance of the RV coefficient. Comput Stat Data Anal. 53, 82–91 (2008)
Laliberté, E., Paquette, A., Legendre, P., Bouchard, A.: Assessing the scale-specific importance of niches and other spatial processes on beta diversity: a case study from a temperate forest. Oecologia. 159, 377–388 (2009)
Lamentowicz, M., Lamentowicz, L., van der Knaap, W.O., Gabka, M., Mitchell, E.A.D.: Contrasting species-environment relationships in communities of testate amoebae, bryophytes and vascular plants along the fen-bog gradient. Microb. Ecol. 59, 499–510 (2010)
Le Dien, S., Pagès, J.: Analyse factorielle multiple hiérarchique. Revue de statistique appliquée. 51, 47–73 (2003)
Lear, G., Anderson, M.J., Smith, J.P., Boxen, K., Lewis, G.D.: Spatial and temporal heterogeneity of the bacterial communities in stream epilithic biofilms. FEMS Microbiol. Ecol. 65, 463–473 (2008)
Legendre, P., Anderson, M.J.: Distance-based redundancy analysis: testing multi-species responses in multi-factorial ecological experiments. Ecol. Monogr. 69, 1–24 (1999)
Legendre, P., Gallagher, E.D.: Ecologically meaningful transformations for ordination of species data. Oecologia. 129, 271–280 (2001)
Legendre, P., Legendre, L.: Numerical Ecology, 3rd English edn. Elsevier, Amsterdam (2012)
Legendre, P., Galzin, R., Harmelin-Vivien, M.L.: Relating behavior to habitat: solutions to the fourth-corner problem. Ecology. 78, 547–562 (1997)
Legendre, P., Oksanen, J., ter Braak, C.J.F.: Testing the significance of canonical axes in redundancy analysis. Methods Ecol. Evol. 2, 269–277 (2011)
McArdle, B.H., Anderson, M.J.: Fitting multivariate models to community data: a comment on distance-based redundancy analysis. Ecology. 82, 290–297 (2001)
McCune, B.: Influence of noisy environmental data on canonical correspondence analysis. Ecology. 78, 2617–2623 (1997)
Miller, J.K.: The sampling distribution and a test for the significance of the bimultivariate redundancy statistic: a Monte Carlo study. Multivar. Behav. Res. 10, 233–244 (1975)
Oksanen, J., Blanchet, F. G., Friendly, M., Kindt, R., Legendre, P., McGlinn, D., Minchin, P. R., O’Hara, R. B., Simpson, G. L., Solymos, P., Stevens, M. H. H., Szoecs, E., Wagner, H. vegan: Community Ecology Package. R package version 2.5-0. (2017)
Peres-Neto, P.R., Legendre, P., Dray, S., Borcard, D.: Variation partitioning of species data matrices: estimation and comparison of fractions. Ecology. 87, 2614–2625 (2006)
Pillai, K.C.S., Hsu, Y.S.: Exact robustness studies of the test of independence based on four multivariate criteria and their distribution problems under violations. Ann. Inst. Stat. Math. 31, 85–101 (1979)
Robert, P., Escoufier, Y.: A unifying tool for linear multivariate statistical methods: the RV-coefficient. Appl. Stat. 25, 257–265 (1976)
ter Braak, C.J.F.: Canonical correspondence analysis: a new eigenvector technique for multivariate direct gradient analysis. Ecology. 67, 1167–1179 (1986)
ter Braak, C.J.F.: The analysis of vegetation-environment relationships by canonical correspondence analysis. Vegetatio. 69, 69–77 (1987)
ter Braak, C.J.F.: Partial canonical correspondence analysis. In: Bock, H.H. (ed.) Classification and Related Methods of Data Analysis, pp. 551–558. North-Holland, Amsterdam (1988)
ter Braak, C.J.F.: Fourth-corner correlation is a score test statistic in a log-linear trait–environment model that is useful in permutation testing. Environ. Ecol. Stat. 24, 219–242 (2017)
ter Braak, C.J.F., Schaffers, A.P.: Co-correspondence analysis: a new ordination method to relate two community compositions. Ecology. 85, 834–846 (2004)
ter Braak, C., Cormont, A., Dray, S.: Improved testing of species traits–environment relationships in the fourth corner problem. Ecology. 93, 1525–1526 (2012)
van den Brink, P.J., ter Braak, C.J.F.: Multivariate analysis of stress in experimental ecosystems by Principal Response Curves and similarity analysis. Aquat. Ecol. 32, 163–178 (1998)
van den Brink, P.J., ter Braak, C.J.F.: Principal response curves: analysis of time-dependent multivariate responses of biological community to stress. Environ. Toxicol. Chem. 18, 138–148 (1999)
Le, S., Josse, J., Husson, F.: FactoMineR: an R package for multivariate Analysis. J Stat Soft. 25, 1–18 (2008).
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Borcard, D., Gillet, F., Legendre, P. (2018). Canonical Ordination. In: Numerical Ecology with R. Use R!. Springer, Cham. https://doi.org/10.1007/978-3-319-71404-2_6
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