STUDYING BETA DIVERSITY: ECOLOGICAL VARIATION PARTITIONING BY MULTIPLE REGRESSION AND CANONICAL ANALYSIS

doi:10.17521/cjpe.2007.0124

Abstract

Abstract:

Aims Beta diversity is the variation in species composition among sites in a geographic region. Beta diversity is a key concept for understanding the functioning of ecosystems, for the conservation of biodiversity, and for ecosystem management. This paper describes how to analyze it from community composition and associated environmental and spatial data tables.

Methods Beta diversity can be studied by computing diversity indices for each site and testing hypotheses about the factors that may explain the variation among sites. Or, one can carry out a direct analysis of the community composition data table over the study sites, as a function of sets of environmental and spatial variables. These analyses are carried out by the statistical method of partitioning the variation of the diversity indices or the community composition data table with respect to environmental and spatial variables. Variation partitioning is briefly described in this paper.

Important findings Variation partitioning is a method of choice for the interpretation of beta diversity using tables of environmental and spatial variables. Beta diversity is an interesting “currency" for ecologists to compare either different sampling areas, or different ecological communities co-occurring in an area. Partitioning must be based upon unbiased estimates of the variation of the community composition data table that is explained by the various tables of explanatory variables. The adjusted coefficient of determination provides such an unbiased estimate in both multiple regression and canonical redundancy analysis. After partitioning, one can test the significance of the fractions of interest and plot maps of the fitted values corresponding to these fractions.

Key words: adjusted coefficient of determination, beta diversity, biodiversity, canonical redundancy analysis, community composition, variation partitioning

Pierre Legendre. STUDYING BETA DIVERSITY: ECOLOGICAL VARIATION PARTITIONING BY MULTIPLE REGRESSION AND CANONICAL ANALYSIS[J]. Chin J Plant Ecol, 2007, 31(5): 976-981.

Add to citation manager EndNote|Ris|BibTeX

URL: https://www.plant-ecology.com/EN/10.17521/cjpe.2007.0124

https://www.plant-ecology.com/EN/Y2007/V31/I5/976

Figures/Tables 2

Fig.1 Venn diagram representing the partition of the variation of a response variable Y or a response matrix Y between two sets of explanatory variables X and W (Legendre, 1993) The rectangle represents 100% of the variation in y or Y. Fraction [b] is the intersection (not the interaction) of the amounts of variation explained by linear models of X and W

Table 1 Method for calculating the adjusted fractions of variation [a] to [d] depicted inFig. 1 (Three multiple regressions or canonical analyses are required)

典范分析 Canonical analyses	计算R² Compute R² (eq. 5)	计算 $R a 2$ 和各方差组分 Compute $R a 2$ (eq. 6) and fractions of variation	显著性检验可行性 Tested for significance
Y~X	R² of Y~X	[a+b]= $R a 2$ of Y~X	可行Yes
Y~W	R² of Y~W	[b+c]= $R a 2$ of Y~W	可行Yes
Y~(X,W)	R² of Y~(X,W)	[a+b+c]= $R a 2$ of Y~(X,W)	可行Yes
		[a]=[a+b]-[b]	可行Yes
		[b]=[a+b]+[b+c]-[a+b+c]	不可行No
		[c]=[b+c]-[b]	可行Yes
		Residuals=[d]=1-[a+b+c]	不可行No

Table 1 Method for calculating the adjusted fractions of variation [a] to [d] depicted inFig. 1 (Three multiple regressions or canonical analyses are required)

典范分析 Canonical analyses	计算R² Compute R² (eq. 5)	计算 $R a 2$ 和各方差组分 Compute $R a 2$ (eq. 6) and fractions of variation	显著性检验可行性 Tested for significance
Y~X	R² of Y~X	[a+b]= $R a 2$ of Y~X	可行Yes
Y~W	R² of Y~W	[b+c]= $R a 2$ of Y~W	可行Yes
Y~(X,W)	R² of Y~(X,W)	[a+b+c]= $R a 2$ of Y~(X,W)	可行Yes
		[a]=[a+b]-[b]	可行Yes
		[b]=[a+b]+[b+c]-[a+b+c]	不可行No
		[c]=[b+c]-[b]	可行Yes
		Residuals=[d]=1-[a+b+c]	不可行No

References 30

[1]	Anderson MJ, Legendre P (1999). An empirical comparison of permutation methods for tests of partial regression coefficients in a linear model. Journal of Statistical Computation and Simulation, 62,271-303.
[2]	Bell G (2001). Neutral macroecology. Science, 293,2413-2418. DOI URL PMID
[3]	Borcard D, Legendre P (1994). Environmental control and spatial structure in ecological communities: an example using oribatid mites (Acari, Oribatei). Environmental and Ecological Statistics, 1,37-53.
[4]	Borcard D, Legendre P (2002). All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices. Ecological Modelling, 153,51-68.
[5]	Borcard D, Legendre P, Avois-Jacquet C, Tuomisto H (2004). Dissecting the spatial structure of ecological data at multiple scales. Ecology, 85,1826-1832.
[6]	Borcard D, Legendre P, Drapeau P (1992). Partialling out the spatial component of ecological variation. Ecology, 73,1045-1055.
[7]	Dray S, Legendre P, Peres-Neto PR (2006). Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM). Ecological Modelling, 196,483-493.
[8]	Ezekiel M (1930). Methods of Correlation Analysis. John Wiley and Sons, New York.
[9]	He F (2005). Deriving a neutral model of species abundance from fundamental mechanisms of population dynamics. Functional Ecology, 19,187-193.
[10]	Hubbell SP (2001). The Unified Neutral Theory of Biodiversity and Biogeography. Princeton University Press, Princeton , New Jersey.
[11]	Legendre P (1990). Quantitative methods and biogeographic analysis. In: Garbary DJ, South RGeds. Evolutionary Biogeography of the Marine Algae of the North Atlantic. NATO ASI Series, Vol. G 22. Springer-Verlag, Berlin,9-34.
[12]	Legendre P (1993). Spatial autocorrelation: trouble or new paradigm? Ecology, 74,1659-1673.
[13]	Legendre P, Borcard D, Peres-Neto PR (2005). Analyzing beta diversity: partitioning the spatial variation of community composition data. Ecological Monographs, 75,435-450.
[14]	Legendre P, Gallagher ED (2001). Ecologically meaningful transformations for ordination of species data. Oecologia, 129,271-280. DOI URL PMID
[15]	Legendre P, Legendre L (1998). Numerical Ecology 2nd English edn. Elsevier Science BV, Amsterdam.
[16]	Manly BJF (1997). Randomization, Bootstrap and Monte Carlo methods in biology 2nd edn. Chapman and Hall, London.
[17]	Miller JK (1975). The sampling distribution and a test for the significance of the bimultivariate redundancy statistic: a Monte Carlo study. Multivariate Behavioral Research, 10,233-244.
[18]	Miller JK, Farr SD (1971). Bimultivariate redundancy: a comprehensive measure of interbattery relationship. Multivariate Behavioral Research, 6,313-324.
[19]	Ohtani K (2000). Bootstrapping R ² and adjusted R ² in regression analysis. Economic Modelling, 17,473-483.
[20]	Oksanen J, Kindt R, Legendre P, O'Hara RB (2007). Vegan: community ecology package version 1.8-5. URL http://cran.r-project.org/.
[21]	Peres-Neto PR, Legendre P, Dray S, Borcard D (2006). Variation partitioning of species data matrices: estimation and comparison of fractions. Ecology, 87,2614-2625. URL PMID
[22]	Rao CR (1964). The use and interpretation of principal component analysis in applied research. Sankhyaá, Series A, Indian Journal of Statistics, 26,329-358.
[23]	R Development Core Team (2007). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org.
[24]	ter Braak CJF (1986). Canonical correspondence analysis: a new eigenvector technique for multivariate direct gradient analysis. Ecology, 67,1167-1179.
[25]	ter Braak CJF (1987a). The analysis of vegetation-environment relationships by canonical correspondence analysis. Vegetatio, 69,69-77.
[26]	ter Braak CJF (1987b). Ordination. In: Jongman RHG, ter Braak CJF, van Tongeren OFReds. Data Analysis in Community and Landscape Ecology. Pudoc, Wageningen (reissued in 1995 by Cambridge University Press, Cambridge),91-173.
[27]	ter Braak CJF (1988). CANOCO — a FORTRAN program for Canonical Community Ordination by [partial] [detrended] [canonical] Correspondence Analysis, Principal Component Analysis and Redundancy Analysis (version 2.1). Agricultural Mathematics Group, Ministry of Agriculture and Fisheries, Wageningen.
[28]	ter Braak CJF, Smilauer P (2002). Canoco Reference Manual and CanoDraw for Windows User's Guide: Software for Canonical Community Ordination (version 4.5). Microcomputer Power, Ithaca, New York.
[29]	Whittaker RH (1960). Vegetation of the Siskiyou mountains, Oregon and California. Ecological Monographs, 30,279-338.
[30]	Whittaker RH (1972). Evolution and measurement of species diversity. Taxon, 21,213-251.