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CWM and Rao’s quadratic diversity: a unified framework for functional ecology


Assessing the effects of environmental constraints on community structure often relies on methods that consider changes in species functional traits in response to environmental processes. Various indices have been proposed to measure relevant aspects of community trait composition from different viewpoints and perspectives. Among these, the ‘community-weighted mean trait value’ (CWM) and the Rao coefficient have been widely used in ecological research for summarizing different facets of functional composition and diversity. Analyzing changes in functional diversity of bee communities along a post-fire successional gradient in southern Switzerland we show that these two measures may be used to describe two complementary aspects of community structure, such as the mean and the dispersion of functional traits within a given species assemblage. While CWM can be adequately used to summarize shifts in mean trait values within communities due to environmental selection for certain functional traits, the Rao coefficient can be effectively applied to analyze patterns of trait convergence or divergence compared to a random expectation.

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  1. Ackerly D, Cornwell W (2007) A trait-based approach to community assembly: partitioning of species trait values into within- and among-community components. Ecol Lett 10:135–145

  2. Agresti A (2007) An introduction to categorical data analysis. Wiley, New York

  3. Anderson MJ (2001) A new method for non-parametric multivariate analysis of variance. Aust Ecol 26:32–46

  4. Anderson MJ (2006) Distance-based tests for homogeneity of multivariate dispersions. Biometrics 62:245–253

  5. Biesmeijer JC, Roberts SP, Reemer M, Ohlemueller R, Edwards M, Peeters T, Schaffers A, Potts SG, Kleukers R, Thomas CD, Settele J, Kunin WE (2006) Parallel declines in pollinators and insect-pollinated plants in Britain and The Netherlands. Science 313:351–354

  6. Botta-Dukat Z (2005) Rao’s quadratic entropy as a measure of functional diversity based on multiple traits. J Veg Sci 16:533–540

  7. Champely S, Chessel D (2002) Measuring biological diversity using Euclidean metrics. Environ Ecol Stat 9:167–177

  8. Chevenet F, Dolédec S, Chessel D (1994) A fuzzy coding approach for the analysis of ecological data. Freshw Biol 31:295–309

  9. Clarke KR, Warwick RM (2001) A further biodiversity index applicable to species lists: variation in taxonomic distinctness. Mar Ecol Prog Ser 216:265–278

  10. Díaz S, Lavorel S, de Bello F, Quetier F, Grigulis K, Robson TM (2007) Incorporating plant functional diversity effects in ecosystem service assessments. Proc Natl Acad Sci USA 104:20684–20689

  11. Fontaine C, Dajoz I, Meriguet J, Loreau M (2006) Functional diversity of plant–pollinator interaction webs enhances the persistence of plant communities. PLoS Biol 4:1–7

  12. Garnier E, Cortez J, Billes G, Navas ML, Roumet C, Debussche M, Laurent G, Blanchard A, Aubry D, Bellmann A, Neill C, Toussaint JP (2004) Plant functional markers capture ecosystem properties during secondary succession. Ecology 85:2630–2637

  13. Gower JC, Legendre P (1986) Metric and Euclidean properties of dissimilarity coefficients. J Classif 3:5–48

  14. Grime JP (1998) Benefits of plant diversity to ecosystems: immediate, filter and founder effects. J Ecol 86:902–910

  15. Grime JP (2006) Trait convergence and trait divergence in herbaceous plant communities: mechanisms and consequences. J Veg Sci 17:255–260

  16. Hooper DU, Chapin FS III, Ewel JJ, Hector A, Inchausti P, Lavorel S, Lawton JH, Lodge D, Loreau M, Naeem S, Schmid B, Setälä H, Symstad AJ, Vandermeer J, Wardle DA (2005) Effects of biodiversity on ecosystem functioning: a consensus of current knowledge and needs for future research. Ecol Monogr 75:3–35

  17. Izsák J, Papp L (2000) A link between ecological diversity indices and measures of biodiversity. Ecol Model 130:151–156

  18. Laliberté E, Legendre P (2010) A distance-based framework for measuring functional diversity from multiple traits. Ecology 91:299–305

  19. Lavorel S, Grigulis K, McIntyre S, Garden D, Williams N, Dorrough J, Berman S, Quétier F, Thébault A, Bonis A (2008) Assessing functional diversity in the field-methodology matters! Funct Ecol 22:134–147

  20. Lepš J, de Bello F, Lavorel S, Berman S (2006) Quantifying and interpreting functional diversity of natural communities: practical considerations matter. Preslia 78:481–501

  21. Mason NWH, Mouillot D, Lee WG, Wilson JB (2005) Functional richness, functional evenness and functional divergence: the primary components of functional diversity. Oikos 111:112–118

  22. Michener CD (2000) The bees of the world. John Hopkins University Press, Baltimore

  23. Mokany K, Ash J, Roxburgh S (2008) Functional identity is more important than diversity in influencing ecosystem processes in a temperate native grassland. J Ecol 96:884–893

  24. Moretti M, Legg C (2009) Combining plant and animal traits to assess community functional responses to disturbance. Ecography 32:299–309

  25. Moretti M, Conedera M, Duelli P, Edwards PJ (2002) The effect of wildfire on ground-active spiders in deciduous forests on the Swiss southern slopes of the Alps. J Appl Ecol 39:321–336

  26. Moretti M, Obrist KM, Duelli P (2004) Arthropod biodiversity after forest fires: winners and losers in the winter fire regime of the southern Alps. Ecography 27:173–186

  27. Moretti M, Duelli P, Obrist MK (2006) Biodiversity and resilience of arthropod communities after fire disturbance in temperate forests. Oecologia 149:312–327

  28. Moretti M, de Bello F, Roberts SPM, Potts SG (2009) Taxonomical vs. functional responses of bee communities to fire in two contrasting climatic regions. J Anim Ecol 78:98–108

  29. Mouchet MA, Villéger S, Mason NWH, Mouillot D (2010) Functional diversity measures: an overview of their redundancy and their ability to discriminate community assembly rules. Funct Ecol 24:867–876

  30. Norberg J (2004) Biodiversity and ecosystem functioning: a complex adaptive systems approach. Limnol Oceanogr 49:1269–1277

  31. Pavoine S, Dolédec S (2005) The apportionment of quadratic entropy: a useful alternative for partitioning diversity in ecological data. Environ Ecol Stat 12:125–138

  32. Pavoine S, Ollier S, Pontier D (2005) Measuring diversity from dissimilarities with Rao’s quadratic entropy: are any dissimilarities suitable? Theor Popul Biol 67:231–239

  33. Pavoine S, Vallet J, Dufour A-B, Gachet S, Daniel H (2009) On the challenge of treating various types of variables: application for improving the measurement of functional diversity. Oikos 118:391–402

  34. Petchey OL, Gaston KJ (2002) Functional diversity (FD), species richness, and community composition. Ecol Lett 5:402–411

  35. Petchey OL, Gaston KJ (2006) Functional diversity: back to basics and looking forward. Ecol Lett 9:74–758

  36. Petchey OL, Hector A, Gaston KJ (2004) How do different measures of functional diversity perform? Ecology 85:847–857

  37. Petchey OL, Evans KL, Fishburn IS, Gaston KJ (2007) Low functional diversity and no redundancy in British avian assemblages. J Anim Ecol 76:977–985

  38. Pillar VD, Duarte LdS (2010) A framework for metacommunity analysis of phylogenetic structure. Ecol Lett 13:587–596

  39. Pillar VD, Sosinski EE (2003) An improved method for searching plant functional types by numerical analysis. J Veg Sci 14:323–332

  40. Pillar VD, Duarte LdS, Sosinski EE, Joner F (2009) Discriminating trait-convergence and trait-divergence assembly patterns in ecological community gradients. J Veg Sci 20:334–348

  41. Podani J (2005) Multivariate exploratory analysis of ordinal data in ecology: pitfalls, problems and solutions. J Veg Sci 16:497–510

  42. Poos MS, Walker SC, Jackson DA (2009) Functional-diversity indices can be driven by methodological choices and species richness. Ecology 90:341–347

  43. Rao CR (1982) Diversity and dissimilarity coefficients—a unified approach. Theor Popul Biol 21:24–43

  44. Rao CR (2010) Quadratic entropy and analysis of diversity. Sankhya 72:70–80

  45. Reynolds HT (1984) Analysis of nominal data. Sage Publications, Beverly Hills

  46. Ricotta C (2005) Through the jungle of biological diversity. Acta Biotheor 53:29–38

  47. Ricotta C, Moretti M (2010) Assessing the functional turnover of species assemblages with tailored dissimilarity matrices. Oikos 119:1089–1098

  48. Ricotta C, Szeidl L (2006) Towards a unifying approach to diversity measures: bridging the gap between the Shannon entropy and Rao’s quadratic index. Theor Popul Biol 70:237–243

  49. Ricotta C, Szeidl L (2009) Diversity partitioning of Rao’s quadratic entropy. Theor Popul Biol 76:299–302

  50. Schleuter D, Daufresne M, Massol F, Argillier C (2010) A user’s guide to functional diversity indices. Ecol Monogr 80:469–484

  51. Schmera D, Podani J, Erös T (2009) Measuring the contribution of community members to functional diversity. Oikos 118:961–971

  52. Schumacher J, Roscher C (2009) Differential effects of functional traits on aboveground biomass in semi-natural grasslands. Oikos 118:1659–1668

  53. Shimatani K (2001) On the measurement of species diversity incorporating species differences. Oikos 93:135–147

  54. Thompson K, Petchey OL, Askew AP, Dunnett NP, Beckerman AP, Willis AJ (2010) Little evidence for limiting similarity in a long-term study of a roadside plant community. J Ecol 98:480–487

  55. Vandewalle M, de Bello F, Berg MP, Bolger T, Dolédec S, Dubs F, Feld CK, Harrington R, Harrison PA, Lavorel S, da Silva PM, Moretti M, Niemela J, Santos P, Sattler T, Sousa JP, Sykes MT, Vanbergen AJ, Woodcock BA (2010) Functional traits as indicators of biodiversity response to land use changes across ecosystems and organisms. Biodivers Conserv 19:2921–2947

  56. Vile D, Shipley B, Garnier E (2006) Ecosystem productivity relates to species’ potential relative growth rate: a field test and a conceptual framework. Ecol Lett 9:1061–1067

  57. Villéger S, Mason NWH, Mouillot D (2008) New multidimensional functional diversity indices for a multifaceted framework in functional ecology. Ecology 89:2290–2301

  58. Waldbusser GG, Marinelli RL, Whitlatch RB, Visscher PT (2004) The effects of infaunal biodiversity on biogeochemistry of coastal marine sediments. Limnol Oceanogr 49:1482–1492

  59. Walker B, Kinzig A, Langridge J (1999) Plant attribute diversity, resilience, and ecosystem function: the nature and significance of dominant and minor species. Ecosystems 2:95–113

  60. Weiher E, Keddy PA (1999) Relative abundance and evenness patterns along diversity and biomass gradients. Oikos 87:355–361

  61. Westrich P (1989) Die Wildbienen Baden-Württembergs. Ulmer, Stuttgart

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We thank Andreas Mueller (Swiss Federal Institute of Technology, Zurich) and Stuart P.M. Roberts (University of Reading) for providing bee traits. Insightful comments and suggestions of two anonymous referees on a previous draft of this paper are also warmly acknowledged.

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Correspondence to Carlo Ricotta.

Additional information

Communicated by Roland Brandl.

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Ricotta, C., Moretti, M. CWM and Rao’s quadratic diversity: a unified framework for functional ecology. Oecologia 167, 181–188 (2011).

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  • Ecological gradient
  • Environmental filters
  • Functional convergence/divergence
  • Niche space
  • Trait dispersion
  • Trait mean