A user's guide to functional diversity indices
Contents
1. Here we chose a simple form for f functions that uses information on both mean trait values and trait variance covariance matrices for each species FRy is then obtained as the integral of the maximum of all f functions integrated over studied trait space For further details see Appendix B A script in Mathematica to calculate this index is available online see footnote 2 Functional evenness Functional evenness indices measure whether mean species traits are distributed regularly within the occupied trait space i e with equal distances between nearest neighbors and equal abun dances a high FE index usually means a very regular distribution a low FE index indicates the existence of separate clouds of species and or abundances Func tional evenness indices are generally used to indicate under or overutilization of resources and thus again productivity reliability and vulnerability to invasion Mason et al 2005 This index group includes species abundances in its calculation 1 One dimensional index Based on Bulla s index for the measurement of species evenness Bulla 1994 the FE index Mouillot et al 2005 measures for each trait separately how evenly the trait values of all species present are distributed Table 1 IN 2 1 First the absolute relative distances between the mean species trait values in order of increasing values is calculated and weighted by the sum of the relative species abunda2. Journal of Fish Biology 64 970 983 Escofier B and J Pages 1994 Multiple factor analysis afmult package Computational Statistics and Data Anal ysis 18 121 140 FUNCTIONAL DIVERSITY INDICES 483 Gotelli N J and G R Graves 1996 Null models in ecology Smithsonian Institution Press Washington D C USA Heemsbergen D A M P Berg M Loreau J R Van Hal J H Faber and H A Verhoef 2004 Biodiversity effects on soil processes explained by interspecific functional dissimi larity Science 306 1019 1020 Hill M O and A J E Smith 1976 Principal component analysis of taxonomic data with multi state discrete charac ters Taxon 25 249 255 Hulot F D G Lacroix F O Lescher Moutoue and M Loreau 2000 Functional diversity governs ecosystem re sponse to nutrient enrichment Nature 405 340 344 Kader G D and M Perry 2007 Variability for categorical variables Journal of Statistics Education 15 2 Lande R 1996 Statistics and partitioning of species diversity and similarity among multiple communities Oikos 76 5 13 Lavit C Y Escoufier R Sabatier and P Traissac 1994 The ACT STATIS method Computational Statistics and Data Analysis 18 97 119 Lebart L A Morineau and M Piron 2000 Statistique exploratoire multidimensionnelle Third edition Dunod Paris France Lep J F de Bello S Lavorel and S Berman 2006 Quantifying and interpreting functional diversity o
3. initial community to 2 25 and 250 individuals to test for the influence of the change in the abundance of one species Fig 2 T3 In T4 the influence of the position in trait space of a dominating species 25 individuals was tested Fig 2 T4 In T5 we increased the distance between two dominating species 25 individuals each starting from an initial community in which the 476 abundance of only one species was set to 50 individuals to simulate two identical species All indices were calculated for the initial community and for the different scenarios The change in index values of the scenarios compared to the initial commu nity was evaluated semi quantitatively We used the symbols and to indicate whether the calculated value was higher lower or equal respective ly compared to the initial community The symbols were replicated to give a raw quantitative view of the changes with lt lt 4 with lt lt Indices correlation The correlations among the different indices were tested using random communities We computed 1000 randomizations for 14 different species richness levels as multiples of five from five to 70 and three different numbers of traits three five 10 The total number of treatments was 42 14 X 3 Trait values were generated using uniform distributions between 0 and 1 The intraspecific standard deviation was considered to be 10 o
4. This evaluates the match between the species richness specific tables and the reference structure by using synthetic auxiliary variables Categorical variables All the indices proposed so far except those based on a distance matrix cannot be computed for categorical variables To address this problem it is recommended in the literature to transform the data set from categorical to continuous variables via distance matrices and ordination methods which allow a mix of continuous and categorical variables as inputs Vill ger et al 2008 proposed calculating the Gower distance and then computing a principal coordinate analysis PCoA The trait values thus transformed principal coordinate axis can then be used to calculate the different functional diversity indices Next to using the Gower distance with a PCoA Hill and Smith s method can also be used Hill and Smith 1976 This ordination technique combines discrete and continuous variables in a single analysis If all variables are discrete it is reduced to a simple correspondence analysis Although it is not really correct to transform categorical traits to continuous ones for the use of indices designed for real valued variables we consider these methods because they are commonly applied Since continuous trait based indices cannot be calculat ed for categorical traits it is impossible to assess the true effect of the transformation on the outputs i e to compare the outputs f
5. consumer s guide to evenness indices Oikos 76 70 82 Stevens R D S B Cox R E Strauss and M R Willig 2003 Patterns of functional diversity across an extensive environmental gradient vertebrate consumers hidden treat ments and latitudinal trends Ecology Letters 6 1099 1108 Ecological Monographs Vol 80 No 3 Tilman D J Knops D Wedin P Reich M Ritchie and E Siemann 1997 The influence of functional diversity and composition on ecosystem processes Science 277 1300 1302 Villeger S N W H Mason and D Mouillot 2008 New multidimensional functional diversity indices for a multifac eted framework in functional ecology Ecology 89 2290 2301 Walker B A Kinzig and J Langridge 1999 Plant attribute diversity resilience and ecosystem function the nature and significance of dominant and minor species Ecosystems 2 95 113 Washington H 1984 Diversity biotic and similarity indices a review with special relevance to aquatic ecosystems Water Research 18 653 694 Whittaker R H 1972 Evolution and measurement of species diversity Taxon 21 213 251 APPENDIX A One dimensional functional richness FRy Ecological Archives M080 016 A1 APPENDIX B Multidimensional functional richness FR m Ecological Archives M080 016 A2 APPENDIX C One dimensional functional divergence FD Ecological Archives M080 016 A3
6. criterion is met by the two FE indices but not by all the FR and FD indices FRp and FRy both correlated with FDyar and FRy also correlated with FD and FDg Table 4 This strongly suggests that FDya FRp and FRy rather measure a mixture of functional richness and divergence Ricotta 2005 gave several other criteria for an index of functional diversity which are only applicable to FR indices For instance an index should not decrease when a species is added the monotonicity criterion or should not increase when exactly the same species is added the twinning criterion These two criteria were also tested in August 2010 FUNCTIONAL DIVERSITY INDICES 481 TABLE 4 Indices of functional diversity tested in this study their properties correlation with other indices and their disadvantages Use Cor Cor with indices A with cat with from other Index Source Description MD incl variables SR categories Disadvantage IN Functional richness FRR Mason etal functional range no no yes yes FD no consideration 1 1 1 2 cat 2005 of gaps one dimensional FR this study individual s no no yes yes one dimensional 1 2 1 2 cat functional intra specific range trait variation needed FRy Vill ger et al functional yes no no yes FD FD no consideration 1 3 2008 volume FDe of gaps SR has to exceed N traits FRp Petchey and sum of branch yes no yes yes difficult to 1 4 Gaston length of interpret long 2002 clas
7. five and 10 traits respectively In addition the cos confirms that the different reference structures reflect the structures of the individual species richness specific tables cos ranging from 0 905 to 0 944 from 0 920 to 0 940 and from 0 924 to 0 941 for three five and 10 traits respectively The lowest cos values are systematically found for the lowest levels of species richness five followed by 10 because of the higher variance of most indices at these species richness levels see Fig 3 The reference structures over the different species richness levels are very similar for all trait levels three five or 10 Fig 4 confirming that there are several independent index groups that describe functional diversity However instead of the expected three axes corresponding to FR FE and FD indices five axes are needed to explain the majority of the variance 74 when FRy is included three traits and four axes without this index 77 and 78 for five and 10 traits respectively The first axis is mainly correlated to the FD indices FD FD FDa and FD Table 3 Fig 4 The fifth FD index FD is also correlated to the first axis but with the second axis as well and it is overall more highly correlated to the FR indices FRpr and FRy than to the other FD indices Next to FD the second axis is correlated mainly to some of the FR indices FRp FRy and partly FR The third axis represent functional evenne
8. multidimensional space Table 2 Only in T1 do FD indices react differently as expected FD and FDy increase when a species outside of the initial community is added whereas FDg FDm and FD decrease Indices correlation Species richness has a clear effect on the different indices either in terms of variance or mean values Fig 3 Nevertheless for all trait levels there is a high similarity between the species richness specific tables Ecological Monographs 478 D SCHLEUTER ET AL Vol 80 No 3 1 0 gece 1 0 ae 0 8 TT LTD 0 8 l oc so cc 0 6 o 04 L 06 rm L 0 4 0 2 0 0 da a E S S 600 0 12 i 0 8 a 400 i A faa aa Lu ie E i 0 6 200 o al r af 04ga bo 2 0 00 0 Oe ae sy pO po a E 1 0 Te E g l w 0 6 o 06 O jili LL L L 0 8 0 2 02 0 4 1 0 0 9 1 0 0 7 l 0 8 lili a tint g 0 5 osso ce a ee ae a 0 4 1 10 30 50 70 10 30 50 70 SR SR Fic 3 Relationship of each of the 12 functional diversity indices with species richness SR 14 levels from 5 to 70 here for a data set with three different species traits There are 1000 index values for each species richness level based on randomized trait and abundance data Trait values and species abundances were generated using uniform distributions between 0 and 1 and between 1 and 100 respectively See Table for explanations of functional diversity indices RV ranging from 0 885 to 0 996 from 0 945 to 0 998 and from 0 962 to 0 998 for three
9. of species diversity and weighs the trait based distances between pairs of species dist s s by the product of their relative abundances Leps et al 2006 suggest calculating the species pairwise dissimilarities through the sum of their D SCHLEUTER ET AL Ecological Monographs Vol 80 No 3 overlaps for each trait In this way individual variability is included in this index and categorical and continuous variables can be mixed Rao s quadratic entropy can also be calculated for single traits the average of which leads to the same result as the multivariate approach if the distance measure used is unchanged by averaging e g Euclidean distance Lep et al 2006 Since Walker et al s 1999 functional attribute diversity differs from Rao s quadratic entropy only in the way the distance matrix is calculated it was not tested as a separate index in this study Villeger et al 2008 proposed a new multivariate measure for functional divergence FD Based on the vertex species V of the convex hull see FR this index first determines the center of gravity G of the convex hull Table 1 IN 3 6 It then computes the abundance weighted deviances Ad of each species present from the species mean distance to the center of gravity In a last step the index is restricted between 0 and 1 The FD index is low when species abundances are close to the center of gravity and high when species and or abundances are hig
10. r with the index calculated with untransformed values ranged from 0 13 to 0 60 depending on species richness and FDg r 0 5 For the one dimensional indices neither of the transformation methods performs well RV ranging between 0 605 and 0 894 for the transformation via 480 TABLE 3 D SCHLEUTER ET AL Ecological Monographs Vol 80 No 3 Axis coordinates of the different functional diversity indices for the first five axes of the reference structure calculated with STATIS Lavit et al 1994 over the different species richness levels here for a data set with three traits Index Axis Axis 2 Richness FRR 0 30 0 72 FRy 0 18 0 41 FRy 0 50 0 56 FRp 0 15 0 2 FRim 0 05 0 11 Evenness FE 0 04 0 05 FE 0 05 0 00 Divergence FD yar 0 54 0 43 Do 0 58 0 36 FD 0 72 0 24 FDo 0 80 0 22 FD 0 63 0 37 Axis 3 Axis 4 Axis 5 0 08 0 06 0 09 0 29 0 15 0 50 0 03 0 24 0 16 0 26 0 19 0 65 0 21 0 76 0 01 0 71 0 21 0 17 0 64 0 26 0 19 0 20 0 35 0 19 0 04 0 05 0 13 0 04 0 10 0 05 0 04 0 08 0 04 0 01 0 01 0 06 Note Values lower than 0 4 or higher than 0 4 are boldface For 1000 communities species richness trait values and species abundances were generated using uniform distributions between 0 and 1 and between 1 and 100 respectively See Table 1 for explanations of indices Gower dis
11. species The size of the dot indicates the abundance of the species small dots reflect one individual and with increasing size 2 25 and 250 individuals T5 Fig 2 to compare the observed and expected changes of index values Trait values were integer numbers between 1 and 5 for trait 1 and between 1 and 8 for trait 2 We assumed a standard deviation of 0 2 and a maximum deviation from the mean equal to 0 4 for all species and traits In all tests the initial species richness was 25 apart from T1 where it was 24 In T1 and T2 and if not stated differently in T3 TS species abundance was set to for all species The main aim of T1 and T2 was to test the effect of empty space in the trait space on the behavior of functional richness indices In T1 we tested the effect of adding one species with varying distance between the added species and the existing community Fig 2 T1 In T2 we removed nine species from the initial community We removed these species so that either one trait value was eliminated completely once at the outer edge of the community and once within the trait space Fig 2 T2 scenarios a and b or the removed species were chosen in the middle of the trait space so that each trait value was present at least once Fig 2 T2 scenario c In T3 T5 the influence of varying species abundances on functional evenness and divergence was tested In T3 we increased the abundance of one species stepwise from 1
12. 005 Petchey and Gaston 2006 Podani and Schmera 2007 Vill ger et al 2008 Two main approaches have emerged on the one hand functional groups can be defined based on few behavioral morphological characteristics e g diet af finities food acquisition methods preferred habitat and the observed species are assigned to different functional categories Bremner et al 2003 Stevens et al 2003 Petchey and Gaston 2006 These data can be further processed with conventional species diversity indices functional group richness Shannon index Simpson diversity index etc e g Stevens et al 2003 This approach is suitable for macro ecological studies since Ecological Monographs 470 D SCHLEUTER ET AL Vol 80 No 3 300r the selection and the treatment of the traits e g how 2507 200 150 100 L eee 1969 1976 1980 1984 1988 1992 1996 2000 2004 2008 Year Number of publications Fic 1 Number of publications containing the term functional diversity in title abstract or key words Source Scopus http www scopus com scopus search form url to 31 December 2008 information on species assignment to functional groups is available for a broad range of species and generally easy to obtain Furthermore such studies only need a low level of detail in contrasting species traits On the other hand functional diversity can be calculated based on specific functional traits measured for e
13. 2005 Ricotta 2005 Petchey and Gaston 2006 Vill ger et al 2008 These indices usually describe two broad aspects of functional diversity 1 how much of the functional niche space is filled by the existing species functional richness and 2 how this space is filled functional evenness functional divergence variance Using functional diversity indices however entails several methodological problems The first difficulty is many and which traits to use how to weigh them and how to combine them Leps et al 2006 Petchey and Gaston 2006 Some solutions to these problems have been discussed and proposed by Lep et al 2006 The second set of problems is related to the indices themselves i e do the indices measure exactly what the user wants to describe Are the chosen indices independent from one another Will diversity be measured for a single trait only or for a multivariate trait data set Does the data set contain categorical and continuous variables It is particularly important that these problems are considered carefully because ecolog ical theories are developed and confirmed based on these results Some properties of selected indices were specified by Petchey and Gaston 2006 and Ricotta 2005 but new indices have been published since then e g Cornwell et al 2006 Podani and Schmera 2007 Vill ger et al 2008 and although the importance of intraspecific specializa tion and variability is clearly acknowled
14. 2006 suggest including intraspecific variability in this index by adding the abundance weighted intraspecific variance to the interspecific variance Since FD can also be understood as the relative range of the trait clustering we propose a new one dimen sional index of functional divergence FD The FD index calculates the range of the zth percentiles e g 25th percentile Q and 75th percentile Q3 relative to the overall range of each trait Table 1 IN 3 4 The species abundance is accounted for by replicating the mean trait value of a species i times the species abundance This approach is conceptually different since it is based on the relative span of the trait cluster and not on the variance Low FD values indicate that half of the individuals in the community occupy a very confined functional space independent of the position of that cluster along the trait axis high values instead suggest that the functional space is more densely occupied at both its edges For further details see Appendix C A script in R to calculate this index is available online see footnote 2 2 Multivariate divergence The most common mul tivariate index of FD is Rao s quadratic entropy FDa Rao 1982 Champely and Chessel 2002 Ricotta 2005 Table 1 IN 3 5 This index calculates the abundance weighted variance of the dissimilarities between all species pairs It is based on the Simpson diversity index Simpson 1949 for the calculation
15. Cabido 2001 Petchey and Gaston 2006 Functional diversity is commonly assumed to be a better predictor of ecosystem produc tivity and vulnerability than species diversity Tilman et al 1997 Hulot et al 2000 Diaz and Cabido 2001 Heemsbergen et al 2004 Including species functions in the measurement of biodiversity is a relatively recent approach Since 1990 the number of publications based on functional diversity Manuscript received 2 December 2008 revised and accepted 2 September 2009 Corresponding Editor J M Levine Present address Limnological Institute University of Konstanz 78457 Konstanz Germany E mail DianaSchleuter web de 469 categorical variables functional divergence functional evenness functional richness has been steadily increasing Fig 1 Although the concept of functional diversity itself is relatively simple to understand its increasing importance in biodiversity studies has revealed that measuring it is a complex endeavor while studies focused on species diversity only need to count individuals from different species i e sort them into several categories functional diversity studies have to describe a multidimensional cloud of points in trait space i e each coordinate corresponds to a measured trait each point representing an individual or a species Several methods have recently been proposed to help identify the necessary measures of functional diversity reviewed in Ricotta 2
16. Ecological Monographs 80 3 2010 pp 469 484 2010 by the Ecological Society of America A user s guide to functional diversity indices D SCHLEUTER M DAUFRESNE F MAssoL AND C ARGILLIER L Institut de Recherche en Sciences et Technologies pour l Environnement CEMAGREF Unit de Recherche Hydrobiologie 3275 Route de C zanne CS 40061 13182 Aix en Provence France Abstract Functional diversity is the diversity of species traits in ecosystems This concept is increasingly used in ecological research yet its formal definition and measurements are currently under discussion As the overall behavior and consistency of functional diversity indices have not been described so far the novice user risks choosing an inaccurate index or a set of redundant indices to represent functional diversity In our study we closely examine functional diversity indices to clarify their accuracy consistency and independence Following current theory we categorize them into functional richness evenness or divergence indices We considered existing indices as well as new indices developed in this study The new indices aimed at remedying the weaknesses of currently used indices e g by taking into account intraspecific variability Using virtual data sets we test 1 whether indices respond to community changes as expected from their category and 2 whether the indices within each category are consistent and independent of indices from other ca
17. FD indices while axis 2 with FR indices b Reference for three traits third and fourth axes the third axis mainly correlates with the two FE indices axis 4 represents FRyn c Reference for five traits first and second axes axis mainly correlates with FD indices while axis 2 correlates mainly with FR indices d Reference for 10 traits first and second axes axis mainly correlates with FD indices while axis 2 with FR indices The insets display the eigenvalues here expressed as percentage of contribution for the explanation of the variance y axis of each axis ordered from 1 to 12 x axis Shading of the bars refers to the importance of each axis whereby white bars can be considered as statistical noise The d values give the scaling of the grid See Table for explanations of functional diversity indices specific tables RV values have been computed sepa rately for these two index groups For multidimensional indices we have found a high similarity between the index tables calculated with the raw data and the differently treated data sets The transformation via the Hill and Smith method performs slightly better RV range dependent on species richness between 0 897 and 0 956 for Gower distance and PCoA transformed data and between 0 997 and 0 999 for data transformed with the Hill and Smith method That the results were worse for the transformation via the Gower distance and PCoA is mainly imputable to the calculation of FRy
18. ach species This approach promises a finer resolution Bremner et al 2003 Petchey and Gaston 2006 but trait values are more difficult to obtain than information on functional group memberships For instance it is easier to categorize fish species by their general diet than to obtain measurements on their size gape width stomach length etc Functional traits can be morphological traits that represent adaptations to different diets or habitats physiological traits e g temperature tolerance repro ductive traits e g number of eggs and egg diameter or behavioral traits e g migratory behavior or parental care Bremner et al 2003 Dumay et al 2004 Leps et al 2006 Because most of these measurements are real valued i e not discrete and more than one trait is used to describe the different functions the indices commonly used to measure species diversity cannot be applied e g Simpson diversity index To make use of multiple trait measurements Bremner et al 2003 compared functional trait compositions between sites using principal components PCA or co inertia analyses However this approach is comparative and not based on functional diversity per se and therefore does not give absolute insight into the distribution of traits within a specific site Alternatively species diversity indices have now been transposed to functional diversity measurements and several new indices have been proposed e g Mason et al
19. ario a T2 scenario c In contrast indices such as functional range or volume FRpr FRy only reflect a decrease in functional richness when species are removed at the edge of the community The dendrogram based index for functional richness FRp is furthest from the expected results for functional richness Among the two FE indices the multivariate index FE behaves more adequately than its one dimensional counterpart FE and matches the expected results quite well Thus FE can be considered an appropriate index Table 2 However this index fails to respond adequately in some cases for example when species in the center of the community are added or subtracted T1 scenario a and T2 scenario c Species traits are then not evenly distributed over the entire functional space but concentrated at the edges of the community and thus the evenness is theoretically lower than in the initial community Yet FE remains equal because the distribution of the branch lengths of the minimum spanning tree the distance to the nearest neighbors does not change The one dimensional index FE does not represent the distribution of species in a multidi mensional trait space but it reflects the evenness of the distribution for a single trait well results not shown and can be used if only one trait is considered All FD indices adequately match expectations Even one dimensional indices averaged over all traits accu rately reflect changes in a
20. basics and looking forward Ecology Letters 9 741 758 Petchey O L and K J Gaston 2007 Dendrograms and measuring functional diversity Oikos 116 1422 1426 Podani J and D Schmera 2006 On dendrogram based measures of functional diversity Oikos 115 179 185 Podani J and D Schmera 2007 How should a dendrogram based measure of functional diversity function A rejoinder to Petchey and Gaston Oikos 116 1427 1430 Prinzing A R Reiffers W G Braakhekke S M Hennekens O Tackenberg W A Ozinga J H J Schamin e and J M van Groenendael 2008 Less lineages more trait variation Phylogenetically clustered plant communities are functionally more diverse Ecology Letters 11 809 819 Purvis A and A Hector 2000 Getting the measure of biodiversity Nature 405 212 219 R Development Core Team 2008 R a language and environment for statistical computing R Foundation for Statistical Computing Vienna Austria 484 D SCHLEUTER ET AL Rao C R 1982 Diversity and dissimilarity coefficients a unified approach Theoretical Population Biology 21 24 43 Ricotta C 2005 A note on functional diversity measures Basic and Applied Ecology 6 479 486 Robert P and Y Escoufier 1976 A unifying tool for linear multivariate statistical methods the RV coefficient Applied Statistics 25 257 265 Simpson E H 1949 Measurement of diversity Nature 163 688 Smith B and J B Wilson 1996 A
21. cies richness In a second step the 14 rows X 14 columns matrix of the pairwise vectorial correlation coefficients RV coefficients between the scalar prod ucts matrices was computed The RV coefficient ranges from 0 to 1 and evaluates the extent to which two matrices share a common structure Robert and Escoufier 1976 Then the matrix of RV values was diagonalized and the 14 coefficients of the first D SCHLEUTER ET AL Ecological Monographs Vol 80 No 3 eigenvector were used to weight the 14 matrices of the scalar product between indices A mean table of maximum inertia called the reference structure was subsequently computed as the weighted sum of the matrices of the scalar product between indices By weighting the sum greater importance was given to tables with similar structures whereas lesser importance was given to the other tables Finally a PCA was performed on the reference structure It provided the graphical representation of the common structure of the indices derived from the 14 species richness specific tables Two groups of statistics synthesized the relevance and the efficiency of STATIS The first statistics were the RV coefficients between two species richness specific tables The second statistical procedure used was the squared cosines cos of the angles between the first axis scores of separate PCAs performed on each species richness specific table and the first axis scores of the reference structure
22. d Gaston 2002 clustering methods single linkage modified by Mouchet et al 2008 complete linkage UPGMA WPGMA UPGMC WPGMC Ward s method 1 5 FRym functional richness J max 2 dZ this study multidimensional SES where f Z exp 0 5 Z X E Z X ISe 1 Xis 1 Xss Ars Ars 1 x 2 1 FE functional evenness Ss min is as is Awst1 Ass es Mouillot et al 2005 one dimensional s l n a 5 Xis Xis As 44 As s l where species subscripts s are ranked by ascending order of trait value f for categorical variables a fA min LA Le dist AeJA 1 1 So min e K A X dist e Ae A ISed 1 ISed 1 e EE 2 2 FEm functional evenness Vill ger et al 2008 multidimensional ic Sel 1 eee 2 LISSA 6g 3 1 FDyar functional logarithmic arctan 5 5 In Xs In Xs Mason et al 2003 variance M AT atte A where In X is the mean of In X over all species present Ay _ 3 2 FD functional variance 2 Xs Xo Lep et al 2006 FD modified SES A ly fA 2 3 3 FDeat functional FDeat 1 5 5 Kader and Perry 2007 alikeabili A unalikeability 1 j Y y 3 4 FD functional divergence Q Y Qs i this study one dimensional maXses Xis minses Xs where Y is a dummy variable that takes values X with frequency A AsAs 3 5 FDo Rao s quadratic Az dist s s Rao 1982 Champely entropy seS 8 ES and Ch
23. e cases Petchey and Gaston 2006 but it failed to pass the tests proposed here perhaps due to the number of traits and or the number of species used The behavior of this index is thus difficult to understand and we suggest rethinking its interpretation These results were confirmed by the multi table ordination analysis STATIS instead of three principal component axes which were expected to correspond to the three index groups FR FE and FD we found that five axes were needed to explain most of the variance three for FR and one for FD and FE That three axes were needed to explain the variance of FR indicates that the existing FR indices describe independent aspects of functional diversity FRr and FRy form one group which describes the traits range volume FRy measures the occupation and span of trait space FRp which represents the branch length of a dendrogram is independent of the other FR indices but what it actually measures is difficult to determine The FR index was not represented by a single axis but was partly correlated with FRr FRy and FRim because FR accounts for gaps in trait space as FRy but fails to take into account the multidimensional nature of trait space so that gaps are severely underestimated when the number of traits is high The second criterion for the selection of an adequate index is that the chosen index is independent of indices describing other aspects of functional diversity This
24. e first criterion for the quality of an index accuracy is the match between the way it actually behaves and the verbal definition of its properties FR FE FD This was tested in our study using an artificial data set up to 25 species two traits We manipulated species composition and abundances in five tests T1 August 2010 FUNCTIONAL DIVERSITY INDICES 475 Initial community Scenario a Scenario b Scenario c T1 Addition of one species with increasing distance to the initial s ee Coe community e e e e e e e e e e e e e i e e e e e e e e e e e e e e e e e e e e e e T2 e e e e Ld e e e e Subtraction of h e o o o eee nine species at different positions ews s lt 2 7 with occasional e o o o o eee elimination of an entire trait value T3 e o o o Stepwise increase of the abundance of one species at oe e o the same position in eeeee the community T4 e e e e One species with changing position dominates the community ee T5 e e e e e Two species with increasing distance from one another dominate the eee o o community Fig 2 Illustration of the artificial scenarios used to test the behavior of the different indices There are five different tests T1 T5 whereby an initial community is modified three times in different ways scenarios a c Each square represents one community with two trait axes one dot within a square represents a
25. e with species that differ in trait values The community s range is calculated based on species mean trait values by simply subtracting the lowest from the highest mean trait value at a site The absolute range is calculated accordingly for all sites together If more than one trait is used the mean community range is then the mean of all traits Neither individual variability nor gaps in trait space are accounted for by FRr This is an issue when individual variability between sites differs and when functionally exceptional species are added to the FUNCTIONAL DIVERSITY INDICES 471 community or when species with trait values within the range are missing We therefore propose a new one dimensional functional richness index FRy Table 1 IN 1 2 The FR is based on species trait variability instead of the community s trait range FR is based on the union of the species trait ranges and thus considers individual variability It is calculated as the union of species trait ranges at one site relative to the union of species trait ranges for all sites together see FRR When calculating FR the species trait range can be calculated using two species trait matrices as input tables containing the species minimum and maximum trait values respectively With this method however the range depends on the number of individuals measured We therefore suggest using more conservative values e g the 10th and 90th p
26. entiation and thus competition Mason et al 2005 but they can also indicate a predominance of 474 extreme species As FE FD includes species abundanc es in its calculation 1 One dimensional divergence Functional diver gence can be calculated as the abundance weighted functional variance using mean species values Mason et al 2003 Mason et al 2003 suggested log transforming the trait values before calculating the variance FD yar Table 1 IN 3 1 and using species relative abundances as abundance weight If more than one trait is used FD ya is calculated for each trait separately and then averaged over traits They recommend then using an arctangent transformation in order to restrict the index between 0 and 1 If species and or abundances are clustered around the mean trait value FD is low if they are clustered at the edges of the community FD is high The FD index is not applicable if the data set contains 0 values To account for this problem we propose using simply the abundance weighted variance without log transformation FD Table 1 IN 3 2 Leps et al 2006 Since the variance is scale dependent traits should be standardized e g centering and scaling by standard deviation in case the trait space is multidi mensional and the different traits have different scales Another possibility is to use the coefficients of variation instead 1 e standard deviations divided by means Leps et al
27. ercentiles confidence intervals or the mean species trait value SD For further details see Appendix A A script in R for this index is available online 2 Multidimensional indices The multivariate coun terpart of FRpr is the functional volume FRy Cornwell et al 2006 Table 1 IN 1 3 The FRy calculates the volume of trait space with the convex hull volume which represents the smallest convex hull that encloses all species With a complex algorithm the most extreme points vertices can be determined and the volume encompassed by these vertices is calculated TraitHull programmed in Python Cornwell et al 2006 available online and for R Vill ger et al 2008 available online 4 To calculate this index the number of species must always exceed the number of traits A second existing multidimensional FR index is the dendrogram based index FRp Petchey and Gaston 2002 Table 1 IN 1 4 This index measures the extent of species complementarity based on a trait distance matrix Petchey and Gaston 2002 a property equiva lent to FR Mouillot et al 2005 A dendrogram is computed by hierarchical clustering the functional richness is then the sum of the branch lengths of species present There has been discussion in the literature on which distance measure and cluster method is best at calculating this index Podani and Schmera 2006 2007 Petchey and Gaston 2007 Mouchet et al 2008 Here we used the method developed by Mouc
28. es richness at a site we suggest using the number of trait levels present at a site as a proportion of the number of trait levels for all communities together see FRy Table 1 IN 1 2 The FE index evenness of trait level distribution We suggest using Bulla s index of species evenness Bulla 1994 based on the contribution of a trait level 4 to the overall sample size A total number of individuals see FE Table 1 IN 2 1 The FD index variability of trait level distribution We suggest using the index of unalikeability FDeat by Kader and Perry 2007 Table 1 IN 3 3 which corresponds actually to the Simpson index of species diversity Simpson 1949 but uses trait levels instead of species The FD eat represents the proportion of possible comparisons which are unalike by calculating the contribution of a factor level Aj to the overall sample size A and subtracts the sum of the squares from 1 Note that for categorical variables FE and FD measure approximately the same thing since both reflect the equitability of distribution Smith and Wilson 1996 The proposed and other indices on species diversity have been extensively tested e g Washington 1984 Smith and Wilson 1996 Beisel et al 2003 Recommendations for users Based on the criteria discussed above and the tests performed we recommend using the multidimensional D SCHLEUTER ET AL Ecological Monographs Vol 80 No 3 index FR m to
29. essel 2002 August 2010 FUNCTIONAL DIVERSITY INDICES 473 TABLE 1 Continued IN IA Name Formula Source Ad dG 3 6 FD functional divergence a Vill ger et al 2008 se Ald dG ye multidimensional where Ad ye dG dG SESe Ald Xse5 As A dG dG dG is the distance between species s and the gravity center of the convex hull coordinates G 1 V Xsc vX1s and dG is the mean value of dG over all present species Notes IN starting with 1 2 or 3 and IA starting with FR FE or FD refer to functional richness functional evenness and functional divergence index groups respectively Subscripts s and m of index acronyms refer to single one dimensional or multidimensional indices Abbreviations i s c l and are individual species community level of trait for categorical variables and trait subscripts respectively A abundance of species s A abundance of species s when species are sorted following trait t ascending ranking A total abundance of all individuals A abundance of trait level set of individuals of species s Lre number of levels of categorical trait t covered by community c L total number of levels of trait L total number of cross trait levels Se set of species present in community c S number of species present in community c T number of traits studied V set of vertex species from the convex hull in community c V the corresponding numbe
30. f natural communities Practical considerations matter Preslia 78 481 501 Mason N W H P Irz C Lanoisel e D Mouillot and C Argillier 2008 Evidence that niche specialization explains species energy relationships in lake fish communities Jour nal of Animal Ecology 77 285 296 Mason N W H C Lanoisel e D Mouillot P Irz and C Argillier 2007 Functional characters combined with null models reveal inconsistency in mechanisms of species turnover in lacustrine fish communities Oecologia 153 441 452 Mason N W H K MacGillivray J B Steel and J B Wilson 2003 An index of functional diversity Journal of Vegetation Science 14 571 578 Mason N W H D Mouillot W G Lee and J B Wilson 2005 Functional richness functional evenness and functional divergence the primary components of functional diversity Oikos 111 112 118 Mouchet M F Guilhaumon S Villeger N W H Mason J A Tomasini and D Mouillot 2008 Towards a consensus for calculating dendrogram based functional diversity indi ces Oikos 117 794 800 Mouillot D W H N Mason O Dumay and J B Wilson 2005 Functional regularity a neglected aspect of functional diversity Oecologia 142 353 359 Petchey O L and K J Gaston 2002 Functional diversity FD species richness and community composition Ecology Letters 5 402 411 Petchey O L and K J Gaston 2006 Functional diversity back to
31. f the randomized trait value Species abundances were randomized with a uniform distribution between 1 and 100 for each randomized trait matrix All indices described were calculated for each of the 42000 random communities apart from FRy which was only calculated for the three trait simulation Actually its computation time was very long on the computers used in this study and it increased exponen tially with the number of traits used To provide an overview of the relationships between different community indices we used ordination tech niques e g PCA which are known to perform well when summarizing complex data Lebart et al 2000 Since we had to compare a set of 42 matrices of 1000 rows randomized communities and 12 columns indi ces a single matrix based approach was not appropri ate As a consequence for each of the three trait levels we used a multi table ordination technique e g Escofier and Pages 1994 Lavit et al 1994 Chessel and Hanafi 1996 to assess the common structure of the 14 species richness specific matrices We performed a STATIS analysis Lavit et al 1994 to summarize the link between the different indices while removing the potential effect of species richness This method is based on the optimization of the average ordination of the species richness specific ordinations The first step of STATIS consisted of calculating a matrix of scalar products between indices for each of the 14 levels of spe
32. ged Bolnick et al 2003 it has rarely been considered in the formaliza tion of functional diversity Moreover a direct compar ison of the different indices and their correlations with one another is still missing and the user of functional diversity still faces the problems described here when selecting an index The aims of this study were therefore 1 to describe the main properties of the different functional diversity indices 2 to propose new indices that enhance and supplement existing ones e g accounting for intraspecific variability 3 to test and compare the accuracy of all these indices in defined scenarios 4 to measure the correlations among all these indices 5 to summarize the results of 1 4 in a table to facilitate the selection of an appropriate index for the user METHODS Functional diversity indices The functional diversity of a community approached through the measurements of traits is usually described by three kinds of indices that can be combined to calculate different facets of functional diversity Mason et al 2005 Vill ger et al 2008 examples for application Mason et al 2007 2008 functional richness FR functional evenness FE and functional divergence FD The FR indices generally measure how much niche space is filled while FE and FD indices describe how this space is filled Defining functional diversity indices however is not a simple task since there is no natural way of de
33. he data set transformed using Hill and Smith s method Hill and Smith 1976 The common structure of the three calculation methods was then assessed for each species richness level separately using the RV values i e the correlation coefficient between two tables obtained from the method specific tables All calculations and tests were carried out using the program R R Development Core Team 2008 The STATIS analysis and the transformation via the Hill and Smith method and PCoA were computed using the ade4 package Chessel et al 2004 Dray et al 2007 RESULTS Accuracy of the indices Testing whether the indices behave according to the properties of the index group to which they have been assigned FR FE FD reveals that among the FR indices only two of them FR and FRim accurately reflect the expected changes in functional richness see Table 2 Tl and T2 because these indices consider empty space in the trait distribution of a community However FR is only applicable to one dimensional situations while FRy well reflects functional richness in a multidimensional space The results from scenario T1 c and scenarios T2 a and b show that when one trait value is missing in the entire data set both indices decrease However in a multidimensional space when none of the trait values of the removed species are unique but their combination is FR m is the only index that reflects these gaps properly Fig 2 Table 2 Tl scen
34. her at the vertices of the convex hull Since this index is based on the calculation of the convex hull the same assumptions as for the calculation of FRy must be met Transformation of trait data Transformations of the original data should be avoided whenever possible Nonetheless one should always consider that some indices are not applicable to differently scaled traits e g FD If at least one trait is scaled differently all traits should be transformed otherwise this trait might have too little or too much weight in the index calculation One possibility is to transform a trait that differs in several orders of magnitudes from the others using an algebraic function such as the logarithm e g number of seeds in plants or number of eggs in fishes Another possibility is to standardize the trait values with respect to the others e g centering and scaling the trait by its standard deviation This transformation has to be done for the overall data set and not on potential subsamples e g at different study sites The subsamples should then be drawn from the transformed trait matrix For indices that calculate relative index values for each trait separately e g FRr FR FDs etc and for one dimensional indices which range between 0 and 1 e g FE FDya r the data need not to be transformed Different transformation methods and their pros and cons are summarized in Leps et al 2006 Accuracy of the indices Th
35. het et al 2008 R program available online see footnote 4 that computes dendrograms based on two distance matrices and seven clustering methods which belong to the family of hierarchical agglomerative classifications and then selects the combination including a consensus tree that best represents the species distribution in functional trait space As for FRr FRy does not consider gaps in functional trait space We therefore developed a multivariate 2 http www cemagref fr le cemagref lorganisation les centres aix en provence ur hyax scripts ecology_schleuter2010 3 http www pricklysoft org software traithull html 4 http www ecolag univ montp2 fr software Ecological Monographs 472 D SCHLEUTER ET AL Vol 80 No 3 TABLE 1 Index number IN index abbreviation IA name formula and references for the different functional diversity indices IN IA Name Formula Source max X s min X s N SESe SESe 7 1 1 FRp_ functional range ex wie Mason et al 2005 SEUS sEUSe max ts x dx ses x 1 2 FR functional richness this study one dimensional max 1y x dx SEUS any where 1 x is 1 if x is between min and max else it is 0 for categorical variables L L 1 3 FRy functional volume the volume inside the minimum Cornwell et al 2006 convex hull that encloses all species in functional space 1 4 FRp functional dendrogram distance matrices Euclidean Gower Petchey an
36. idence and implications of individual specialization American Naturalist 161 1 28 Bremner J S I Rogers and C L J Frid 2003 Assessing functional diversity in marine benthic ecosystems a com parison of approaches Marine Ecology Progress Series 254 11 25 Bulla L 1994 An index of evenness and its associated diversity measure Oikos 70 167 171 Champely S and D Chessel 2002 Measuring biological diversity using Euclidean metrics Environmental and Eco logical Statistics 9 167 177 Chessel D A B Dufour and J Thioulouse 2004 The ade4 package I one table methods R News 4 5 10 Chessel D and M Hanafi 1996 Analyses de la co inertie de K nuages de points Revue de Statistique Appliqu e 44 35 60 Cianciaruso M V M A Batalha K J Gaston and O L Petchey 2009 Including intraspecific variability in functional diversity Ecology 90 81 89 Cornwell W K D W Schwilk and D D Ackerly 2006 A trait based test for habitat filtering convex hull volume Ecology 87 1465 1471 Diaz S and M Cabido 2001 Vive la difference plant functional diversity matters to ecosystem processes Trends in Ecology and Evolution 16 646 655 Dray S A B Dufour and D Chessel 2007 The ade4 package II two table and K table methods R News 7 47 52 Dumay O P S Tari J A Tomasini and D Mouillot 2004 Functional groups of lagoon fish species in Languedoc Roussillon southern France
37. measure FR or FR if only a single trait is considered Both indices account for intraspecific variability and consider the existence of empty space within the functional trait space and therefore reflect the true functional richness of the community better Besides these indices are orthogonal to FE and FD indices If the user decides to compute FR through FRy the data set should not be transformed even in the case of differently scaled traits since the convex hull calculated with transformed data does not properly represent the functional richness for geometrical rea sons but correlates negatively with the expected results and the other indices for functional richness results not shown The FR indices are naturally correlated to species richness When indices of different communities are compared with one another or when differences between communities are explained with predictor variables the effect of species richness should therefore be removed from the observed pattern in order to describe patterns of pure functional diversity Since the observed rela tionships are not simply linear and differ between the FR indices and because of unequal variances we recommend using null models to remove the effect of species richness rather than using the residuals from a constructed model method Gotelli and Graves 1996 example for application Mason et al 2007 Prinzing et al 2008 Both FE and FE can be used to calculate FE The u
38. nces Second if the distance is greater than 1 S 1 with S representing species richness it is replaced by 1 S 1 the distance which is obtained for an optimal even distribution Mouillot et al 2005 FE is 1 if the distance between nearest neighbor species is exactly 1 S 1 and all species have the same abundances The more a community differs from the optimal distribution in terms of abundance and trait difference the lower the FE When trait space is multidimensional the community s evenness is the average of the FE calculated for each trait 2 Multidimensional index The multivariate equiva lent of FE is FE Vill ger et al 2008 Table 1 IN 2 2 Instead of using the absolute distances between the species trait values for each trait separately this index uses the abundance weighted distances between all species pairs to calculate first the minimum spanning tree MST that links all the species in a multidimen sional trait space The index then measures the regularity of the MST branch lengths according to FE i e comparison with the optimal branch length distribution Functional divergence The FD indices finally mea sure the variance of the species functions and the position of their clusters in trait space a high FD is caused by the clustering of species and or abundances at the edges of the traits space The FD indices find application for indicating the degree of resource differ
39. opose three additional indices two FR indices and one FD index which are also described and tested in this study Functional richness The FR indices measure how much of the niche space is occupied by the species present They are usually interpreted by ecologists as an indicator for potentially used unused niche space and thus e g for productivity buffering against environ mental fluctuations or vulnerability to invasion Mason et al 2005 Functional richness is naturally positively correlated with the number of species present the more species there are the larger the functional space occupied when species traits are somewhat randomly distributed However two communities with the same number of species may have different FR when functional traits of species are more closely clustered in one community than in the other Functional richness is not weighted by species abundance 1 One dimensional indices Mason et al 2005 suggested using the functional range FRR as a measure of FR Table 1 index number IN 1 1 Functional range is the relative range of a trait that is filled by a community at a site compared to the range of the trait for all communities together In this way FRpr is restricted between 0 and 1 and becomes comparable for differently scaled traits Please note however that the value for an individual site is not absolute but might change when the overall range changes e g by the addition of a new sit
40. or transformed and untrans formed trait values As a consequence we only pseudo tested the effects of these transformations using continuous traits as input In this way the normally calculated indices can serve as a reference value As described above we computed 1000 randomizations for six different species richness levels in multiples of 10 from 10 to 60 with three traits We then calculated all indices 1 with the original data set 2 with the data set transformed via the Gower distance and PCoA Podani August 2010 TABLE 2 FUNCTIONAL DIVERSITY INDICES 477 Results for the five index tests T1 T5 as illustrated in Fig 2 expected changes for the index categories boldface and observed changes of index values for the different scenarios compared to the initial community and in relation to one another Tl T2 13 T4 Ts Index b c a b Richness FRR b c a b G a b c FRis FRy FRp FRim Evenness FE FEm Divergence FDyar FD FD FD FD pe rel WSs po en iit vit E e Notes Symbols are lower than initial community higher than initial community no change The number of and signs indicates the increase compared to the other scenarios See Table 1 for explanations of indices and Schmera 2006 Vill ger et al 2008 and 3 with t
41. r of vertices x value of trait in individual i from species s X mean value of trait in species s X mean value of all traits in species s arranged in a vector Z sample vector of all traits used for the purpose of computing integrals over trait space E variance covariance matrix of traits dist s s distance between species pairs based on mean trait values for continuous variables Euclidean distance is used for discrete variables the Gower distance is used set of edges connecting species pairs in the minimum spanning tree e subscript of an edge dist e distance between endpoint species of edge e Ae sum of the abundances of the endpoint species of edge e Q1 lower quartile Q3 upper quartile UPGMA unweighted pair group method using arithmetic averages WPGMA weighted pair group method using arithmetic averages UPGMC unweighted pair group centroid method WPGMC weighted pair group centroid method counterpart to FRy FRy Table 1 IN 1 5 The FRim index is specifically designed to account for individual variability and for gaps in the multidimensional functional trait space The idea is to compute an equivalent range union as for FRj across species present in a community To do so each species s is assigned a function on trait space f that indicates whether a particular point in trait space is or is not occupied by species s that is whether species s trait values encompass this point
42. scribing richness evenness or diver gence when individuals are not assigned to classes i e species but rather described by their traits First any index should reflect the verbal definition of its proper ties Second FR FE and FD indices aim at measuring different aspects of functional diversity and should therefore be uncorrelated independent in a random community August 2010 There are nine indices available in the literature to calculate functional diversity on the basis of measured traits which we describe and test in this study three FR indices first described by Petchey and Gaston 2002 Mason et al 2005 Cornwell et al 2006 two FE indices Mouillot et al 2005 Vill ger et al 2008 and four FD indices first described by Rao 1982 Mason et al 2003 Leps et al 2006 Vill ger et al 2008 Each index group contains one and multidimensional indices Despite their multiplicity these indices still miss some important points e g FR indices do not consider individual variability Indeed individual variability in functional diversity has been approached through the expansion of existing indices for the use of individuals Leps et al 2006 Cianciaruso et al 2009 However indices that specifically account for the use of intraspecific variation using means and intraspecific variability as input not individual trait values have only been proposed for two FD indices Leps et al 2006 To fill these gaps we pr
43. se of FE is however narrowly restricted to one dimensional data sets Similarly we cannot recommend the use of a specific FD index because all indices reflect the expected changes well irrespective of whether they are one or multidimensional However it should be remembered that not all FD indices are independent of FR indices Table 4 seemingly because FDyar and some FR indices measure a mixture of FR and FD properties Further it should be mentioned that indices based on a distance matrix allow for the use of categorical and continuous variables simultaneously and they can include intraspecific variability when calculated via the pairwise trait overlap as suggested by Leps et al 2006 Specific disadvantages of all indices are further summa rized in Table 4 Concerning categorical variables we recommend at present avoiding their use if possible or using indices based on a distance matrix or the one dimensional indices of species diversity Table 1 IN 1 2 2 1 3 3 In this case however the user should keep in mind that the indices designed for categorical variables especially for FE and FD do not have the same meaning as their continuous counterparts The results for the different traits should therefore be averaged over continuous and categorical variables separately rather than directly averaged over all traits In real world measurements the user comes to face more problems apart from choosing the right index Aug
44. sification computation time FR m this study functional yes no no yes long computation 1 5 integral time intra specific trait variation needed Functional evenness FE Mouillot et al evenness of trait no yes yes no or one dimensional 2 1 2005 values FE Vill ger et al evenness of mini yes yes yes no 3 2 2 2008 mum spanning tree branch lengths Functional divergence FD a Mason etal logarithmic var no yes yes no FRr FRy no 0 values possible 3 1 3 3 cat 2003 iance of traits FD Lep et al variance of traits no yes yes no oi 3 2 3 3 cat 2006 FD this study relative range no yes no no FRy 3 4 of the distri butional center FDo Rao 1982 variance of yes yes yes no FRy 31D Champely distances be and Chessel tween species 2002 FD Vill ger et al mean deviation yes yes no SR has to exceed 3 6 2008 of the distance N traits of the center see FRy of gravity Notes The table can be used to choose an index for the data set in question dimensionality type of variable correlation with indices of a different category Abbreviations are A abundance Adapt adaptation cat categorical Cor correlated IN index number MD multidimensional N number SR species richness See Table for explanations of indices the scenario test Fig 4 Table 2 and complied well with nearly all FR indices except FRp In this study we restricted our tests to classical versions of the indices without specific pa
45. ss FE and FE and the remaining two FR indices FRj and FRp are best represented by the fourth and fifth axes respectively Table 3 Fig 4 Note that FRy is not fully independent of FDg and FD r ranging from 0 09 to 0 59 for FD and from 0 21 to 0 71 for FDg depending on the species richness level and the number of traits despite distance in the F1 X F2 factorial plane of the STATIS analysis In addition the correlation between FR and FRy decreases with increasing species richness from r 0 61 in a community with five species to r 0 00 in a community with 25 species Categorical variables One and multidimensional indices react differently to the transformation from discrete to continuous vari ables Therefore correlations between the method August 2010 FUNCTIONAL DIVERSITY INDICES 479 a 3 traits S d 0 2 b 3 traits d 0 2 FD FD SFD o FE Foy FE FR Axis 1 LFR FR Fo FR 25 Eigenvalues FR FR n 0 On c 5 traits N d 02 d 10 traits N d 0 2 4 x i B FD ei FoF Axis 1 Axis 1 FRE fas FE pala V 30 Eigenvalues FRY 30 Eigenvalues ig FR g FRA FR I FR 0 0 Fic 4 Reference structures over the different species richness specific ordinations gained from the STATIS analysis Lavit et al 1994 a Reference for three traits first and second axes axis mainly correlates with
46. tance and PCoA and between 0 688 and 0 897 for transformation via the Hill and Smith method DISCUSSION Performance of the indices The functional diversity indices described and tested in this study performed quite differently The quality of the indices aimed at describing the same aspect of functional diversity differed markedly especially within the group of the FR indices In our view the main criterion for the selection of an index is whether it accurately measures what it is intended to describe This was the case for all FD indices irrespective of whether they were one or multidimensional as well as for the two FE indices However while the multidimensional index FEm per formed quite well in describing the evenness in the two dimensional space its one dimensional counterpart FEs was only able to measure evenness when a single trait was considered In contrast the FR indices differed most strongly in their quality the only indices that reflected the expected changes in FR when species were removed within the functional trait space were the two indices that consider gaps in the functional trait space FR s and FRy The FRp and FRy indices only partly reflected what is considered functional richness only if there was a continuously filled trait space and species were removed or added at the edges of the community while the results obtained from the index FRp did not match the expected values This index may work in som
47. tegories We also test the accuracy of methods proposed for the use of categorical traits Most classical functional richness indices either failed to describe functional richness or were correlated with functional divergence indices We therefore recommend using the new functional richness indices that consider intraspecific variability and thus empty space in the functional niche space In contrast most functional evenness and divergence indices performed well with respect to all proposed tests For categorical variables we do not recommend blending discrete and real valued traits except for indices based on distance measures since functional evenness and divergence have no transposable meaning for discrete traits Nonetheless species diversity indices can be applied to categorical traits using trait levels instead of species in order to describe functional richness and equitability Key words morphological traits species richness INTRODUCTION Biodiversity is commonly expressed through indices based on species richness and species abundances Whittaker 1972 Lande 1996 Purvis and Hector 2000 Recently however studies focused on diversity have begun to incorporate the concept of functional diversity In contrast to species diversity functional diversity measures the distribution and the range of what organisms do in communities and ecosystems and thus considers the complementarity and redundancy of co occurring species Diaz and
48. the ordination technique is missing in this approach we expect less loss of information Another possibility is the approach suggested by Leps et al 2006 who recommend calculating dissimilarity between species pairs via the sum of their overlaps for each trait continuous and categorical In both approaches Gower distance and summed overlaps continuous and categorical variables share the same meaning and indicate just to which extent two species are identical Otherwise discrete variables cannot be transformed to continuous variables and processed further with indices designed for this type of variable since functional evenness and divergence have no transposable meaning for discrete traits 1 e no spatial evenness and diver gence of discrete variables but evenness or variability of trait level distribution Since categorical and real valued variables are of completely different character it is difficult to find an index in which they can be mixed up for an exception see Material and methods Functional diversity indices Functional divergence Mul tivariate indices FDg A second approach would therefore be to use indices aimed at measuring the specific properties of categorical variables At present we propose using the one dimensional indices as follows designed to calculate species diversity using trait levels instead of species The FR index relative richness of trait levels Corresponding to the relative speci
49. tterns of data distribution such as skewness It would be of further interest to include hypotheses on the data randomiza tions and test how variations of a certain index e g selection of the distance measure used in FDqg react under certain circumstances However we only expect slight fine tuning and not fundamental deviations from the results presented here The use of categorical variables is problematic since most tested indices can only be applied to continuous variables However our pseudo test with only continu ous variables already revealed that the two transforma tions proposed Gower distance followed by a PCoA and the Hill and Smith method led to a loss of information When tested for multivariate traits this was especially true for the Gower distance transforma tion method followed by a PCoA for one dimensional indices both index matrices calculated on the basis of transformed data correlated only weakly with the matrix based on the untransformed traits This effect might be even worse when real discrete variables are included in the data set In this case the transformation methods via ordination techniques may not be the best choice for the treatment of categorical variables One way to overcome this problem could be to use a multivariate index based 482 on a distance matrix e g FDg and to transform the trait matrix only via the Gower distance Podani and Schmera 2006 Since the second transformation step
50. ust 2010 many of which e g weighing of traits are discussed in Leps et al 2006 and Petchey and Gaston 2006 One frequently occurring problem is that normally not all trait values can be measured for each individual This is however not a problem for the calculation of the indices since the calculation of all indices including the newly proposed indices that include intraspecific vari ability is based on mean trait values and their variability And how many individuals should be measured for each species In general the more the better However to restrict sampling effort a reference value could be the amount of individuals that are necessary to describe the species trait distribution Cianciaruso et al 2009 ACKNOWLEDGMENTS D Schleuter was funded by the National Research Agency of France ANR within the project Freshwater Fish Diversity ANR 06 BDIV 010 We thank Linda Northrup for correcting the English F Guilhaumon N W H Mason D Mouillot J Veslot and S Vill ger for discussion and help with some of the indices and three anonymous reviewers for their valuable comments LITERATURE CITED Beisel J N P Usseglio Polatera V Bachmann and J C Moreteau 2003 A comparative analysis of evenness index sensitivity International Review of Hydrobiology 88 3 15 Bolnick D I R Svanb ck J A Fordyce L H Yang J M Davis C D Hulsey and M L Forister 2003 The ecology of individuals inc
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