An index for measuring functional extension and evenness in trait space

Abstract Most existing functional diversity indices focus on a single facet of functional diversity. Although these indices are useful for quantifying specific aspects of functional diversity, they often present some conceptual or practical limitations in estimating functional diversity. Here, we present a new functional extension and evenness (FEE) index that encompasses two important aspects of functional diversity. This new index is based on the straightforward notion that a community has high diversity when its species are distant from each other in trait space. The index quantifies functional diversity by evaluating the overall extension of species traits and the interspecific differences of a species assemblage in trait space. The concept of minimum spanning tree (MST) of points was adopted to obtain the essential distribution properties for a species assembly in trait space. We combined the total length of MST branches (extension) and the variation of branch lengths (evenness) into a raw FEE0 metric and then translated FEE0 to a species richness‐independent FEE index using a null model approach. We assessed the properties of FEE and used multiple approaches to evaluate its performance. The results show that the FEE index performs well in quantifying functional diversity and presents the following desired properties: (a) It allows a fair comparison of functional diversity across different species richness levels; (b) it preserves the essence of single‐facet indices while overcoming some of their limitations; (c) it standardizes comparisons among communities by taking into consideration the trait space of the shared species pool; and (d) it has the potential to distinguish among different community assembly processes. With these attributes, we suggest that the FEE index is a promising metric to inform biodiversity conservation policy and management, especially in applications at large spatial and/or temporal scales.

. Examples of counter-intuitive results for some popular functional diversity indices, calculated with the FD package in R (Laliberté et al. 2014). The dashed boxes indicate the 2-dimensional trait space (unit square). Intuitively, between the two communities (white and black) in panel a, the 'black' community has higher functional diversity than the 'white' one. However, the three representative indices of functional richness, evenness, and divergence (FRic, FEve, and FDiv) imply the opposite relationship (panel a). Between the panels b and c, all species are same except the one in the center. Compared to panel c, the trait space in panel b is not evenly taken by the central species. Thus, functional diversity in panel b is expected to be slightly lower than in panel c. However, none of the five functional diversity indices we tested show this expected relationship. In the three panels, indices showing counter-intuitive results are highlighted in red. In contrast to the counter-intuitive results shown in the figure, values of the FEE index proposed in this paper are consistent with intuition. Figure S2. Five scenario series tests (T1 to T5) with artificial communities proposed by Schleuter et al. (2010) to their evaluate functional diversity indices. The species pool contains 40 species (black crosses and red dots). Red dots show the species presented in a community, and their size represents species abundance. The dashed box in each panel presents the 2-dimensional trait space (unit square). Values of our FEE index, which accounts for species abundances (Equation 3), are given under each panel. Table S1 summarizes the trends of indices (FEE and five FD indices) for each series, and the expected behavior of three components of functional diversity according to Schleuter et al. (2010) andFontana et al. (2016). Figure S3. Tests with artificial communities to evaluate FEE (accounting for species abundances according to Equation 3) and other single-facet indices with respect to the criteria summarized by Mason et al. (2003) and Ricotta (2005). Evaluation results are summarized in Table S2. The species pool contains 40 species (black crosses and red dots). Red dots show the species present in a community, and their size represents species abundance. The dashed box in each panel represents the 2-dimensional trait space (unit square). Values of our FEE index (in bold) and the single-facet indices are given in the table to the right of the figure. Table S1. Expected and observed trends for functional diversity indices for five scenario series tests (T1-T5 in Figure S2), adapted from Figure 2 in Schleuter et al. (2010). Symbols in the table summarize the expected trends for three functional diversity components (richness, evenness, and divergence) according to Schleuter et al. (2010)  FRic, FEve, and FDiv, respectively, are compared to the expected trends for richness, evenness, and divergence (blue and red symbols). No expected trends for FDis and Rao's Q are available, but their observed trends for T1-T5 are presented for completeness. Actual values for FEE are shown in Figure S2 (actual values for other indices are not shown, but are summarized qualitatively in this table). The order of symbols in each list (from top to bottom) corresponds to the scenarios (from left to right) in each series in Figure S2. Symbols: ∘ indicates the diversity of the reference community in each series (left column in Figure S2), or communities whose diversity is equal to that of the reference community; + indicates that diversity is higher than in the reference community; − indicates that diversity is lower than in the reference community. Following the notation of Schleuter et al. (2010) Table S2. Evaluation of our FEE index and some single-facet indices with respect to the criteria proposed by Mason et al. (2003) (criteria 1-10 in our table) and Ricotta (2005) (criteria 11-14 in our table). Evaluation results in the table are informal (rigorous and general proofs are not available) and are mainly based on the results presented in Figure S3. FEE results in this table and Figure S3 account for species abundances according to Equation (3). In the 'Result' column, 'S' indicates satisfied and 'N/A' indicates that the criterion is not applicable to FEE. In the 'Notes' column, text in square brackets refers to previously published indices. All other text in the 'Notes' column refers to our FEE index.
Criterion from the literature Result Note 1. Be constrained to the 0-1 range (for convenience) and use that range well.

S
See the relevant distributions in Figure 3.
[Rao's Q only occupies a small part of the range in our tests (Figure 3).] 2. Reflect the range of character values present, since that is the point of the index. S 3. Reflect the contribution of each species in proportion to its abundance; a community is not functionally diverse if all species with extreme trait values are rare.

S
Comparing Figure S3a and S3g (or Figure S3a and S3b) shows that introducing a rare species on the edge of trait space has only a small effect. Thus, FEE satisfies this criterion in a broad sense (if not strictly 'in proportion' to abundance).
[FRic and FEve do not meet this criterion.] 4. Decrease when the abundance of a minor species with an extreme trait value decreased.

S
See difference between Figure S3a and S3d (or between Figure S3b and S3e, or between Figure S3c and S3f).
[FRic does not meet this criterion.] 5. Not change appreciably when a very rare species disappears.

S
Comparing Figure S3a and S3g (or Figure S3a and S3b) shows that removing rare species has only a small effect. However, in these comparisons, the effect is noticeable because the removed species occupied a distinct part of trait space.
[FRic does not meet this criterion well.] 6. Be unaffected by the units in which the trait is measured. This is essential for traits that could be measured on more than one scale (e.g., mm, cm, or m).

S
Trait values in the species pool are normalized to the 0-1 range.
7. Be symmetrical with regard to small and large character values.

S
Using the cumulative distribution function from a null model to translate FEE0 to FEE (Equation 2) leads to a broad and symmetric distribution (see the probability density function of FEE in Figure 3).
[FRic, FEve, FDiv, FDis, and Rao's Q do not meet this criterion.] 8. Be unaffected by the units in which the abundance is measured. It is unacceptable to have the index value dependent on the unit chosen (e.g., mg, g, or kg).

S
The abundance-adjusted version of FEE relies on unitless relativeabundance weights (Equation 3). The definition of these weights is flexible; e.g., they could be based on biomass, percent cover, individual density, etc.
9. Be unaffected by species richness. The number of taxonomic species per se is not relevant to functional diversity.
S FEE is intrinsically independent of species richness (n) (Figure 3); i.e., FEE is independent of n when communities are randomly sampled from the species pool, although FEE may be correlated with n under some assembly processes (Figure 4 and Table S3).
[FRic, FEve, FDiv, FDis, and Rao's Q do not strictly meet this criterion, although some of these are only weakly correlated with n; see Figure  3.] 10. Be unaffected when a species is split in two (i.e., one species is replaced by two species with the same traits and the same total abundance as the original). N/A We adopt the 'functional species' concept of Ricotta (2005), such that two species with identical trait values would be considered the same species.
11. Set monotonicity (a subset of a community is less diverse than the entire community).
N/A This criterion focuses on species or functional richness, which is not applicable to indices (such as FEE) that account for abundance (Ricotta 14. Concavity (the average diversity of a set of communities is less than the diversity of the aggregated pool).
N/A See Notes for #11 above. Table S3. The coefficients ( 10 -4 ) of species richness in the linear regression models of functional diversity indices under the three community assembly processes: neutral, niche filtering (NF), and limiting similarity (LS). Artificial community data were analyzed both as presence-absence (p-a) data and abundance (ab.) data. The table shows the coefficients and their associated significance levels (***, < 0.001; **, < 0.01; *, < 0.05; no asterisk, ≥ 0.05). Boxplots for the FEE results are shown in Figure 4b-c. The FDiv index is not included here because it is unavailable for the one-trait case (Laliberté et al. 2014