A geometric approach to beta diversity: Comment

Song et al., (2025) introduced a geometric approach intended to quantify beta diversity. In principle, new methods to assess patterns of biological diversity can open new avenues to understand how biological communities are structured and why. However, to fulfil this expectation, any new measure of biological diversity (i) needs to be linked to a clear ecological concept, and (ii) its mathematical properties need to be coherent with that ecological concept. Song et al. claim that their new measure, betavolume, is a measure of beta diversity. In this paper I show that betavolume does not account for beta diversity, community dissimilarity or community differentiation. Therefore, the measure proposed by Song et al. should not be termed beta diversity or any other related concept. This does not negate the possibility that betavolume could quantify another relevant attribute of biological communities, but proving this would require a (still missing) rigorous assessment of what ecological concept is betavolume actually accounting for.

Beta diversity has historically proven a difficult concept to operate with, and as a result multiple measures and methods have been introduced. In fact, many of the methods used to assess beta diversity patterns, like many similarity/dissimilarity indices (Jaccard, 1912; Simpson, 1943; Sørensen, 1948), predated the term beta diversity, which was introduced by Whittaker in 1960. Even in his seminal paper, Whittaker (1960) used several methods to assess beta diversity patterns, including similarity indices. Whittaker defined beta diversity as “the extent of change in community composition, or degree of community differentiation,” and proposed that the simplest measurement of beta diversity is beta = gamma/alpha, where alpha diversity is the average site diversity and gamma diversity is the diversity of the regional pool of sites. This simple mathematical formula is thus the strict sense definition of beta diversity (gamma/alpha), in which beta quantifies the effective number of communities in a region (Tuomisto, 2010). This applies to any diversity measure computed as numbers equivalents (Hill's numbers, see Jost, 2007), and the multiplicative nature of the partition of gamma diversity into alpha and beta components (because gamma = alpha × beta) ensures that beta diversity measures based on Hill's numbers are independent of alpha diversity (Baselga, 2010a; Jost, 2007). This property is crucial, because it allows making analysis of the ecological correlates of beta diversity and inferences about the underlying processes driving changes in species composition without the confounding effects of alpha diversity gradients (Baselga, 2010a; Jost, 2010). There is also a broad sense definition of beta diversity (i.e., the variation of biological communities, in line with the previous original definition by Whittaker) that is linked to the concepts of community change, community dissimilarity or differentiation (Anderson et al., 2011; Koleff et al., 2003; Legendre & De Cáceres, 2013). The strict and broad sense definitions do not conflate. We can bridge from the strict sense mathematical formulation to other measures of community differentiation because beta diversity (gamma/alpha), does not only measure how biological diversity scales from local to regional scales, or the effective number of distinct communities, but also how different are the local communities. This is because regional diversity can only be higher than local diversity if communities are different across sites. In fact, beta diversity has a monotonic relationship with several dissimilarity indices. For example, when diversity is measured as species richness, Sorensen and Jaccard dissimilarities are just monotonic transformations of strict sense beta diversity (Chao et al., 2012; Jost, 2007). These dissimilarity measures can thus be understood as standardizations of beta diversity to remove its dependence from the number of sites involved in the calculation (Chao et al., 2012). In other words, any valid measure of beta diversity, even in the broad sense, must comply with the expectation that the metric should increase as communities become more different in community composition. We also know that beta diversity accounts for various sources of differentiation (Koleff et al., 2003), including the replacement of species from site to site (spatial turnover) or the ordered loss of species in strict subsets (nestedness) (Baselga, 2010b). Total dissimilarity, measured with Sorensen or Jaccard dissimilarity indices (which are monotonic transformations of beta diversity) can be partitioned into components accounting for these two sources of differentiation (Baselga, 2010b, 2012), further clarifying the relationship between beta diversity and indices accounting for spatial turnover only, as Simpson dissimilarity (Koleff et al., 2003; Simpson, 1960). Simpson dissimilarity is thus just one of the components of beta diversity, and the remaining fraction of total dissimilarity is the nestedness-resultant component of beta diversity (Baselga, 2010b, 2012). In sum, the integrated progress made over the past two decades has fostered a robust understanding of the concepts and metrics underlying community differentiation and beta diversity.

So, how does betavolume fit within the conceptual framework of beta diversity or community differentiation? First, Song et al. state that the introduction of betavolume is justified by the need to account for complex interactions as it “explicitly incorporates species associations.” Given that betavolume exclusively relies on occurrence data (presence/absence tables), in the very same way that beta diversity does, it is safe to say that betavolume does not provide accurate information about species interactions (Blanchet et al., 2020), and there is no explicit indication of how complexity or nonadditivity is accounted for. Second, despite what Song et al. claim after their Equation (4), betavolume cannot be interpreted as the effective number of communities. This can be inferred from the fact that identical communities yield a value of betavolume = 0 (which would imply zero distinct communities, i.e., no species at all) or from the detailed analyses I provide in the next section, which shows that betavolume does not reach its maximum when communities are completely different (i.e., the largest possible number of distinct communities). Third, Song et al. start their argumentation positing that a community with species A, B is inherently different from communities with only A or only B. This suggests there is a gap in previous measures of beta diversity, but in fact strict sense beta diversity (gamma/alpha) does also consider that these communities are different. From this premise, they jump to the conclusion that beta diversity must be higher in their Metacommunity II (Site 1 [A], Site 2 [B], and Site 3 [A, B]) than in Metacommunity I (Site 1 [A] and Site 2 [B], thus with no species overlap). The reasoning is problematic, because the conclusion does not logically follow the premise. The fact that [A, B] is different from [A] or [B] does not necessarily imply that Metacommunity II must have higher beta diversity. In fact, strict sense beta diversity (gamma/alpha) would consider [A, B] different from [A] or [B], but would still quantify beta diversity in Metacommunity II to be lower than in Metacommunity I. So, instead of a conclusion derived from the premise, the statement that beta diversity must be higher in Metacommunity II is a second independent premise, for which the authors do not provide any justification. In fact, community differentiation is not only lower in Metacommunity II, but much lower than reflected by the effective number of distinct communities (beta diversity). This is because, as mentioned above, beta diversity depends on the number of sites (N), as it varies between 1 (minimum differentiation or 1 distinct community) and N (maximum differentiation or N distinct communities). Therefore, to compare beta diversity between metacommunities with different N as Song et al. are doing, one needs to standardize beta diversity, obtaining a dissimilarity index. For example, the min-max normalization of beta diversity yields the Sorensen dissimilarity index, as β_SOR = (beta − 1)/(N − 1). In the case of betavolume we would need the same standardization, as it varies between zero and N. Such a standardization (betavolume/N, in this case, given that its minimum is zero) would yield values of 0.71 for Metacommunity I and 0.67 for Metacommunity II, suggesting that even betavolume does not consider community differentiation to be higher in Metacommunity II than in Metacommunity I. In sum, the logic behind the motivation to introduce betavolume is not robust. In any case, to understand the performance of betavolume it is necessary to perform a systematic assessment of its behavior under a complete gradient of community differentiation.

First, I started with simple scenarios, including pairs of communities with 10 species in total (gamma diversity), analyzing how beta diversity (gamma/alpha) and betavolume perform in gradients of (1) increasing species replacement (Figure 1a) and (2) increasing richness difference in strictly nested subsets (Figure 1c). In both gradients, species overlap systematically decreases from Pair 1 to Pair 6. For all these presence/absence tables, I computed beta diversity (gamma/alpha) and betavolume. To compute betavolume, I used the function betavolume in R package BetaVolume (Song, 2025), setting the argument weights to TRUE. All other computations were also performed in R (R Development Core Team, 2024). In both cases, starting from two identical communities, beta diversity monotonically increases as either replacement or richness difference increases (i.e., species overlap decreases in both gradients). In contrast, betavolume initially increases until the number of unique species approximates the number of shared species, but then decreases despite replacement or richness difference keep increasing (Figure 1b,d). These are clear gradients of community differentiation, so we can conclude that betavolume does not account for beta diversity or community differentiation.

The previous simplified gradients of replacement and richness difference provide a clear insight of the performance of betavolume, but do not cover all potential scenarios involving all possible combinations of species overlap and richness difference between two sites. Thus, to consider the complete space of variation of beta diversity and betavolume computed for two sites, I built a table with all possible combinations of the three matching components (a, b, c), where a is the number of species present in both sites, b is the number of species present in the first site but not in the second, and c is the number of species present in the second site, but not in the first. Matching components varied between 0 and 100, and I considered all combinations given that gamma diversity (i.e., both sites taken together) equals 100 species (i.e., a + b + c = 100). After removing the combinations in which one of the sites has no species (a + b = 0 or a + c = 0), this results in 5149 cases. For each of them, a presence/absence table was built according to the corresponding matching components, and beta diversity (gamma/alpha) and betavolume were computed. A similar procedure can be followed to assess all possible levels of community differentiation for three sites, although here the number of matching components (i.e., the regions in a Venn diagram for three sets) is higher (seven, a to g), because we need to consider species shared by the three sites (a), species shared by pairs of sites (b, c, d), and species unique to one of the sites (e, f, g). The number of combinations for gamma diversity = 100 species would be unfeasible to handle (>1 billion), so, I restricted gamma diversity to 14 species, and computed all possible combinations of the 7 matching components using function compositions() in R package partitions (Hankin, 2006). After removing combinations where any of the sites had zero species, this resulted in 38,403 cases. For four sites the number of matching components (i.e., the regions in a Venn diagram for four sets) increases to 15 (a to o), and the number of combinations is too high even for low gamma diversity. Therefore, I computed only the combinations of matching components involving the extreme cases of species overlap or unique species, and then randomly resampled these combinations twice to simulate intermediate scenarios, for a total of 65,429 cases. As for the scenarios with two sites, for these datasets corresponding to three or four sites, a presence/absence table was built according to the corresponding matching components for each case, and beta diversity (gamma/alpha) and betavolume were computed. The code for all the analyses described above is available in Baselga (2025).

The results for two sites reveal that the two curves describing the relationship between beta diversity and betavolume under gradients of replacement or nested richness difference (Figure 1b,d) delimit the space of variation between both variables (Figure 2). Inspection of the most extreme cases reveals again that betavolume reaches its maximum at intermediate values of community differentiation, when a, b, and c components take similar values, and then decrease with larger values of community differentiation (i.e., lower values of species overlap). As a measure of community differentiation, it is not justified why two communities with 50 unique species each (red case in Figure 2) should have the same value than two communities with 63 species in common (yellow case in Figure 2) or a lower value than two communities with 39 species in common (black case in Figure 2). In the same way, the fact that two communities with zero species overlap and different richness (one with 99 unique species, the other with only one unique species, orange case in Figure 2) takes the same value of betavolume than two almost identical communities (99 species in common, light blue case in Figure 2) is at odds with the concepts of beta diversity, effective number of communities, or community differentiation.

The results for three and four sites reveal similar patterns (Figure 3), with the additional insight that the maximum of betavolume shifts towards lower values of community differentiation (i.e., beta diversity standardized by its maximum, the number of sites). By inspecting what combination of matching components yields the maximum betavolume, these cases correspond to an even distribution of species among matching components (i.e., all matching components take similar values). This implies that with increasing N, betavolume would reach its maximum at increasingly lower values of community differentiation, because for N sites there are N matching components for species shared among N − 1 sites, and N(N − 1)/2 matching components for species shared among N − 2 sites, and so on. So, for example, for 10 sites, there are 10 matching components accounting for the number of species that are shared by nine sites, and 45 matching components accounting for the number of species shared by eight sites. As a result, when N increases, betavolume unavoidably shifts its maximum towards increasing levels of species overlap (and thus decreasing levels of community differentiation). In the reported results for two, three, and four sites, betavolume takes its maximum at decreasing values of standardized beta diversity [(beta − 1)/(N − 1)], specifically 0.44, 0.31 and 0.24, for N = 2, 3, and 4, respectively.

The systematic assessment of the performance of betavolume reveals that it does not quantify the effective number of communities (i.e., strict sense beta diversity), or community differentiation (i.e., broad sense beta diversity). In contrast, betavolume is maximized by evenly splitting species in matching components instead of by decreasing levels of species overlap. This implies that, for two communities, betavolume reaches is maximum (i.e., betavolume = 2) when approximately a third of the species is unique to community 1, a third of the species is unique to community 2, and a third of the species is present in both communities. When the distribution of species across matching components differs from this even splitting, it results in lower values of betavolume. For a system involving 100 species and two sites, this implies that betavolume is maximum when 40 species are shared between the two sites, and 30 species are unique to each site. Betavolume decreases when the matching components deviate from this, but in a way that does not follow any defined ecological concept. For example, betavolume takes the same value (betavolume = 1.12) in contrasting situations: (1) when 70 species are unique to Site 1, 30 species are unique to Site 2, and 0 species are shared between both sites (i.e., no species overlap), (2) when 70 species are unique to Site 1, 0 species are unique to Site 2, and 30 species are shared between both sites (i.e., intermediate species overlap), and (3) when 30 species are unique to Site 1, 0 species are unique to Site 2, and 70 species are shared between both sites (i.e., large species overlap). A second feature of betavolume is that its maximum shifts to lower values of community differentiation as the number of sites increases. This implies that as we increase the number of sites from two to three and four, for example, the maximum possible value of betavolume is reached when one-third of the species is unique to each site (for N = 2), to when one-seventh of the species is unique to each site (for N = 3), to when one-fifteenth of the species is unique to each site (for N = 4). The shift towards lower levels of community differentiation would be even larger for higher numbers of sites (N), because the number of matching components increases exponentially with N (i.e., as 2^N − 1). So, what does a value of betavolume = 1.12 say about a set of communities? Or what does the variation between betavolume = 1.12 and 2 mean in ecological terms? Responding to these questions would be crucial because a measure of biological diversity is useful only if a given value or its variation from one value to another can be interpreted according to a clear ecological concept.

The strict sense definition of beta diversity (gamma/alpha) quantifies the effective number of communities in a region (Jost, 2007; Tuomisto, 2010), while the broad sense definition of beta diversity (i.e., the variation of biological communities) is linked to the concepts of community change, dissimilarity, or differentiation (Anderson et al., 2011; Koleff et al., 2003; Legendre & De Cáceres, 2013). As shown above, betavolume is not linked to any of these concepts, so it is not justified to call it a measure of beta diversity. The next question, which remains open, is whether betavolume is useful to understand any other aspect of how biological diversity is distributed across sites and why. Song et al. do not provide a compelling justification of why this new measure improves our understanding of biological communities. First, as argued above, their theoretical argumentation is based on unexisting gaps (i.e., community with species A and B is different from communities with only species A or B, which is perfectly accounted by strict sense beta diversity) and broken logical chains (i.e., it does not follow from the former that beta diversity is higher in Metacommunity II than Metacommunity I in their Figure 1). Second, Song et al. do not provide a systematic assessment that illustrates how the measure performs in relation to a relevant ecological concept. There is no obvious reason why maximizing the evenness in the distribution of species across matching components (species shared by sets of n sites, with n ranging from 1 to N) is a relevant attribute of communities. There is not a clear understanding, either, of how betavolume varies when species are not evenly distributed across matching components. The invocation of species interactions or nonadditivity is farfetched because the link between betavolume and these ecological attributes was never actually proven. Third, the empirical analyses provided by Song et al. only evidence that betavolume does not account for beta diversity and, as a result, the patterns differ between both measures. It could not be any other way, given the lack of correlation between betavolume and beta diversity. Song et al. interpret the unimodal latitudinal variation of betavolume as the result of the contrasting effects of nestedness and spatial turnover, which would be jointly accounted for by betavolume, according to its proponents. However, to support this claim it would be necessary to know how both variables (turnover and nestedness) mathematically contribute to betavolume. This, in contrast, is well known for beta diversity: turnover and nestedness-resultant components additively contribute to beta diversity (Baselga, 2010b; Legendre, 2014). Given that the turnover and nestedness components of beta diversity are computed by additively partitioning a dissimilarity measure that is a linear transformation of strict sense beta diversity, it seems misleading to interpret betavolume (which is uncorrelated to beta diversity) as the combination of turnover and nestedness-resultant dissimilarity. I would argue that the only way to unveil the contrasting latitudinal patterns of nestedness-resultant dissimilarity and spatial turnover is to partition beta diversity into both components and separately assess how they vary with latitude (Baselga et al., 2012; Castro-Insua et al., 2016; Leprieur et al., 2011). Thus, the inability of betavolume to separate between components of community variation linked to species replacement or to nestedness is another major limitation. Based on this inability, Song et al. jump to the conclusion that the replacement/nestedness problem is inherently ill-defined. This is equivalent to stating that, because I am color-blind, the problem of separating red from green is inherently ill-defined. The framework is clearly defined: beta diversity (i.e., the effective number of communities) can be standardized by the number of sites (Chao et al., 2012), and the resulting dissimilarity indices (i.e., Sorensen or Jaccard) can be partitioned into replacement and nestedness-resultant components (Baselga, 2010b, 2012), which has proven a useful tool to understand how and why communities change in space and time (e.g., Blowes et al., 2019; Chase et al., 2020; Mori et al., 2018; Socolar et al., 2016; Soininen et al., 2018).

To conclude, I would like to emphasize that my criticisms of Song et al.'s contribution should not be understood in terms of right or wrong methods, nor as opposition to the development of novel metrics. However, new measures should be grounded to clear ecological concepts, and the links between measures and concepts should be rigorously assessed. My assessment shows that (1) betavolume does not measure beta diversity or community differentiation, (2) it is not clear if it is linked to any relevant ecological attribute, (3) it is not justified interpreting patterns of variation in betavolume in terms of beta diversity or its turnover and nestedness components, and (4) because betavolume is based purely on occurrence data (in the very same way as beta diversity), its link to species interactions (or any emergent properties related to ecosystem function or stability) should be proven.

The author declares no conflicts of interest.

Code (Baselga, 2025) is available in Figshare at https://doi.org/10.6084/m9.figshare.29340269.