The delineation of biogeographical regions (or bioregionalization) consists of grouping localities according to their compositional dissimilarity, and hence in distinguishing among regional faunas and floras with distinct biogeographical histories (Kreft & Jetz, 2010). Delineating biogeographical regions provides important information for conservation planning and presents an opportunity to explore the relative roles of ecological, evolutionary and historical factors in shaping regional pools of species over large spatial scales (Ladle & Whittaker, 2011). Recently, Kreft & Jetz (2010) proposed a quantitative framework to delineate biogeographical regions, based on clustering and ordination techniques. Specifically, they pointed out that measures of species turnover (or species replacement) that are weakly influenced by species richness differences are more informative for the purpose of bioregionalization than classical metrics, such as the Jaccard and Sørensen dissimilarity indices. Kreft & Jetz (2010) therefore recommended the use of the βsim index, which is known to be weakly affected by differences in species richness (see Koleff et al., 2003; Baselga, 2010; Mouillot et al., 2013). For instance, Mouillot et al. (2013) showed that the βsim index minimized the potential confounding effect of the relative magnitude of sampling areas when delineating biogeographical regions, as a sampling design that comprises wide variation in sampling area can itself induce large differences in species richness. The βsim is formulated as follows:
where a is the number of species common to both sites, b is the number of species that occur in the first site but not in the second, and c is the number of species that occur in the second site but not in the first. The βsim index varies between 0 (low dissimilarity, identical or nested taxa lists) and 1 (high dissimilarity, no shared taxa).
The βsim index has recently been criticized by Carvalho et al. (2012), who argued that it overestimates species replacement because it measures replacement relative to the species-poorer site and not as a proportion of all species. Therefore, Carvalho et al. (2012) recommended the use of the β-3 index, which was initially proposed by Cardoso et al. (2009):
According to Cardoso et al. (2009), the β-3 index, which varies between 0 (identical taxa lists) and 1 (no shared taxa), is insensitive to differences in species richness between localities. Similarly to βsim, β-3 is also equal to 0 when the two compared assemblages are nested (e.g. a = 10, b = 0 and c = 5).
In response to Carvalho et al. (2012), Baselga (2012) argued that the β-3 index underestimates species replacement because it accounts for the total number of species in the denominator and not for the total number of species that would potentially be replaced. Baselga (2012) therefore proposed a modified version of the β-3, namely the βjtu index, which is formulated as follows:
The βjtu index measures the proportion of species that would be replaced between assemblages if both had the same number of species and, hence, accounts for species replacement without the influence of differences in richness. The βjtu varies between 0 (low dissimilarity, identical or nested taxa lists) and 1 (high dissimilarity, no shared taxa). Baselga (2012) showed that the closely related βjtu and βsim provided roughly similar results.
Here we used simple numerical examples and an empirical case study (European freshwater fish fauna; Leprieur et al., 2009) to provide a clear understanding of the potential pitfalls associated with the use of the β-3 index in the context of bioregionalization.
Let us consider nine localities (A to I) and the comparisons between the locality A and the localities B to I (see Table 1). The number of species unique to A was kept constant (b = 10) while the number of species unique to the other localities (c) increased from 10 to 40. In the first four comparisons, the number of shared species (a) was equal to 10 while no species were shared among localities for the last four comparisons. First, comparisons between A and B, C, D, E revealed that the β-3 index decreased from 0.66 to 0.33 with increasing differences in species richness, while the number of shared species (a) was constant across comparisons (Table 1). By contrast, the βsim and βjtu indices showed constant pairwise dissimilarity values along this richness gradient (βsim = 0.5 and βjtu = 0.66). Second, comparisons between A and F, G, H, I showed that the β-3 index decreased from 1 (maximum value) to 0.40 with increasing differences in species richness, while no species were shared between the compared localities (Table 1). Again by contrast, the βsim and βjtu indices showed constant and maximal pairwise dissimilarity values even though no species were shared between localities (βsim = 1 and βjtu = 1), and this was the case whatever their differences in species richness.
The fact that β-3 decreased with increasing differences in species richness, even when no species were shared, may clearly be misleading in the context of bioregionalization. For instance, the β-3 indicated that A had as much dissimilarity in species composition with I as with D (β-3 = 0.4, see Table 1). This means that I and D were equally likely to be grouped with A within a hierarchical clustering procedure. Yet, no species were shared between A and I while 10 species were shared between A and D. A required property of a compositional dissimilarity index, namely the ‘complementarity’ property, is that localities without species in common have the largest dissimilarity (e.g. Clarke et al., 2006; Legendre & De Cáceres, 2013). As indicated by Legendre & De Cáceres (2013), compositional dissimilarity indices that violate the ‘complementarity’ property are not suitable for beta diversity studies. This simple numerical example emphasizes that the β−3 index does not respect the ‘complementarity’ property. In contrast to what Cardoso et al. (2009) stated, the β-3 index is not always maximal (i.e. equal to 1) when the two communities being compared share no species (a = 0, see Table 1 and comparison A–I for example). Indeed, an additional condition for the β-3 to be equal to one (maximum) is that the number of species unique to each community must be equal (b = c, see Table 1 and comparison A–F). All evidence indicates that the natural world is characterized by multi-scale gradients of species richness (Field et al., 2009) and so this above condition is almost never fulfilled.
Using the occurrences of 136 native freshwater fish species in 26 major European river basins (see Leprieur et al., 2009; and see Appendix S1a in Supporting Information), we compared the results of clustering obtained using the βsim, βjtu and β-3 indices. For each compositional dissimilarity matrix, we applied a hierarchical clustering analysis (HCA) to produce a dendrogram representing the relative distance between river basins based on the composition of their fish fauna. To do so, we used the unweighted pair-group method using arithmetic averages (UPGMA) linkage method as recommended by Kreft & Jetz (2010). Based on a recently proposed goodness-of-fit measure (the 2-norm; Mérigot et al., 2010), preliminary analyses confirmed that UPGMA provided a more faithful representation of the initial dissimilarity matrix than other linkage methods [unweighted pair-group method using centroids (UPGMC), weighted pair-group method using arithmetic averages (WPGMA), Ward's method, single linkage, complete linkage]. Note here that the dendrogram based on βjtu is not shown because the βsim and βjtu indices provided similar results. Following Kelley et al. (1996), we then used a Kelley–Gardner–Sutcliffe (KGS) penalty function to determine the optimal number of groups of river basins. Last, we performed a Mantel test (999 permutations) to assess the linear relationship between the compositional dissimilarity matrices based on βsim, βjtu and β-3 and the absolute differences in species richness between river basins.
The dendrogram based on βsim (Fig. 1a, Appendix S1b) showed a clear grouping of the four major river basins of the Iberian Peninsula (Ebro, Douro, Tagus and Guadalquivir), hence indicating that the Iberian Peninsula has a unique freshwater fish fauna (Fig. 1a, Appendix S1b). Supporting this result, we found that the average level of species turnover between the 4 Iberian river basins and the 22 other European river basins was very high (average βsim = 0.814). Similarly, the Pô river basin (Italian Peninsula) displayed a distinct freshwater fish fauna according to the dendrogram based on βsim (Fig. 1a, Appendix S1b). In contrast, according to the dendrogram based on β-3, the Iberian river basins were not grouped together, with the exception of the Douro and Tagus river basins (Fig. 1b, Appendix S1c). For instance, the Ebro river basin was found to be as dissimilar in species composition with the Tagus and Douro river basins as it was with the western and central European basins (e.g. Danube, see Fig. 1b). The Guadalquivir and Pô basins were grouped together when they are geographically distant and separated by two major geographical barriers, the Pyrennees and the Alps (Fig. S1c). Indeed, the β-3 index indicated that the Guadalquivir and Pô river basins displayed a medium level of species turnover (β-3 = 0.52), while the βsim index indicated a high level of species turnover (βsim = 0.83). This result based on β-3 could clearly lead to misleading interpretations in the context of bioregionalization as the Guadalquivir and Pô river basins only share 2 species and the number of species unique to each basin is 10 and 26, respectively.
Unlike the results based on βsim, those based on β-3 are not consistent with previous studies showing that the Iberian and Italian peninsulas displayed distinct freshwater fish faunas and a high level of endemism (e.g. Griffiths, 2006; Leprieur et al., 2009). In Europe, spatial discontinuity in fish faunal composition is mainly related to the Pyrenees and Alps, which prevented exchanges of freshwater fish between the Iberian and Italian peninsulas, and the rest of Europe, respectively, in response to past climatic fluctuations (Griffiths, 2006). Despite these dicrepancies, both the βsim and β-3 indices showed the grouping of the river basins of continental Europe (i.e. the group 3, see Fig. 1 and Appendix S1). This result is related to the fact that both the βsim and β-3 indices indicate a low level of species turnover when the degree of nestedness is high (Baselga, 2010; Carvalho et al., 2012), which is the case for the river basins of continental Europe (see Leprieur et al., 2009, for more details).
The Mantel test showed a significant negative correlation between β-3 and differences in species richness between river basins (rM = −0.4314, P < 0.001), indicating that species turnover between river basins decreases with increasing difference in their species richness. By contrast, neither βsim nor βjtu was associated with differences in species richness between river basins (Mantel test: rM = −0.05 and −0.021 for βsim and βjtu, respectively, P > 0.05). Because the above results may be related to a small sample size (n = 26), we also assessed the relationship between βsim, βjtu, β-3 and species richness differences using the data provided by Heikinheimo et al. (2007) on the distribution of European land mammals (124 species in 2183 grid cells). We found a strong negative correlation between β-3 and differences in species richness between grid cells (Mantel test: rM = −0.55, P < 0.001). By contrast, both βsim and βjtu were weakly associated with differences in species richness (Mantel test: rM = −0.163 and −0.157 for βsim and βjtu, respectively, P < 0.001). These results using empirical case studies are not fundamentally surprising (see the numerical examples in Table 1) as the denominator of β-3 reflects species richness differences between localities (i.e. accounts for both b and c, see equation 2). While Cardoso et al. (2009) and Carvalho et al. (2012) claimed that the β-3 index is insensitive to differences in species richness between localities, the current analyses show that this is not the case.
Overall, both the numerical example and the case study emphasize that the β-3 index tends to underestimate the level of spatial species turnover by accounting for species richness differences in the denominator (see also Baselga, 2012), which can lead to spurious associations between localities based on their species composition (e.g. the Guadalquivir and Pô river basins). Furthermore, this index violates the ‘complementarity’ property, which is a prerequisite when analysing patterns and processes of beta diversity (Legendre & De Cáceres, 2013). Based on these results, the β-3 index should not be used to delineate biogeographical regions. By contrast, we recommend the use of the βsim and βjtu indices because they have desirable properties for bioregionalization studies. These indices are indeed weakly sensitive to species richness differences and they also respect the ‘complementarity’ property.