Additive partitioning of a beta diversity index is controversial

A recent study (1) reports the spatial clustering patterns of the global distribution of human infectious diseases. However, there are some controversial issues that arise from the beta diversity partitioning method used in the paper. To further debate these issues, we follow the same notations in the paper (1) by writing the Sorensen beta diversity index as ${\beta }_{sor}=\left(b+c\right)/\left(2a+b+c\right)$. Here, a is the number of diseases common to a pair of sites, whereas b and c are the number of diseases that are unique to each of the two sites being compared, respectively.

Murray et al. (1) argue that the partitioned components ${\beta }_{sim}$ and ${\beta }_{nes}$ based on Baselga’s method (2) [${\beta }_{sor}={\beta }_{sim}+{\beta }_{nes}$, where ${\beta }_{sim}=\min \left(b,c\right)/\left(a+\min \left(b,c\right)\right)$ and ${\beta }_{nes}=\left(|b-c|/\left(2a+b+c\right)\right)\times \left(a/\left(a+\min \left(b,c\right)\right)\right)$] can indicate turnover and nestedness, respectively. However, Schmera and Podani (3) have already shown that ${\beta }_{nes}$ derived from Baselga’s partitioning method actually has no connections to any other nestedness indices. Moreover, Carvalho et al. (4) list theoretical and empirical arguments against the use of Baselga’s method (2). In particular, they (4) claim that ${\beta }_{sim}$ overrepresents the replacement component due to the scaling difference of the partitioned components and the Sorensen index in Baselga’s method (one can see that the denominators of ${\beta }_{sim}$, ${\beta }_{nes}$, and ${\beta }_{sor}$ are different).

Based on the above discussion, it is questionable whether Baselga’s method is an adequate way to partition a beta diversity index into separate components with clear ecological interpretations. There is another competitive partitioning framework, which is developed by Podani and Schmera (5). Under this alternative framework, Jaccard beta diversity index [${\beta }_{jac}$ = $\left(b+c\right)/\left(a+b+c\right)$] can be partitioned into relativized species replacement R_rel and relativized richness difference D_rel components as ${\beta }_{jac}$ = R_rel + D_rel, where R_rel = $\left(2\min \left(b,c\right)\right)/\left(a+b+c\right)$ and D_rel = $|b-c|/\left(a+b+c\right)$, respectively (4, 5). The corresponding Sorensen index can be simply partitioned by adjusting the denominator as $2a+b+c$. Consequently, this framework provides a partitioning of beta diversity with direct connection to nestedness and without any scaling problems.

In summary, the definition of nestedness and the scaling issue of partitioned components lead to the controversy about the additive partitioning of a beta diversity index. The authors of the paper (1) should report further results on the comparison of the above two different partitioning methods (2, 5). The two frameworks can present distinct difference when analyzing ecological communities or biogeographic patterns due to their contrasting theoretical foundations.