Simple null model analysis subsumes a new species co‐occurrence index: A comment on Mainali et al. (2022)

Recently, Mainali et al. (2022, termed MSSF henceforth) proposed a new index of pairwise species cooccurrence based on the log odds ratio α of conditional occurrence probability. Under the assumption that the total numbers of occurrences of both species are fixed in a number of sites, they demonstrate that the associated random distribution of paired occurrences follows a noncentral hypergeometric distribution, with α being the single unknown parameter. MSSF call the maximum likelihood estimator of α the ‘affinity’ of both species and suggest that it might serve as an index of species cooccurrence. Note that in most of the biodiversity literature, α refers to local (withinsite) species richness. The affinity index α is centred around zero (random association), with positive index values indicating increased numbers of joint species occurrences. For pairs of sites that share species, it behaves similarly to Jaccard's index, with positive values indicating a relatively large number of shared species (species aggregation) and negative values indicating a relatively small number shared species (species segregation). As recognized by MSSF, an analogous probabilistic approach was proposed almost a decade ago by Veech (2013). Recently and independently, the hypergeometric distribution was introduced by Carmona and Pärtel (2020) to estimate dark diversity and by Zhou et al. (2022) in medical bibliometrics. Arita (2016) first noted that Veech's (2013) probabilistic approach is identical to Fisher's exact test for a 2 × 2 matrix of species cooccurrence incidences. Importantly, the affinity index is defined only for individual pairs of species or pairs of sites. But cooccurrence data are usually organized as a matrix, with the occurrence of multiple species (=rows) recorded at multiple sites (=columns). For such matrices, indices of species cooccurrences quantify the level of βdiversity (betweensite differences in species composition). It is unclear how the affinity index would be used with such a matrix, which contains many pairs of species and many pairs of sites. Taking average affinity scores across all possible pairs of species or sites potentially obscures patterns from multiple species (and site) interactions (Chao et al., 2008) and loses information on the ordered loss of species among sites (the degree of nestedness, Baselga, 2010). Like other pairwise indices (e.g. Jaccard, Sørensen and Simpson), the affinity index may be difficult to use or interpret within a biogeographic and ecological context based on multiple sites. MSSF argue that the known sensitivity of common indices of pairwise cooccurrence to the number of occurrences invalidates the use of these traditional indices. They also make the bold claim that ‘half a century of development in analyses of cooccurrence has been marred by failures’ and that this new index will ‘resolve all the aforementioned challenges’. However, these sweeping claims are undermined by three problems: