Checkerboard score

In biodiversity studies, the checkerboard score or C-score is a statistic which determines the randomness of the distribution of two or more species through a collection of biomes. The statistic, first published by Stone and Roberts in 1990,[1] expands on the earlier work of Diamond[2] that defined a notion of "checkerboard distributions" as an indicator of species competition.

A low c-score indicates a higher randomness, i.e. a greater likelihood that the distribution of one species has not been directly affected by the presence of other species.

## Definition and calculation

Given two species sp1, sp2 and n islands, an incident matrix is built. In the [math]\displaystyle{ 2 \times n }[/math] incident matrix, each row represents one of the two species and each column represents a different island. The matrix is then filled with each cell being set to either 0 or 1. Cell with the value of 0 means that a given species doesn't exist in the given island whilst the value of 1 means that the species do exist in the given island.

The calculation of the co-occurrence of two species sp1, sp2 in the given set of islands is done as follows:

[math]\displaystyle{ C_{ij} = (r_i- S_{ij}) (r_j - S_{ij} ) }[/math]

Cij \- C-score for the two species sp1, sp2 in the given set of islands
Sij \- The number of co-occurrences of sp1, sp2
ri \- Number of islands in which sp1 has 1
rj \- Number of islands in which sp2 has 1

The checkerboard score (c-score) for the colonisation pattern is then calculated as the mean number of checkerboard units per species-pair in the community:

For M species, there are P = M(M-1)/2 species-pairs, so C-score is calculated:

[math]\displaystyle{ C =\sum_{j=0}^{M} \sum_{i\lt j}C_{ij} / P }[/math]

The C-score is sensitive to the proportion of islands that are occupied, thereby confounding comparisons between matrices or sets of species pairs within them. An extension of the C-score therefore standardizes by the number of islands each species-pair occupies using:[3]

[math]\displaystyle{ C_{ij} = (r_i- S_{ij}) (r_j - S_{ij} )/(r_i + r_j - S_{ij}) }[/math]

## References

1. ↑ Stone, Lewis; Roberts, Alan (1990). "The checkerboard score and species distributions". Oecologia 85 (1): 74–79. doi:10.1007/BF00317345. ISSN 0029-8549. PMID 28310957. Bibcode: 1990Oecol..85...74S.
2. ↑ Diamond, JM (1975). "Assembly of species communities". in Cody, ML; Diamond, JM. Ecology and evolution of communities. Belknap Press. pp. 342–444. https://archive.org/details/ecologyevolution00gres.
3. ↑ Novak, Mark; Moore, Jonathan W.; Leidy, Robert A. (2011). "Nestedness patterns and the dual nature of community reassembly in California streams: a multivariate permutation-based approach". Global Change Biology 17 (12): 3714–3723. doi:10.1111/j.1365-2486.2011.02482.x. Bibcode: 2011GCBio..17.3714N.

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