C-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, expands on the earlier work of Diamond
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 $2 \times n$ 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:

:$C_ = (r_i- S_) (r_j - S_ )$

: - C-score for the two species ''sp₁'', ''sp₂'' in the given set of islands
: - The number of co-occurrences of ''sp₁'', ''sp₂''
: - Number of islands in which ''sp₁'' has 1
: - 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 species-pairs, so C-score is calculated:

:$C =\sum_^ \sum_C_ / P$

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:

:$C_ = (r_i- S_) (r_j - S_ )/(r_i + r_j - S_)$

References

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Ecology
Biodiversity