Checkerboard score

Given two species sp₁, sp₂ and n islands, an incident matrix is built. In the ${\displaystyle 2\times n}$ 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:

${\displaystyle C_{ij}=(r_{i}-S_{ij})(r_{j}-S_{ij})}$

C_ij - C-score for the two species sp₁, sp₂ in the given set of islands

S_ij - The number of co-occurrences of sp₁, sp₂

r_i - Number of islands in which sp₁ has 1

r_j - 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:

${\displaystyle C=\sum _{j=0}^{M}\sum _{i<j}C_{ij}/P}$

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:[3]

${\displaystyle C_{ij}=(r_{i}-S_{ij})(r_{j}-S_{ij})/(r_{i}+r_{j}-S_{ij})}$