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, 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 formula_1 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:
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:
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: