The first part of the coefficient is the same as S14, i.e. the geometric mean of the proportions of co-occurrence for each of the two species; the second part is a correction for small sample size.
As a probabilistic coefficient for presence-absence data, Krylov (1968) proposed to use the probability associated with the chi-square statistic of the above 2 x 2 frequency table to test the null hypothesis that two species are distributed independently of each other among the various sites. Rejecting H0 leads gives support to the alternative hypothesis of association between the two species. In the case of a 2 x 2 contingency table, and using Yate s correction factor for small samples, the X2 formula is:
The number of degrees of freedom for the test of significance is v = (no. rows - 1) x (no. columns - 1) = 1. The X2 statistic could also be tested by permutation (Section 1.2). Given that associations should be based on positive relationships between pairs of species (negative relationships reflecting competition), Krylov proposed to set S (y1, y2) = 0 when the expected value of co-occurrence, E = (a + b) (a + c) / n, is larger than or equal to the observed frequency (E > a). Following the test, two species are considered associated if the probability (p) associated to their X2 value is smaller than a pre-established significance threshold, for example a = 0.05. The similarity measure between species is the complement of this probability:
S25(y1, y2) = 1 - p(X2), with v = 1, when (a + b)(a + c) / n < a S25(y1, y2) = 0 when (a + b)(a + c) / n > a (7.61)
When the number of sites n is smaller than 20 or a, b, c or d are smaller than 5, Fisher's exact probability formula should be used instead of X2. This formula can be found in most textbooks of statistics
The same formula can be derived from Pearson's 6 (phi) (eq. 7.9), given that
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