Measures for presence/absence and ordinal data

The measures below are suitable for presence/absence (binary) data and ordered classes (e.g. "absent", "low abundance", "moderate abundance", "high abundance"). Note that ordered classes would be represented by dummy variables in order for these measures to be used. 

The coefficients suggested are usually similarity coefficients (S). Their one-complement (1-S) is their corresponding dissimilarity or, in the case of metric coefficients, distance coefficient. The "Sn notation was adopted by Legendre and Legendre (1998) and is reproduced below to assist cross-referencing.

Jaccard coefficient  Metric The Jaccard similarity coefficient assess the degree of overlap between two objects, ignoring double zeros (e.g. double absences). It is the quotient of the number of double presences ("1,1"s) and the sum of double presences and differences ("1,0"s and "0,1"s). When dealing with OTUs or species, its one complement may be used to assess turnover.

S8, S9, S13, S14  Semimetric Semimetric variants of the Jaccard coefficient give differing degrees of weight to double presences. Sørensen/Dice coefficient (S8), for example, is identical to Jaccard's coefficient, however, the number of double presences is doubled prior to calculating similarities. Increasing the weight of double presences is asserting that the co-occurrence/coincidence of variables is more informative than either differences or double absences. Coefficient S9 triples the weight of double presences.

Kulczynski coefficient  Nonmetric  It is the quotient of double presences ("1,1"s) and the sum of differences ("1,0"s and "0,1"s). This coefficient is non-linear: it does not respond linearly to a linear change in variable values. As such,Legendre and Legendre (1998) caution against its use.

S27  Probabilistic This measure, described by Legendre and Legendre (1998) and related to the work of Raup and Crick (1979) and McCoy et al. (1986), is the one complement of the probability that the number of double presences shared between two sites is greater than can be expected by chance. The null probabilities are determined by permutation. (page 273