binomial_deviance {abdiv}R Documentation

Binomial deviance and CY index of dissimilarity


The binomial deviance dissimilarity and the CY (or Cao) index of dissimilarity were created to compare species counts at sites with moderate to large differences.


binomial_deviance(x, y)

cy_dissimilarity(x, y, base = 10, min_value = 0.1)


x, y

Numeric vectors


Base of the logarithm


Replacement for zero or near-zero values. Values less than min_value are replaced with min_value.


Both of these measures were designed to be used with whole-numbered counts, and may not make sense for comparing normalized vectors or vectors of species proportions.

For two vectors x and y, the binomial deviance dissimilarity is

d(x,y) = \sum_i{ \frac{1}{n_i} \left ( x_i \log{\frac{x_i}{n_i}} + y_i \log{\frac{y_i}{n_i}} - (x_i + y_i) log{2} \right ) },

where n_i = x_i + y_i. This value is the weighted average of the deviance for each species, under a binomial model where the expected counts are n_i / 2 at each site. It was proposed by Anderson and Millar in 2004. Relation to other definitions:

The CY index was proposed by Cao, Williams, and Bark in 1997. For two vectors x and y, the CY index is

d(x,y) = \frac{1}{N} \sum_i \left ( \frac{ (x_i + y_i) \log_{10} ( \frac{x_i + y_i}{2} ) - x_i \log_{10}(y_i) - y_i \log_{10}(x_i) }{ x_i + y_i } \right ),

where N is the total number of species in vectors x and y. Double zeros are not considered in the measure.

When either x_i or y_i are zero, they need to be replaced by another value in the CY index to avoid infinities. Cao suggested replacing zero values with 0.1, which is one log lower than the minimum value for whole-numbered counts. Here, we use a min_value argument to allow the user set a lower limit on the values. For vectors of species counts, this function follows the formulation of Cao by default.

Relation of the CY index to other definitions:


The Binomial deviance or CY index of dissimilarity. The CY index is undefined if all elements of x and y are zero, in which case we return NaN.


Anderson MJ, Millar RB. Spatial variation and effects of habitat on temperate reef fish assemblages in northeastern New Zealand. Journal of Experimental Marine Biology and Ecology 2004;305:191–221.

Cao Y, Williams WP, Bark AW. Similarity measure bias in river benthic Aufwuchs community analysis. Water Environment Research 1997;69(1):95-106.

[Package abdiv version 0.2.0 Index]