binomial_deviance {abdiv}  R Documentation 
Binomial deviance and CY index of dissimilarity
Description
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.
Usage
binomial_deviance(x, y)
cy_dissimilarity(x, y, base = 10, min_value = 0.1)
Arguments
x , y 
Numeric vectors 
base 
Base of the logarithm 
min_value 
Replacement for zero or nearzero values. Values less than

Details
Both of these measures were designed to be used with wholenumbered 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:
Equivalent to vegdist() with method = "binomial".
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 wholenumbered 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:
Equivalent to the
vegdist()
function withmethod = "cao"
, ifbase = exp(1)
.
Value
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
.
References
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):95106.