Bray–Curtis dissimilarity

In ecology and biology, the Bray–Curtis dissimilarity, named after J. Roger Bray and John T. Curtis,[1] is a statistic used to quantify the compositional dissimilarity between two different sites, based on counts at each site. As defined by Bray and Curtis, the index of dissimilarity is:

${\displaystyle BC_{ij}=1-{\frac {2C_{ij}}{S_{i}+S_{j}}}}$

Where ${\displaystyle C_{ij}}$ is the sum of the lesser values (see example below) for only those species in common between both sites. ${\displaystyle S_{i}}$ and ${\displaystyle S_{j}}$ are the total number of specimens counted at both sites. The index can be simplified to 1-2C/2 = 1-C when the abundances at each site are expressed as proportions, though the two forms of the equation only produce matching results when the total number of specimens counted at both sites are the same. Further treatment can be found in Legendre & Legendre.[2]

For a simple example, consider two aquariums:

Tank one: 6 goldfish, 7 guppies and 4 rainbow fish.

Tank two: 10 goldfish and 6 rainbow fish.

To calculate Bray-Curtis, let’s first calculate ${\displaystyle C_{ij}}$, the sum of only the lesser counts for each species found in both sites. Goldfish are found on both sites; the lesser count is 6. Guppies are only on one site, so they can’t be added in here. Rainbow fish, though, are on both, and the lesser count is 4. So ${\displaystyle C_{ij}=6+4=10}$.

${\displaystyle S_{i}}$ ( total number of specimens counted on site i) ${\displaystyle =6+7+4=17}$, and

${\displaystyle S_{j}}$ (total number of specimens counted on site j) ${\displaystyle =10+6=16}$.

This leads to ${\displaystyle BC_{ij}=1-(2\times 10)/(17+16)=0.39}$.

The Bray–Curtis dissimilarity is directly related to the quantitative Sørensen similarity index ${\displaystyle QS_{ij}}$ between the same sites:

${\displaystyle {\overline {BC}}_{ij}=1-QS_{ij}}$.

The Bray–Curtis dissimilarity is bounded between 0 and 1, where 0 means the two sites have the same composition (that is they share all the species), and 1 means the two sites do not share any species. At sites with where BC is intermediate (e.g. BC = 0.5) this index differs from other commonly used indices.[3]

The Bray–Curtis dissimilarity is often erroneously called a distance ("A well-defined distance function obeys the triangle inequality, but there are several justifiable measures of difference between samples which do not have this property: to distinguish these from true distances we often refer to them as dissimilarities"[4]). It is not a distance since it does not satisfy triangle inequality, and should always be called a dissimilarity to avoid confusion.