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Beta diversity measures

The following beta diversity measures are available:

Bray-Curtis: $B = \frac{\sum\limits_{i=1}^{n} \left\vert x_i^A - x_i^B \right\vert}{\sum\limits_{i=1}^{n} (x_i^A -x_i^B )}$
Jaccard: $J = 1 - \frac{\sum\limits_{i=1}^{n} \mathrm{min}(x_i^A, x_i^B)}{\sum\limits_{i=1}^{n} \mathrm{max}(x_i^A, x_i^B)}$
Euclidean: $E = \sum\limits_{i=1}^{n} \sqrt{\left( x_i^A - x_i^B \right)^2}$

where

is the number of OTUs and

and

are the abundances of OTU

in samples

and

, respectively.

The following distances are also available:

Unweighted UniFrac: $d^{(U)} = \frac{\sum\limits_{i=1}^{n} b_i \left\vert I(p_i^A >0) - I(p_i^B > 0) \right\vert}{\sum\limits_{i=1}^{n} b_i}$
Weighted UniFrac: $d^{(W)} = \frac{\sum\limits_{i=1}^{n} b_i \left\vert p_i^A - p_i^B \right\vert}{\sum\limits_{i=1}^{n} b_i (p_i^A + p_i^B)}$
Weighted UniFrac not normalized: $d^{(w)} = \sum\limits_{i=1}^{n} b_i \left\vert p_i^A - p_i^B \right\vert$
D_0 UniFrac: The generalized UniFrac distance $d^{(0)} = \frac{\sum\limits_{i=1}^{n} b_i \left\vert \frac{p_i^A - p_i^B}{p_i^A + p_i^B} \right\vert}{\sum\limits_{i=1}^{n} b_i}$
D_0.5 UniFrac: The generalized UniFrac distance $d^{(0.5)} = \frac{\sum\limits_{i=1}^{n} b_i \sqrt{p_i^A + p_i^B} \left\vert \f... ...^B}{p_i^A + p_i^B} \right\vert}{\sum\limits_{i=1}^{n} b_i \sqrt{p_i^A + p_i^B}}$

where

is the number of branches in the phylogenetic tree,

is the length of branch

;

and

are the proportion of taxa descending from branch

for samples

and

, respectively; and the indicator functions

and

assume the value of

if any taxa descending from branch

is present in samples

and

, respectively, or 0 otherwise.

The unweighted UniFrac distance gives comparatively more importance to rare lineages, while the weighted UniFrac distance gives more important to abundant lineages. The generalized UniFrac distance $d^{(0.5)}$ offers a robust tradeoff [Chen et al., 2012].

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Beta diversity measures