A Commentary on Diversity Measures UniFrac in Very Small Sample Size

Abstract

In phylogenetic, the diversity measures as UniFrac, weighted UniFrac, and normalized weighted UniFrac are used to estimate the closeness between two samples of genetic material sequences. These measures are widely used in microbiology to compare microbial communities. Furthermore, when the sample size is large enough, very good results have been obtained experimentally. However, some authors do not suggest using them when the sample size is very small. Recently, it has been mentioned that the weighted UniFrac measure can be seen as the Kantorovich-Rubinstein metric between the corresponding empirical distributions of samples of genetic material. Also, it is well known that the Kantorovich-Rubinstein metric complies the metric definition. However, one of the main reasons to establish it is that the sample size is large enough. The goal of this article is to prove that the diversity measures UniFrac are not metrics when the sample size is very small, which justifies why it must not be used in that case, but yes the Kantorovich-Rubinstein metric.

Keywords: phylogenetic; phylogenetic tree; pseudometric; semimetric.

Figure 1.

(a) Tree for samples $A$ and $B$. (b) Tree for samples $A$ and $B$ with the label leaves for $l_i$ with $i=1,2,3$ ($b_1=l_1,b_2=l_2,a_2=l_3$).

Figure 2.

Tree for samples $A$ and $B$.

Figure 3.

(a) Tree for samples $A$ and $B$. (b) Tree for samples $A$ and $C$. (c) Tree for samples $C$ and $B$.

Figure 4.

Tree for samples $A$ and $B$.

Figure 5.

(a) Tree for samples $A$ and $B$. (b) Tree for samples $A$ and $C$. (c) Tree for samples $C$ and $B$.

Figure 6.

(a) Tree by samples $A$ and $B$. (b) Tree by samples $A$ and $C$. (c) Tree by samples $C$ and $B$.