Normalized Phylogenetic Similarity Measures

For traditional abundance-based diversity, the most commonly used similarity measures include N-assemblage generalizations of the Jaccard et al. (1966) and MorisitaHorn (Morisita 1959) measures. The latter three measures were integrated into a class of CqN measures by Chao et al. (2008). Jost (2006, 2007), Chao et al. (2008, 2012), and Chiu et al. (2014) have demonstrated that all the above measures are monotonic transformations of beta diversity based on the ordinary Hill numbers. This is an advantage of using the framework of Hill numbers: a direct link exists between diversity and similarity (or differentiation) among assemblages.

Chiu et al. (2014) extended this framework by proposing four classes of similarity (or differentiation) measures that are monotonic functions of phylogenetic beta diversity. The basic idea is that the phylogenetic beta diversity, a ratio of gamma and alpha phylogenetic Hill numbers, is independent of alpha and measures the pure differentiation among assemblages. The phylogenetic beta component always lies in the range [1, N] for any measures of species importance and all orders q ≥ 0. Since the range depends on N, the phylogenetic beta diversity cannot be used to compare phylogenetic differentiation among assemblages across multiple regions with different numbers of assemblages. To remove the dependence on N, several transformations can be used to transform the phylogenetic beta component onto [0, 1] to measure local overlap, regional overlap, homogeneity and turnover. We give a summary of these four transformations below and tabulate formulas and the relationship with previous measures in Table 1 for the two most important classes. The formulas for the special cases for q = 0, 1 and 2 are also displayed there.

1. A class of branch overlap measures from a local perspective:

This gives the effective average proportion of shared branches in an individual assemblage. This class of similarity measures extends the CqN overlap measure derived in Chao et al. (2008) to a phylogenetic version. The corresponding differentiation measure 1CqN (T ) quantifies the effective average proportion of non-shared branches in an individual assemblage.

(1a) For q = 0, this similarity measure is referred to as the “phylo-Sørensen” N-assemblage overlap measure because for N = 2, it reduces to the measure PhyloSør (phylo-Sørensen) developed by Bryant et al. (2008) and Ferrier et al. (2007).

(1b) For q = 1, this measure C1N (T ) is called the “phylo-Horn” N-assemblage overlap measure because it extends Horn (1966) two-assemblage measure to incorporate phylogenies for N assemblages.

(1c) For q = 2, C2 N (T ) is called the “phylo-Morisita-Horn” N-assemblage similarity measure because it extends Morisita-Horn measure (Morisita 1959) to incorporate phylogenies for N assemblages. The differentiation measure 1C2 N (T ) when the species importance measure is relative abundances reduces to the measure proposed by de Bello et al. (2010). However, their measure is valid only for ultrametric trees (p. 7 of de Bello et al. 2010). Here, the measure can be applied to non-ultrametric trees to obtain

where Qγ and Qα are respectively gamma and alpha quadratic entropy, and T is the mean branch length in the pooled assemblage. A general form for any species importance measure (including absolute abundances) is

Table 1 Two major classes of phylogenetic similarity measures based on the transformations of phylogenetic beta diversity when species importance measures are incidences (for q = 0), relative abundances or absolute abundance (for q = 1 and 2). The corresponding differentiation measures are the one-complements of the similarity measures. When all lineages are completely distinct (this includes T ® 0 , ignoring phylogeny), these phylogenetic measures reduce to the corresponding non-phylogenetic versions. All measures can also be applied to non-ultrametric trees if T is substituted for T

The above expression shows that the similarity index C2 N (T ) , as in all other

abundance-sensitive similarity measures, is unity if and only if zij = zik(i.e.,

species importance measures are identical for any node i in the branch set and for

any two assemblages j and k). This reveals that the similarity index C2 N (T ) quantifies the node-by-node resemblance among the N abundance sets {zik; i∈BT̅}, k = 1, 2, …, N from a local perspective. See Fig. 2 of Chiu et al. (2014) for a simple example of the framework.

2. A class of branch overlap measures from a regional perspective:

This class of measures quantifies the effective proportion of shared branches in the pooled assemblage. The corresponding differentiation measure 1-UqN (T ) quantifies the effective average proportion of non-shared branches in the pooled assemblage.

(2a) For q = 0, this measure is called the “phylo-Jaccard” N-assemblage measure because for N = 2 the measure 1-U02 (T ) reduces to the Jaccard-type UniFrac measure developed by Lozupone and Knight (2005) and the PDdissimilarity measure developed by Faith et al. (2009).

(2b) For q = 1, this measure is identical to the “phylo-Horn” N-assemblage overlap measure C1N (T ) ; see Table 1.

(2c) For q = 2, we refer to the measure U2̅ N(T) as a “phylo-regional-overlap”

measure. When the species importance measure is relative abundance, we have the following formula for non-ultrametric trees:

where T denotes the mean branch length in the pooled assemblage. A

general form for any species importance measure (including absolute abundances) is

The numerator is the same as that in C2 N (T ) , revealing that the similarity

index U2 N (T ) also quantifies the node-by-node resemblance among the N abundance sets {zik; i∈BT̅}, k = 1, 2, …, N; but here the denominator (for the purpose of normalization) is different and takes a regional perspective.

3. A class of phylogenetic homogeneity measures

This measure is linear in the proportion of regional phylogenetic diversity contained in a typical assemblage.

(3a) For q = 0, it reduces to the “phylo-Jaccard” measure U0̅ N (T), i.e.,

S0 N (T ) = U0 N (T ) .

(3b) For q = 1, this measure does not reduce to the “phylo-Horn” overlap measure.

(3c) For q = 2, this measure is identical to C2 N (T ) , the “phylo-Morisita-Horn” similarity measure, i.e., S2 N (T ) = C2 N (T ).

4. A class of measures of the complement of “phylogenetic turnover rate”:

This measure in linear in the phylogenetic beta diversity and the corresponding

differentiation measure éë q D (T ) -1ùû / ( N -1) quantifies the relative branch

turnover rate per assemblage.

(4a) For q = 0, the measure V0 N (T ) is identical to the “phylo-Sørensen” measure, i.e., V0 N (T ) = C0 N (T ) .

(4b) For q = 1, this measure does not reduce to the “phylo-Horn” overlap measure.

(4c) For q = 2, this measure is identical to U2̅ N (T), the “phylo-regional-overlap” measure. That is, V2 N (T ) = U2 N (T ) .

As with the phylogenetic diversity measures, all the above similarity or differentiation measures are functions of two parameters: the sensitivity parameter q and the time perspective T. Thus, for each measure, we suggest using the two types of profiles described in section “Phylogenetic Hill numbers and related measures” for the two major similarity measures CqN (T ) and UqN(T) (or their complements) to convey complete information about the similarity or differentiation of a set of assemblages. An example showing the two types of profiles is given in section “An example”.

The lineage excess q D (T ) q D (T ) and the phylogenetic diversity excess

q PD (T ) q PD (T ) can be interpreted as the effective number of regional lineages (or regional phylogenetic diversity) not contained in a typical local assemblage. However, they cannot be directly applied to compare the similarity or differentiation across multiple regions because both depend not only on the number of assemblages, but also on their corresponding alpha diversity. Following Chao et al. (2012) and Chiu et al. (2014, their Appendix D) proved that we can eliminate these dependences by using an appropriate normalization. After proper normalizations, the two measures lead to the same four classes of normalized similarity and differentiation measures as those obtained from the phylogenetic beta diversity. This is another advantage of using the framework of phylogenetic Hill numbers. That is, a consensus can be achieved on phylogenetic similarity and differentiation measures, including N-assemblage phylogenetic generalizations of the classic Jaccard, Sørensen, Horn and Morisita-Horn measures, regardless of whether one prefers multiplicative or additive decompositions.