Biodiversity Conservation and Phylogenetic Systematics

162

1

1

2

2

1

2

-=

()-

()-

Î>

Î=

åå

åå

CT

Lzz

NLz

N

i

i

mk

N

im ik

i

i

k

N

ik

T

T

().

B

B

(11c)

The above expression shows that the similarity index CT 2 N(), as in all other
abundance- sensitive similarity measures, is unity if and only if zzij= ik (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 CT 2 N()
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:

UT

DT N

N

qN

q q q

() q

/()/

/

=

éë ùû -()

• ()
  
  
  • 
    
    
    
    • 
      
    
    
    • 
      
    
    
    
    
  
  
  
  

11

11

(^11)
1
b
(12a)
This class of measures quantifies the effective proportion of shared branches in
the pooled assemblage. The corresponding differentiation measure 1 - UTqN()
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 - UT 02 () reduces to the Jaccard-type
UniFrac measure developed by Lozupone and Knight ( 2005 )andthePD-
dissimilarity measure developed by Faith et al. ( 2009 ).
(2b) For q = 1, this measure is identical to the “phylo-Horn” N-assemblage over-
lap measure CT 1 N(); see Table 1.
(2c) For q = 2, we refer to the measure U̅ 2 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:

1

(^211)
2
-=

-

-

=

-

()- ()-

UT

NDT

N

QQ

NTQ

N()

()

bga ,

g

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

general form for any species importance measure (including absolute abun-

dances) is

1

(^21)
2
-= 2

()-

()-

Î>

Î

+

åå

å

UT

Lzz

N NLz

i

i

mk

N

im ik

i

ii

T

T

().

B

B

A. Chao et al.