Conceptual problems
Functional evenness (FEve) is based on a minimum spanning tree (MST) of
a complete, undirected network of S vertices (species) with edges
weighted by distance. An MST links all vertices through \(S-1\) edges
such that there are no cycles, i.e., there is only one pathway between
any two species. For S vertices there are \(S^{S-2}\) possible
spanning trees. The MST is the tree with the minimum possible total sum
of the distances between all pairs of connected vertices (species).
Importantly, several MSTs with the same minimum total distance may exist
for a given network, if there are edges with the same distances. Such
equal distances are highly likely if most of the traits are categorical
or meristic (counts). At the extreme, if all edges are of equal
distance, there are \(S^{S-2}\ \)MSTs.
If FEve is to be used as a measure of some property of biodiversity, its
conceptual basis needs to be described and justified. In particular,
what is the reasoning for the use of MST-edges in combination with
abundances as a functional characteristic? Which functional
characteristic is addressed by this combination? In what sense is it a
measure of evenness? In addressing these questions, we uncover two
conceptual problems: (1) the possibility of non-uniqueness of MSTs and,
(2) its use as an index of evenness.
Given a particular MST with S nodes (species \(s_{i}\),\(i=1,2,\ldots,S\)), FEve is calculated as follows. First, each edge
linking species \(s_{i}\) and \(s_{j}\) with functional distance\(d_{\text{ij}}=dist\left(s_{i},s_{j}\right)\) between them is
weighted by the sum of their abundances (\(w_{i}\) and \(w_{j}\)):
\(\text{EW}_{\text{ij}}=\frac{\text{dist}\left(s_{i},s_{j}\right)}{w_{i}+w_{j}}=\ \frac{d_{\text{ij}}}{w_{i}+w_{j}}\). (1)
Second, those weighted edges are normalized by the sum of the\(\text{EW}_{\text{ij}}\) values for the corresponding MST:
\(\text{PEW}_{\text{ij}}=\frac{\text{EW}_{\text{ij}}}{\sum_{\left(i,j\right)=1}^{S-1}\text{EW}_{\text{ij}}}\), (2)
where \(\left(i,j\right)\) designates an edge between species\(s_{i}\) and \(s_{j}\). (Because of this normalization, either relative
or absolute abundances can be used.) Finally, FEve is calculated as:
\(FEve=\frac{\left[\sum_{\left(i,j\right)=1}^{S-1}{\min\left(\text{PEW}_{\text{ij}},\frac{1}{S-1}\right)}-\ \frac{1}{S-1}\right]}{\left[1-\ \frac{1}{S-1}\right]}\), (3)
which takes values between 0 and 1 (the denominator is the theoretically
possible maximum value of the numerator). According to Villéger et al.
(2008), ”our new functional evenness index measures both the regularity
of branch lengths in the MST and evenness in species abundances.” From
context, it is also clear that the authors intended the MST branches
(edges) to connect nearest neighbors.
The authors do not explicitly identify the characteristics and objects
that are the focus of their metric. We do so as follows. The combination
of an edge plus abundances (PEWij values) serves
as the characteristic of interest, with pairs of species being the
objects (eq. 1). Evenness of these objects (eq. 2) is the focus of the
metric. Evenness is quantified as a deviation of the relative
representations from their associated uniform distribution (the
numerator in eq. 3).
This approach has several conceptual problems. First, abundance-edge
combinations (EWij values) do not necessarily
represent evenness relationships between nearest neighbors. The internal
nodes of an MST have at least two connecting edges. If one edge is
smaller than another (e.g., \(d_{\text{ij}}<d_{\text{ik}}\)), its
abundance-weighted representation can be larger than that of the second
edge (\(E_{\text{ij}}>E_{\text{ik}}\)) when the sum of abundances of
the corresponding species is sufficiently larger
(\(w_{i}+w_{k}\gg w_{i}+w_{j}\)). Depending on how species
abundances are distributed along MST nodes, it is possible that none of
those abundance-edge pairs on the MST represent nearest neighbors.
Therefore, estimation of functional evenness with FEve does not really
mirror a concept of measuring functional variability based on the
functionally nearest types (species).
Second, the authors state that ”to transform species distribution in a
T-dimensional functional space to a distribution on a single axis, we
choose to use the minimum spanning tree”. No reasoning is given for why
such a transformation is required. Nor is it explained in what way an
MST can be considered as yielding a distribution on a single axis, given
that nodes can connect to more than two others.
Third, the functional relevance of combining MST-distances (edge values)
with abundances is simply assumed. The use of abundances assumes that
any and all functional traits have a similar per capita functional
effect.
Fourth, the potential for a single set of points to have multiple MSTs
is ignored. While one can demonstrate that the distribution of edge
values is the same for all alternative MSTs, this property is lost when
they are combined with species abundances. As we show in the next
section, such combinations can lead to more than one value of evenness
for the same data set.