Inferences from constructed examples
Multiple MSTs can result in multiple, different values of FEve index for
the same community if the species have unequal species abundances, which
severely limits the utility of the metric. The following example
demonstrates such a situation. Let the community consist of three
equally distant species (\(s_{1},s_{2}\), and \(s_{3}\)) in a given
trait spaces (i.e.\({\text{dist}\left(s_{1},s_{2}\right)=d}_{12}=d_{13}=d_{23}=d\))
with abundances \(w_{1}=1\), \(w_{2}=2\) and \(w_{3}=3\),
respectively (Fig. 1, community network). There are three MSTs with the
same minimum total distance (\(2d\)): MST1 with one edge
connecting \(s_{1}\) and \(s_{2}\), and one edge connecting \(s_{2}\)and \(s_{3}\) (Fig. 1, MST1); MST2 with
edges connecting \(s_{1}\) and \(s_{2}\), and \(s_{1}\) and \(s_{3}\)(Fig. 1, MST2); and MST3 with edges
connecting \(s_{1}\) and \(s_{3}\), and \(s_{2}\) and \(s_{3}\) (Fig. 1,
MST3). The three trees result in different estimates for
FEev (Fig. 1). Thus, for any community there is a likelihood for
multiple FEev estimates making interpretation of any estimates suspect.
This problem does not arise if all distances for a given network are
different; then there will be only one, unique MST. Such differences in
distances are likely if all or most of the traits are quantitative.
However, such unique values for FEve do not solve the underlying
conceptual problems.
Now consider the three MSTs in Fig. 1 to be three different communities
and the distances between the species no longer identical, but just
very, very slightly different so that each MST is unique for that
community (e.g., for MST1 d 12 =d 23 = 1 and d 13 = 1.0001;
for MST2 d 12 =d 13 = 1 and d 23 = 1.0001;
for MST3 d 13 =d 23 = 1 and d 12 = 1.0001).
Intuition says that the three communities have nearly the same evenness,
and yet they have very different values of FEve.
Additional, hidden pitfalls come about from how FEve is often
calculated. Rather than using the original matrix of pairwise distances,
PCoA or Multidimensional Scaling (MDS) is used first to transform the
distance matrix, and then only the first two or three axes of the
transformed space are considered when calculating species’ distances
(e.g., Mouillot, Villéger, Scherer-Lorenzen, & Mason, 2011; Taudiere,
& Violle, 2016). This transformation generally results in a
distribution of nodes with no equal distances so that the corresponding
MST is unique. However, because of the dimensional reduction, the new
pairwise distances are only approximations of the original ones, and the
corresponding FEve estimate depends on accuracy of PCoA performance
(goodness of fit of the approximations to the original distances). While
one could argue that the problems with FEve can be solved by always
using untransformed distances, doing so does not guarantee a solution to
the other problems listed above.
There is one circumstance that non-unique MSTs result in the same FEve
values. This can happen if all species have exactly the same abundances.
This equality occurs because any two MSTs of a given network have the
same distribution of the edge weights. However, the meaning of FEve in
such an instance is unclear as the purpose of the metric is to measure
variability of abundances in trait space.
A central reason for the problems raised above is that FEve uses only a
fraction of the information contained in the matrix of species
distances. Only \(S-1\) of the\(\frac{\left(S-1\right)\times S}{2}\) pairwise distances are used
in the calculation of FEve; the much larger portion of the distances are
simply ignored. This can cause the same FEve scores for communities with
different patterns of species dispersal in trait space (Fig. 2). In our
example, this result occurs because the distance between species 1 and 3
is ignored. In addition, it is possible to have a community where FEve =
1 even when neither species abundances nor distances between species are
evenly distributed (Fig. 3). In general, complete evenness (FEve = 1) is
realized if and only if all \(\text{PEW}_{\text{ij}}\) values are equal
(eqs. 2, 3), which does not necessarily imply that all distances or all
abundances are equal. This behaviour contradicts the claim of Villéger,
Mason, and Mouillot (2008; p. 2293) that, ”FEve decreases either when
abundance is less evenly distributed among species or when functional
distances among species are less regular.” Their claim is correct as an
absolute statement only if the other factor (abundances or distances)
are held constant, which will not occur when comparing actual
communities.