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.