### Abstract

- Top of page
- Abstract
- Distance measures and clustering algorithms
- Numbers of traits
- FD of a sub-assemblage
- Conclusions
- Acknowledgements
- References

Patterns and changes in functional diversity can inform about spatial and temporal variation in trait diversity, about the processes that drive assembly, and whether assemblages are likely to contain redundant species. We recently provided a new measure (termed FD) and detailed its advantages over previous ones. Since then an increasing amount of research effort has been directed towards both developing appropriate measures of functional diversity and critiquing previous ones, including FD. Podani and Schmera (2006) attempt to do both, though here we argue that they accomplish neither. First, they suggest that a particular distance measure and clustering method are appropriate. We suggest that this is not the case, and show that they may have little effect on quantitative patterns in FD. Second, they suggest that values of functional diversity must be insensitive to the number of functional traits used. We do not agree because we can envisage no relevant ecological question. Third, they observe that we originally defined an FD of zero for an empty assemblage, whereas it is more appropriate for single species assemblages to have FD of zero. We agree. Their solution, however, is to create a measure of functional diversity which violates set monotonicity. Our solution is a revised version of FD for which single species assemblages have FD=0, and which does not violate set monotonicity. In conclusion, we are confident that FD behaves appropriately and note that it remains the measure of functional diversity with greatest power to explain variation in ecosystem functioning.

We recently developed a new measure of functional diversity that has clear advantages over functional group richness (FGR), the previously dominant measure of functional diversity (Petchey and Gaston 2002b). We termed this FD and defined it as the total branch length of the functional dendrogram that can be constructed from information about species’ functional traits. The functional dendrogram can be thought of as a description of the functional relationships shared by the species it includes, in the same way as a phylogenetic tree describes phylogenetic relationships (although without any associated inference about evolutionary relationships). Just as one can measure phylogenetic diversity as the total branch length of a phylogeny, one can measure functional diversity as the total branch length of a functional dendrogram.

Since our first publication, further indices of functional diversity have been published, some of which are intended as improvements on FD, and there has been discussion of the strengths and weaknesses of this particular measure (reviewed by Petchey and Gaston 2006). Most recently, Podani and Schmera (2006) made a number of points about dendrogram-based measures of functional diversity, including FD. Some of these are incorrect, some need to be more clearly formulated, and others can be addressed in better ways. The purpose of this article is to refute, clarify, and provide more appropriate solutions to a number of the issues raised by Podani and Schmera (2006).

### Distance measures and clustering algorithms

- Top of page
- Abstract
- Distance measures and clustering algorithms
- Numbers of traits
- FD of a sub-assemblage
- Conclusions
- Acknowledgements
- References

To construct the functional dendrogram from which FD is measured one needs to calculate the pairwise distances between species, and then use a clustering algorithm (Petchey and Gaston 2002b). We purposefully excluded from our original definition of FD a particular measure of pairwise distances or clustering algorithm. Different measures of distance and different clustering algorithms are more or less appropriate depending on the type of trait data used (e.g. binary, nominal, ordinal, continuous) and how species are distributed in trait space (Pielou 1984). We did not, however, set a particularly good example by using Euclidean distances in our empirical examples for mixed types of traits (some nominal, some ordinal, some continuous traits). Podani and Schmera (2006) correctly point this out, and that Gower distance is a measure of distance that can cope with mixed trait data and missing values.

Though technically correct, the implications of using Gower distance instead of Euclidean appear to be relatively minor. The most heterogeneous dataset in our original paper (from Chapin et al. 1996) contains several types of trait. However, there is a very strong correlation between FD measured using Euclidean distance and FD measured using Gower distance (Fig. 1). This results in almost identical relationships between FD and species richness for both distance measures, so that general conclusions about the form of this relationship are insensitive to the distance measure used, and in fact also to the clustering algorithm, as we stated in our original article (Petchey and Gaston 2002b).

Even if it gave different results to Euclidean distance, Gower distance should not be considered optimal in all circumstances, and we advocate a case by case assessment of which distance measure is most suitable. Contrary to Podani and Schmera (2006), who suggest UPGMA as a standard, this also goes for the choice of clustering algorithm. An objective approach to choosing distance measure and clustering algorithm is to select distance measures that can accommodate the types of traits used (e.g. any of several distance measures if traits are all continuous) and clustering algorithms (e.g. UPGMA, UPGMC, WPGMC, single linkage, complete linkage, etc) and to then maximise the correlation between pairwise distances in trait space and pairwise distances across the dendrogram (the cophenetic correlation) (Blackburn et al. 2005). This could, for example, result in Eulicdean distance being chosen over Gower distance (in the case where all traits are continuous).

In a related vein, careful thought needs to be given to the coding of traits. For example, in an unpublished study of avian functional diversity we have a nominal trait called ‘main foraging method’ with the categories ‘pursuit, gleaning, pouncing, grazing, digging, scavenging, and probing’. This could be coded as a single trait, based on the predominant foraging method. However, the foraging methods are not exclusive, with individuals or species potentially employing more than one. Consequently, such information is more appropriately coded as seven independent binary traits. Such cases are likely to be widespread.

### Numbers of traits

- Top of page
- Abstract
- Distance measures and clustering algorithms
- Numbers of traits
- FD of a sub-assemblage
- Conclusions
- Acknowledgements
- References

Podani and Schmera (2006) and others (Mason et al. 2005) suggest that Euclidean distance is an inappropriate distance metric for measuring FD because it is affected by the number of functional traits. Including more traits will, in general, tend to lead to longer Euclidean distances, which will in turn lead to larger values of FD. In practice, this does not seem to be a genuine problem, because we cannot think of any ecological questions that require a comparison of the functional diversity of two assemblages when different numbers of traits are measured for each. Comparisons of FD only seem meaningful when identical traits are measured in the assemblages being considered. Hence, whether the number of traits affect the Euclidean distances or the values of FD is irrelevant.

Another highly questionable comparison is of the FD of two assemblages when the identity of the traits being used is different, whether or not the number of traits remain constant or varies. We are unclear as to what can be learnt from such a comparison. When Podani and Schmera (2006) compare the FD of Patagonian forbs with that of insectivorous birds the FD values for the two assemblages are derived from different niche dimensions (i.e. different niche spaces) with different axes/traits. The comparison seems analogous to asking whether a kilometre is longer than an hour.

### FD of a sub-assemblage

- Top of page
- Abstract
- Distance measures and clustering algorithms
- Numbers of traits
- FD of a sub-assemblage
- Conclusions
- Acknowledgements
- References

How can we calculate the functional diversity of a local assemblage? We originally suggested calculating the functional dendrogram of the regional species pool from which the local assemblage is drawn, and that the FD of that assemblage is the branch length across the dendrogram that is required to connect the species found in the local assemblage to the root of the tree (Petchey and Gaston 2002b). This gives a positive value of FD to an assemblage which comprises a single species (and all single species assemblages have the same value), and an assemblage which contains no species (if strictly speaking such a thing can exist) has a FD of zero. As Podani and Schmera (2006) correctly point out, this is inappropriate.

We have since developed an improved version of FD in which the total branch length is that required only to connect all species to each other across the dendrogram (Fig. 2) (Petchey and Gaston 2006). This may or may not result in species also being connected to the root of the tree (Fig. 2). One consequence of this improvement is that, as Podani and Schmera (2006) propose, an assemblage comprising a single species has an FD of zero. As an aside, the branch length required to connect all species to the root of a phylogenetic tree seems entirely appropriate for measuring the amount of evolutionary history represented in an assemblage (Faith 1992).

Podani and Schmera (2006) suggest an alternate solution to ensure that a local assemblage with no species has an FD of zero. Their method is to recalculate the functional dendrogram from a subset of the trait matrix, where the subset includes only the species in the assemblage (Fig. 2). When there is only one species in the trait matrix, there is no dendrogram, and this seems to be interpreted as meaning that FD is zero.

Unfortunately, this approach to solving one problem creates another. Recalculating the functional dendrogram for each assemblage can result in loss of a species leading to an increase in FD, and conversely in addition of a species leading to a reduction in FD (Podani and Schmera 2006). Technically speaking, this means that the diversity measure is not set monotone, a property of diversity measures that is generally considered to be desirable. Podani and Schmera (2006) observe that ‘… total branch length of trees if used as a measure of FD does not satisfy the set monotonicity property (Ricotta 2005)…’ This is incorrect: it is not the use of total branch length, but the recalculation of the functional dendrogram for each assemblage, that causes the violation of set monotonicity’. Our definition of FD uses total branch length without violating set monotonicity.