Material and methods
We tested the effect of missing data and the difference between the methods of imputation on the computation of three functional diversity indices at the community level using grassland communities' data. These indices were the community-weighted mean value of the trait (functional identity), its functional range, as well as its functional divergence. The functional range of the traits (difference between the minimum and the maximum) is important to understand the rules of plant community assemblage (Petchey and Gaston 2002, 2006; Mouchet et al. 2010). The functional divergence corresponds to the repartition of the abundance regarding functional identity within a plant community (Mason et al. 2005; Mouchet et al. 2010). We chose the functional divergence index proposed by Schleuter et al. (2010) among the several indices available for the calculation of functional divergence.
The functional traits were extracted from the LEDA trait database (Kleyer et al. 2008), Fig. 3A). We limited the trait selection to four traits (SLA, SM, H, and LDMC) often used in grassland studies. The SLA, H, and SM are, for instance, the traits proposed on the leaf-height-seed (LHS) model of Westoby et al. (2002), which is useful to assess the live strategy of the species. Moreover, LDMC and SLA are important traits in the leaf economic spectrum and are often linked with ecosystem function.
Figure 3. General procedure the assessment of the effects of the imputation methods for the calculation of functional diversity indices. (A) a database without missing data was created from the LEDA database (four traits for 526 species; some replacement of missing values by the dissimilarity methods where necessary); (B) 50 relevés were randomly selected from a large set of relevés (this process was repeated 100 times); (C) 50 relevés and the trait database were crossed and functional diversity indices were computed; (D) missing data were inserted in the trait database with several percentages; (E) missing data were replaced with the dissimilarity and the relationships methods; (E) these corrected databases were crossed with the 50 relevés, and functional diversity indices were computed; (F) the indices computed from database without missing values were compared to the indices computed from the databases with replaced missing values using a Pearson's correlation test.
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The grassland botanical relevés originated from three datasets: one from the Swiss Alps (Peter et al. 2008a,b), one from the Vosges mountains in northeastern France (Plantureux and Thorion 2005), and another from a broader range of regions in France from Atlantic to continental conditions (Michaud et al. 2012). The grassland relevés used to represent a large gradient of ecologic filters (climatic and agricultural management).
Our first attempt involved only relevés where all the species have a value for the four traits in the database. However, only four relevés fell within this constraint. Therefore, to start our test with enough data for the species present in the relevés, the missing trait values in the LEDA database had to be imputed. Imputation was used on 20 species for H (3% of the data), on 136 species for LDMC (22%), 69 species for SM (11%), and 96 species for SLA (15%). The dissimilarity method was used, as it proved satisfactory for the H, LDMC and the SLA in the first part of the study. SM, for which the dissimilarity method was less accurate, had only 11% of missing values. Species unidentified in the surveys and species with missing data for the four traits were omitted. Only the relevés where the abundance of these unidentified species was inferior to 5% of the total abundance were kept. After these modifications, 722 relevés were available with 606 species.
The use of the dissimilarity imputation before the insertion of missing data induced some circularity in the evaluation of the imputation method. However, we think that the circularity is low. This circularity would be very problematic if a trait value was imputed twice the same way. In our work, this probability of double imputation is very low. Indeed, the imputation of one value depends on all the different trait values of the other species and also the missing data on the entire functional trait database. Indeed, the calculation of the dissimilarity would differ between two calculations if the missing data are not exactly on the same trait values. The selection of the close species in the dissimilarity method is related to the calculation of the Gower dissimilarity and so to the distribution of missing data in the functional trait database. Secondly, the calculation of the median of the trait value of the close species depends also on the presence of missing data for the functional trait value of these species.
Different other option could have to use: only use the dominant species in the survey (80% of the abundance) or virtually assemble species. The use of only dominant species would leave out the minor species. If we only interest of the dominant species, the percentage of missing data would be quite low and so the necessity of imputation would be less important.
The creation of artificial species assemblages with only species having a value for the four traits in the database would have yield unrealistic differences in functional diversity indices of the communities, because the majority of these species would have been common and thus ubiquist species. Thus, we consider that replacing some missing trait values in true communities to create a complete database as comparison point for our study was the most appropriate option.
Among these 722 relevés, for each simulation, we randomly selected 50 different relevés. This random selection was made 100 times to have 100 sets of 50 plant communities (Fig. 3B). Each set of relevés was crossed with the functional trait database.
We deliberately inserted missing data in the trait database, by randomly deleting some trait values (Fig. 3D), and so created datasets with different percentages of missing data (1%, 5%, 10%, 20%, 30%, 40%, and 50%). For each percentage, the insertion of missing data was made 100 times (one insertion per set of 50 communities). These missing data were then replaced using the dissimilarity, the relationships, or the MICE method (Fig. 3E) to create functional trait databases with imputed data. We did not examine imputation by the median or the average on the calculation of functional diversity indices, because at the species level, one of the two ecological methods was always better or as good as the two mathematical methods (Table 2). The 50 communities were crossed with these trait databases with different percentages of replaced missing data, and functional diversity indices were computed (Fig. 3F). For the MICE method, the functional diversity indices were computed for each of the five imputations and the average value of these five estimations of the diversity indices was used for the comparison. The indices calculated from the values of the datasets with imputed values were compared to those calculated from the original database (without missing data) using a Pearson's correlation test. From this comparison, we assessed the effect of replacing missing data on the ranking between the functional diversity indices of 50 grasslands. The P value was calculated for each correlation between the two rankings for 100 sets of 50 grasslands. In most studies on functional diversity, the ranking between communities is more important than the absolute value of the functional diversity. We thus focused on the effect of replacing missing data on this ranking. For the discussion, we use the following threshold: If the correlation P value was not significant for five or more of the 100 sets of communities, the results obtained by the imputation methods were considered unsuitable (by similitude with significant threshold at 5%). The percentage of missing data for which this threshold was exceeded was estimated by linear estimation between the simulations with the different levels of missing data.
We also conducted the simulation on the ranking of the communities for their functional diversity indices after deleting the species with a missing value (deletion option, also known as “complete-case analysis”).
Discussion of the effects of the imputation methods on functional diversity indices
The results clearly show the superiority of the tested imputation methods over the deletion of species with missing trait values for the estimation of functional diversity indices of grassland communities. They also show that single imputation methods that can be interpreted in ecological terms or Multivariate Imputation by Chained Equations can be used to replace missing data in a functional trait database to calculate functional diversity indices, with only few effects on the ranking between communities. None of these methods was able to perform best for all the traits and indices tested in this study. With the Multivariate Imputation by Chained Equations, the ranking of the grasslands was robust for all indices for the Height, the SLA, and the LDMC. But the accuracy of the MICE method was not better than the one of the single imputation methods based on ecological hypothesis for the functional identity and functional richness indices. For the Height, LDMC, and SLA, the relationship method performed as well that the MICE. For the SM, the dissimilarity method was the most accurate for the functional identity and range.
Consistently with the results at the species level, the distribution of the trait values seems to be a key parameter in explaining the robustness of the indices to imputation. Indeed, the indices calculated with the SM were more robust when imputation was conducted with the dissimilarity method. The SM exhibited an unbalanced distribution in the database with 606 species in contrast to the other traits. The results for the SM indicate that the MICE method also has to be used with caution for traits with an unbalanced distribution, although this was not obvious at the species level.
Using the dissimilarity method for the SM (unbalanced distribution) and the relationships method for the other traits (balanced distribution), the ranking between grasslands remained robust with up to 30% of the data missing for the functional identity (community-weighted mean), the functional range, as well as the functional divergence. We propose this percentage of missing data as a limit for the utilization of these single imputation methods. In our simulations, we randomly inserted the missing data by deletion. Each species had thus the same probability to have a missing value. The situation usually encountered in ecological studies is that the most common and dominant species have less missing data than the rare and subordinate species. Indices that are more influenced by dominant species than by minor ones (community mean value and function divergence) might therefore be, for the same percentage of missing data, less affected than in our study. For this type of indices, the 30% threshold is therefore conservative. In grassland plant communities, extreme trait values could be carried by dominant as well as by minor species, so that the effect of the repartition of the missing data is probably unsteady for the functional range index. The errors induced by the imputation of missing values has yet to be compared with other errors, such as those induced by the intraspecific variability of functional traits (Albert et al. 2010a,b).
The 8–19% of missing data threshold for the deletion method cannot be compared with the 20% of abundance threshold propose by Garnier et al. (2004). Indeed, they proposed to measure the functional traits of dominant species only (no traits measured for the minor species). In our study, missing data occurred for both dominant and minor species and could affect one or several traits per species.
As discussed in the first part of this study, using the dissimilarity method might underestimate the functional distance between the species. We could therefore suppose that this method could be problematic previous to calculation of the functional range of the communities. However, the imputation was computed on the functional trait database with the 606 species. Species with extreme trait values in a community might not be functionally isolated in the database, so that the imputed values are not necessarily forced toward the median of the community. The ranking of the communities for their functional range was similarly affected by the percentage of replaced data with the dissimilarity as with the relationships or the MICE methods.
Multivariate functional diversity indices like those propose by Villeger et al. (2008) were not tested. Thus, the replacement method proposed here cannot be compared with the method of the Gower dissimilarity follow by a PCoA. However, Gower dissimilarity can only be computed between two species with at least one common trait documented and the PCoA can only be implemented if all the pairwise distances between species are known. This method will therefore only be useful for a low percentage of missing data or/and a large number of traits.