Fuentes-Rodríguez: Macroinvertebrate diversity in farm ponds
Appendix S1 Description of the procedure implemented to generate Moran’s eigenvector maps (MEM)
spatial variables.
MEMs are an extension of the previously developed approach known as principal
coordinates of neighbor matrices (PCNM; Borcard & Legendre, 2002; Borcard et al.,
2004). Both methods produce a set of orthogonal spatial variables (eigenvectors), that
can be used as predictors in multiple regression or redundancy analysis (RDA), but
MEM variables are more flexible in case of geographically irregular distribution of sites
(Sattler et al., 2010), as it was the case in our ponds. The more general framework
defined by MEM differ from PCNM in two ways: (1) different connectivity matrices
can be applied to define the links among sampling sites, (2) different spatial weighting
functions among sampling sites can be used. Both connectivity and weighting matrices
define the final spatial weighting matrix, which may be used as a predictor to model
spatial variation of biological data. The choice of both connectivity and weighting
matrices is a critical step since it affects the outcome results of the spatial analysis. As
suggested by Dray, Legendre & Peres-Neto (2006), we applied a data-driven approach
to select the spatial weighting matrix consisting in the following steps: (1) to define a
set of possible spatial weighting matrices (obtained from the different combinations of
connectivity matrices and weighting functions); (2) to compute MEM for each of these
models, (3) to compute RDA of the multivariate biological data with each MEM model
and retain those eigenvectors that result in the most appropriate model according to the
corrected Akaike Information Criterion (AICc, Burnham & Anderson, 2002) and finally
(4) to select the model with the lowest AICc.
Five ways of defining neighbor networks were examined (Dray, Legendre & PeresNeto, 2006): Delaunay triangulation (tri), Gabriel graph (gab), relative neighbourhood
graph (rel), minimum spanning tree (mst) and distance criterion (dnn). For the last case,
Fuentes-Rodríguez: Macroinvertebrate diversity in farm ponds
two sites i and j were considered as neighbourghs if dij < k, where dij is the Euclidean
distance between sites and k is the threshold distance. Ten values of the parameter k
were considered, which were evenly distributed between the shortest value that keeps
all sites connected and the highest distance at which an empirical multivariate
variogram (Wagner, 2003) is significant (Dray, Legendre & Peres-Neto, 2006).
To construct the matrix of weights, we assumed that similarity in assemblages’
composition is higher for pairs of ponds that are spatially closer. Thus, ecological
similarity was supposed to decrease with distance according to the function f = 1 –
(dij/max(dij))y, where max(dij) is the maximum distance defined within a given
neighbour network and y is a parameter. We considered the sequence of integers
between 1 (linear decay of similarity with distance) and 10 (different concave-down
spatial relationships) for y. We computed MEM for the five types of binary connectivity
networks (bin) and for the combinations of these connectivity matrices with the
weighting function and identified the most appropriate spatial weighting matrix
according to AICc.
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Fuentes-Rodríguez: Macroinvertebrate diversity in farm ponds
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