Material and Methods
The database and the macroecological variables used for the analyses were the same considered in Vieira et al (Vieira et al. 2018), January evapotranspiration (ETJan), June evapotranspiration (ETJune); annual rainfall variation (ARV), primary productivity (PP), annual temperature variation (TempVar) and annual rainfall variation (ARV) and can be accessed by https://doi.org/10.1371/journal.pone.0204114.s005. The stationarity’s quantification of the fish richness-macroecologial variables relationship was done using a Geographically Weighted Regression (GWR) following the protocol indicated in Figure 1. This analysis establishes local estimates of adjustments and regression coefficients using subsets of the database considered and defined according to a Spatial Weighting Function. This function attributes a weight (or importance) to each site, which will be used in the coefficient estimates of a focal point. Thus, sites close to each other (given a connectivity criterion) will have greater importance than sites further away from the focal point, considering the close and far threshold defined by the chosen bandwidth, which in our case is represented by the connectivity between sites. This allows the specification of heterogeneity in relationships and identify regions where the model is more robust, as well as which variables are more important to explain the observed pattern. In this study, connectivity between sampling sites was defined in three ways: i) Euclidian distance between all sites; ii) Euclidian distance between all sites (W Global) present in a same hydrographic basin (W Basin), so the sites located in different basins have a connectivity of zero; and iii) Euclidian distance between all sites present in the same ecoregion (W FEOW), thus the sites in different ecoregions have a connectivity of zero. The number of sites used in local estimates was defined as fixed and the radius (664.05 km to Global; 403.14 km to Basin and 486.706 km to FEOW) that minimized spatial self-correlation was chosen (Figure 1). Ecoregions were considered as defined by Abell et al. (2008).
A way to quantify non-stationarity is using local estimate techniques such as the Geographically Weighted Regression – GWR. It calculates all regression coefficients for each point present in a database, contrasting from the Ordinate Least Square – OLS, which calculates the average parameters for the entire database considered in the analysis. The calculation is done by partitioning the data set into subsets, given a connectivity criterion between points(C. Brunsdon, Fotheringham, and Charlton 1998; Chris Brunsdon et al. 1998). A way of quantifying stationarity level is by using local estimate techniques such as Geographically Weighted Regression – GWR, which calculates and adjusts the coefficients of determination for each data point in the database. If the relationship is stationary, the GWR presents the same coefficients throughout the entire geographical extension. Although GWR presents advantages over OLS regression models, it should not be used as an alternative, but as a supplement to OLS(Osborne et al. 2007). While OLS offers an average global estimate of the relationships, GWR shows the peculiarities present in the database, therefore improving the power to predict and explain mechanisms and processes(Osborne et al. 2007).
To quantify spatial autocorrelation, there was considered the W Global matrix (connectivity criterion) and 17 distance classes (each composed by an equal number of sites). For the W Basin and W FEOW matrices, classes that maintained equal distances between the classes’ centroids were defined. Afterwards, a GWR was generated for each class (using its respective W matrix as the sites’ connectivity criterion) and the Akaike Information Criterion (AIC) was calculated for each model. The Moran’s I and AIC distance class values were plotted on a graph, and the classes with the lowest value for the AIC and Moran’s I equal or close to zero were selected. This procedure was performed for each W matrix, allowing the selection of three GWR models, one W Global model, one W Basin model, and one W FEOW model (Figure 1). The autocorrelation of each model was evaluated with a Moran’s scatterplot. For the best GWR, the global adjustment for the model (r²) was calculated, the spatial autocorrelation of the residuals was measured, and the variable determination coefficients were specified. The GWR was run on the program Spatial Analysis for Macroecology (SAM(Rangel, Diniz-Filho, and Bini 2010)) using the Gaussian Spatial Weighting Function, all models present the Moran’s I value and the Akaike Information Criterion.