Ecological Archives E086-171-A2

Raffael Ernst and Mark-Oliver Rödel. 2005. Anthropogenically induced changes of predictability in tropical anuran assemblages. Ecology 86:3111–3118.

Appendix B. A description of the statistical analyses and tests for spatial autocorrelation.

Statistical Analyses

Results of covariance comparisons.―Most statistics are based on the assumption that the values of observations in each sample are independent of one another. Positive spatial autocorrelation may violate this if the samples were taken from nearby areas. We thus tested the assumption of independence of sample units by means of covariance comparisons (Moran’s I and Geary’s C) using the Excel 97/2000 Visual Basic (VB) Add-in ROOKCASE (Sawada 1999) as well as Mantel-tests (see below). For a discussion of statistical analyses of spatial data in ecology see Liebhold and Gurevitch (2002).

The general method of describing autocorrelation in a variable is to compute some index of covariance for a series of lag distances from each point. The resulting correlogram illustrates autocorrelation at each lag distance. Membership in a given distance class is defined by assigning a weight to each pair of points in the analysis. Typically this weight is a simple indicator function. Moran’s I compares the value of a variable at any one location with the value at all other locations. It is similar to the correlation coefficient and varies between –1.0 and + 1.0. When autocorrelation is high, the coefficient is high. A high I value indicates positive autocorrelation. Geary’s C is inversely related to Moran’s I. C values typically range between 0 and 2. If the values of any one zone are spatially unrelated to any other zone, the expected value of C will be 1. Geary’s C does not provide identical inference because it emphasizes the differences in values between pairs of observations, rather than the covariation between pairs. Moran’s I gives a more global indicator, whereas the Geary coefficient is more sensitive to differences in small neighborhoods. In ecological applications, Moran’s I tends to perform better than Geary’s C, capturing known patterns more cleanly and providing more interpretable results (Legendre and Fortin 1989).

Our analysis of covariance revealed that SUs (25-m subunits of each transect) were not spatially autocorrelated based on the assumptions outlined above. All values for Moran’s I were slightly above or below 0, whereas all values for Geary’s C were slightly above or below 1, indicating no significant spatial autocorrelation (Fig. B1).

Mantel test.―Mantel tests are widely used in ecological studies (Douglas and Endler 1982, Burgman 1987, Böhning-Gaese and Oberrath 1999), particularly when assessing the relationship between multivariate community structure and environmental variables (Gascon 1991, Diniz-Filho and Bini 1996, Parris and McCarthy 1999). The simple Mantel correlation addresses the basic ecological question of whether environmentally similar samples are also similar with respect to species composition. This is computed as the correlation between compositional and environmental dissimilarity matrices. Likewise, it is possible to correlate either compositional or environmental dissimilarity to geographic distance, and to test whether there is spatial structure (autocorrelation) in these data (Urban et al. 2002). This autocorrelation is averaged over all distances (Legendre and Fortin 1989).

The Mantel test evaluates the null hypothesis of no relationship between two dissimilarity (distance) or similarity matrices (Mantel 1967). It is an alternative to regressing distance matrices that circumvents the problem of partial dependence in these matrices. This dependence of pairwise elements does not allow for parametric testing of the correlation from matrix comparisons. Therefore Mantel tests are appropriate when more than one distance matrix is to be compared from the same set of sample units. The correlation can be tested by rearranging the rows and columns (simultaneously) of one of the matrices randomly and calculating a new correlation between one of the original matrices and the permuted one. After each permutation the Z statistic is calculated and the resulting values provide an empirical distribution that is used for the significance test. A randomization (Monte Carlo) test can be used for the calculation of the P value. In the case of this study we chose 10,000 permutations for the randomization test.

Original matrices.Generally three types of “primary” matrices were constructed for each separate analysis (1) species by sample unit (SU) matrices with relative species abundance as cell entry; (2) habitat parameters by SU matrices with respective habitat parameter value as cell entry; and (3) SU by SU matrices with true geographic distances between SU in meters as cell entry. This means that a total of 17 distinct matrices (18 + 3 when counting those that are identical; note that geographic distance matrix (GDM) was identical across seasonal analyses within a respective disturbance regime) had to be constructed in order to perform 21 separate pairwise matrix-tests.

With the exception of GDMs which are distance matrices by definition, all “primary” matrices were transformed into distance matrices using the particular distance indices as outlined. The analyses were performed as follows: (A) using the entire data set disregarding season or disturbance status (3 distinct matrices, 3 pairwise tests; species distribution matrix (SDM) vs. environmental characteristic matrix (ECM); SDM vs. GDM; GDM vs. ECM); (B) using the entire data set disregarding season but considering disturbance status (2 × 3 distinct matrices = 2 × 3 pairwise tests; SDM vs. ECM; SDM vs. GDM; GDM vs. ECM for both primary and secondary forest). In B each matrix was distinct from the matrices used in previous analyses and “primary forest matrices” were distinct from “secondary forest matrices”, i.e. they had different cell entries because relative species abundance values had to be calculated separately for each SDM, as the number of individuals and transect hours varied. ECMs represented a “subset” of the matrix used in previous analyses, i.e. split into primary vs. secondary forest. The same holds true for GDM, only distances between SUs within a habitat complex appeared in these matrices; (C) using dry-season data only, evaluating primary vs. secondary forest (2 × 2 distinct matrices; + 2 GDMs from previous analyses because GDM does not vary across seasons = 2 x 3 pairwise tests; SDM vs. ECM; SDM vs. GDM; GDM vs. ECM for both primary and secondary forest). SDM and ECM were distinct matrices, whereas GDMs were identical with GDMs from previous analyses since geographic distance does not vary seasonally; (D) using wet-season data only, evaluating primary vs. secondary forest (2 × 2 distinct matrices; + 2 GDMs from previous analyses because GDM does not vary across seasons = 2 × 3 pairwise tests; SDM vs. ECM; SDM vs. GDM; GDM vs. ECM for both primary and secondary forest). SDM and ECM again were distinct matrices, whereas GDMs remained identical.

SDMs and ECMs always differ between primary and secondary forest and between analysis complexes (A) and (D). GDM, however, only differs between primary forest and secondary forest and between analysis complex (A) and (B) but not between complexes (B) and (D), as geographic distances are fix and therefore not subject to seasonal change. Relative species abundances (individuals/transect hour), however, have to be calculated in every new case since the number of specimens and transect hours varies as do the recorded habitat parameters.

Distance measures.―Matrices of distance used in Mantel-tests were based on relative species abundance data and constructed using the Sørensen quantitative (Bray-Curtis) index.

It is most useful for ecological community data and appears to be a robust measure of the ecological distance between sites (Faith et al. 1987). As compared to Euclidean distance it retains sensitivity in more heterogeneous data sets and gives less weight to outliers (Roberts 1986). For distances of environmental characteristics we used the relativized Euclidian distance (RED), since Euclidian distances are suitable for abiotic data (Legendre and Legendre 1998). RED is conceptually similar to Euclidean distance, except that the data are normalized, thus putting differently scaled variables on the same footing, eliminating any signal other than relative abundance. Geographic distance matrices were based on the actual distances (in meters) between single transect segments (SUs).

 
   Fig. B1. Results of Moran’s I (rhomb) and Geary’s C (triangle) computation using Euclidean coordinates for each SU (coordinates in meters, lag distance 25 m) within single transects and axis residuals of a three-dimensional NMDS habitat model based on original habitat variables (three explanatory axes per transect). Coefficients were calculated for each dimension = axis residual separately, hence producing three values per coefficient per transect.  Means and standard deviations (SD) thus represent the mean and SD of the calculated values for a given transect. P1-P6 = primary-forest transects, S1-S4 = secondary-forest transects.

 

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