Spatial variables

where n represents the number of sampling sites and xj equals the number of in-dividuals in sample j = 1, 2, 3 ... n. When rescaled to vary between -1 and 1 (Smith-Gill 1975) using the critical values from a chi-square distribution with n - 1 degrees of freedom, the standardised Morisita Index (I_Mstd) is expressed as a likeli-hood ratio where values > 0 indicate that conspecific abundance is aggregated (i.e.

underdispersed), values < 0 indicate a regular (i.e. overdispersed) arrangement in space, and values of -0.5 and 0.5 represent 95% confidence limits around the mean of a randomly dispersed distribution (Veech 2005). Mean values of the index were cal-culated at each sampling scale for non-singleton species and compared to expected values generated by 999 individual-based randomisations (see 2.2.5).

2.2.7 Spatial variables

Space can either be considered as a factor responsible for ecological patterns, or as a confounding variable which inhibits the interpretation of other processes of inter-est. A number of methods have been proposed for explicitly incorporating spatial information into ecological models: Legendre & Troussellier (1988) used Mantel and partial Mantel tests with a matrix of geographic distances to explore the spatial structure of ecologial data. Multiple spatial structures have been modelled simulta-neously by including geographic coordinates as well as all terms of a polynomial or trend-surface equation as explanatory variables within a linear modelling framework (Borcard & Legendre 1994; Borcard et al. 1992; Legendre 1990). The inclusion of

quadratic and cubic terms of the coordinates and their interactions allows linear gra-dients in species distributions as well as more complex patterns, such as patches to be detected. Trend-surface analysis is however limited to the identification of coarse spatial structures that can be represented by simple shapes (eg. planes, parabolas or saddles); attempting to model finer structures rapidly leads to over-parameterisation (Borcard & Legendre 2002). Other shortcomings stem from the correlation between spatial variables leading to a lack of statistical independence and the arbitrary choice of degree for the polynomial function (Dray et al. 2006).

A more recent spatial eigenfunction approach, principal coordinates of neigh-bour matrices (PCNM; Borcard & Legendre 2002; Borcard et al. 2004), identifies and quantifies spatial structures across various scales and addresses many of the shortcomings associated with trend-surface analysis; a set of eigenvectors is gener-ated by principle coordinate analysis (PCoA) from a truncgener-ated matrix of geographic distances between sampling sites. Because eigenvectors are orthogonal and there-fore linearly independent they can be employed as spatial variables to represent distinct spatial patterns at multiple scales. Dray et al. (2006) demonstrated that the positive and negative eigenvalues produced by the PCNM approach are linearly related to Moran’s index of spatial autocorrelation, and thus respectively represent the strength of positive or negative spatial autocorrelation. Eigenvectors associated with the highest eigenvalues correspond to the broadest scales of spatial variabil-ity (e.g. global trends), while eigenvectors with eigenvalues closer to zero capture fine-scale trends (e.g. local patchiness) (Griffith & Peres-Neto 2006).

Moran’s eigenvector maps (MEMs; Dray et al. 2006) provide a generalised frame-work within which to generate a variety of orthogonal spatial variables (eigenvec-tors), of which PCNMs are a particular case. Moran’s eigenvector maps are derived from a spatial weighting matrix W=[w_ij] which is defined as the Hadamard (element-wise) product of a connectivity matrix B=[b_ij] by a weighting matrix A=[a_ij]. The connectivity matrix B is binary, with sites i and j defined as either neighbours (1) or non-neighbours (0). Whereas PCNM classifies two sites as neighbours if their

euclidean distance (dij) is less than the longest distance (t) required to connect all sites by a minimum spanning tree algorithm, MEM generalises this approach by allowing the connectivity matrix to be defined by a variety of connectivity schemes (e.g. Delaunay triangulation, Gabriel graph, relative neighbour, sphere of influ-ence and distance threshold) (see Fortin & Dale 2005; Legendre & Legendre 1998).

The PCNM approach weights the connection between neighbours using the function 1 − (d_ij/4t)². In the MEM approach, the spatial weighting matrix W can be mod-ified by assigning different weighting functions A to various types of connections B to relate the strength of the spatial process (e.g. dispersal) to various spatial structures (i.e. the neighbourhood).

The distribution of sites within TNP MPA and Algoa Bay deviated from that of a regular sampling design. In such cases, the choice of spatial weighting matrix W becomes a critical step, which can greatly influence the outcome of spatial anal-yses. In the absence of a clear theoretical justification for the form of the spatial weighting matrix, this study adopted the recommendations of Dray et al. (2006) and employed a data-driven approach to test the relative performance of various con-nectivity schemes and weighting functions. A bias-corrected multivariate analogue of the Akaike Information Criterion (AIC_c; Burnham & Anderson 2002; Godinez-Dominguez & Freire 2003) was used to select from competing spatial models. Due to the linear relationship between MEMs and Moran’s I, this approach results in the selection of spatial variables (eigenvectors) which maximise Moran’s index of spatial autocorrelation in the response variable.

In order to generate a set of orthogonal spatial variables (MEMs), neighbour-hood relationships within locations were first defined by five different connectiv-ity schemes: Delaunay triangulation (del), Gabriel graph (gab), relative neighbour (rnb), sphere of influence (soi) and minimum spanning tree (mst). Based on the neighbourhood relationship, sites were defined as either neighbours (1) or non-neighbours (0). To model the processes of spatial autocorrelation decaying with dis-tance, spatial weights were assigned to neighbourhood links by evaluating three

de-creasing monotonic functions (sensu Dray et al. 2006): linear (f1 = 1−dij/max(dij)), concave-down (f₂ = 1 − (d_ij/max(d_ij))^a), and concave-up (f₃ = 1/(d_ij)^b). Following the approach of (Caruso et al. 2012), a sequence of integers between 1 and 10 were considered as parameter values for a and b, where the case of a = 1 in f₂is equivalent to a linear weighting function, thus making f₁ redundant. Alternative sets of spatial variables (MEMs), each represented by a spatial weighting matrix, were generated by combining the five binary connectivity matrices (del, gab, rnb, soi and mst) with alternative weighting functions f2 and f3. An additional spatial weighting matrix created using the original PCNM approach of Borcard & Legendre (2002) (pcnm) was also included. Community abundance data (the response matrix) were Hellinger transformed to reduce the influence of rare species and shared absences (Legendre

& Gallagher 2001). Because MEM analysis is inefficient in modelling linear spatial trends, community abundance data were detrended by multiple linear regression on X and Y geographical coordinates to remove the effects of linear gradients. (see Bor-card & Legendre 2002, for details). The best spatial weighting matrix and subset of MEMs to be used in the final spatial model were selected by identifying the model with the lowest AIC_c score. This model included the subset of spatial variables which maximised spatial autocorrelation in the multivariate response variable.