Wagner H.H., Dray S. (2015). Generating spatially constrained null models for irregularly spaced data using Moran spectral randomization methods. Methods in Ecology and Evolution, 6(10):1169--1178. ISSN 2041210X. 10.1111/2041-210X.12407.

Spatial autocorrelation jeopardizes the validity of statistical inference, for example correlation and regression analysis. Restricted randomization methods can account for the effect of spatial autocorrelation in the observed data by building it into an empirical null model for hypothesis testing. This can be achieved, for example, based on conditional simulation, which fits a highly parameterized geostatistical model to the observed spatial structure, or, for data observed on a regular transect or grid, with Fourier spectral randomization methods that can flexibly model spatial structure at any scale. This study uses Moran eigenvector maps to extend spectral randomization to irregularly spaced samples. We present different algorithms to perform restricted randomization to suit different types of research questions: individual randomization of each variable, joint randomization of a group of variables while keeping within-group correlations fixed, and randomization with a fixed correlation between original data and randomized replicates (e.g. as input for simulation studies). The performance of the proposed Moran spectral randomization methods for regularly and irregularly spaced samples is assessed with correlation analysis of simulated data. Moran spectral randomization closely matched the spatial structure of original simulated data sets, with identical or nearly identical Moran's I values and power spectra, depending on the algorithm. In correlation analysis of two spatially autocorrelated variables, Moran spectral randomization produced correct type I error rates for stationary spatial data, even for very small and highly irregular samples, but was sensitive to linear trend. When one or both variables lacked spatial structure, Moran spectral randomization tests were more conservative than correlation t-tests. The proposed Moran spectral randomization method requires a minimum of parameterization and is able to address multivariate data with spatial structure at multiple scales, with the option of controlling levels of correlation with the original data. It can provide technically unlimited numbers of randomizations even for small samples while closely maintaining the spatial characteristics of uni- or multivariate data at all spatial scales. The method is applicable for correlation analysis of stationary, autocorrelated spatial or temporal series. Further research should assess whether the method can be extended to multiple regression analysis.

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