Regression analysis of spatial data

Abstract

Many of the most interesting questions ecologists ask lead to analyses of spatial data. Yet, perhaps confused by the large number of statistical models and fitting methods available, many ecologists seem to believe this is best left to specialists. Here, we describe the issues that need consideration when analysing spatial data and illustrate these using simulation studies. Our comparative analysis involves using methods including generalized least squares, spatial filters, wavelet revised models, conditional autoregressive models and generalized additive mixed models to estimate regression coefficients from synthetic but realistic data sets, including some which violate standard regression assumptions. We assess the performance of each method using two measures and using statistical error rates for model selection. Methods that performed well included generalized least squares family of models and a Bayesian implementation of the conditional auto-regressive model. Ordinary least squares also performed adequately in the absence of model selection, but had poorly controlled Type I error rates and so did not show the improvements in performance under model selection when using the above methods. Removing large-scale spatial trends in the response led to poor performance. These are empirical results; hence extrapolation of these findings to other situations should be performed cautiously. Nevertheless, our simulation-based approach provides much stronger evidence for comparative analysis than assessments based on single or small numbers of data sets, and should be considered a necessary foundation for statements of this type in future.

Strengths and Weaknesses of Quantitative Climate Reconstructions Based on Late-Quaternary Biological Proxies

“…Model validation ideally requires an independent testset but when the two sets of variables (Figs. 1 and 2) being considered (geographical distributions of taxa and broadscale climatic variables) show strong spatial auto-correlation (Beale et al 2010;de Knegt et al 2010), it is impossible to test a MCR model with an independent test-set (Telford & Birks 2005;) because the geographical position of any test-set used will fall within the geographical range of the primary biological data used in the model (Araújo et al 2005a). To address this issue in an MCR approach applied to beetle data, Bray et al (2006) proposed "to avoid circular reasoning in these experiments (to evaluate the accuracy and sensitivity of the MCR method), care was taken to use only samples of modern beetles that have not previously been used in the construction of the primary MCR database".…”

“…Lichstein et al 2002;Araújo et al 2005;Betts et al 2006;Rangel et al 2006;Dormann et al 2007;Dormann 2007aDormann , 2007bDormann , 2007cBeale et al 2010). As in palaeoecology, there is currently a lively debate about the need or otherwise to account for the effects of spatial autocorrelation (e.g.…”

Infrastructure features outperform environmental variables explaining rabbit abundance around motorways

“…To account for this in the models, we used generalized least squares (GLS) regression models with spherical structure for spatial autocorrelation based on the semivariogram values (Beale, Lennon, Yearsley, Brewer, & Elston, 2010; Dormann et al., 2007). …”

Linking vegetation patterns to environmental gradients and human impacts in a mediterranean-type island ecosystem

“…This correspondence is notable because the link between spatial autocorrelation and dispersal limitation is often suggested (e.g. Beale et al 2010), but it is very difficult to quantify at the landscape scale. Other possible explanations for these scales of spatial autocorrelation could relate to past land use or fire history, though we were unable to find records of other possible impacts with a similar spatial scale.…”

“…To account for this spatial autocorrelation in the regression models we used simultaneous autoregression (SAR; Haining 2003). Recent reviews have shown this approach to be successful when used on simulated ecological data, in comparison to other methods (Dormann 2007;Kissling and Carl 2008;Beale et al 2010).…”