Covariate influence in spatially autocorrelated occupancy and abundance data

The autologistic model and related auto-models, commonly applied as autocovariate regression, offer distinct advantages for analysing spatially autocorrelated species distribution or abundance data, allowing simple and direct modelling of the dependence of nearby observations. However, Carl and Kühn (Ecological Modelling, 2007, 207, 159) and Dormann (Ecological Modelling, 2007, 207, 234) questioned their validity: the former analysing simulated and empirical abundance data, the latter presenting theoretical arguments and analysis of simulated data. Further simulation studies were carried out by Dormann et al. (Ecography, 2007, 30, 609) and Beale et al. (Ecology Letters, 2010, 13, 246). These studies reached negative conclusions concerning autocovariate regression, based on numerical evidence of ‘bias’ comprising examples where autocovariate regression yielded much smaller covariate parameter magnitudes than associated linear regressions. In all but the first study, this numerical evidence is erroneous due to use of invalid neighbourhood weighting schemes and, in the auto-Poisson case, invalid application to cooperative interactions. Here we show that even when these technical errors are corrected, a more fundamental conceptual error remains: all four studies are founded on a mathematically incorrect notion of bias, involving direct comparison of parameter estimates across models differing in mathematical structure. We develop a set of simulation-based measures of covariate influence that are directly comparable across models. We apply these to examples from the abovementioned studies where covariate parameter estimates are much smaller for auto-models than for the associated linear models. We find that in these cases, the effect of smaller auto-model parameters is similar to (and consistent with) the corresponding linear model effects, due to a phenomenon within auto-models that we refer to as ‘covariate amplification’. Thus, simple comparison of parameter magnitudes between structurally different models can be highly misleading. We demonstrate that the recent critique of auto-models is entirely unfounded. Correctly applied and interpreted, autocovariate regression provides a practical approach to inference for spatially autocorrelated species distribution or abundance data, while overcoming well-known limitations of generalized linear models.