Response to Comment on “methods to Account for Spatial Autocorrelation in the Analysis of Species Distributional Data: a Review”

I was amused to read, in a criticism of another publication, that the criticised authors did not adhere to the ‘‘principles of ecological niche modelling’’ (McNyset and Blackburn 2006, p. 782). There aren’t any such ‘‘principles’’. Ecological niche modelling really is statistics on species distribution data. All methods employed need to be statistically consistent, i.e. meet the assumptions made. (It would be a nice bonus if the analysis would also make ecological sense.) In the case discussed below, the crucial assumption is independence of data points, and it is usually violated in spatial distribution data, by spatial autocorrelation. Autologistic regression (Augustin et al. 1996), and its generalisation the autocovariate regression, has up to now been the most popular way to deal with spatial autocorrelation in biogeographical analysis of species distribution (Dormann 2007b). In two recent papers analysing simulated data, this method has, however, performed worse than other methods (Dormann 2007a, Dormann et al. 2007), calling into question the validity of its use. In a comment, Betts et al. (2009, this issue) warn that these conclusions may be incorrect, and that autocovariate regression can be used when applied and interpreted properly. In the following paragraphs, I like to draw attention to two issues relevant in this debate: the use of the word ‘‘space’’, and the ecological interpretation of the values of the autocovariate (and other measures of aggregation or separation from spatial models in general), i.e. what the data tell us about the species. I was seriously confused, when I realised that the autocovariate regression as applied in biogeography may be biased. In our review (Dormann et al. 2007), autocovariate regression was the only method that yielded a consistent bias in the parameter estimation, an observation confirmed by further simulations specifically looking at autologistic regression (Dormann 2007a). However, as Betts et al. pointed out correctly, all these analyses were carried out on the same underlying spatial structure, and if the environmental variable determining the virtual species’ distribution (called ‘‘rain’’) was confounded with its spatial aggregation (what Betts et al. termed ‘‘collinearity of environment with space’’), all these results may be fundamentally flawed. Since several previous studies have shown autologistic regression to be a reliable method (Wu and Huffer 1997, Huffer and Wu 1998, Hoeting et al. 2000, Hooten et al. 2003, Wintle and Bardos 2006), this explanation seems reasonable. So, why don’t I simply concede my error and let everyone use autocovariate regression in peace? There are several lines of argument that indicate that my results may not, after all, be wrong. A brief list: 1) environment and ‘‘space’’ are not correlated, unless we accept Betts et al.’s definition of ‘‘space’’. 2) Confirmation for the method comes only from studies that use an iterative implementation and missing data (see the studies cited above). 3) The idea of the autocovariate has an intrinsic circularity (since the new explanatory variable is constructed from the data to be explained: Dormann 2007a). 4) Two other methods using variables to represent spatial effects in a somewhat comparable way to the autocovariate are unbiased under the simulated conditions (spatial eigenvector mapping SEVM: Griffith and Peres-Neto 2006, Dormann et al. 2007, and a new wavelet method: Carl and Kuhn 2007, Carl et al. 2008). Here, I shall only address the first topic, which I regard as most important to the differences between my view and that of Betts et al. (2009). Spatial models attempt to correct for the non-independence of data points near each other, which may be connected through ecological and non-ecological processes (yielding spatial autocorrelation). To do so, they usually work through some quantification of spatial distances between sites (even if the model formula uses locations, i.e. longitude and latitude as inputs). Location, as one might have naively suspected, is thus not the same as ‘‘space’’ in the sense of spatial models. To avoid confusion, I prefer to use ‘‘neighbourhood’’ instead of ‘‘space’’ to refer to distance-related influences, irrespective of their location. The effect of a point’s neighbourhood on this point can thus be referred to as the effect of ‘‘space’’ (in the words of Betts et al.) or of the neighbourhood. How can this be Ecography 32: 379 381, 2009 doi: 10.1111/j.1600-0587.2009.05907.x # 2009 The Author. Journal compilation # 2009 Ecography