Hi Tim, there are several ways of dealing with spatial autocorrelation in ecological models (see e.g. Dormann 2007: Methods to account for spatial autocorrelation in the analysis of species distributional data: a review; and Beale et al. 2010: Regression analysis of spatial data). As always, this is an area of active research, so the right or wrong thing to do is not as clear as it may seem. Some have even concluded that "changes in coefficients between spatial and non-spatial methods depend on the method used and are largely idiosyncratic", so that "researchers may have little choice but to be more explicit about the uncertainty of models and more cautious in their interpretation" (Bini et al. 2009: Coefficient shifts in geographical ecology: an empirical evaluation of spatial and non-spatial regression). Thus, new methods are emerging at a faster rate than people studying and comparing their properties. Nonetheless, I think some observations are useful: 1) there are methods explicitly designed to detect spatial autocorrelation such as Moran's autocorrelograms or variograms (available in several R packages). As already pointed out, the autocorrelation function is "well behaved" with linear, equally spaced series in time or space; 2) minimal adequate model selection with the AIC is sensitive to residual autocorrelation; it tends to generate unstable and overfitted models. Thus, when applying any model selection procedure, you should account for the uncertainty in the process by averaging the model set (or model predictions) with respect to the relative support of each model (e.g. Akaike weights). Since you have a large sample, you could account for residual spatial autocorrelation using eigenvector filtering, which produces synthetic variables that capture spatial patterns and can be included in linear models as explanatory variables - a more intuitive approach if you don't want to mess with random factors (see e.g. Diniz-Filho et al. 2008: Model selection and information theory in geographical ecology). This can be implemented with packages spacemakeR and vegan; 3) autocorrelation in model residuals is not the only - nor most important - problem in biological modeling; model misspecification is the major issue. Residual autocorrelation often arises due to not including relevant explanatory variables, interaction terms, assuming an inappropriate response shape and/or an inadequate variance structure, or any combination of these. All these things need to be checked for proper model validation, for instance by partial regression plots (or added-varialbe plots), which help you see the shape of the response to each explanatory variable after account for the variation in the remaining explanatory set. At the same time, you could plot the residuals againts both model predictions and explanatory variables. Since residuals are the stochastic component of the model ("noise"), its relation with the systematic components should be random; clear patterns in these plots are indications of misspecification. Finally, all this "model tinkering" is based on two fundamental premises: you want to model a mean tendecy of response and the pattern of variation around it. These are strictly statistical properties of data - they have nothing to do with biology. If you don't really believe the biological process you are studying implies a mean response, but rather e.g. a maximum one (such as in population or abundance limitation), than all these methods will actually induce you to misspecify the model, but there are alternatives (see e.g. Cade et al. 2005 - Quantile regression reveals hidden bias and uncertainty in habitat models).
2011/8/25, Tim Seipel <t.sei...@env.ethz.ch>: > > Dear List, > I am trying to determine the best environmental predictors of the > presence of a species along an elevational gradient. > Elevation ranges from 400 to 2050 m a.s.l. and the ratio of presences to > absences is low (132 presences out 2800 samples) > > So to start I fit the full model of with the variable of interest. > > sc.m<-glm(PA~sp.max+su.mmin+su.max+fa.mmin+fa.max+Slope+Haupt4+Pop_density+Dist_G+Growi_sea+,data=sc.pa,'binomial') > > First, I performed univariate and backward selection using Akaike > Information Criteria, and the fit was good and realistic given my > knowledge of the environment though the D^2 was low 0.08. My final model > was: > --------------------------------- > glm(formula = PA ~ Slope + sp.mmin + su.max + fa.mmin + Haupt4, > family = "binomial", data = sc.pa) > > Deviance Residuals: > Min 1Q Median 3Q Max > -0.5415 -0.3506 -0.2608 -0.1762 3.0768 > > Coefficients: > Estimate Std. Error z value Pr(>|z|) > (Intercept) -73.45212 23.13842 -3.174 0.00150 ** > Slope -0.03834 0.01174 -3.265 0.00109 ** > sp.mmin -15.34594 5.30360 -2.893 0.00381 ** > su.max 5.09712 1.70332 2.992 0.00277 ** > fa.mmin 13.52262 4.64021 2.914 0.00357 ** > Haupt42 -0.72237 0.27710 -2.607 0.00914 ** > Haupt43 -0.95730 0.37762 -2.535 0.01124 * > Haupt44 -0.25357 0.24330 -1.042 0.29731 > --- > Null deviance: 958.21 on 2784 degrees of freedom > Residual deviance: 896.10 on 2777 degrees of freedom > AIC: 912.1 > > ---------------------- > > I then realized that my residuals were all highly correlated (0.8-0.6) > when I plotted them using acf() function. > > So to account for this I used glmmPQL to fit the full model: > > model.sc.c <- glmmPQL(PA ~ > sp.mmin+su.mmin+su.max+fa.mmin+Slope+Haupt4+Pop_density+Dist_G+Growi_sea, > random= > ~1|group.sc, data=sc.dat, family=binomial, correlation=corAR1()) > > However, the algorithm failed to converge and all the p-vaules were > either 0 or 1 and coefficient estimates approached infinity. > Additionally the grouping factor of the random effect is slightly > arbitrary and accounts a tiny amount of variation. > > --- > So know I feel stuck between a rock and a hard place, on the one hand I > know I have a lot of autocorrelation and on the other hand I don't have > a clear way to include it in the model. > > I would appreciate any advice on the matter. > > Sincerely, > > Tim > > [[alternative HTML version deleted]] > > _______________________________________________ > R-sig-ecology mailing list > R-sig-ecology@r-project.org > https://stat.ethz.ch/mailman/listinfo/r-sig-ecology > -- Pedro A. C. Lima Pequeno Programa de Pós-graduação em Ecologia Instituto Nacional de Pesquisas da Amazônia Manaus, AM, Brasil _______________________________________________ R-sig-ecology mailing list R-sig-ecology@r-project.org https://stat.ethz.ch/mailman/listinfo/r-sig-ecology