Quantile regression reveals hidden bias and uncertainty in habitat models

We simulated the effects of missing information on statistical distributions of animal response that covaried with measured predictors of habitat to evaluate the utility and performance of quantile regression for providing more useful intervals of uncertainty in habitat relationships. These procedures were evaulated for conditions in which heterogeneity and hidden bias were induced by confounding with missing variables associated with other improtant processes, a problem common in statistical modeling of ecological phenomena. Simulations for a large (N = 10 000) finite population representing grid locations on a landscape demonstrated various forms of hidden bias that might occur when the effect of a measured habitat variable on some animal was confounded with the effect of another unmeasured variable. Quantile (0 ≤ τ ≤ 1) regression parameters for linear models that excluded the important, unmeasured variable revealed bias relative to parameters from the generating model. Depending on whether interactions of the measured and unmeasured variables were negative (interference interactions) or positive (facilitation interactions) in simulations without spatial structuring, either upper (τ > 0.5) or lower (τ < 0.5) quantile regression parameters were less biased than mean rate parameters. Heterogeneous, nonlinear response patterns occurred with correlations between the measured and unmeasured variables. When the unmeasured variable was spatially structured, variation in parameters across quantiles associated with heterogeneous effects of the habitat variable was reduced by modeling the spatial trend surface as a cubic polynomial of location coordinates, but substantial hidden bias remained. Sampling (n = 20-300) simulations demonstrated that regression quantile estimates and confidence intervals constructed by inverting weighted rank score tests provided valid coverage of these parameters. Local forms of quantile weighting were required for obtaining correct Type I error rates and confidence interval coverage. Quantile regression was used to estimate effects of physical habitat resources on a bivalve (Macomona liliana) in the spatially structured landscape on a sandflat in a New Zealand harbor. Confidence intervals around predicted 0.10 and 0.90 quantiles were used to estimate sampling intervals containing 80% of the variation in densities in relation to bed elevation. Spatially structured variation in bivalve counts estimated by a cubic polynomial trend surfa