Quantile regression models of animal habitat relationships
Typically, all factors that limit an organism are not measured and included in statistical models used to investigate relationships with their environment. If important unmeasured variables interact multiplicatively with the measured variables, the statistical models often will have heterogeneous response distributions with unequal variances. Quantile regression is an approach for estimating the conditional quantiles of a response variable distribution in the linear model, providing a more complete view of possible causal relationships between variables in ecological processes. Chapter 1 introduces quantile regression and discusses the ordering characteristics, interval nature, sampling variation, weighting, and interpretation of estimates for homogeneous and heterogeneous regression models. Chapter 2 evaluates performance of quantile rankscore tests used for hypothesis testing and constructing confidence intervals for linear quantile regression estimates (0 ≤ τ ≤ 1). A permutation F test maintained better Type I errors than the Chi-square T test for models with smaller n , greater number of parameters p , and more extreme quantiles τ. Both versions of the test required weighting to maintain correct Type I errors when there was heterogeneity under the alternative model. An example application related trout densities to stream channel width:depth. Chapter 3 evaluates a drop in dispersion, F -ratio like permutation test for hypothesis testing and constructing confidence intervals for linear quantile regression estimates (0 ≤ τ ≤ 1). Chapter 4 simulates from a large ( N = 10,000) finite population representing grid areas on a landscape to demonstrate 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 (spatially and not spatially structured). Depending on whether interactions of the measured habitat and unmeasured variable were negative (interference interactions) or positive (facilitation interactions), either upper (τ > 0.5) or lower (τ < 0.5) quantile regression parameters were less biased than mean rate parameters. Sampling ( n = 20 - 300) simulations demonstrated that confidence intervals constructed by inverting rankscore tests provided valid coverage of these biased parameters. Quantile regression was used to estimate effects of physical habitat resources on a bivalve mussel ( Macomona liliana ) in a New Zealand harbor by modeling the spatial trend surface as a cubic polynomial of location coordinates.