LAUSR

Abstract The relationships between an environmental variable and an ecological response are usually estimated by models fitted through the conditional mean of the response given environmental stress. For example, nonparametric loess and parametric piecewise linear regression model (PLRM) are often used to represent simple to complex nonlinear relationships. In contrast, piecewise linear quantile regression models (PQRM) fitted across various quantiles of the response can reveal nonlinearities in its range of variation across the explanatory variable. We assess the number and positions of candidate breakpoints using loess and compare the relative efficiencies of PLRM and PQRM to quantitatively determine the breakpoints' location and precision. We propose a nonparametric method to generate bootstrap confidence intervals for breakpoints using PQRM and prediction bands for loess and PQRM. We illustrated the applications using data from two aquatic studies suspected to exhibit multiple environmental breakpoints: relating a fish multimetric index of community health (MMI) to agricultural activity in wetlands' adjacent drainage basins; and relating cyanobacterial biomass to total phosphorus concentration in Canadian lakes. Two statistically significant breakpoints were detected in each dataset, demarcating boundaries of three linear segments, each with markedly different slopes. PQRM generated less biased, more accurate, and narrower confidence intervals for the breakpoints and narrower prediction bands than PLRM, especially for small samples and large error variability. In both applications, the relationship between the response and environmental variables was weak/nonsignificant below the lower threshold, strong through the midrange of the environmental gradient, and weak/nonsignificant beyond the upper threshold. We describe several advantages of PQRM over PLRM in characterizing environmental relationships where the scatter of points represents natural environmental variation rather than measurement error. The proposed methodology will be useful for detecting multiple breakpoints in ecological applications where the limits of variation are as important as the conditional mean of a function.