In a recent Concepts paper in Ecology, Thomson et al. emphasized that assumptions of conventional correlation and regression analyses fundamentally conflict with the ecological concept of limiting factors, and they called for new statistical procedures to address this problem. The analytical issue is that unmeasured factors may be the active limiting constraint and may induce a pattern of unequal variation in the biological response variable through an interaction with the measured factors. Consequently, changes near the maxima, rather than at the center of response distributions, are better estimates of the effects expected when the observed factor is the active limiting constraint. Regression quantiles provide estimates for linear models fit to any part of a response distribution, including near the upper bounds, and require minimal assumptions about the form of the error distribution. Regression quantiles extend the concept of one-sample quantiles to the linear model by solving an optimization problem of minimizing an asymmetric function of absolute errors. Rank-score tests for regression quantiles provide tests of hypotheses and confidence intervals for parameters in linear models with heteroscedastic errors, conditions likely to occur in models of limiting ecological relations. We used selected regression quantiles (e.g., 5th, 10th, . . . , 95th) and confidence intervals to test hypotheses that parameters equal zero for estimated changes in average annual acorn biomass due to forest canopy cover of oak (Quercus spp.) and oak species diversity. Regression quantiles also were used to estimate changes in glacier lily (Erythronium grandiflorum) seedling numbers as a function of lily flower numbers, rockiness, and pocket gopher (Thomomys talpoides fossor) activity, data that motivated the query by Thomson et al. for new statistical procedures. Both example applications showed that effects of limiting factors estimated by changes in some upper regression quantile (e.g., 90-95th) were greater than if effects were estimated by changes in the means from standard linear model procedures. Estimating a range of regression quantiles (e.g., 5-95th) provides a comprehensive description of biological response patterns for exploratory and inferential analyses in observational studies of limiting factors, especially when sampling large spatial and temporal scales.
A wedge-shaped pattern of variation in stream fish standing stock estimates relative to a habitat variable, in which range of standing stocks increases as a function of the variable, is consistent with the concept that the habitat variable is a limiting factor for fish populations. This pattern of variation complicates interpretation of parameter estimates and significance of ordinary least-squares (OLS) regression models of conditional mean standing stock; slopes of these regression models may have little or no relation to slopes of models describing standing stock limits. We modeled standing stock limits by testing for homoscedastic error distributions, screening plots of coordinate pairs for evidence of a wedge-shaped pattern of data, and estimating 90th regression quantiles for simple linear models. Application of this technique to data sets supporting 35 previously published OLS regression models of stream fish standing stocks led to rejection of homoscedasticity (P < 0.10) in 13 of the 35 data sets. Eight of these heteroscedastic data sets had wedge-shaped patterns of variation in standing stock and slopes of 90th regression quantiles that differed from slopes of OLS regression models. For three of these eight data sets, tests rejecting homoscedasticity were more significant than tests rejecting zero slope parameters in OLS regression models. In a separate exercise, analysis of simulated standing stock data generated from known distributions indicated that our technique can detect heteroscedastic error distribution patterns and yield 90th regression quantile models of standing stock limits from data sets characterized by OLS regression as having no correlation between mean standing stock and a habitat variable. Identification of correlations between habitat variables and standing stock by OLS regression is a common method of determining whether a variable is to be used for habitat assessment. Application of our technique to data sets that display wedge-shaped patterns of variation should help identify variables that may be limiting standing stock from data sets that do not yield significant OLS regression models of mean standing stock.Numerous ordinary least-squares (OLS) regres-et al. (1988) indicate a continuing trend for greater sion models have been published that predict mean model precision to be associated with smaller data standing stock of fishes in streams from measures sets. All of the significant (P < 0.05) OLS models of macrohabitat variables such as stream width, described by Layher and Maughan (1985), Lanka average velocity, substrate composition, and total et al. ), Layher and Maughan (1987a, 1987b), dissolved solids. Fausch et al. (1988 reviewed 99 McClendon and Rabeni (1987), Pajak and Neves of these models and found that for small data sets (1987), Wesche et al. (1987), and Hubert and Rahel (especially those with less than 20 df) the models (1989) that had r 2 greater than 0.60 had 22 df or seemed to be precise predictors of mean standing fewer. stock and had coefficients of determinati...
We used regression quantiles to model potentially limiting relationships between the standing crop of cutthroat trout Oncorhynchus clarki and measures of stream channel morphology. Regression quantile models indicated that variation in fish density was inversely related to the width:depth ratio of streams but not to stream width or depth alone. The spatial and temporal stability of model predictions were examined across years and streams, respectively. Variation in fish density with width:depth ratio (10th-90th regression quantiles) modeled for streams sampled in 1993-1997 predicted the variation observed in 1998-1999, indicating similar habitat relationships across years. Both linear and nonlinear models described the limiting relationships well, the latter performing slightly better. Although estimated relationships were transferable in time, results were strongly dependent on the influence of spatial variation in fish density among streams. Density changes with width:depth ratio in a single stream were responsible for the significant (P 0.10) negative slopes estimated for the higher quantiles (80th). This suggests that stream-scale factors other than width:depth ratio play a more direct role in determining population density. Much of the variation in densities of cutthroat trout among streams was attributed to the occurrence of nonnative brook trout Salvelinus fontinalis (a possible competitor) or connectivity to migratory habitats. Regression quantiles can be useful for estimating the effects of limiting factors when ecological responses are highly variable, but our results indicate that spatiotemporal variability in the data should be explicitly considered. In this study, data from individual streams and stream-specific characteristics (e.g., the occurrence of nonnative species and habitat connectivity) strongly affected our interpretation of the relationship between width:depth ratio and fish density.
We used quantile regression to compare the body condition of walleye Sander vitreus and white bass Morone chrysops before (1980–1988) and after (1989–2004) the establishment of alewives Alosa pseudoharengus in Lake McConaughy, Nebraska. Higher quantiles (percentiles = 100% × quantiles (0, 1)) of weight (W) at the same total length (TL) were indicative of better body condition in an allometric growth model that included separate slopes and intercepts for the before and after groups. All quantiles of walleye weights by TL increased in the years after alewife introduction, ranging from 1.01 to 1.12 times weights in the years before alewife introduction, with greatest increases for the lower (<0.50) quantiles and greater TLs. Quantiles up to 0.25 (the lowest 25th percentiles) of white bass weights were reduced in years after alewife introduction for TLs less than 300 mm, ranging from 0.78 to 0.98 times weights in the years before alewife introduction. However, quantiles greater than or equal to 0.50 (the upper 50th percentiles) of white bass weights increased for all TLs, ranging from 1.01 to 1.06 times the pre‐1988 weights. A three‐group analysis, which improved the model fit for longer white bass, indicated a reduction (0.80–1.0) in white bass body condition across all TLs in the first 2 years (1989–1990) after alewife introduction, whereas body condition actually improved (1.02–1.12) across all TLs in later years (1991–2004). Thus, after the establishment of alewives in 1988, walleye body condition improved for all fish at all lengths (the greatest improvement occurring among fish in poorer condition), whereas white bass body condition was initially reduced for all fish at all lengths for 2 years and improved in subsequent years. The approach that we developed for comparing fish body condition before and after a management action in Lake McConaughy could be applied to other weight–length data sets typically evaluated with relative weight indices.
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