Modeling Stream Fish Habitat Limitations from Wedge-Shaped Patterns of Variation in Standing Stock

Abstract

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...

A Gentle Introduction to Quantile Regression for Ecologists

“…A regression model with heterogeneous variances implies that there is not a single rate of change that characterizes changes in the probability distributions. Focusing exclusively on changes in the means may underestimate, overestimate, or fail to distinguish real nonzero changes in heterogeneous distributions (Terrell et al 1996;Cade et al 1999).The Dunham et al (2002) analyses relating the abundance of Lahontan cutthroat trout (Oncorhynchus clarki henshawi) to the ratio of stream width to depth illustrates the value of the additional information provided by quantile regression (Figure 1). The ratio was used as a predictor variable because it was an easily obtained measure of channel morphology that was thought to be related to the integrity of habitat in small streams like those typically inhabited by cutthroat trout.…”

Modelling spatial variations in spawning habitat suitability for the sole Solea solea using regression quantiles and GIS procedures

“…Unfortunately this is a common problem with ecological data. For example, Terrell et al (1996) evaluated 35 datasets of stream fish standing stock and found that 13 violated the assumption of homoscedastic error variance required for leastsquares regression. Regression quantiles are also capable of modelling variability in the response distribution through the estimation of a range of quantiles.…”

“…Moreover, least-squares modelling techniques that estimate changes through the centre of the response will consistently underestimate the potential response to the habitat factors considered in the model, and provide only the general direction and shape of the response. A number of methodologies have been explored that are capable of describing the upper bounds of scatterplots, and include partitioned regression (Thomson et al 1996) and regression quantiles (Terrell et al 1996, Cade et al 1999. Whilst still in the developmental stage, Terrell & Carpenter (1997) call for the use of such techniques to improve the efficacy of HSI models.…”

“…Firstly, to define and delineate sole spawning habitat by developing an empirical habitat suitability model within a GIS environment. Secondly, to compare a model developed from central estimates with one based on estimates of the upper bounds of the response through the use of regression quantiles, as described by Terrell et al (1996) and Cade et al (1999). And thirdly, to evaluate the predictive power of the final models of spawning habitat suitability by using an independent dataset describing the distribution of sole eggs within the study area.…”

Ecotoxicological Assessment Model of Cultural Plant-Soil Complex Treated with Waste Water

“…Most contaminants interact with the soil physiochemical structure to a different extent which modifies their influence on soil respiration. As a result of these complex interactions, bivariate scattergrams of soil respiration values versus contaminant concentrations often display a characteristic 'wedge-shape' pattern that suggests contaminants act to limit maximum respiration values (32). the same dose-response curve was obtained when the influence of ni-solutions with different concentration was evaluated (16).…”