Brian S. Cade

Determinants of woody cover in African savannas

Savannas are globally important ecosystems of great significance to human economies. In these biomes, which are characterized by the co-dominance of trees and grasses, woody cover is a chief determinant of ecosystem properties. The availability of resources (water, nutrients) and disturbance regimes (fire, herbivory) are thought to be important in regulating woody cover, but perceptions differ on which of these are the primary drivers of savanna structure. Here we show, using data from 854 sites across Africa, that maximum woody cover in savannas receiving a mean annual precipitation (MAP) of less than ∼650 mm is constrained by, and increases linearly with, MAP. These arid and semi-arid savannas may be considered ‘stable’ systems in which water constrains woody cover and permits grasses to coexist, while fire, herbivory and soil properties interact to reduce woody cover below the MAP-controlled upper bound. Above a MAP of ∼650 mm, savannas are ‘unstable’ systems in which MAP is sufficient for woody canopy closure, and disturbances (fire, herbivory) are required for the coexistence of trees and grass. These results provide insights into the nature of African savannas and suggest that future changes in precipitation may considerably affect their distribution and dynamics.

ESTIMATING EFFECTS OF LIMITING FACTORS WITH REGRESSION QUANTILES

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 ...

Model averaging and muddled multimodel inferences

Three flawed practices associated with model averaging coefficients for predictor variables in regression models commonly occur when making multimodel inferences in analyses of ecological data. Model-averaged regression coefficients based on Akaike information criterion (AIC) weights have been recommended for addressing model uncertainty but they are not valid, interpretable estimates of partial effects for individual predictors when there is multicollinearity among the predictor variables. Multicollinearity implies that the scaling of units in the denominators of the regression coefficients may change across models such that neither the parameters nor their estimates have common scales, therefore averaging them makes no sense. The associated sums of AIC model weights recommended to assess relative importance of individual predictors are really a measure of relative importance of models, with little information about contributions by individual predictors compared to other measures of relative importance based on effects size or variance reduction. Sometimes the model-averaged regression coefficients for predictor variables are incorrectly used to make model-averaged predictions of the response variable when the models are not linear in the parameters. I demonstrate the issues with the first two practices using the college grade point average example extensively analyzed by Burnham and Anderson. I show how partial standard deviations of the predictor variables can be used to detect changing scales of their estimates with multicollinearity. Standardizing estimates based on partial standard deviations for their variables can be used to make the scaling of the estimates commensurate across models, a necessary but not sufficient condition for model averaging of the estimates to be sensible. A unimodal distribution of estimates and valid interpretation of individual parameters are additional requisite conditions. The standardized estimates or equivalently the t statistics on unstandardized estimates also can be used to provide more informative measures of relative importance than sums of AIC weights. Finally, I illustrate how seriously compromised statistical interpretations and predictions can be for all three of these flawed practices by critiquing their use in a recent species distribution modeling technique developed for predicting Greater Sage-Grouse (Centrocercus urophasianus) distribution in Colorado, USA. These model averaging issues are common in other ecological literature and ought to be discontinued if we are to make effective scientific contributions to ecological knowledge and conservation of natural resources.

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 d...

Influences of Spatial and Temporal Variation on Fish-Habitat Relationships Defined by Regression Quantiles

Abstract 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 st...

PERMUTATION TESTS FOR LEAST ABSOLUTE DEVIATION REGRESSION

A permutation test based on proportionate reduction in sums of absolute deviations when passing from reduced to full parameter models is developed for testing hypotheses about least absolute deviation (LAD) estimates of conditional medians in linear regression models. Sampling simulations demonstrated that the permutation test on full model LAD estimates had greater relative power (1.06-1.43) than normal theory tests on least squares estimates for asymmetric, chi-square error distributions and symmetric, double exponential error distributions for models with one (n = 35 and n = 63) and three (n 63) independent variables. Relative power was .77 -.84 for normal error distributions. Power simulations demonstrated the low sensitivity of LAD estimates and permutation tests to outlier contamination and heteroscedasticity that was a linear function of X, and increased sensitivity to heteroscedasticity that was a function of X2 for simple regression models. Three permutation procedures for testing partial models in multiple regression were compared: permuting residuals from the reduced model, permuting residuals from the full model, and permuting the dependent variable. Permuting residuals from the reduced model maintained nominal error rates best under the null hypothesis for all error distributions and for correlated and uncorrelated independent variables. Power under the alternative hypotheses for this partial testing procedure was similar to full LAD regression model tests. Four example applications of LAD regression are provided.

Conditional vulnerability of plant diversity to atmospheric nitrogen deposition across the United States

Significance Human activities have elevated nitrogen (N) deposition and there is evidence that deposition impacts species diversity, but spatially extensive and context-specific estimates of N loads at which species losses begin remain elusive. Across a wide range of climates, soil conditions, and vegetation types in the United States, we found that 24% of >15,000 sites were susceptible to N deposition-induced species loss. Grasslands, shrublands, and woodlands were susceptible to species losses at lower loads of N deposition than forests, and susceptibility to species losses increased in acidic soils. These findings are pertinent to the protection of biodiversity and human welfare and should be considered when establishing air quality standards. Atmospheric nitrogen (N) deposition has been shown to decrease plant species richness along regional deposition gradients in Europe and in experimental manipulations. However, the general response of species richness to N deposition across different vegetation types, soil conditions, and climates remains largely unknown even though responses may be contingent on these environmental factors. We assessed the effect of N deposition on herbaceous richness for 15,136 forest, woodland, shrubland, and grassland sites across the continental United States, to address how edaphic and climatic conditions altered vulnerability to this stressor. In our dataset, with N deposition ranging from 1 to 19 kg N⋅ha−1⋅y−1, we found a unimodal relationship; richness increased at low deposition levels and decreased above 8.7 and 13.4 kg N⋅ha−1⋅y−1 in open and closed-canopy vegetation, respectively. N deposition exceeded critical loads for loss of plant species richness in 24% of 15,136 sites examined nationwide. There were negative relationships between species richness and N deposition in 36% of 44 community gradients. Vulnerability to N deposition was consistently higher in more acidic soils whereas the moderating roles of temperature and precipitation varied across scales. We demonstrate here that negative relationships between N deposition and species richness are common, albeit not universal, and that fine-scale processes can moderate vegetation responses to N deposition. Our results highlight the importance of contingent factors when estimating ecosystem vulnerability to N deposition and suggest that N deposition is affecting species richness in forested and nonforested systems across much of the continental United States.

Trophic magnification of PCBs and its relationship to the octanol-water partition coefficient

We investigated polychlorinated biphenyl (PCB) bioaccumulation relative to octanol-water partition coefficient (K(OW)) and organism trophic position (TP) at the Lake Hartwell Superfund site (South Carolina). We measured PCBs (127 congeners) and stable isotopes (δ¹⁵N) in sediment, organic matter, phytoplankton, zooplankton, macroinvertebrates, and fish. TP, as calculated from δ¹⁵N, was significantly, positively related to PCB concentrations, and food web trophic magnification factors (TMFs) ranged from 1.5-6.6 among congeners. TMFs of individual congeners increased strongly with log K(OW), as did the predictive power (r²) of individual TP-PCB regression models used to calculate TMFs. We developed log K(OW)-TMF models for eight food webs with vastly different environments (freshwater, marine, arctic, temperate) and species composition (cold- vs warmblooded consumers). The effect of K(OW) on congener TMFs varied strongly across food webs (model slopes 0.0-15.0) because the range of TMFs among studies was also highly variable. We standardized TMFs within studies to mean = 0, standard deviation (SD) = 1 to normalize for scale differences and found a remarkably consistent K(OW) effect on TMFs (no difference in model slopes among food webs). Our findings underscore the importance of hydrophobicity (as characterized by K(OW)) in regulating bioaccumulation of recalcitrant compounds in aquatic systems, and demonstrate that relationships between chemical K(OW) and bioaccumulation from field studies are more generalized than previously recognized.

Fundamental limits to the accuracy of deuterium isotopes for identifying the spatial origin of migratory animals

Deuterium isotope analyses have revolutionized the study of migratory connectivity because global gradients of deuterium in precipitation (δDP) are expressed on a continental scale. Several authors have constructed continental scale base maps of δDP to provide a spatial reference for studying the movement patterns of migratory species and, although they are very useful, these maps present a static, 40-year average view of the landscape that ignores much underlying inter-annual variation. To more fully understand the consequences of this underlying variation, we analyzed the GNIP deuterium data, the source for all current δDP maps, to estimate the minimum separation in δDP (and latitude) necessary to conclude with a given level of confidence that distinct δDP values represent different geographic sites. Extending analyses of δDP successfully to deuterium in tissues of living organisms, e.g., feathers in migratory birds (δDF), is dependent on the existence of geographic separation of δDP, where every geographic location has a distribution of values associated with temporal variability in δDP. Analyses were conducted for three distinct geographic regions: North America, eastern North America (east of longitude 100°W), and Argentina. At the 80% confidence level, the minimum separation values were 12, 7, and 14° of latitude (equivalent to 53, 31, and 32‰) for North America, eastern North America, and Argentina, respectively. Hence, in eastern North America, for example, one may not be able to accurately assign individual samples to sites separated by less than about 7° of latitude as the distributions of δDP were not distinct at latitudes <7° apart. Moreover, two samples that differ by less than 31‰ cannot be confidently said to originate from different latitudes. These estimates of minimum separation for δDP do not include other known sources of variation in feather deuterium (δDF) and hence are a first order approximation that may be useful, in the absence of more specific information for the system of interest, for planning and interpreting the results of new stable isotope studies.

Genetic diversity and species diversity of stream fishes covary across a land-use gradient

Genetic diversity and species diversity are expected to covary according to area and isolation, but may not always covary with environmental heterogeneity. In this study, we examined how patterns of genetic and species diversity in stream fishes correspond to local and regional environmental conditions. To do so, we compared population size, genetic diversity and divergence in central stonerollers (Campostoma anomalum) to measures of species diversity and turnover in stream fish assemblages among similarly sized watersheds across an agriculture–forest land-use gradient in the Little Miami River basin (Ohio, USA). Significant correlations were found in many, but not all, pair-wise comparisons. Allelic richness and species richness were strongly correlated, for example, but diversity measures based on allele frequencies and assemblage structure were not. In-stream conditions related to agricultural land use were identified as significant predictors of genetic diversity and species diversity. Comparisons to population size indicate, however, that genetic diversity and species diversity are not necessarily independent and that variation also corresponds to watershed location and glaciation history in the drainage basin. Our findings demonstrate that genetic diversity and species diversity can covary in stream fish assemblages, and illustrate the potential importance of scaling observations to capture responses to hierarchical environmental variation. More comparisons according to life history variation could further improve understanding of conditions that give rise to parallel variation in genetic diversity and species diversity, which in turn could improve diagnosis of anthropogenic influences on aquatic ecosystems.