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 approximately 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 approximately 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.

Regression is a common statistical method employed by scientists to investigate relationships between variables. Quantile regression (Koenker and Bassett 1978) is a method for estimating functional relations between variables for all portions of a probability distribution. Although it has begun to be used in ecology and biology (Table 1), many ecologists remain unaware of it, as it was developed relatively recently and is rarely taught in statistics courses at many universities. We present this introduction both to encourage additional applications in ecology and to educate those who are already contemplating using the method.Typically, a response variable y is some function of predictor variables X, so that y = f(X). Most regression applications in the ecological sciences focus on estimating rates of change in the mean of the response variable distribution as some function of a set of predictor variables; in other words, the function is defined for the expected value of y conditional on X, E(y|X). Mosteller and Tukey (1977) noted that it was possible to fit regression curves to other parts of the distribution of the response variable, but that this was rarely done, and therefore most regression analyses gave an incomplete picture of the relationships between variables. This is especially problematic for regression models with heterogeneous variances, which are common in ecology. 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. Quantile regression estimates indicated a nonlinear, negative relationship with the upper 30% (≥ 70th percentiles, P ≤ 0.10 for H 0 : ␤ 1 = 0) of cutthroat densities across 13 streams and 7 years. A weighted least squares regression estimated zero change (90% confidence intervals of -0.014 to 0.012, P = 0.901 for H 0 : ␤ 1 = 0) in mean densities with stream width to depth. If the authors REVIEWS REVIEWS REVIEWS A gentle introduction to quantile regression for ecologists Brian S Cade 1,2 and Barry R Noon 3Quantile regression is a way to estimate the conditional quantiles of a response variable distribution in the linear model that provides a more complete view of possible causal relationships between variables in ecological processes. Typically, all the factors that affect ecological processes are not measured and incl...

Quantile regression is a way to estimate the conditional quantiles of a response variable distribution in the linear model that provides a more complete view of possible causal relationships between variables in ecological processes. Typically, all the factors that affect ecological processes are not measured and included in the statistical models used to investigate relationships between variables associated with those processes. As a consequence, there may be a weak or no predictive relationship between the mean of the response variable (y) distribution and the measured predictive factors (X). Yet there may be stronger, useful predictive relationships with other parts of the response variable distribution. This primer relates quantile regression estimates to prediction intervals in parametric error distribution regression models (eg least squares), and discusses the ordering characteristics, interval nature, sampling variation, weighting, and interpretation of the estimates for homogeneous and heterogeneous regression models.

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.