A Gentle Introduction to Quantile Regression for Ecologists

Abstract

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...

Determinants of fruit and vegetable intake in England: a re-examination based on quantile regression

“…They hold particular promise in applications to nutrition problems where dietary excess and/or inadequacy questions beg particular attention to the tails of distributions, although there seem to be only a small number of applications so far (9,10) . Accessible introductions to QR methods are available in Koenker and Hallock (11) and Cade and Noon (12) . Some additional statistical explanation is presented in the Appendix.…”

Benefits of riparian forest for the aquatic ecosystem assessed at a large geographic scale

“…As there was a range of unmeasured factors that were limiting for the response, quantile regression was used to obtain a better estimation and interpretation of the relationship under these circumstances, as this technique can more clearly distinguish changes in heterogeneous distributions (Cade and Noon, 2003). It is also useful to estimate a range of quantiles rather than basing an analysis on only the mean or a single selected quantile (Cade and Noon, 2003).…”

“…As there was a range of unmeasured factors that were limiting for the response, quantile regression was used to obtain a better estimation and interpretation of the relationship under these circumstances, as this technique can more clearly distinguish changes in heterogeneous distributions (Cade and Noon, 2003). It is also useful to estimate a range of quantiles rather than basing an analysis on only the mean or a single selected quantile (Cade and Noon, 2003). Here, we investigated the significance and concordance in the responses for the median and upper quantiles (50th, 75th, 90th and 95th), to see whether a general response for means as a conformity in the response of extreme values and thus an overall response was present.…”

“…In addition, the responses were determined with different statistical approaches; the quantile regression analysis corroborated the responses, showing the presence of the response over the entire gradient, and not merely for the mean values. Quantile regression provides a more complete picture of the relationships between variables missed by other regression techniques, and is especially efficient for datasets with many unmeasured explanatory factors (Cade and Noon, 2003), as is the case for our river quality data. Where the general linear regression pointed out a correlation between forest cover and the macroinvertebrate integrity in ecoregion HER9, the detailed analysis of the response in quantile regression revealed the difference in the responses between the ecoregions.…”

Field surveys can support ecological risk assessment

“…The first approach estimates safe concentrations of individual metals using quantile regression analysis (Cade and Noon 2003) using large field data sets (e.g., >250 sites; Crane et al 2007). Iwasaki and Ormerod (2012) used this approach to suggest the likely ranges of environmental quality standards for metals using field data on macroinvertebrates.…”