A gentle introduction to quantile regression for ecologists

Abstract

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.

The Speed of Invasion: Rates of Spread for Thirteen Exotic Forest Insects and Diseases

“…Quantile regression can fit a model to portray the change in the 50th (median) or any other quantile of the distribution of distance as a function of time. While the 50th quantile provides a measure of the central trend of the data, the upper quantiles, i.e., the 90th or 95th, can be considered the unconstrained rate of spread [34,35].…”

Relationships between annual plant productivity, nitrogen deposition and fire size in low-elevation California desert scrub

“…In order to get a more complete assessment of how fire size responded to our predictor variables of interest, we used a modelling technique called quantile regression to assess the relationships between fire size distribution and the precipitation, N deposition and biomass variables. Quantile regression (Koenker and Bassett 1978) estimates the effects of explanatory variables for different portions of the distribution of a response variable, rather than just modelling the mean response, and has been shown to be a useful technique for analysing a variety of ecological datasets (Cade and Noon 2003), including identifying relationships between wildfire size and climate variables (Slocum et al 2010). A series of modelling functions is estimated at different levels of t, with t representing the fractions of expected response values (e.g.…”

“…Functions can be estimated for a range of t between zero and one, resulting in multiple rates of change (slopes) between an explanatory variable and the response variable, therefore providing a more complete picture of how the distribution of the response is changing given changes in a set of predictor variables. Quantile regression is a semi-parametric modelling technique because no parametric distributional form is assumed for the random error part of the model, whereas a parametric form is assumed for the deterministic part (Cade and Noon 2003). Graphical visualisations of two model variablesfire size and distance to road -suggested that they should be log-transformed before analysis to meet this assumption.…”

Development of integrated habitat indices for bigeye tuna, Thunnus obesus, in waters near Palau

“…Quantile regression is a robust technique that can provide a good approach to detecting how a species may respond to environmental fluctuations (Eastwood et al 2003). When the distribution of parameters is free and even though part of the limiting factors is used, it can more accurately estimate the responses of a species with respect to habitat variables, especially for the upper quantiles in regression models (Cade and Noon 2003).…”