Logistic quantile regression for bounded response variables
Bounded response variables (e.g., proportions) in regression models create statistical issues because their conditional distributions are often skewed and heterogeneous, and nonlinear functional forms are required to guarantee the response function remains within the data range. I show how a simple nonlinear logit transformation of a bounded continuous response variable, ln[(y – ymin/(ymax - y)], can readily be accommodated in the usual linear quantile regression model to provide estimates for a logistic quantile model for bounded responses. Estimates for the nonlinear functions are obtained without bias by back- transformation because of the equivariance properties of quantiles. When the bounded responses are counts, these discrete random variables with a limited range of integer values are unlikely to be approximated well by any conventional parametric count distributions (Poisson, negative binomial, and their zero-inflated counterparts). Here, an additional quantile specific transformation to randomly jitter the discrete responses into continuous variables prior to taking the logit transformation can be used to estimate a logistic quantile regression model for bounded counts. We demonstrate features of this model with an analysis of California spotted owl (Strix occidentalis occidentalis) fledgling counts (0 – 3) on the Lassen National Forest.
We modeled 20 years (1991 – 2010) of California spotted owl fledgling counts related to climatic (temperature and precipitation from PRISM), demographic (age of adults, number of young fledged in prior years), and landscape habitat (percent forest cover, etc.) characteristics at owl nesting territories (n = 795 territory-year observations on 87 territories). The full logistic quantile regression model for fledgling counts on owl territories estimated negative effects of prior fledgling production, negative effects of subadult compared to adult parents, negative effects of increasing precipitation and positive effects of increasing minimum temperatures during early nesting (Mar-Apr), negative followed by positive nonlinear effects as winter (Nov-Feb) precipitation increased, negative effects of increasing precipitation in the previous growing season (May-Oct), and positive followed by negative nonlinear effects as cover of large trees (>25m height on 500 acres) on territories increased. Climate explained 9%, landscape habitat explained an additional 5%, and parent age explained an additional 6% of the annual variance in the discrete cdf of fledgling counts (20% of annual variance explained).