A simple significance test for quantile regression.

Where OLS regression seeks to model the mean of a random variable as a function of observed variables, quantile regression seeks to model the quantiles of a random variable as functions of observed variables. Tests for the dependence of the quantiles of a random variable upon observed variables have only been developed through the use of computer resampling or based on asymptotic approximations resting on distributional assumptions. We propose an exceedingly simple but heretofore undocumented likelihood ratio test within a logistic regression framework to test the dependence of a quantile of a random variable upon observed variables. Simulated data sets are used to illustrate the rationale, ease, and utility of the hypothesis test. Simulations have been performed over a variety of situations to estimate the type I error rates and statistical power of the procedure. Results from this procedure are compared to (1) previously proposed asymptotic tests for quantile regression and (2) bootstrap techniques commonly used for quantile regression inference. Results show that this less computationally intense method has appropriate type I error control, which is not true for all competing procedures, comparable power to both previously proposed asymptotic tests and bootstrap techniques, and greater computational ease. We illustrate the approach using data from 779 adolescent boys age 12-18 from the Third National Health and Nutrition Examination Survey (NHANES III) to test hypotheses regarding age, ethnicity, and their interaction upon quantiles of waist circumference.