A Permutation Test for Quantile Regression

A drop in dispersion, F -ratio like, permutation test (D) for linear quantile regression estimates (0 ≤ τ ≤ 1) had relative power ≥1 compared to quantile rank score tests (T ) for hypotheses on parameters other than the intercept. Power was compared for combinations of sample sizes (n = 20 − 300) and quantiles (τ = 0.50 − 0.99) where both tests maintained valid Type I error rates in simulations with p = 2 and 6 parameters in homogeneous and heterogeneous error models. The D test required two modifications of permuting residuals from null, reduced parameter models to maintain correct Type I error rates when null models were constrained through the origin or included multiple parameters. A double permutation scheme was used when null models were constrained through the origin and all but 1 of the zero residuals were deleted for null models with multiple parameters. Although there was considerable overlap in sample size, quantiles, and hypotheses where both the D and rank score tests maintained correct Type I error rates, we identified regions at smaller n and more extreme quantiles where one or the other maintained better error rates. Confidence intervals on parameters for an ecological application relating Lahontan cutthroat trout densities to stream channel width:depth were estimated by test inversion, demonstrating a smoother pattern of slightly narrower intervals across quantiles than those provided by the rank score test.