Permutation Tests for Least Absolute Deviation Regression

A permutation test based on proportionate reduction in sums of absolute deviations when passing from reduced to full parameter models is developed for testing hypotheses about least absolute deviation (LAD) estimates of conditional medians in linear regression models. Sampling simulations demonstrated that the permutation test on full model LAD estimates had greater relative power (1.06-1.43) than normal theory tests on least squares estimates for asymmetric, chi-square error distributions and symmetric, double exponential error distributions for models with one (n = 35 and n = 63) and three (n 63) independent variables. Relative power was .77 -.84 for normal error distributions. Power simulations demonstrated the low sensitivity of LAD estimates and permutation tests to outlier contamination and heteroscedasticity that was a linear function of X, and increased sensitivity to heteroscedasticity that was a function of X2 for simple regression models. Three permutation procedures for testing partial models in multiple regression were compared: permuting residuals from the reduced model, permuting residuals from the full model, and permuting the dependent variable. Permuting residuals from the reduced model maintained nominal error rates best under the null hypothesis for all error distributions and for correlated and uncorrelated independent variables. Power under the alternative hypotheses for this partial testing procedure was similar to full LAD regression model tests. Four example applications of LAD regression are provided.