Robust Regression Estimators When There are Tied Values

There is a vast literature on robust regression estimators, the bulk of which assumes that the dependent (outcome) variable is continuous. Evidently, there are no published results on the impact of tied values in terms of the Type I error probability. A minor goal in this paper to comment generally on problems created by tied values when testing the hypothesis of a zero slope. Simulations indicate that when tied values are common, there are practical problems with Yohai’s MM-estimator, the Coakley–Hettmansperger M-estimator, the least trimmed squares estimator, the Koenker–Bassett quantile regression estimator, Jaeckel’s rank-based estimator as well as the Theil–Sen estimator. The main goal is to suggest a modification of the Theil–Sen estimator that avoids the problems associated with the estimators just listed. Results on the small-sample efficiency of the modified Theil-Sen estimator are reported as well. Data from the Well Elderly 2 Study are used to illustrate that the modified Theil-Sen estimator can make a practical difference.