Comparing robust regression lines associated with two dependent groups when there is heteroscedasticity

Abstract

The paper deals with three approaches to comparing the regression lines corresponding to two dependent groups when using a robust estimator. The focus is on the Theil–Sen estimator with some comments about alternative estimators that might be used. The first approach is to test the global hypothesis that the two groups have equal intercepts and slopes in a manner that allows a heteroscedastic error term. The second approach is to test the hypothesis of equal intercepts, ignoring the slopes, and testing the hypothesis of equal slopes, ignoring the intercepts. The third approach is to test the hypothesis that the regression lines differ at a specified design point. This last goal corresponds to the classic Johnson and Neyman method when dealing with independent groups and when using the ordinary least squares regression estimator. Based on extant studies, there are guesses about how to proceed in a manner that will provide reasonably accurate control over the Type I error probability: Use some type of percentile bootstrap method. (Methods that assume the regression estimator is asymptotically normal were not considered for reasons reviewed in the paper.) But there are no simulation results providing some sense of how well they perform when dealing with a relatively small sample size. Data from the Well Elderly II study are used to illustrate that the choice between the ordinary least squares estimator and the Theil–Sen estimator can make a practical difference.