Within groups analysis of covariance: multiple comparisons at specified design points using a robust measure location when there is curvature

Consider the situation where measurements are taken at two different times and let M-j(x) be some conditional robust measure of location associated with the random variable Y at time j, given that some covariate X=x. The goal is to test H-0: M-1(x)=M-2(x) for each x is an element of x(1), horizontal ellipsis , x(K) such that the probability of one or more Type I errors is less than alpha, where x(1), horizontal ellipsis , x(K) are K specified values of the covariate. The paper reports simulation results comparing two methods aimed at accomplishing this goal without specifying some parametric form for the regression line. The first method is based on a simple modification of the method in Wilcox [Introduction to robust estimation and hypothesis testing. 3rd ed. San Diego, CA: Academic Press; 2012, Section 11.11.1]. The main result here is that the second method, which has never been studied, can have higher power, sometimes substantially so. Data from the Well Elderly 2 study, which motivated this paper, are used to illustrate that the alternative approach can make a practical difference. Here, the estimate of M-j(x) is based in part on either a 20% trimmed mean or the Harrell-Davis quantile estimator, but in principle the more successful method can be used with any robust location estimator.