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626 Introduction to Robust Estimation and Hypothesis Testing so the groups could be compared, for example, based on the multivariate OP measure of location. For the kth group, let ( 0k , 1k ) represent some measure of location associated with ( 01k , 11k ), . . . , ( 0n k k , 1n k k ) where 0ik and 1ik are the intercept and slope, respectively, associated with the kth group (k = 1, 2). For example, 1k might be the median of the slopes associated with group k. Yet another approach is to test H 0 : ( 01 , 11 ) = ( 02 , 12 ). (11.33) So the p-variate data has been reduced to two variables, and these two variables could be compared, for example, using the methods in Sections 6.8 or 6.9. A broader, more involved approach toward longitudinal data, based on a marginal model and the MM-estimator in Section 10.9.1, is summarized by Heritier et al. (2009, Section 6.2). Included is an inferential technique that is based in part on appropriate estimates of the standard errors; the test statistic is assumed to be approximately standard normal. Evidently, there are no results on the ability of this approach to control type I errors when dealing with skewed distributions or heteroscedasticity. In simpler situations, skewness is a serious concern when using M-estimators and a hypothesis testing method is based on a (nonbootstrap) technique that is a function of estimated standard errors. So caution seems warranted for the situation at hand. For robust methods based on a random effects model, see Mills, Field, and Dupuis (2002), Sinha (2004), and Noh and Lee (2007). The basic strategy is to estimate parameters assuming observations are randomly sampled from a class of distributions that includes normal distributions as a special case. For example, Mills et al. assume that sampling is from a mixture of normal and t distributions, which results in a bounded influence function. Certainly these methods are an improvement on methods that assume normality. Again, what is unclear is the extent skewness and heteroscedasticity affect efficiency and type I error probabilities. How well do these methods handle contamination bias as described in Section 10.14.1? If practical problems are found, perhaps some bootstrap methods can provide more satisfactory results, but this remains to be determined. 11.12.1 R Functions long2g, longreg, longreg.plot, and xyplot The R function long2g(x, x.col, y.col, s.id, grp.id, regfun = tsreg, MAR = T, tr = 0.2) compares two groups based on estimates of the slope and intercept for each participant. The data are assumed to be stored in a matrix or data frame as illustrated by the orthodontic data in the previous section. The arguments x.col and y.col indicate the columns of x where the covariate and outcome variables are stored, respectively. The argument s.id is the column containing the subject's identification and grp.id is the column indicating group membership, www.elsevierdirect.com