302 Introduction to Robust Estimation and Hypothesis Testing replacing the Welch-type method with some other procedure, but this remains to be seen. (In the two-sample case, with the amount of trimming less than 20%, a generalization of the Yuen­Welch test seems to have merit; see Othman, Keselman, Wilcox, Fradette, & Padmanabhan, 2002.) As was done in Section 5.3.2, the method begins by obtaining a bootstrap sample from each ¯ of the J groups: X 1 j , . . . , X n j j . Next, set C i j = X i j - X t j , i = 1, . . . , n j . Then C 1 j , . . . , C n j j represents a sample from a distribution that has a trimmed mean of zero, so the hypothesis of equal trimmed means among these J distributions is true. Let F t be the value of F t (described in Table 7.1), when applied to the C i j values. Repeat this process B times, each time obtaining bootstrap samples and computing F t using the C i j values that result. Label the resulting test statistics F t1 , . . . , F t B . Each time this process is applied, the null hypothesis is true, by construction, so the values F t1 , . . . , F t B provide an estimate of an appropriate critical value. Letting F t · · · F t be the F t1 , . . . , F t B values written in ascending order, an (1) (B) estimate of the critical value is F t (m) , where u = (1 - )B, rounded to the nearest integer. That is, reject the null hypothesis of equal trimmed means if F t , computed as described in Table 7.1, is greater than or equal to F t . (u) 7.1.8 R Functions t1waybt and btrim The R function t1waybt(x,tr=0.2,alpha=0.05,grp=NA,nboot=599). tests the hypothesis of equal trimmed means using the bootstrap-t method. As with t1way and box1way, the argument x can be any R variable that is a matrix or has list mode. If unspecified, the amount of trimming defaults to tr=0.2, and the argument alpha, corresponding to , defaults to 0.05. Again the argument grp can be used to test the hypothesis of equal trimmed means for some subgroup of interest. If unspecified, all J groups are used. The default value for B is nboot=599 which appears to give good results, in terms of controlling the probability of a type I error, when = 0.05 and n j 10, j = 1, . . . , J . Little is known about how the method performs when < 0.05. Cribbie et al. (in press) found that with 20% trimming, a parametric bootstrap technique performs a bit better than the method in Section 7.1.1. Limited checks indicate that the bootstrap-t method used here is better than the parametric bootstrap method in terms of avoiding type I error probabilities larger than the nominal level. But extensive comparisons have not been made. I Example Again consider the data in Section 7.1.2. Assuming the data are stored in the variable w, the command t1waybt(w) reports that the 0.05 critical value is 4.97. The value of the www.elsevierdirect.com