For robust measures of location associated with J dependent groups, various methods have been proposed that are aimed at testing the global hypothesis of a common measure of location applied to the marginal distributions. A criticism of these methods is that they do not deal with outliers in a manner that takes into account the overall structure of the data. Location estimators have been derived that deal with outliers in this manner, but evidently there are no simulation results regarding how well they perform when the goal is to test the some global hypothesis. The paper compares four bootstrap methods in terms of their ability to control the Type I error probability when the sample size is small, two of which were found to perform poorly. The choice of location estimator was found to be important as well. Indeed, for several of the estimators considered here, control over the Type I error probability was very poor. Only one estimator performed well when using the first of two general approaches that might be used. It is based on a variation of the (affine equivariant) Donoho—Gasko trimmed mean. For the second general approach, only a skipped estimator performed reasonably well. (It removes outliers via a projection method and averages the remaining data.) Only one bootstrap method was found to perform well when using the first approach. A different bootstrap method is recommended when using the second approach.