A goal of Phase I analysis of multivariate data is to identify multivariate outliers and step changes so that the Phase II estimated control limits are sufficiently accurate. High breakdown estimation methods based on the minimum volume ellipsoid (MVE) or the minimum covariance determinant (MCD) are well suited for detecting multivariate outliers in data. As a result of the inherent difficulties in their computation, many algorithms have been proposed to detect multivariate outliers. Due to their availability in standard software packages, we consider the subsampling algorithm to obtain the MVE estimators and the FAST-MCD algorithm to obtain the MCD estimators. Previous studies have not clearly determined which of these two available estimation methods is best for control chart applications. The comprehensive simulation study presented in this paper gives guidance for the correct use of each estimator. Control limits are provided. High breakdown estimation methods based on the MCD and MVE approaches can be applied to a wide variety of multivariate quality control data. Copyright © 2006 John Wiley & Sons, Ltd.