A Distribution-Free Control Chart for Retrospective Location Analysis of Subgrouped Multivariate Data

Abstract

In multivariate quality control, a proper Phase I analysis is essential to the success of Phase II monitoring. Even self-starting methods, which seek to minimize the Phase I process, usually recommend a single retrospective analysis at some point in the control charting process. This is true regardless of the underlying distribution of a process, which cannot often be assumed to be multivariate normal. A literature review reveals no distribution-free Phase I multivariate techniques in existence, so this research seeks to fill that gap by developing a distribution-free method of establishing an in-control reference sample for subgrouped multivariate processes in Phase I. The resulting multivariate sample, representing the in-control state of a process, can then be used to estimate the appropriate parameters for the Phase II multivariate quality control monitoring method of choice. The proposed method, which assumes constant covariance within subgroups, uses data depth in conjunction with robust estimators to detect both isolated and sustained shifts in subgroup location. Using Monte Carlo simulation, the proposed method is compared to the traditional Hotelling's T2 chart with a Phase I upper control limit. Although Hotelling's T2 chart is preferred when data are multivariate normally distributed, the proposed method is shown to perform significantly better than Hotelling's T2 chart when a process distribution is heavy-tailed or skewed.