In many quality control applications, use of a single (or several distinct) quality characteristic(s) is insufficient to characterize the quality of a produced item. In an increasing number of cases, a response curve (profile) is required. Such profiles can frequently be modeled using linear or nonlinear regression models. In recent research others have developed multivariate T 2 control charts and other methods for monitoring the coefficients in a simple linear regression model of a profile. However, little work has been done to address the monitoring of profiles that can be represented by a parametric nonlinear regression model. Here we extend the use of the T 2 control chart to monitor the coefficients resulting from a parametric nonlinear regression model fit to profile data. We give three general approaches to the formulation of the T 2 statistics and determination of the associated upper control limits for Phase I applications. We also consider the use of non-parametric regression methods and the use of metrics to measure deviations from a baseline profile. These approaches are illustrated using the vertical board density profile data presented in Walker and Wright (Comparing curves using additive models.

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Profile monitoring is a relatively new technique in quality control best used where the process data follows a profile (or curve) at each time period. Little work has been done on the monitoring on nonlinear profiles. Previous work has assumed that the measurements within a profile are uncorrelated. To relax this restriction we propose the use of nonlinear mixed models to monitor the nonlinear profiles in order to account for the correlation structure. We evaluate the effectiveness of fitting separate nonlinear regression models to each profile in Phase I control chart applications for data with uncorrelated errors and no random effects. For data with random effects, we compare the effectiveness of charts based on a separate nonlinear regression approach versus those based on a nonlinear mixed model approach. Our proposed approach uses the separate nonlinear regression model fits to obtain a nonlinear mixed model fit. The nonlinear mixed model approach results in charts with good abilities to detect changes in Phase I data and has a simple to calculate control limit.

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.

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