Jeffrey B. Birch

Statistical monitoring of nonlinear product and process quality profiles

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 T2 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 T2 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 T2 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. Journal of Quality Technology 2002; 34:118–129). Copyright

MONITORING CORRELATION WITHIN LINEAR PROFILES USING MIXED MODELS

Profile monitoring is a relatively new set of techniques in quality control used when the product or process quality is best represented by a function (or a curve) at each time period. The idea is often to model the profile via some parametric method and then monitor the estimated parameters over time to determine if there have been changes in the profiles. Previous modeling methods have not incorporated a correlation structure within the profiles. We propose the use of linear mixed models to monitor the linear profiles in order to account for any correlation structure within a profile. We conclude that, when the data are balanced, there appears to be no advantage in modeling correlation and/or including random effects because a simpler analysis that ignores the correlation structure will perform just as well as the more complicated analysis. When the data are unbalanced or when there are missing data, we find that the linear mixed model approach is preferable to an approach that ignores the correlation structure. Our focus is on Phase I control-chart applications.

Robust Window Operators

It is a common practice in computer vision and image processing to convolve rectangular constant coefficient windows with digital images to perform local smoothing and derivative estimation for edge detection and other purposes. If all data points in each image window belong to the same statistical population, this practice is reasonable and fast. But, as is well known, constant coefficient window operators produce incorrect results if more than one statistical population is present within a window, for example, if a gray-level or gradient discontinuity is present. This paper shows one way to apply the theory of robust statistics to the data smoothing and derivative estimation problem. A robust window operator is demonstrated that preserves gray-level and gradient discontinuities in digital images as it smooths and estimates derivatives.

Profile Monitoring via Nonlinear Mixed Models

Profile monitoring is a relatively new technique in quality control best used where the process data follow a profile (or curve) at each time period. Little work has been done on the monitoring of 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. Our studies show the nonlinear mixed model approach to be clearly superior to fitting separate nonlinear regression models. As a consequence, 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.

High breakdown estimation methods for Phase I multivariate control charts

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

Statistical monitoring of heteroscedastic dose-response profiles from high-throughput screening

In pharmaceutical drug discovery and agricultural crop product discovery, in vivo bioassay experiments are used to identify promising compounds for further research. The reproducibility and accuracy of the bioassay is crucial to be able to correctly distinguish between active and inactive compounds. In the case of agricultural product discovery, a replicated dose-response of commercial crop protection products is assayed and used to monitor test quality. The activity of these compounds on the test organisms, the weeds, insects, or fungi, is characterized by a dose-response curve measured from the bioassay. These curves are used to monitor the quality of the bioassays. If undesirable conditions in the bioassay arise, such as equipment failure or problems with the test organisms, then a bioassay monitoring procedure is needed to quickly detect such issues. In this article we illustrate a proposed nonlinear profile monitoring method to monitor the variability of multiple assays, the adequacy of the dose-response model chosen, and the estimated dose-response curves for aberrant cases in the presence of heteroscedasticity. We illustrate these methods with in vivo bioassay data collected over one year from DuPont Crop Protection.

Influence measures in ridge regression

In regression, it is of interest to detect anomalous observations that exert an unduly large influence on the least squares analysis. Frequently, the existence of influential data is complicated by the presence of collinearity (see, e.g., Lawrence and Marsh 1984). Very little work has been done, however, on the possible effects that collinearity can have on the influence of an observation. In this article, we show that when ridge regression is used to mitigate the effects of collinearity, the influence of some observations can be drastically modifield. Approximate deletion formulas for the detection of influential points are proposed for ridge regression.

Distribution of hotelling's T2 statistic based on the successive differences estimator

In the historical (or retrospective or Phase I) multivariate data analysis, the choice of the estimator for the variance–covariance matrix is crucial to successfully detecting the presence of special causes of variation. For the case of individual multivariate observations, the choice is compounded by the lack of rational subgroups of observations with the same distribution. Other research has shown that the use of the sample covariance matrix, with all of the individual observations pooled, impairs the detection of a sustained step shift in the mean vector. For example, research has shown that, with the use of the sample covariance matrix, the probability of a signal actually decreases below the false-alarm probability with a sustained step shift near the middle of the data and that the signal probability decreases with the size of the shift. An alternative estimator, based on the successive differences of the individual observations, leads to an increasing signal probability as the size of the step shift increases and has been recommended for use in Phase I analysis. However, the exact distribution for the resulting T2 chart statistics has not been determined when the successive differences estimator is used. Three approximate distributions have been proposed in the literature. In this paper, we demonstrate several useful properties of the T2 statistics based on the successive differences estimator and give a more accurate approximate distribution for calculating the upper control limit for individual observations in a Phase I analysis.

Algorithm 905: SHEPPACK: Modified Shepard Algorithm for Interpolation of Scattered Multivariate Data

Scattered data interpolation problems arise in many applications. Shepard’s method for constructing a global interpolant by blending local interpolants using local-support weight functions usually creates reasonable approximations. SHEPPACK is a Fortran 95 package containing five versions of the modified Shepard algorithm: quadratic (Fortran 95 translations of Algorithms 660, 661, and 798), cubic (Fortran 95 translation of Algorithm 791), and linear variations of the original Shepard algorithm. An option to the linear Shepard code is a statistically robust fit, intended to be used when the data is known to contain outliers. SHEPPACK also includes a hybrid robust piecewise linear estimation algorithm RIPPLE (residual initiated polynomial-time piecewise linear estimation) intended for data from piecewise linear functions in arbitrary dimension m. The main goal of SHEPPACK is to provide users with a single consistent package containing most existing polynomial variations of Shepard’s algorithm. The algorithms target data of different dimensions. The linear Shepard algorithm, robust linear Shepard algorithm, and RIPPLE are the only algorithms in the package that are applicable to arbitrary dimensional data.

Model robust regression: combining parametric, nonparametric, and semiparametric methods

The proper combination of parametric and nonparametric regression procedures can improve upon the shortcomings of each when used individually. Considered is the situation where the researcher has an idea of which parametric model should explain the behavior of the data, but this model is not adequate throughout the entire range of the data. An extension of partial linear regression and two other methods of model-robust regression are developed and compared in this context The model-robust procedures each involve the proportional mixing of a parametric fit to the data and a nonparametric fit to either the data or residuals. Asymptotically optimal estimates for the mixing parameters are given, along with their convergence rates. Performance is based on bias and variance considerations, and theoretical mean squared error formulas are used to compare procedures. Simulation results establish the accuracy of the theoretical formulas and illustrate the potential benefits of the model-robust procedures. Two examples are given: Example 1 uses generated data from an underlying model with defined misspecification to show the theoretical benefits of the model-robust procedures, and Example 2 supplies an interesting application.