Willis A. Jensen

Effects of parameter estimation on control chart properties : A literature review

Control charts are powerful tools used to monitor the quality of processes. In practice, control chart limits are often calculated using parameter estimates from an in-control Phase I reference sample. In Phase II of the monitoring scheme, statistics based on new samples are compared with the estimated control limits to monitor for departures from the in-control state. Many studies that evaluate control chart performance in Phase II rely on the assumption that the in-control parameters are known. Although the additional variability introduced into the monitoring scheme through parameter estimation is known to affect the chart performance, many studies do not consider the effect of estimation on the performance of the chart. This paper contains a review of the literature that explicitly considers the effect of parameter estimation on control chart properties. Some recommendations are made and future research ideas in this area are provided.

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.

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

A case study on monitoring polynomial profiles in the automotive industry

In some statistical process control applications, the quality of a process or product can be characterized by a relationship between a response variable and one explanatory variable, which is referred to as profile. We give an example here of a profile that can be described using a polynomial model. This example comes from the automotive industry, where one of the most important quality characteristics of an automobile engine is the relationship between the torque produced by an engine and the engine speed in revolutions per minute. We find for this data set that a second-order polynomial works well. In addition, we show that there is autocorrelation within each profile, thus an ordinary least-square method that ignores the autocorrelation is inappropriate. We propose a linear mixed model method as an alternative approach. After the reduction of the data to a series of parameter estimates, we then conduct a step-by-step Phase I analysis of the polynomial profiles monitoring using a T2-based procedure to check the stability of the process and whether or not there are outlying profiles. The remaining profiles are used to form the estimated mean vector and variance–covariance matrix to be used in Phase II studies. Finally, a brief discussion is presented to show how one can use these parameters in Phase II. Copyright

Design issues for adaptive control charts

Adaptive control charts allow the components of the quality-monitoring scheme to vary in order to obtain improved performance over non-adaptive control charts. Research has centered on components such as the sample size, time between samples, warning limits, and control limits and has recommended a variety of schemes, many of which are optimal in some sense. In practice, there are many other adaptive schemes that are near optimal, which will still yield considerable improvement over non-adaptive control charts. In addition, the impact of parameter estimation on adaptive control chart performance must be taken into consideration. Based on the simulation results shown here, adaptive control charts should only be used for mature processes, where a sufficient amount of Phase I data have been obtained to ensure that the estimated control limits are accurate. When evaluating control chart performance, we consider initial state performance measures for simplicity and note that the conclusions obtained here apply to steady-state performance measures. The evaluation of performance measures is easily handled by the Markov chain approach detailed in the Appendix. Copyright

A Semiparametric Mixed Model Approach to Phase I Profile Monitoring

Profile monitoring is an approach in quality control best used where the process data follow a profile (or curve). The majority of previous studies in profile monitoring focused on the parametric (P) modeling of either linear or nonlinear profiles, with both fixed and random effects, under the assumption of correct model specification. More recently, in the absence of an obvious P model, nonparametric (NP) methods have been employed in the profile monitoring context. For situations where a P model is adequate over part of the data but inadequate of other parts, we propose a semiparametric procedure that combines both P and NP profile fits. We refer to our semiparametric procedure as mixed model robust profile monitoring (MMRPM). These three methods (P, NP and MMRPM) can account for the autocorrelation within profiles and treat the collection of profiles as a random sample from a common population. For each approach, we propose a version of Hotellings T2 statistic for use in Phase I analysis to determine unusual profiles based on the estimated random effects and obtain the corresponding control limits. Simulation results show that our MMRPM method performs well in making decisions regarding outlying profiles when compared to methods based on a misspecified P model or based on NP regression. In addition, however, the MMRPM method is robust to model misspecification because it also performs well when compared to a correctly specified P model. The proposed chart is able to detect changes in Phase I data and has easily calculated control limits. We apply all three methods to the automobile engine data of Amiri et al.5 and find that the NP and the MMRPM methods indicate signals that did not occur in a P approach. Copyright

Statistics to Facilitate Innovation*: A Panel Discussion

ABSTRACT Innovation is defined as the process of moving an initial invention or creative idea through research and development to the eventual market introduction. It is an important consideration for organizations to stay competitive and to continue to evolve in todays fast-paced environment. Statistics can play a large role in encouraging and facilitating innovation, through idea evaluation, collection of customer feedback, assessment of prototypes, and evaluation of the quality of products and processes. We define innovation and consider questions connecting innovation and statistics. The answers by a panel of industry leaders include discussion of the relationships between innovation, statistical thinking, and statistical engineering.

Approximations of tolerance intervals for normally distributed data

Tolerance intervals are lesser known relatives of confidence and prediction intervals. They can be very useful in many situations to make product or process quality assessments. Even for normally distributed data, their calculation is less trivial than confidence and prediction intervals, which makes them underutilized in practice. In addition, they are not always readily available in statistical software packages. As a result, there have been several approximate methods proposed in the literature to calculate them. After a review of tolerance intervals and their variations, we investigate those approximations and compare them with the exact values for one-sided and two-sided intervals. We first propose a modification of an approximation for a one-sided interval. Then we propose new approximations for two-sided intervals based on Bonferronis Inequality. We find that these approximations are extremely satisfactory for practical applications. Copyright

Response Surface Methodology: Process and Product Optimization Using Designed Experiments 4th edition

(2017). Response Surface Methodology: Process and Product Optimization Using Designed Experiments 4th edition. Journal of Quality Technology: Vol. 49, No. 2, pp. 186-188.