Profile Monitoring via Linear Mixed Models

Profile monitoring is a relatively new technique in quality control used when the product or process quality is best represented by a profile (or a curve) at each time period. The essential 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 the correlation structure within the profiles. We propose the use of linear mixed models to monitor the linear profiles in order to account for the correlation structure within a profile. We consider various data scenarios and show using simulation when the linear mixed model approach is preferable to an approach that ignores the correlation structure. Our focus is on Phase I control chart applications.

[1]  David A. Harville,et al.  Extension of the Gauss-Markov Theorem to Include the Estimation of Random Effects , 1976 .

[2]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[3]  G. Reinsel,et al.  Models for Longitudinal Data with Random Effects and AR(1) Errors , 1989 .

[4]  Christine Waternaux,et al.  Methods for Analysis of Longitudinal Data: Blood-Lead Concentrations and Cognitive Development , 1989 .

[5]  Miloslav Suchánek,et al.  Multivariate control charts: Control charts for calibration curves , 1994 .

[6]  Malay Ghosh,et al.  Small Area Estimation: An Appraisal , 1994 .

[7]  G. Verbeke,et al.  A Linear Mixed-Effects Model with Heterogeneity in the Random-Effects Population , 1996 .

[8]  William H. Woodall,et al.  A Comparison of Multivariate Control Charts for Individual Observations , 1996 .

[9]  E. Vonesh,et al.  Linear and Nonlinear Models for the Analysis of Repeated Measurements , 1996 .

[10]  Kathryn S. Dawson,et al.  Two Graphical Techniques Useful in Detecting Correlation Structure in Repeated Measures Data , 1997 .

[11]  Edward W. Frees,et al.  Influence Diagnostics for Linear Longitudinal Models , 1997 .

[12]  E Lesaffre,et al.  Local influence in linear mixed models. , 1998, Biometrics.

[13]  T. Lewis,et al.  Outliers in multilevel data , 1998 .

[14]  E Lesaffre,et al.  Assessing the goodness-of-fit of the Laird and Ware model--an example: the Jimma Infant Survival Differential Longitudinal Study. , 1999, Statistics in medicine.

[15]  Lan Kang,et al.  On-Line Monitoring When the Process Yields a Linear Profile , 2000 .

[16]  Marie Davidian,et al.  Consequences of misspecifying assumptions in nonlinear mixed effects models , 2000 .

[17]  M. Berger,et al.  Detection of Influential Observations in Longitudinal Mixed Effects Regression Models , 2001 .

[18]  Richard K. Burdick.  Linear Models in Statistics , 2001, Technometrics.

[19]  Ana Ivelisse Avilés,et al.  Linear Mixed Models for Longitudinal Data , 2001, Technometrics.

[20]  Nicholas T. Longford,et al.  Simulation‐based diagnostics in random‐coefficient models , 2001 .

[21]  F. J. Pierce,et al.  Contemporary Statistical Models for the Plant and Soil Sciences , 2001 .

[22]  R. Xu.  Measuring explained variation in linear mixed effects models , 2003, Statistics in medicine.

[23]  N. José Alberto Vargas,et al.  Robust Estimation in Multivariate Control Charts for Individual Observations , 2003 .

[24]  Mahmoud A. Mahmoud,et al.  On the Monitoring of Linear Profiles , 2003 .

[25]  Nicole A. Lazar,et al.  Statistical Analysis With Missing Data , 2003, Technometrics.

[26]  John Haslett,et al.  Application of ‘delete = replace’ to deletion diagnostics for variance component estimation in the linear mixed model , 2004 .

[27]  Douglas C. Montgomery,et al.  Using Control Charts to Monitor Process and Product Quality Profiles , 2004 .

[28]  Louise Ryan,et al.  Cholesky Residuals for Assessing Normal Errors in a Linear Model With Correlated Outcomes , 2004 .

[29]  William H. Woodall,et al.  Phase I Analysis of Linear Profiles With Calibration Applications , 2004, Technometrics.

[30]  E. Demidenko,et al.  Mixed Models: Theory and Applications (Wiley Series in Probability and Statistics) , 2004 .

[31]  Fugee Tsung,et al.  Using Profile Monitoring Techniques for a Data‐rich Environment with Huge Sample Size , 2005 .

[32]  Robert A. Kozak,et al.  MIXED-MODEL DEVELOPMENT FOR REAL-TIME STATISTICAL PROCESS CONTROL DATA IN WOOD PRODUCTS MANUFACTURING , 2005 .

[33]  Eugene Demidenko,et al.  Influence analysis for linear mixed‐effects models , 2005, Statistics in medicine.

[34]  William H. Woodall,et al.  High breakdown estimation methods for Phase I multivariate control charts , 2007, Qual. Reliab. Eng. Int..