A semiparametric nonlinear mixed model approach to phase I profile monitoring

ABSTRACT When process data follow a particular curve in quality control, profile monitoring is suitable and appropriate for assessing process stability. Previous research in profile monitoring focusing on nonlinear parametric (P) modeling, involving both fixed and random-effects, was made under the assumption of an accurate nonlinear model specification. Lately, nonparametric (NP) methods have been used in the profile monitoring context in the absence of an obvious linear P model. This study introduces a novel technique in profile monitoring for any nonlinear and auto-correlated data. Referred to as the nonlinear mixed robust profile monitoring (NMRPM) method, it proposes a semiparametric (SP) approach that combines nonlinear P and NP profile fits for scenarios in which a nonlinear P model is adequate over part of the data but inadequate of the rest. These three methods (P, NP, and NMRPM) account for the auto-correlation within profiles and treats the collection of profiles as a random sample with a common population. During Phase I analysis, a version of Hotelling’s T2 statistic is proposed for each approach to identify abnormal profiles based on the estimated random effects and obtain the corresponding control limits. The performance of the NMRPM method is then evaluated using a real data set. Results reveal that the NMRPM method is robust to model misspecification and performs adequately against a correctly specified nonlinear P model. Control charts with the NMRPM method have excellent capability of detecting changes in Phase I data with control limits that are easily computable.

[1]  Wen Wan,et al.  A semiparametric technique for the multi‐response optimization problem , 2011, Qual. Reliab. Eng. Int..

[2]  Fugee Tsung,et al.  Monitoring Profiles Based on Nonparametric Regression Methods , 2008, Technometrics.

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

[4]  J B Birch,et al.  A semiparametric approach to analysing dose-response data. , 2000, Statistics in medicine.

[5]  Jeffrey B. Birch,et al.  Model robust regression: combining parametric, nonparametric, and semiparametric methods , 2001 .

[6]  B. Ripley,et al.  Semiparametric Regression: Preface , 2003 .

[7]  Peihua Qiu,et al.  Nonparametric Profile Monitoring by Mixed Effects Modeling , 2010, Technometrics.

[8]  Abdel-Salam G. Abdel-Salam,et al.  Several nonparametric and semiparametric approaches to linear mixed model regression , 2015 .

[9]  Lewis B. Sheiner,et al.  Evaluation of methods for estimating population pharmacokinetic parameters. I. Michaelis-menten model: Routine clinical pharmacokinetic data , 1980, Journal of Pharmacokinetics and Biopharmaceutics.

[10]  Christopher Zorn.  Generalized Estimating Equation Models for Correlated Data: A Review with Applications , 2001 .

[11]  Ying Wei,et al.  PROFILE CONTROL CHARTS BASED ON NONPARAMETRIC L-1 REGRESSION METHODS. , 2012, The annals of applied statistics.

[12]  P. Albert,et al.  Models for longitudinal data: a generalized estimating equation approach. , 1988, Biometrics.

[13]  Abdel-Salam G. Abdel-Salam,et al.  A Semiparametric Mixed Model Approach to Phase I Profile Monitoring , 2013, Qual. Reliab. Eng. Int..

[14]  D. Bates,et al.  Mixed-Effects Models in S and S-PLUS , 2001 .

[15]  W. Woodall,et al.  Statistical monitoring of heteroscedastic dose-response profiles from high-throughput screening , 2007 .

[16]  Mohammadsadegh Mobin,et al.  A Semi parametric approach to dual modeling , 2012 .

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

[18]  Christine M. Anderson-Cook,et al.  A semi-parametric approach to robust parameter design , 2008 .

[19]  D. Bates,et al.  Nonlinear mixed effects models for repeated measures data. , 1990, Biometrics.

[20]  Rassoul Noorossana,et al.  Profile Monitoring Using Nonparametric Bootstrap T 2 Control Chart , 2012, Commun. Stat. Simul. Comput..

[21]  Willis A. Jensen,et al.  Profile Monitoring via Nonlinear Mixed Models , 2009 .

[22]  E. Vonesh,et al.  Goodness-of-fit in generalized nonlinear mixed-effects models. , 1996, Biometrics.

[23]  Changliang Zou,et al.  CONTROL CHART FOR MONITORING NONPARAMETRIC PROFILES WITH ARBITRARY DESIGN , 2009 .

[24]  Rassoul Noorossana,et al.  Statistical Analysis of Profile Monitoring , 2011 .

[25]  T. Snijders.  Mixed-effects model , 2003 .

[26]  D. Bates,et al.  Approximations to the Log-Likelihood Function in the Nonlinear Mixed-Effects Model , 1995 .

[27]  Jeffrey B. Birch,et al.  A semi-parametric approach to dual modeling when no replication exists , 2010 .

[28]  D. Harville.  Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems , 1977 .

[29]  H. Shahriari,et al.  Robust estimation of complicated profiles using wavelets , 2017 .

[30]  D. Bates,et al.  Newton-Raphson and EM Algorithms for Linear Mixed-Effects Models for Repeated-Measures Data , 1988 .

[31]  Douglas C. Montgomery,et al.  Some Current Directions in the Theory and Application of Statistical Process Monitoring , 2014 .

[32]  M. Wegener,et al.  Examining heterogeneity in implied equity risk premium using penalized splines , 2008 .

[33]  Russell D. Wolfinger,et al.  Laplace's approximation for nonlinear mixed models. , 1993 .

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

[35]  William H. Woodall,et al.  Current research on profile monitoring , 2007 .

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

[37]  D. Hawkins,et al.  NONPARAMETRIC CONTROL CHART FOR MONITORING PROFILES USING CHANGE POINT FORMULATION AND ADAPTIVE SMOOTHING , 2009 .