This study aims to extend the investigation for efficiency of MRR2 to model the
profile data in linear mixed model at Phase I profile monitoring. At first stage of this
study, a parametric model is fitted to the correlated and uncorrelated profile data using
mixed model linear regression. At second stage, a non-parametric fit is evaluated for the
residuals from the fitted LMMR. Consequently, the final semi-parametric model is
obtained by combining the parametric fit (linear mixed model) and non-parametric fits
based on residuals profile. The objectives are to provide a better fit to the profile data to
adequately monitoring the future observations.

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The application of profile monitoring is adapted somewhat from the calibration
application presented by Stover and Brill (1998). They worked on the multilevel ion
chromatography (IC) calibration experiment and used quality control charts to determine
the stability of instrument response and its minimum calibration frequency. They
considered calibration data as a linear profile and monitored profiles using multivariate
T2 chart based on the first principal component.
The research on profile monitoring has spread in various directions for the
methods proposed at Phase I, Phase II, or both. (Woodall, 2007; Woodall &
Montgomery, 2014; Woodall et al., 2004) have highlighted some broad areas of interest
in profile monitoring where regression technique remains the core of estimation for
profile parameters. These broad areas include parametric (linear or non-linear), non-
parametric (linear or non-linear) and some robust (semi-parametric) methods for profile
monitoring. Each method uses a different approach for control charting at Phase I and/ or
Phase II. The focus of this study is on presenting a semi-parametric approach for Phase I
monitoring using the parametric fit LMMR combined with a non-parametric fit on
residuals; MRR2.
2.1 Parametric Profile Monitoring
For many calibration problems, a profile can be represented as a linear regression
model. The two common parametric regression techniques are ordinary least squares
(OLS) and the generalized least squares (GLS) method. Either of two methods can be
used to estimate in-control profile parameters (for intercept, slope, and error variance) at


Phase I using a historical dataset. At Phase I, the control charts are used to detect the
outlying profiles before setting the in-control limits. The commonly proposed control
charts for monitoring parametric profiles at Phase II include Hoteling’s T2, EWMA and R
charts. Subsequently using these charts, the process is monitored for future observations.
2.1.1 Parametric Profile Monitoring for Fixed Effects
Most of the literature on profile monitoring assumes a correctly specified model
and the parameters are known for setting the control limits at Phase I. The methods are
thus proposed for Phase II monitoring. Kang and Albin (2000) proposed two methods for
Phase II monitoring of linear profiles for fixed effects; one using multivariate T2 chart (by
parameterizing the profile using least squares estimation), and second the univariate
EWMA and R charts (based on residuals of each profile from the reference line). Both
charts performed well to monitor the performance of mass flow controller (MFC) device
in manufacturing.
Kim, Mahmoud, and Woodall (2003) suggested bivariate T2 chart and a Shewhart
chart at Phase I to check the stability of variations in the regression parameters and along
the regression line respectively. To overcome the problem of ignored correlation, they
used transformed x-values (coded sum to be zero) at Phase II. The three univariate
EWMA charts (for y-intercept, slope, and error variance) were used jointly; combined
signaling with the first chart to signal. Their methods turned out to be more efficient as
compared to the ones proposed by Kang and Albin (2000).
Mahmoud and Woodall (2004) suggested using F test for Phase I profile
monitoring in multiple regression case by pooling all the samples and using indicator


variables in combination with a univariate T2 control chart to monitor process variance
(variations about the regression line). All the regression lines are identical if the null
hypothesis of equality of parameters is rejected, and the process is said to be out-of-
control. They compared their method with other T2 alternatives and suggested to develop
a robust linear profile method.
S Gupta, Montgomery, and Woodall (2006) replaced EWMA-3 charts by
Shewhart X-bar charts in the method introduced by Kim et al. (2003), to monitor
regression intercept and slope. In addition, they suggested S2 chart to monitor error
variance instead of R chart. However, their analysis resulted out in line with the fact that
EWMA charts perform better than X-bar charts. Mahmoud, Parker, Woodall, and
Hawkins (2007) proposed a change point method using the heteroscedastic segmented
regression model (on pooled samples) to detect parameter (intercept, slope, and residuals)
changes in Phase I of a simple linear profile.
Williams, Woodall, and Birch (2007) worked on monitoring the non-linear
parametric profiles with fixed effects and proposed three T2 statistics to monitor a
multivariate quality characteristic vector. The three statistics are differentiated as per the
estimation for variance-covariance matrix. They used estimated variance-covariance
matrix by: 1). Sample variance-covariance matrix, 2). Based on successive differences,
3). Robust estimator ‘minimum volume ellipsoide’. They proposed using both ;#3627408455;;#3627408448;;#55349;;#56393;;#3627408440;2 and
;#3627408455;;#3627408439;2 to detect both the multivariate outliers and step shifts. However, they suggested to be
cautious while using both statistics simultaneously. More work and discussion on
monitoring of linear profiles can be found in (Shilpa Gupta, 2010; Rassoul Noorossana,


Eyvazian, ; Vaghefi, 2010; Zhang, Li, ; Wang, 2009).
2.1.2 Parametric Profile Monitoring for Mixed Effects
Jensen, Birch, and Woodall (2008) proposed a linear mixed model approach at
Phase I to monitor the correlation structure within linear profiles. They used the distance
between estimated profile parameters and the center of profiles (profile average) to
identify outlying profiles. Their proposed T2 statistic uses sample mean vector and
estimated variance-covariance matrix using two methods; 1). Deviation of each profile
from the profile average, 2). Successive differences between profile parameters. Their
simulation results suggest that the mixed model approach for monitoring the correlation
or random effects is preferable when the data are unbalanced or when there are missing
data. For balanced designs, a simpler analysis ignoring correlation will perform just as
well as a complicated analysis.
Jensen and Birch (2009b) compared the nonlinear fixed effects model with the
nonlinear mixed effects profile monitoring for Phase I. The first method suggests a
parametric non-linear estimation for each profile separately without considering the
random effects, while the mixed model suggests a simultaneous estimation for all the
profiles with adding random effects. They found the estimation and efficiency of
nonlinear mixed model better as compared to the individual nonlinear models with fixed
effects; where the mixed model provides a relaxation for modeling the autocorrelation
between errors.


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