ABSTRACT

The decomposition of the Hotelling’s T2 statistic into orthogonal components is
considered to be one of the most effective methods for detecting variable(s) responsible for an
out-of-control signal. In this work, an extension of the T2 decomposition from three variables
(p=3) to four (p=4) variables, where the number of decompositions increased from 3! = 6 to 4! =
24 and the decomposition terms also increased from 18 to 96 terms having 32 distinct terms were
provided. These distinct terms are the ones that were examined for possible contribution to the T2
signal. A dataset obtained from an Indomie company in Northern Nigeria was used to assess the
validity of constructed model by demonstrating the invariance property of the Hotelling’s T2
statistic. The model was also used to identify the variable(s) that significantly contribute to an
out-of-control signal. By comparing the critical values with the corresponding T2 values, we
were able to detect variation between the four (4) variables in their mean and also their variancecovariance
structure.
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TABLE OF CONTENTS

Declaration ……………………………………………………………………………………………………………………… i
Certification……………………………………………………………………………………………………………………. ii
Dedication …………………………………………………………………………………………………………………….. iii
Acknowledgement ………………………………………………………………………………………………………….. iv
Abstract …………………………………………………………………………………………………………………………. v
Table of Contents……………………………………………………………………………………………………………. vi
List of Tables ……………………………………………………………………………………………………………….. viii
List of Figures ……………………………………………………………………………………………………………….. ix
List of Appendices …………………………………………………………………………………………………………… x
List of Abbreviations ………………………………………………………………………………………………………. xi
CHAPTER ONE: GENERAL INTRODUCTION ………………………………………………………………… 1
1.0 Introduction………………………………………………………………………………………………………….. 1
1.1 Motivation of the Study ………………………………………………………………………………………….. 2
1.2 Assumptions of Statistical Process Control (SPC) ……………………………………………………….. 3
1.3 Aim and Objectives of the Study ……………………………………………………………………………… 5
1.4 Significance of the Study ………………………………………………………………………………………… 5
1.5 Traditional Statistical Process ………………………………………………………………………………….. 5
1.5.1 Univariate Control Charts ……………………………………………………………………………………….. 6
1.6 Multivariate Statistical Process Control (MSPC) …………………………………………………………. 8
1.6.1 Advantages of MSPC …………………………………………………………………………………………….. 9
1.6.2 Disadvantages of MSPC ……………………………………………………………………………………….. 10
1.7 Application of Multivariate Quality Control …………………………………………………………….. 10
CHAPTER TWO: LITERATURE REVIEW ………………………………………………………………………… 12
2.0 Introduction………………………………………………………………………………………………………… 12
2.1 Multivariate Chart ……………………………………………………………………………………………….. 12
2.1.1 Hotelling’s T2 Control Chart ………………………………………………………………………………….. 12
2.1.2 Multivariate Exponentially–Weighted Moving Average Control Chart…………………………. 16
2.1.3 Multivariate Cumulative Sum Control Chart …………………………………………………………….. 17
2.2 Identifying Out- of- Control Variable ……………………………………………………………………… 17
2.2.1 Using Bonferroni Control Limits. …………………………………………………………………………… 18
2.2.2 Application of Principal Components ……………………………………………………………………… 20
2.2.3 Application of T2 Decomposition……………………………………………………………………………. 21
vii
2.2.4 Application of Neural Networks …………………………………………………………………………….. 23
2.2.5 Using Graphical Techniques ………………………………………………………………………………….. 24
2.2.6 Cause-Selecting Control Chart and Regression Adjusted Variables. ……………………………… 25
CHAPTER THREE: METHODOLOGY ………………………………………………………………………………. 27
3.0 Introduction………………………………………………………………………………………………………… 27
3.1 Interpretation of Out-of-Control …………………………………………………………………………….. 27
3.2 The Decomposition of Hotelling’s T2 Statistic ………………………………………………………….. 30
3.3 Model for the T2 Decomposition Using Four Variables ………………………………………………. 31
3.4 Computing the MYT Decomposition Terms …………………………………………………………….. 35
3.6 Method of Data Collection and Data Analysis ………………………………………………………….. 39
CHAPTER FOUR: RESULTS AND DISCUSSIONS ……………………………………………………………… 41
4.0 Introduction………………………………………………………………………………………………………… 41
4.1 Normality Test…………………………..……………………………………………………41
4.2 Hotelling’s Control Chart (Phase I)…………………………………………………………………………. 42
4.3 Hotelling’s Control Chart (Phase II) ……………………………………………………………………….. 42
4.4 Computation of the T2 Decomposition Terms …………………………………………………………… 43
4.5 Hotelling’s T2 Control Chart after Taking out Abnormal Observations …………………………. 49
4.6 The Invariance Property of the Hotelling’s T2 Statistic. ………………………………………………. 50
CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATION ……………………….. 52
5.0 Introduction………………………………………………………………………………………………………… 52
5.1 Summary ……………………………………………………………………………………………………………. 52
5.2 Conclusion …………………………………………………………………………………………………………. 54
5.3 Recommendation…………………………………………………………………………………………………. 54
5.4 Contribution To Knowledge ………………………………………………………………………………….. 55
5.5 Further Research …………………………………………………………………………………………………. 55
REFERENCES ……………………………………………………………………………………………………………… 56
APPENDICES ………………………………………………………………………………………………………………. 64
viii
LIST OF TABLES
Table 3.1:Unique Decomposition Terms (cited from Mason et al., 1997) …………………………… 34
Table 4.1:MYT decomposition terms of three signaling points. ………………………………………. 444
ix

 

 

CHAPTER ONE

GENERAL INTRODUCTION
1.0 INTRODUCTION
Statistical Process Control (SPC) has played a significant role in controlling the product
quality for decades since Shewhart (1931) illustrated the technique of the control charts by
applying statistical concepts in the manufacturing process.
According to MacCarthy and Wasusri (2002), statistical process control is a powerful
tool for monitoring and control processes and has been widely used in manufacturing and nonmanufacturing
processes.
With the advancement in technology, there has been an increase in customer expectations
and the need to monitor correlated variables simultaneously. Process monitoring in which
several variables are of interest is called Multivariate Statistical Process control (MSPC).
Multivariate control charts is widely used in practice to monitor the simultaneous
performance of several related quality characteristics. The origin of multivariate control chart
can be attributed to Hotelling (1947). A multivariate control scheme has a better sensitivity than
one based on the univariate control charts in monitoring multivariate quality process. (Lu et al.,
1998)
Woodall and Montgomery (1999) stated that multivariate process control is one of the
most rapidly developing sections of statistical process control. The demand to implement MSPC
in a production process for quality improvements increases daily. Statistical methods play a
very important role in quality improvement in manufacturing industries (Woodall, 2000).
The quality of any product is usually determined by several correlated quality variables.
One of the popular multivariate control charts is based on Hotelling’s T2 statistic which is used
2
to simultaneously monitor those quality variables and taking their correlations into
consideration. There are a lot of literatures focusing on multivariate control charting methods
based on Hotelling’s T2 statistic in detecting mean shift such as Sullivan and Woodall (1996),
Mason and Young (1999), Tong et al., (2005), and many others.
1.1 MOTIVATION OF THE STUDY
When Hotelling T2 detects a change in the mean vector, corrective action is required. A
T2 value, however, does not provide direct information about which variable is responsible for
the overall out-of-control condition. This information is of practical importance because quality
engineers/analysts need to know which variable requires adjustments after the process is declared
out-of-control. The most challenging issue about multivariate quality control chart is the ability
to identify the variable which is responsible for an out-of-control condition.
Many literatures have discussed and presented methods which can be used to identify
out-of-control variable or variables and much credit has been to the method proposed by Mason
et al.,(1995), for more information see the works of Bersimis et al.,(2007). This method involves
the decomposition of the T2 statistic into orthogonal components which reflects the contribution
of each variable in an observation vector. Much application of the decomposition technique has
been on two and three variables as seen in the case of Yarmohammadi and Ebrahimi (2010),
Ulen and Demir (2013), Sani and Abubakar (2013) and so many others.
Holmes and Mergen (1993), Sullivan and Woodall (1996), and Vargas (2003) have noted
that the Phase I Hotelling’s T2 control chart for individual observations is less sensitive in
detecting trend or process mean shifts. In this work we would apply both Phase I and Phase II of
the Hotelling’s T2 control chart and also provide the T2 decomposition model for p=4, using
3
Mason, Young, Tracy (MYT) decomposition technique which would be used for identifying outof-
control variable .
1.2 ASSUMPTIONS OF STATISTICAL PROCESS CONTROL (SPC)
The standard assumptions in SPC are that the observed process values are normally,
independently and identically distributed (iid) with fixed mean (μ) and standard deviation σ
when the process is in control. Before MSPC can be implemented, the p variables must be
related to each other.
Correlation analysis is a technique used to show the strength of the relationship between
pairs of variables. Del Castillo (2002) defined correlation as the departure of two or more
variables from independence. Montgomery (2001), defined correlation as a degree to which two
or more quantities are associated.
When two or more random variables are defined on a probability space, it is useful to
measure the relationship between the variables. A common measure of the relationship between
two variables is called covariance. The covariance between random variables X and Y, denoted
as COV(X, Y) or XY  is
XY  E[(X X )(Y Y )]
Covariance provides an idea of the strength of correlation. In the case of two variables X
and Y, the correlation is considered to be very strong if X is far from its mean and Y is also far
from its mean. Hence, the covariance between the two variables X and Y describes the variation
between the two variables.
In the multivariate case, the population covariance is represented in a matrix denoted as
Σ. The covariance matrix is also called the variance-covariance matrix. The variance-covariance
matrix is a symmetrical matrix that contains the covariance among a set of random variables. The
4
main diagonal elements of the matrix are the variances of the random variables, and the offdiagonal
elements are the covariance between the p variables (Neter et al.,1996).
The p  p variance-covariance matrix, S is as follows;
S=
2
1 12 1
2
21 2 2
2
1 2
P
p
p p p
S S S
S S S
S S S
 
 
 
 
 
 


   

In a two-dimensional plot, the degree of correlation between the values on the axes is
quantified by the so-called correlation coefficient. The most common correlation coefficient is
the Moment Correlation, which is found by dividing the covariance of the two variables by the
product of their standard deviation. This correlation coefficient (r) is a measure of the degree of
linear relationship between two variables X and Y. The square of (r) is called the coefficient of
determination and denotes the portion of total variance explained by the regression model
(Walpole and Myers, 1993). The sample correlation coefficient is calculated by
( )( )
( 1)
i i
xy
x y
x x y y
r
n s s
 



where x and y are the sample means, x S and y S are the sample standard deviations of i x and i y
respectively.
The correlation coefficient
cov( , ) xy
xy
x y x y
x y 

   
 
The correlation coefficient may take value between -1.0 and +1.0.
5
1.3 AIM AND OBJECTIVES OF THE STUDY
Much research has been done on multivariate process control for variable data in various
situations. Moreover, interpretation of out-of-control signals and how to identify the quality
characteristics contributing to out-of-control signals have also been discussed. However,
identifying influential variable(s) that contribute to out-of-control signal is still a difficult task
especially when the quality characteristics are beyond three.
The aim of this research work is to:
Determine variable(s) that significantly contributes to an out-of-control signal in a multivariate
quality control chart. This is achieved through the following objectives:
1. Application of Hotelling’s T2 components using MYT decomposition technique.
2. Identifying out-of-control condition and out-of-control variable(s)
3. Illustration of invariance property of the Hotelling’s T2 statistic from the derived
components.
1.4 SIGNIFICANCE OF THE STUDY
The significance of this study is geared toward detecting out-of-control variable(s) in a
multivariate quality control chart. In achieving this, we explored and compared various methods
and techniques use in checking out-of-control condition in multivariate control chart. This work
is also aimed at helping practitioners in the field of quality control to be able to determine
variables that causes out-of-control signal in a monitoring process.
1.5 TRADITIONAL STATISTICAL PROCESS
Control charts were developed in 1931 by Shewhart to be used for process monitoring.
Control charts are widely used for detecting assignable and chance causes of variation. Some
definitions of control charts are presented as follows.
6
According to Shewhart (1931), “the control chart may serve these purposes, first, it is
used to define the goal or standard for a process that management strives to attain and secondly,
it may be used as an instrument for attaining that goal and thirdly, it may serve as means to
judging whether the goal has been reached.” Control chart may also be viewed as a statistical
tool as defined by Duncan in 1956.
Feigenbaum (1983) defined control charts as “…a graphical comparison of the actual
product characteristics with limits reflecting the ability to produce as shown by past experience
on the product characteristics.”
Therefore, control chart is a graphical display used to monitor a process. It usually
consists of a horizontal centerline corresponding to the in-control value of the parameter that is
being monitored and the upper and lower control limits. Control limits are not determined
arbitrarily, nor are they related to specification limits but rather by statistical criteria. Sample
points that fall within the control limits are said to be in-control while those points that fall
beyond the control limits are said to be an out-of-control process.
The traditional statistical process control is generally referred to as the univariate control
charts. This is due to the fact that it considers only a variable for monitoring quality
characteristics.
1.5.1 Univariate Control Charts
One major setback of the Shewhart chart is that it regards only the last data point and
does not carry a memory of the previous data. As a result, small changes in the mean of a
random variable are less likely to be detected rapidly. Exponentially weighted moving average
(EWMA) chart improves upon the detection of small process shifts. Rapid detection of small
7
changes in the quality characteristic of interest and ease of computations through recursive
equations are some of the many good properties of EWMA chart that make it attractive.
EWMA chart was first introduced by Roberts (1959) to achieve faster detection of small
changes in the mean. The EWMA is a statistic for monitoring the process that averages the data
in a way that gives less weight to data as they are further removed in time. EWMA is defined as:
Zi  Xi (1 )Zi 1     with 0  1, 0 0 Z 
It is used as the basis of a control chart. The procedure consists of plotting the EWMA statistic
i Z versus the sample number on a control chart with center line 0 CL  .
The upper control limit (UCL) is
2
0 [1 (1 ) ]
2
i
x UCL K

  

   

and lower control limits (LCL) is
2
0 [1 (1 ) ]
2
i
x LCL K

  

   

where
0  =mean
K=constant
X  =standard deviation
 =smoothing parameter
The term[1(1)2i ]approaches unity as i get larger, so after several sampling intervals, the
control limits will approach the steady state values
0 x 2 UCL K

 

 

8
0 x 2 LCL K

 

 

The CUSUM (cumulative sum) chart is an effective way of monitoring small deviations
in the process mean when small deviations are of interest. The CUSUM chart, originally
developed by Page (1954), incorporates all information in the sequence of sample values and
plots the cumulative sums of the deviations from a target value using samples from prior
observations.
1.6 MULTIVARIATE STATISTICAL PROCESS CONTROL (MSPC)
According to Montgomery and Klatt (1972), a lot of attention has been given to the
design of control charts where only one quality characteristic is of interest. However, based on
the two authors industrial products and processes are characterized by more than one
measurable quality characteristic and their joint effect describes product quality.
Process monitoring in which several variables are of interest is called MSPC.
Multivariate charts are better than the simultaneous operation of several univariate control
charts. Process monitoring using control charts can be seen in two- stage process, Phase I and
Phase II (Woodall, 2000). Each phase has a role in monitoring the quality of a product. In Phase
I, charts are used for retrospectively testing whether the process was in control using historical
dataset. This Phase aids the practitioners in bringing a process to an in-control state. In Phase II,
the main concern is to further monitor the historical data set when subsequent samples are
drawn.
The parameter of the run-length distribution is often used for measuring the performance
of the control chart methods, where the run length is the number of samples taken before an outof-
control condition occurs.
9
Hotelling (1947) was the pioneer to develop a quality control chart for several related
variables and the control chart is well known as the Hotelling T2 control chart. The Hotelling T2
control chart is rated as the most widely used multivariate control chart that deals with changes
in the mean vector of p correlated quality characteristics (Aparisi and Haro, 2001). Hotelling’s T2
control chart is a direct analogue of the Shewhart X control chart.
The main tool used for monitoring MSPC is through the use of the quality control chart.
MSPC procedure involves fulfilling four conditions:
1. One should be able to state if the process is in control or not.
2. Should be able to know if there was/is a false signal.
3. Should be able to know the relationship among variables, attributes taken into
consideration.
4. If the process is out of control, one should know the reasons why it is out of control
(Bersimis et al., 2007).
1.6.1 Advantages of MSPC
MSPC has several advantages as compared to its univariate equivalent. As considered by
Hotelling (1947), Alt (1985), and Lowry and Montgomery (1995).
According to Hotelling (1947), MSPC has the ability to combine measures in several
dimensions into a single measure of performance. In addition, MSPC offers an easier graphical
tool to the practitioner. The practitioner can only use one chart instead of multiple univariate
charts to evaluate the product or system quality as a whole rather than the sum of many
individual parts (Hotelling, 1947 and Montgomery, 2001).
Montgomery (2001) gave a demonstration that multivariate control charts will produce an
acceptable type I error or in-control run length while maintaining the original data of means and
10
variances and correlations. Multivariate statistics also consider the relationship between the
variables since the variance-covariance matrix is part of the computations (Hotelling, 1947). And
hence, multivariate charts can detect changes in the relationships among variables being
monitored, which would not be noticeable from separate univariate chart (Lowry and
Montgomery, 1995).
Also, MSPC provides the appropriate control region for the application. If the assumption
of independence does not hold, then the assumed performance of the traditional Shewhart
approaches can be misleading. The MSPC can guarantee error protection from a variety of
different types of shifts in the process. Another advantage of the MSPC is that it moves away
from the application of run rules (Sullivan and Woodall, 1996).
1.6.2 Disadvantages of MSPC
Much satisfying evidence has been presented concerning the benefits of applying the
MSPC, the following limitations were noticed.
According to Mason et al., (1997a), Ryan (2000) and Montgomery (2001), multivariate
control charting procedures are computationally intensive. And hence, it works well when the
number of variables is not too large, that is when p 10. As the number of variables grows,
multivariate control chart lose its efficiency with regard to process shift detection. Also, a
multivariate control chart procedure does not directly provide the information an operator need
when the control chart signals an out-of-control condition. It doesn’t give information on which
variable or set of variables is out-of-control (Hawkins, 1991).
1.7 APPLICATION OF MULTIVARIATE QUALITY CONTROL
Control charts are originally developed for individual processes and have been applied
within a number of areas, including:
11
1. Hospital infection control (Sellick,1993)
2. Prediction of business failures (Theodossiou, 1993)
3. Monitoring the impact of human disturbance of ecological systems (Anderson and
Thompson, 2004)
4. Quality Management of higher education (Mergen et al.,2000)
5. Corroborating bribery ( Charnes and Gitlow, 1995)
6. Improving athletic performance (Clark and Clark, 1997)
7. Improving the quality of Pharmaceutical products (Ulen and Demir, 2013)
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