## 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.

vi

## 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)

12

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