Contributions to Profile Monitoring andMultivariate Statistical Process Control

James D. Williams

Dissertation submitted to the faculty of the

Virginia Polytechnic Institute & State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophyin

Statistics

Jeffrey B. Birch, Co-Chairman

William H. Woodall, Co-Chairman

Christine M. Anderson-Cook

Dan J. Spitzner

G. Geoffrey Vining

December 1, 2004

Blacksburg, Virginia

KEYWORDS: Bioassay, False Alarm Rate, Functional Data, Heteroscedasticity,

Hotelling’s T 2 Statistic, Lack-of-Fit, Minimum Volume Ellipsoid, Nonlinear Regres-

sion, Sample Size, Successive Differences, Vertical Density Profile.

c© 2004 by James D. WilliamsALL RIGHTS RESERVED

Contributions to Profile Monitoring andMultivariate Statistical Process Control

James D. Williams

Abstract

The content of this dissertation is divided into two main topics: 1) nonlinear profile

monitoring and 2) an improved approximate distribution for the T 2 statistic based

on the successive differences covariance matrix estimator.

Nonlinear Profile Monitoring

In an increasing number of cases the quality of a product or process cannot ad-

equately be represented by the distribution of a univariate quality variable or the

multivariate distribution of a vector of quality variables. Rather, a series of measure-

ments are taken across some continuum, such as time or space, to create a profile. The

profile determines the product quality at that sampling period. We propose Phase I

methods to analyze profiles in a baseline dataset where the profiles can be modeled

through either a parametric nonlinear regression function or a nonparametric regres-

sion function. We illustrate our methods using data from Walker and Wright (2002)

and from dose-response data from DuPont Crop Protection.

Approximate Distribution of T 2

Although the T 2 statistic based on the successive differences estimator has been

shown to be effective in detecting a shift in the mean vector (Sullivan and Woodall

(1996) and Vargas (2003)), the exact distribution of this statistic is unknown. An

accurate upper control limit (UCL) for the T 2 chart based on this statistic depends on

knowing its distribution. Two approximate distributions have been proposed in the

literature. We demonstrate the inadequacy of these two approximations and derive

useful properties of this statistic. We give an improved approximate distribution and

recommendations for its use.

iii

Acknowledgments

The first person to whom I owe an eternal debt of gratitude is my precious wife, Gina,

who not only gave birth to two beautiful children since we moved to Blacksburg, but

bore a disproportionate load of raising our three children while I worked towards

finishing this degree. Since our marriage over five years ago, I have been a full-time

graduate student. I am extremely thankful for how supportive she has been through

these tough graduate school years.

From the beginning I knew that Dr. Jeffrey B. Birch would not only be an

inspirational teacher and mentor, but a good friend as well. After taking three classes

from him, I was impressed with his ability to teach and inspire students to rise to

their potential. I have tried to pattern my own teaching style according to his. Dr.

Birch has been like a second father to me. He puts his own work on hold to hear my

thoughts on a moments notice. I owe him a huge debt of gratitude for the countless

selfless hours he spend teaching me, guiding me, counselling me, and simply listening

to me. There are too many things to thank him for than can be adequately listed

here.

Dr. William H. Woodall has been an inspiration to me as well. At one point

while deriving the results from Chapter 4, I asked him if he thought it would be

alright if I used a simulation study to prove a theorem. His response was, “You

could do that, but it would be better if you proved it analytically.” He left it at that,

and I walked away scratching my head. It took me several months to figure it out,

iv

but the analytical proof was finally completed. In addition to inspiring me to be a

better researcher, Dr. Woodall helped me get through my final semester at Virginia

Tech by selecting me to be supported in part by National Science Foundation Grant

DMII-0354859.

I also thank Dr. G. Geoffrey Vining for “that hallway conversation” during my

first semester here at Virginia Tech, which lifted my sights and gave me a new vision

of what I can become in the statistical profession. I also thank him for the many

hours he spent preparing my dossier for the Virginia Tech College of Science Most

Outstanding Graduate Student Award.

During my tenure here at Virginia Tech, I counselled with many faculty and

graduate students who greatly helped me. I thank Dr. George Terrell for always

holding an “open door policy” and for giving many insightful hints that lead to

big steps forward in completing my proofs. I thank Dr. J. P. Morgan for helping

me to get started on my proofs. I also thank Mahmoud A. Mahmoud, Landon Sego,

Willis Jensen, and Mike Joner for many insightful conversations in our quality control

research group meetings.

Most importantly, I lift a voice of gratitude to my Heavenly Father for hearing

and answering my many sincere prayers for help in finishing this work. During the

more difficult days I found myself on my knees multiple times in my graduate student

carrell pleading for help. I acknowledge the hand of divinity in guiding my thoughts

to find solutions when my mortal mind could not.

— James D. Williams

v

Contents

List of Figures ix

List of Tables xi

Glossary of Acronyms xii

Common Notation xiii

1 Introduction 1

1.1 Multivariate Statistical Process Control . . . . . . . . . . . . . . . . . 1

1.1.1 Phase I Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Phase II Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Research Scope . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Nonlinear Profile Monitoring . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Distribution of the T 2 Statistic . . . . . . . . . . . . . . . . . . . . . 4

1.4 Example of Monitoring Dose-Response Profiles . . . . . . . . . . . . . 5

1.5 Proposals for Future Work . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Literature Review 7

2.1 Nonlinear Profile Monitoring . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Distribution of the T 2 Statistic . . . . . . . . . . . . . . . . . . . . . 11

vi

3 Nonlinear Profile Monitoring 13

3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Nonlinear Model Estimation . . . . . . . . . . . . . . . . . . . 14

3.1.2 Multivariate T 2 Control Chart . . . . . . . . . . . . . . . . . . 16

3.1.3 Control Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.4 Monitoring the Variance . . . . . . . . . . . . . . . . . . . . . 21

3.1.5 Nonparametric Approach . . . . . . . . . . . . . . . . . . . . . 22

3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Distribution of the T 2 Statistic Based on the Successive Differences

Covariance Matrix Estimator 37

4.1 The T 2 Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.1 Asymptotic Marginal Distribution . . . . . . . . . . . . . . . . 39

4.1.2 Approximate Marginal Distribution . . . . . . . . . . . . . . . 40

4.2 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 Control Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.2 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Example of Monitoring Dose-Response Profiles from High Through-

put Screening 56

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Bioassay Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

vii

5.3 Methods: Homoscedastic Case . . . . . . . . . . . . . . . . . . . . . . 60

5.3.1 Dose-Response Model . . . . . . . . . . . . . . . . . . . . . . . 60

5.3.2 Phase I Profile Analysis . . . . . . . . . . . . . . . . . . . . . 61

5.3.3 Phase II Profile Monitoring . . . . . . . . . . . . . . . . . . . 68

5.4 Proposed Methods: Heteroscedastic Case . . . . . . . . . . . . . . . . 72

5.4.1 Phase I Profile Analysis . . . . . . . . . . . . . . . . . . . . . 72

5.4.2 Phase II Monitoring . . . . . . . . . . . . . . . . . . . . . . . 77

5.5 Analysis of Dose-Response Profiles . . . . . . . . . . . . . . . . . . . 77

5.5.1 Analysis Assuming Homoscedasticity . . . . . . . . . . . . . . 78

5.5.2 Analysis Accounting for Heteroscedasticity . . . . . . . . . . . 85

5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 Future Work and Conclusion 97

6.1 Profile Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Distribution of T 2D,i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3 Combining Multivariate T 2 Control Charts . . . . . . . . . . . . . . . 100

6.3.1 A Proposed Simulation Study . . . . . . . . . . . . . . . . . . 102

6.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

A Appendix A: Result from Chapter 3 107

B Appendix B: Results from Chapter 4 109

B.1 Asymptotic Distribution of T 2D,i . . . . . . . . . . . . . . . . . . . . . 109

B.2 Maximum Value of T 2D,i Statistics . . . . . . . . . . . . . . . . . . . . 110

References 124

Vita 130

viii

List of Figures

3.1 Vertical Density Profile (VDP) of 24 Particleboards . . . . . . . . . . 24

3.2 “Bathtub” Function Fit to Board 1 . . . . . . . . . . . . . . . . . . . 26

3.3 Nonlinear Regression Parameter Estimates a1, a2, b1, b2, c, and d by

Board for the VDP Data . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 The T 2 Control Charts for the VDP Data . . . . . . . . . . . . . . . 29

3.5 Spline Fit of Board 1 and Average Spline for the VDP Data . . . . . 31

3.6 Control Charts on Metrics . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1 Q-Q plots of empirical quantiles of the T 2D,i statistic (i = 1 and 2)

versus a χ2(p) distribution . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Boxplots of T 2D,i for p = 2 and m = 5. . . . . . . . . . . . . . . . . . . 42

4.3 Q-Q plots of empirical quantiles of the scaled T 2D,i statistics for combi-

nations of p = 4 and m = 30, 60 . . . . . . . . . . . . . . . . . . . . . 46

4.4 Q-Q plots of empirical quantiles of the T 2D,i statistic for combinations

of p = 8 and m = 30, 60 . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5 Overall Probability of a False Alarm for p = 2, 3, 4, 5 . . . . . . . . . 50

4.6 Overall Probability of a False Alarm for p = 6, 7, 8, 9 . . . . . . . . . 51

4.7 T 2D,i statistics and UCL values for the Quesenberry (2001) data . . . . 53

5.1 Estimated profiles for all 44 weeks, in a trellis plot. . . . . . . . . . . 80

5.2 Estimated profiles for all 44 weeks, overlaid. . . . . . . . . . . . . . . 81

ix

5.3 Variance chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Lack-of-fit chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.5 First T 2 chart for the mean profiles. . . . . . . . . . . . . . . . . . . . 83

5.6 Second T 2 chart for the mean profiles. . . . . . . . . . . . . . . . . . 84

5.7 Estimated in-control profiles. . . . . . . . . . . . . . . . . . . . . . . . 85

5.8 Fitted variance profiles for all 44 weeks. . . . . . . . . . . . . . . . . . 87

5.9 Fitted variance profiles for all 44 weeks, overlaid. . . . . . . . . . . . 88

5.10 T 2 chart based on successive differences (a) and the MVE (b). . . . . 89

5.11 Fitted mean profiles based on estimated weights for all 44 weeks. . . . 90

5.12 Lack-of-fit chart based on the weighted sums of squares. . . . . . . . 91

5.13 First T 2 chart for the mean profiles, heteroscedastic case . . . . . . . 91

5.14 Second T 2 chart for the mean profiles, heteroscedastic case . . . . . . 92

5.15 Estimated in-control mean profiles for the analysis assuming heteroscedas-

ticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

x

List of Tables

3.1 Estimated Parameter Values and T 2 Statistics for the VDP Data . . . 27

4.1 The T 2D,i statistics scaled according to Sullivan and Woodall (1996),

Mason and Young (2002), and Equation (4.6) for a data set. . . . . . 41

4.2 The T 2D,i statistics and UCLvec values based on the Quesenberry (2001)

data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1 Estimated parameter values, Wi, LOFi, and T2D,i statistics for the

DuPont Crop Protection data . . . . . . . . . . . . . . . . . . . . . . 79

5.2 S2ij values for every dose and week combination of the DuPont Crop

Protection data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3 Estimated θ1,i (Slope) values, their standard errors, and 99.88% one-

sided lower Wald confidence limit. . . . . . . . . . . . . . . . . . . . . 96

xi

Glossary of Acronyms

ANSS Average Number of Samples to Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

ARL Average Run Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

ATS Average Time to Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

GLIM Generalized Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

HDS Historical Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

HTS High Throughput Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

i.i.d. independent and identically distributed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

IPP Inter-profile Pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

LCL Lower Control Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

LOF Lack-of-fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

MCUSUM Multivariate Cumulative Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

MEWMA Multivariate Exponentially Weighted Moving Average . . . . . . . . . . . . . . . 100

MSE Mean Squared Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

MVE Minimum Volume Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

OD Optical Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

PC Percent Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

PE Pure Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

POX Power of x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

SPC Statistical Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

UCL Upper Control Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

VDP Vertical Density Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

WNLS Weighted Nonlinear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

xii

Common Notationyijk The response of replication k of the jth dose of profile i

f(·) Nonlinear regression function for the meanX Historical dataset matrix

βi Parameter vector for profile i

Di Matrix of derivatives of f(·) with respect to βidi The number of doses in profile i (as in Chapter 5)

rij The number of replications in dose j of profile i

m The number of samples in the historical dataset

p The number of parameters (dimension of βi)

α The probability of a false alarm (for an individual observation)

αoverall The probability that a control chart will have at least one false alarm

θi Variance function parameter vector for profile i

θ̂GLIM

i Estimate of θi obtained by generalized linear models techniques

S2ij Sample variance estimator for dose j in profile i

SP Sample (pooled) variance-covariance matrix estimator

SD Successive differences variance-covariance matrix estimator

SIPP Intra-profile pooling variance-covariance matrix estimator

SMV E Minimum volume ellipsoid variance-covariance matrix estimator

T 2P,i T2i statistic based on SP

T 2D,i T2i statistic based on SD

T 2IPP,i T2i statistic based on SIPP

T 2MV E,i T2i statistic based on SMV E

T 2P T2 chart based on T 2P,i statistics

T 2D T2 chart based on T 2D,i statistics

T 2IPP T2 chart based on T 2IPP,i statistics

T 2MV E T2 chart based on T 2MV E,i statistics

MV (m, i) The maximum value of the T 2D,i statistic

F(df1, df2) The F-distribution with df1 and df2 numerator and denominatordegrees of freedom, respectively

F(q, df1, df2) The qth quantile of an F-distribution with df1 and df2 numeratorand denominator degrees of freedom, respectively

BETA(p1, p2) The beta distribution with first and second shape parameters p1 andp2, respectively

BETA(q, p1, p2) The qth quantile of a beta distribution with first and second shapeparameters p1 and p2, respectively

χ2(p) The χ2 distribution with p degrees of freedom

χ2(q, p) The qth quantile of a χ2 distribution with p degrees of freedom

xiii

Chapter 1

Introduction

1.1 Multivariate Statistical Process Control

Multivariate Statistical Process Control (SPC) is a broad field of research and ap-

plications devoted to the improvement of products and processes. In monitoring

the quality of a product or process, quite often more than one quality characteris-

tic is measured on each manufactured item, thus producing a multivariate response.

These quality measurements are usually correlated with each other. Multivariate

SPC methods are designed to account for the correlations among the variables and

to simultaneously monitor the variables through time.

There are two phases of Multivariate SPC, Phase I and Phase II. A successful

Phase II analysis depends on a successful Phase I analysis. Although the two phases

are both dedicated to identifying out-of-control situations, each phase has a unique

objective.

1.1.1 Phase I Analysis

The Phase I analysis begins with an analysis of a historical data set (HDS) or a set of

baseline data consisting of multivariate observations taken on the product or process

1

consecutively for a specified period of time. In a Phase I analysis one seeks to identify

a stable subset of the HDS with which to estimate the in-control mean vector and the

in-control variance-covariance matrix for use in a Phase II analysis. Some examples

of unstable product or process conditions that one looks for are multivariate outliers,

shifts in the mean vector, or shifts in the variance-covariance matrix. Once out-of-

control data have been identified and eliminated, then a stable subset of the HDS is

used to estimate in-control parameters.

When evaluating the performance of competing control charts in a Phase I anal-

ysis, one usually sets the false alarm rates of the competing charts to be the same

by adjusting their respective control limits. Once the competing charts are set on

“equal footing,” then one calculates the probability of signal for given shifts in the

mean vector or covariance matrix, or some other out-of-control situation. The chart

that has the highest probability of signal for a given shift is the most desirable chart.

1.1.2 Phase II Analysis

The Phase II analysis depends on the success of the Phase I analysis in estimating

in-control mean, variance, and covariance parameters. The estimates of the in-control

parameters become the target values in Phase II analysis control charts. The lower

control limit (LCL) and upper control limit (UCL) also depend on the in-control

parameter estimates. The purpose of a Phase II analysis is, thus, continuous process

monitoring through time. Control limits are designed with the purpose of achieving

an acceptable false alarm rate. If an out-of-control signal is found, then assignable

causes of the signal are sought.

When evaluating the performance of competing control charts in a Phase II analy-

sis, one usually adjusts the control limits so that the in-control average time to signal

(ATS) of the competing charts are equal. Other similar performance measures may

2

also be used, such as the average run length (ARL) or the average number of samples

to signal (ANSS). One then calculates the ATS (or other performance measure) value

for a given shift from control for various out-of-control conditions. The chart that

has the lowest ATS value is the desirable chart since the objective is quick detection

of an out-of-control situation.

1.1.3 Research Scope

The scope of the research for this dissertation will be mostly restricted to Phase I

analyzes. Correspondingly, the performance measure employed to evaluate competing

control charts is the probability of signal given an out-of-control condition. In Chapter

5 we give both Phase I and Phase II profile monitoring methods to monitor the dose-

response quality profiles of herbicides in high throughput screening.

This research is divided into two general areas: (1) profile monitoring and (2)

properties of the multivariate T 2 control chart based on the successive differences

variance-covariance matrix estimator. In Chapter 2 we review published literature on

these two topics. Chapters 3 and 4 are dedicated to developing new statistical theory

and methodology in these two areas. Chapter 5 contains an example of nonlinear

profile monitoring applied to dose-response bioassay data. Chapter 6 contains a

proposal for future work.

1.2 Nonlinear Profile Monitoring

In many quality control applications, use of a single (or several distinct) quality

characteristic(s) is insufficient to characterize the quality of a produced item. In

an increasing number of cases, a response curve (profile), is required. Such profiles

can frequently be modeled using linear or nonlinear regression models. In recent

research others have developed multivariate T 2 control charts and other methods for

3

monitoring the coefficients in a simple linear regression model of a profile. However,

little work has been done to address the monitoring of profiles that can be represented

by a parametric nonlinear regression model.

In Chapter 3 we extend the use of the T 2 control chart to monitor the coeffi-

cients resulting from a nonlinear regression model fit to profile data. We give four

general approaches to the formulation of the T 2 statistics and determination of the

associated upper control limits for Phase I applications. We also consider the use of

nonparametric regression methods and the use of metrics to measure deviations from

a baseline profile. These approaches are illustrated using the vertical board density

profile data presented in Walker and Wright (2002).

1.3 Distribution of the T 2 Statistic

In the historical or retrospective data analysis of Phase I, especially with individual

observations, the choice of the estimator for the variance-covariance matrix is crucial

to successfully detecting the presence of special causes of variation. The traditional

estimator based on pooling all the historical observations is not useful in detecting a

shift in the mean vector, because such a shift near the middle of the data actually

reduces the probability that the corresponding T 2 chart will signal. The estimator

based on successive differences is useful in detecting such shifts, but the exact dis-

tribution for the corresponding T 2 chart statistic has not been determined. Three

approximate marginal distributions have been proposed.

In Chapter 4, several useful properties of the T 2 statistic based on the successive

differences estimator are demonstrated and an improved distribution for calculating

the UCL for individual observations in a Phase I analysis is given. This improved

method for calculating the UCL is evaluated by comparing the estimated false alarm

rate with the desired false alarm rate. It is shown that the proposed method achieves

4

a false alarm rate which is closer to the desired false alarm rate than the competing

methods when the sample size is small. For large sample sizes, the method based on

the asymptotic distribution is shown to be superior. We give recommendations when

to use each.

1.4 Example of Monitoring Dose-Response Pro-

files

In pharmaceutical drug discovery and agricultural crop product development, in vitro

bioassay experiments are used to identify promising compounds for future research.

The reproducibility and accuracy of the bioassay is crucial to be able to correctly dis-

tinguish between active and inactive compounds. In the case of agricultural product

development, compound activity for a given test organism, such as weeds, insects, or

fungi, is characterized by a dose-response curve measured from the bioassay. These

curves determine the quality of the bioassay procedure. When undesirable conditions

in the bioassay arise, such as equipment failure or hormetic effects, then a bioassay

monitoring procedure is needed to quickly detect such causes. In Chapter 5 we illus-

trate the proposed nonlinear profile monitoring methods to monitor the variability

of assay, the adequacy of the dose-response model chosen, and the estimated dose-

response curves for aberrant cases. We illustrate these methods with in vitro bioassay

data from DuPont Crop Protection collected over one year.

1.5 Proposals for Future Work

The concept of combining two or more control charts into one overall monitoring

procedure has been around for several years. The purpose of running two or more

charts simultaneously is to leverage the strengths of each chart. For example, one

5

chart may be sensitive to certain out-of-control situations and insensitive to others,

whereas the second chart may have the opposite sensitivity properties. The idea

is that the combination of the two charts will form an overall procedure which has

sensitivity to both classes of out-of-control situations. It is conjectured that the chart

combination’s loss in power to detect any specific out-of-control situation is small

compared to the large gain in overall performance over a wider class of problems.

As noted earlier, the purpose of a Phase I analysis is to identify a subset of stable

data from the HDS with which to estimate the in-control mean vector and variance-

covariance matrix. Two out-of-control situations that we seek to find are multivariate

outliers and shifts in the mean vector. It has been shown by Vargas (2003) that the

T 2 chart based on the minimum volume ellipsoid (MVE) mean and covariance matrix

estimators is very effective in detecting multivariate outliers for a Phase I analysis.

However, this chart is not very effective in detecting a shift in the mean vector.

Sullivan and Woodall (1996) studied the performance of the T 2 chart based on the

successive difference variance-covariance matrix estimator and found that this chart

is very effective in detecting a shift in the mean vector. However, this chart is not

very effective in detecting a multivariate outlier.

In order to have a multivariate control chart that performs well in both detecting

multivariate outliers and shifts in the mean vector, a new chart that combines the

chart proposed by Vargas (2003) and Sullivan and Woodall (1996) could be explored.

The details of how this chart is constructed are given in Chapter 6. The design for

a simulation study is proposed to evaluate the power properties of this new chart

compared to the Vargas (2003) chart alone, the Sullivan and Woodall (1996) chart

alone, and the T 2 control chart based on the usual sample variance-covariance matrix

estimator. The simulation consists of various out of control situations and the charts

can be evaluated based on the probability of detecting the out-of-control observations.

6

Chapter 2

Literature Review

2.1 Nonlinear Profile Monitoring

In SPC applications, manufactured items are sampled over time and quality character-

istics are measured. Often a product’s quality can be determined through measuring

several characteristics at each sampling interval. Multivariate T 2 control charts and

other methods have been developed for this scenario. See, for example, Fuchs and

Kenett (1998) and Mason and Young (2002). Increasingly, however, a sequence of

measurements of one or more quality characteristics are taken across some continuum

producing a curve or surface that represents the quality of the item. This curve or

surface is referred to as a profile. Very little work has been done in developing sta-

tistical process control methodology for monitoring profile data. For an overview of

profile monitoring techniques see Woodall, Spitzner, Montgomery, and Gupta (2004).

Profile data consist of a set of measurements with a response variable y and one

or more explanatory variables xj, j = 1, . . . , k, which are used to assess the quality of

a manufactured item. For example, the density profile of a particleboard is measured

on a vertical cross-section, which reveals patterns in board density across the depth of

the board. Another example is the estimated dose-response curve of a manufactured

7

drug. Once a batch of the drug is produced, several different doses of the drug are

administered to subjects and the responses measured. The resultant dose-response

curve summarizes the quality of the particular batch of the drug, indicating the maxi-

mal effective response, minimal effective response, and the rate in which the response

changes between the two. In these examples, a single measurement is insufficient to

adequately assess quality. Instead, a relationship between two variables, referred to

as the profile, should be monitored over time. Profile data is multivariate, but it is

not appropriate to apply standard multivariate control chart methods since this leads

to overparameterization. It is more efficient to model the structure of the data.

Profiles can take on several different functional forms, depending on the specific

application. For many calibration problems, the profile can be represented by a

simple linear regression model (see, e.g., Mahmoud and Woodall (2004)). Kang and

Albin (2000) proposed two methods, including a multivariate T 2 control chart, to

monitor such profiles. Specifically, we let the subscript i index each individual profile

(i = 1, . . . , m) in the historical Phase I data. In the simple linear regression case, the

ith profile is modeled as

yij = βi0 + βi1xij + ²ij, (2.1)

where yij is the jth measurement (j = 1, . . . , n), ²ij is the j

th random error, and xij is

the jth value of the explanatory variable corresponding to the ith profile. It is assumed

that the values of xij are the same for all i. This assumption is often reasonable since

in many engineering applications product or process profiles are measured at fixed

values of the explanatory variable at each sampling stage. Kang and Albin’s (2000)

multivariate T 2 chart is used to monitor simultaneously the β0, the y-intercept, and

β1, the slope. Kim, Mahmoud, and Woodall (2003) proposed an alternative approach

with better statistical properties such that individual control charts can be used for

the y-intercept and slope independently.

8

In general we refer to any profile that can be modeled by the linear regression

function

yij = βi0 + βi1xij1 + βi2xij2 + · · ·+ βikxijk + ²ij (2.2)

as a linear profile, where xijl, l = 1, . . . , k, are k predictor variables. The predictor

variables can be the original variables themselves, any function of the variables, or

any combination of both. In matrix notation, we let yi = [yi1, yi2, . . . , yin]′ be the

vector of responses for profile i, βi = [βi0, βi1, . . . , βik]′ be the vector of parameters to

be monitored, x′ij = [1, xij1, xij2, . . . , xijk] be the vector of explanatory variables for

item i, and ²i = [²i1, ²i2, . . . , ²in]′ be the corresponding vector of random errors. After

collecting the x′ij vectors into an n× p matrix, where p = k + 1, as

Xi =

x′i1x′i2...

x′in

,

then model (2.2) can be written in matrix form as

yi = Xiβi + ²i, i = 1, . . . ,m.

We assume that Xi is the same for each profile and that the vectors ²i are independent

and identically distributed (i.i.d.) multivariate normal random vectors with mean

vector zero and covariance matrix σ2I.

Jensen, Hui, and Ghare (1984) proposed a control chart based on the F -distribution

to monitor the k +1 parameters (coefficients) from a multiple linear regression model

for Phase II applications. Given the parameter vector estimator for item i, β̂i, and

the target parameter vector β0, one plots on their control chart the well-known F

statistic

Fi = (β̂i − β0)′X′iXi(β̂i − β0)/(k + 1)s2i

against i, where s2i =∑n

i=1(yi − ŷ)2/(n − p). A Phase I procedure for this generallinear case has yet to be developed.

9

In many cases, however, profiles cannot be well-modeled by a linear regression

function. Walker and Wright (2002) proposed a nonparametric approach for com-

paring profiles using additive models. Such models do not have a specific functional

form and have no model parameters to estimate, but rather one employs smooth-

ing techniques such as local polynomial regression or spline smoothing to model a

profile. Nonparametric regression techniques provide great flexibility in modeling the

response. One disadvantage of nonparametric smoothing methods is that the subject-

specific interpretation of the estimated nonparametric curve may be more difficult,

and may not lead the user to discover as easily assignable causes that lead to an

out-of-control signal.

Often, however, scientific theory or subject-matter knowledge leads to a natural

nonlinear function that well-describes the profiles. Hence, an alternative method is

to model each profile by a nonlinear regression function. A nonlinear profile of an

item can be modeled by the nonlinear regression model given generally by

yij = f(xij,βi) + ²ij, (2.3)

where xij is a k× 1 vector of regressors for the jth observation of the ith profile, ²ij isthe random error, βi is a p×1 vector of parameters for profile i, and f(·) is a functionwhich is nonlinear in the parameter vector βi. The random errors ²ij are assumed

to be i.i.d. normal random variables with mean zero and variance σ2. In many

applications, there is only one regressor (k = 1), but there are multiple parameters

to monitor (p > 1). An example of this form of the model is the 4-parameter logistic

model, often used to model dose-response profiles of a drug, given by

yij = Ai +Di − Ai

1 +

(xijCi

)Bi + ²ij, (2.4)

where yij is the measured response of the subject exposed to dose xij for batch i,

i = 1, . . . , m, j = 1, . . . , d, where d is the number of doses. In equation (2.4), we

10

have k = 1 and p = 4, giving four parameters to monitor, each parameter having a

specific interpretation. For example, Ai is the upper asymptote parameter, Di is the

lower asymptote parameter, Ci is the ED50 parameter (the dose required to elicit a

50% response), and Bi is the rate parameter for the ith batch. Another example is

the “bathtub” function described in Section 3.2 where the density of particleboard is

measured across the vertical profile. Note that for any given application, the specific

form of the nonlinear function, f , in equation (2.3) must be specified by the user.

In Phase I analysis, we are concerned with distinguishing between in-control con-

ditions and the presence of assignable causes so that in-control parameters may be

estimated for further product or process monitoring in Phase II analysis. In Chapter

3 we discuss some procedures for Phase I analysis for monitoring items or processes

whose quality is reflected by a nonlinear profile.

2.2 Distribution of the T 2 Statistic

Multivariate SPC is prevalent in many aspects of industry and manufacturing wher-

ever several different measures of a product or process are taken at each sampling

stage to assess quality. A common statistical method used to simultaneously monitor

the multiple quality characteristics is use of the Hotelling T 2 statistic.

For a retrospective Phase I analysis of an HDS the objective is twofold: (1) to

identify and eliminate multivariate outliers, and (2) to identify shifts in the mean

vector which might distort the estimation of the in-control mean vector and variance-

covariance matrix. The T 2 control chart is a tool to detect multivariate outliers and

mean shifts. The T 2 statistic one plots in the chart can be based on the usual sample

variance-covariance matrix estimator or some other alternative. One alternative is

the estimator based on the successive differences of observation vectors. As shown in

both Sullivan and Woodall (1996) and Vargas (2003) the T 2 statistic based on the

11

successive differences variance-covariance matrix estimator is effective in detecting

sustained step and ramp shifts in the mean vector. Sullivan and Woodall (1996)

found that the T 2 statistic based on the usual sample variance-covariance matrix

estimator is not only less effective in detecting a shift in the mean vector, but as the

magnitude of the shift increased, the power to detect the shift decreased. They found

that the sample variance-covariance matrix has the effect of “pooling” the data all

together such that a large step shift “inflates” the variance, thus making detection of

the shift more difficult.

Imperative to constructing any multivariate control chart is knowing the marginal

distribution of the test statistic. The UCL of the control chart is calculated from a

specified quantile of this marginal distribution. If the marginal distribution of the

test statistic is unknown or untractable, then the UCL is calculated from either an

approximate marginal distribution, where available, or from a Monte Carlo simula-

tion. If an approximate marginal distribution is used, the researcher should be aware

of the cases under which the approximation performs well and when it does not.

Unfortunately, the exact small-sample marginal distribution of the T 2 statistic

based on the successive differences variance-covariance matrix estimator is unknown.

Two approximate marginal distributions have been proposed, one by Sullivan and

Woodall (1996) and the other by Mason and Young (2002). Another possible approx-

imate marginal distribution is the asymptotic distribution. In Chapter 4 we give the

asymptotic marginal distribution and give recommendations for its use. We propose

an improved small-sample marginal distribution and demonstrate that the proposed

approximate distribution gives rise to UCLs that perform much better than the other

approaches for small sample sizes. We will discuss some useful properties of the dis-

tribution of the T 2 statistic based on the successive differences variance-covariance

matrix estimator and compare the performance of the two approximate distributions.

12

Chapter 3

Nonlinear Profile Monitoring

In Section 3.1 we give a brief review of nonlinear regression. We introduce the multi-

variate T 2 statistic in the context of monitoring nonlinear profiles. We then introduce

four formulations of the T 2 statistic and discuss the determination of the UCL for the

corresponding charts. In addition, a control chart to monitor the variance σ2 in the

context of monitoring profile data is proposed. Finally, we discuss a nonparametric

regression approach to monitoring the profiles. In Section 3.2 we illustrate the T 2

control charts and the nonparametric approaches using the vertical density profile

data of Walker and Wright (2002). In Section 3.3 we discuss the effects that auto-

correlation in the error terms may have on the analysis. Finally, in Section 3.4 we

discuss potential alternative methods and give directions for future research topics in

nonlinear profile monitoring.

3.1 Methodology

We begin a Phase I analysis with a baseline dataset consisting of m items sampled

over time. For each item i we observe a response yij and a set of predictor vari-

ables xij, i = 1, . . . ,m, j = 1, . . . , n, resulting in the quality profile for item i, i.e.,

(yi1,xi1), (yi2,xi2), . . . , (yin,xin). In this section we develop the methodology to ana-

lyze the profiles to gain understanding of the product or process in a Phase I setting.

13

3.1.1 Nonlinear Model Estimation

For simplicity of notation, we write the scalar model given in equation (2.3) in matrix

form by stacking the n observations within each profile as yi = (yi1, yi2, . . . , yin)′,

f(Xi, βi) = (f(xi1,βi), f(xi2,βi), . . . , f(xin,βi))′, and ²i = (²i1,²i2, . . . ,²in)′. The vec-

tor form is then given by

yi = f(Xi,βi) + ²i, i = 1, . . . , m. (3.1)

For the nonlinear regression model given in (3.1), estimates of βi for each sample

must be obtained. This is usually accomplished by employing the Gauss-Newton pro-

cedure and iterating until convergence to obtain the maximum likelihood estimates.

Specifically, for each sampling stage we define the m × p matrix of derivatives off(Xi,βi) with respect to βi as

Di =∂f(Xi, βi)

∂βi=

∂f(xi1,βi)∂βi1

∂f(xi1,βi)∂βi2

. . . ∂f(xi1,βi)∂βip

∂f(xi2,βi)∂βi1

∂f(xi2,βi)∂βi2

. . . ∂f(xi2,βi)∂βip

......

. . ....

∂f(xin,βi)∂βi1

∂f(xin,βi)∂βi2

. . . ∂f(xin,βi)∂βip

. (3.2)

We let f(Xi, β̂

(h)

i

)=

(f(xi1, β̂

(h)

i ), f(xi2, β̂(h)

i ), . . . , f(xin, β̂(h)

i ))′

, where β̂(h)

i is the

estimate of β at iteration h and we let D̂(h)i be the matrix of derivatives of f given

in equation (3.2) evaluated at β̂(h)

i . Then an iterative solution for β̂i is given by

β̂(h+1)

i = β̂(h)

i +(D̂i

′(h)D̂

(h)i

)−1D̂′(h)i

(yi − f(Xi, β̂(h)i )

).

Upon convergence of the algorithm, the estimated covariance matrix of β̂i is the

estimated Fisher information matrix and is given as

ˆV ar(β̂i

)= σ̂2i (D̂

′iD̂i)

−1,

where σ̂2i =∑m

j=1(yij−f(xij, β̂i))2/(n−p) and D̂i is the derivative matrix in equation(3.2) evaluated at the converged parameter vector estimate β̂i. See Myers (1990, chap.

14

9) or Schabenberger and Pierce (2002, chap. 5) for a concise discussion of nonlinear

regression model estimation. A more detailed treatment can be found in Gallant

(1987) and Seber and Wild (1989).

Unlike linear regression, the small-sample distribution of parameter estimators in

nonlinear regression is unobtainable, even if the errors ²ij are assumed to be i.i.d.

normal random variables. Instead, asymptotic results must be applied. Seber and

Wild (1989, chap. 12) give the asymptotic distribution of β̂i and the necessary

assumptions and regularity conditions for the asymptotic distribution to be obtained.

Given their assumptions and assuming that n−1D̂′iD̂i converges to some nonsingular

matrix Ωi as n →∞, the asymptotic distribution of β̂i is given by

√n(β̂i − βi) D−→ Np(0, σ2Ω−1i ) (3.3)

where βi is the asymptotic expected value of β̂i and Np indicates a p-dimensional mul-

tivariate normal distribution. For practical purposes, the distribution given by (3.3)

is incalculable since the matrix Ωi is unknown. Instead the approximate asymptotic

distribution of β̂i is commonly used, given by

β̂i·∼ Np(βi, σ2i (D′iDi)−1). (3.4)

The standard estimator of V ar(β̂i) is σ̂2i D̂

′iD̂i. For the most traditional “in-control”

case, we have βi = β for all profiles i = 1, . . . , m, where β is the in-control parameter

vector. Consequently, the Ωi and Di matrices are the same across all m items since

all items have the same underlying model, f , the same x-values are observed, and the

same values of βi. However, the D̂i matrices are not equal since the β̂i values vary

from profile to profile.

15

3.1.2 Multivariate T 2 Control Chart

In order to develop the methodology to monitor nonlinear profiles, we first consider the

general framework of the multivariate T 2 statistic. Given a sample of m independent

observation vectors to be monitored, wi (i = 1, . . . ,m), each of dimension p, the

general form of the T 2 statistic in Phase I for observation i is

T 2i = (wi − w̄)′ S−1 (wi − w̄) , (3.5)

where w̄ = 1m

∑mj=1 wi and S is some estimator of the variance-covariance matrix of

wi (Mason and Young 2002, chap. 2). We then plot the T2i statistics, i = 1, . . . ,m,

against i in a T 2 control chart, and out-of-control signals will be given for any T 2i value

exceeding the UCL. For determining the statistical properties of the T 2 chart it is

usually assumed that each of the wi vectors follows a multivariate normal distribution

with common mean vector µ and covariance matrix Σ. This assumption is critical

to finding the marginal distribution of T 2i , as discussed in Section 3.1.3.

In the nonlinear regression model given in equation (2.3), βi is a p × 1 vectorof parameters that determines the curve f(Xi,βi). We employ the multivariate T

2

statistic to assess stability of the the p parameters simultaneously, i.e., to evaluate

the assumption βi = β, i = 1, . . . , m. We do not employ individual control charts for

each of the p nonlinear regression parameters since this may give misleading results

due to the built-in correlation structure of the parameter estimators in nonlinear

regression.

Once β̂i is obtained from each sample in the baseline dataset, we calculate the

average vector¯̂β and some corresponding estimate of the covariance matrix, replace

wi with β̂i and w̄ with¯̂β in equation (3.5) to obtain

T 2i =(β̂i − ¯̂β

)′S−1

(β̂i − ¯̂β

). (3.6)

A large value of T 2i indicates an unusual β̂i, suggesting that the profile for item i

16

is out-of-control. In contrast to the traditional use of the T 2 statistic to monitor a

multivariate quality characteristic vector, we employ the T 2 statistic to monitor the

coefficient vectors of the nonlinear regression fit to each individual profile.

There are several choices for the estimator S. Here we discuss the effects of four

choices and later discuss under what conditions, if any, each should be used.

The first choice we consider for S is the sample covariance matrix, given by

SP =1

m− 1m∑

i=1

(β̂i − ¯̂β

)(β̂i − ¯̂β

)′. (3.7)

Consequently, the T 2i statistics take on the form

T 2P,i =(β̂i − ¯̂β

)′S−1P

(β̂i − ¯̂β

).

We use the subscript “P ” to emphasize the fact that SP is the sample variance-

covariance matrix based on pooling all the data in the HDS. We refer to the T 2 chart

based on values of T 2P,i as the T2P chart. Use of the T

2P,i values was mentioned by

Brill (2001) in the context of monitoring nonlinear profiles of a chemical product.

The advantage of this statistic is that it is very well understood and widely used.

However, as was shown by Sullivan and Woodall (1996) and Vargas (2003), a T 2

statistic based on SP is ineffective in detecting sustained shifts in the mean vector

during the Phase I period. In fact, it was shown that as the step shift size increased,

the power to detect the shift actually decreased.

An alternative choice of S is one based on successive differences, proposed origi-

nally by Hawkins and Merriam (1974) and later by Holmes and Mergen (1993). To

obtain the estimator, we define vi = β̂i+1 − β̂i for i = 1, . . . , m − 1 and stack thetranspose of these m− 1 difference vectors into the matrix V as

V =

v′1v′2...

v′m−1

.

17

The estimator of the variance-covariance matrix is

SD =V′V

2(m− 1) . (3.8)

Sullivan and Woodall (1996) showed that SD is an unbiased estimator of the true

covariance matrix if the process is stable in Phase I. The resulting T 2i statistics are

given by

T 2D,i =(β̂i − ¯̂β

)′S−1D

(β̂i − ¯̂β

). (3.9)

We refer to the T 2 chart based on values of T 2D,i as the T2D chart. Sullivan and Woodall

(1996) showed that the T 2D chart was effective in detecting both a step and ramp shift

in the mean vector during Phase I. They also showed that the T 2D,i values are invariant

to a full-rank linear transformation on the observations.

Neither SP or SD, however, directly incorporate the information on the variation

in the regression parameter estimators from the nonlinear regression estimation of

β̂i. Consequently, a third choice for S is to pool the m covariance matrices resulting

from the estimation of each β̂i vector. We refer to this method of calculating S as

intra-profile pooling (IPP). If we let D̂i be the derivative matrix for item i evaluated

at converged parameter vector estimate β̂i, then a third estimator of the variance-

covariance matrix is

SIPP =1

m

m∑i=1

ˆV ar(β̂i

)=

1

m

m∑i=1

σ̂2i (D̂′iD̂i)

−1. (3.10)

Consequently, the T 2 statistics are given as

T 2IPP,i =(β̂i − ¯̂β

)′S−1IPP

(β̂i − ¯̂β

), i = 1, . . . , m. (3.11)

We refer to the T 2 chart based on values of T 2IPP,i as the T2IPP chart.

Our fourth choice for S is a robust estimator of the variance-covariance ma-

trix known as the minimum volume ellipsoid (MVE) estimator, first proposed by

18

Rousseeuw (1984). In our application of the MVE method, we find outlier-robust es-

timates for both the in-control parameter vector and the variance-covariance matrix

based on finding the ellipsoid with the smallest volume that contains at least half

of the β̂i vectors, i = 1, . . . , m. The MVE estimator of β is the mean vector of the

smallest ellipsoid, and the estimator of the variance-covariance matrix is the sample

covariance matrix of the observations within the smallest ellipsoid multiplied by a

constant to make the estimator unbiased for multivariate normal data. In a simu-

lation study, Vargas (2003) studied the power properties of several different choices

of S in the context of the T 2 statistic given in (3.5) and found that the T 2 statistic

based on the MVE estimators of β and the variance-covariance matrix is very pow-

erful in detecting multivariate outliers. We denote the MVE estimators of β and the

covariance matrix by β̂MV E and SMV E, respectively. Hence, the fourth choice of T2

is

T 2MV E,i =(β̂i − β̂MV E

)′S−1MV E

(β̂i − β̂MV E

), i = 1, . . . , m. (3.12)

We refer to the T 2 chart based on values of T 2MV E,i as the T2MV E chart.

3.1.3 Control Limits

The distribution of the T 2i statistics for monitoring nonlinear profiles is more complex

than in the linear profile case. Recall that the distribution of the parameter estimators

in nonlinear regression is difficult to obtain for small sample sizes. Instead we employ

the asymptotic distribution (as n → ∞) of β̂i, i = 1, . . . , m. Hence, in order todetermine the marginal distribution of T 2i in this case, we assume that the sample size,

n, from each item in the baseline data set is of sufficient size such that the distributions

of β̂i, i = 1, . . . , m are approximately multivariate normal. The subsequent UCLs for

the multivariate T 2 control charts are determined based on this normality assumption.

In order to control the overall probability of a false alarm, based on some ap-

19

propriate UCL, the joint distribution of the T 2i values is required. However, these

values are correlated since¯̂β and S are used in all T 2i statistics (i = 1, . . . , m), thus

making the joint distribution of the T 2i values difficult to obtain. As an alternative,

Mahmoud and Woodall (2004) suggested using an approximate joint distribution as-

suming that the T 2i statistics are independent. We let α be the probability of a false

alarm for any individual T 2i statistic. Then the approximate overall probability of a

false alarm for a sample of m items is given by αoverall = 1 − (1 − α)m. Thus, fora given overall probability of a false alarm, we use α = 1 − (1 − αoverall)1/m in thecalculation of UCLs. In their simulation study, Mahmoud and Woodall (2004) found

that this approximation used to determine the UCLs performed well.

As noted in Tracy, Young, and Mason (1992), Gnanadesikan and Kettenring

(1972) suggested that for a stable process the marginal distribution of the T 2P,i statistic

is proportional to a beta distribution, i.e.,

T 2P,im

(m− 1)2 ∼ BETA(

p

2,m− p− 1

2

).

A formal proof can be found in Chou, Mason, and Young (1999). Note that it

is assumed that the distribution of β̂i is approximately normal, as given in equa-

tion (3.4). Therefore, an approximate UCL is (m−1)2

mBETA(1 − α, p

2, m−p−1

2), where

BETA(1 − α, p2, m−p−1

2) is the 1 − α quantile of a beta distribution with first and

second shape parameters p/2 and (m− p− 1)/2, respectively.Sullivan and Woodall (1996) proposed an approximate marginal distribution for

the T 2D,i statistic as

T 2D,im

(m− 1)2 ∼ BETA(

p

2,f − p− 1

2

), (3.13)

where f = 2(m−1)2

3m−4 . Mason and Young (2002, pp. 26-27) gave a correction to the

distribution proposed by Sullivan and Woodall (1996), by replacing each m in equa-

tion (4.4) with f . Hence, an approximate UCL is (f−1)2

fBETA(1− α, p

2, f−p−1

2). Our

20

simulation results, not shown here, showed that neither one of these approximations

worked well in obtaining the appropriate UCL, so in our example in Section 3.2 we

used simulation to find the appropriate UCL. In Chapter 4 we give an approximate

distribution that yields improved UCL values.

Since the small-sample exact distribution of the T 2IPP,i statistic is difficult to obtain

we use instead the asymptotic distribution. As proven in Appendix A, the asymptotic

distribution is χ2(p). The asymptotic distribution of T 2IPP,i must be used in practice

since the small-sample distribution is unknown. Hence, an approximate UCL is χ21−α,p.

The exact marginal distribution of the T 2MV E,i statistic is unknown and intractable.

Hence, in order to find the UCL for T 2MV E,i we used simulation.

3.1.4 Monitoring the Variance

In addition to checking the stability of each profile in the baseline dataset, it is

important to check the stability of the variability about each profile. This is anal-

ogous to monitoring the process variance in the standard univariate case. In the

case of monitoring profiles, we seek to monitor the variability about each profile, or

the within-profile variability. Our measure of within-profile variability is the mean

squared error (MSE) defined as MSEi =∑n

j=1(yij − ŷij)2/(n − p), where ŷij is thepredicted value of yij based on the nonlinear regression model in equation (2.3).

Wludyka and Nelson (1997) recommended a method to monitor variances based on

an analysis-of-means-type test utilizing S2i = MSEi. In their paper, S2i is plotted

against i with associated lower- and upper-control limits equal to (Lα,m,n−p)mS2 and

(Uα,m,n−p)mS2, respectively, where L and U are critical values given in their paper

and S2 is the average of the S2i values, i = 1, . . . , m. For large n, their approximate

upper and lower control limits are S2 ± hα,m,∞σ̂ where h is a critical value given inNelson (1983) and σ̂ = S2

√2(m− 1)/m(n− p). The S2i statistics are plotted on

21

a separate control chart to monitor the variance of the error terms and lack of fit

simultaneously with a T 2 control chart for the nonlinear regression parameters. We

recommend use of this method when within-profile error terms are independent. An

approach to monitor the variance when there is replication is given in Chapter 5.

3.1.5 Nonparametric Approach

When a parametric form of a profile would be overly complex, nonparametric proce-

dures may be more appropriate. These include fitting each profile via some smoothing

method, such as local polynomial regression, spline smoothing, or wavelets. Walker

and Wright (2002) give a spline-fitting approach to the vertical density profile (VDP)

of particleboard, which we use as an illustration in Section 3.2. However, these au-

thors discussed using splines to assess variation, not to monitor profiles in a Phase I

analysis to check for process stability. Winistorfer, Young, and Walker (1996) illus-

trated the use of splines to model the VDP of oriented strandboard generated from

a 32 factorial design with 3 replicates. However, their spline-fitting method is used

in the context of comparing profiles among differing experimental conditions, not

monitoring profiles in a Phase I or Phase II analysis.

For the case of a single explanatory variable, we denote the nonparametric fit of

profile i by ẏij, for the corresponding value of the explanatory variable equal to xj,

j = 1, . . . , n. The general nonparametric approach to monitoring profiles in Phase I

analysis is to establish a “baseline” curve with which to compare all other curves. A

natural choice of baseline profile is the average estimated profile across all m profiles,

denoted by ỹj =∑m

i=1 ẏij/m, j = 1, . . . , n. Once a baseline curve is found, some

appropriate distance metric can be used to measure how “different” each individual

curve is from the baseline. Researchers at Boeing (1998, pp. 140-144) proposed the

following three metrics:

22

1. Mi1 = sign(ẏij∗ − ỹij∗)×maxj |ẏij − ỹij|2. Mi2 =

∑nj=1 |ẏij − ỹij|

3. Mi3 =∑n

j=1 |ẏij − ỹij|/m,

where j∗ is the value of j that produces the maximum absolute deviation. The three

metrics, Mi1, Mi2, and Mi3, are referred to as the maximum deviation, sum of absolute

deviations, and the mean absolute deviation, respectively. Further, it may be of

interest to compute the absolute value of Mi1, which obviously reflects the magnitude

of the dissimilarity between ẏij and ỹij disregarding the direction of dissimilarity. We

denote this metric by Mi4. Other metrics are proposed in Gardner, et. al. (1997), who

note that metrics can be defined to detect changes in profiles resulting from particular

known process faults. One of these metrics is the sum of squared differences between

each estimated profile and the average profile, denoted Mi5 =∑n

j=1(ẏij − ỹij)2. Fora given metric, one plots the metric value for profile i against i (i = 1, . . . , m) and

checks for unusual observations. Researchers at Boeing (1998) suggested using a

standard univariate I-chart on the metrics to establish control limits. The method of

smoothing splines with several dissimilarity metrics is illustrated in Section 3.2.

3.2 Example

To illustrate the application of the various approaches we use the vertical density pro-

file data from Walker and Wright (2002), available at the website http://bus.utk.edu/

stat/walker/VDP.Allstack.TXT. In the manufacture of particleboard, the density

properties of the finished boards are quality characteristics that are monitored through

time. It is well known that the density (in pounds per ft3) near the core of a parti-

cleboard is much less than the density at the top and bottom faces of a board (see

Young, Winistorfer, and Wang (1999)). The standard sampling procedure calls for

a laser-aided density measuring device to scan fixed vertical depths of a board and

23

record the density at each depth. Since the depths are fixed for each board, we de-

note the depth xij by simply xj. Density measurements for this dataset were taken

at depths of xj = (0.002)j inches, j = 0, 1, 2, . . . , 313. Correspondingly, a sequence

of ordered pairs, (xi, yij), j = 1, . . . , n, results for board i and forms a vertical density

profile (VDP) of the board. A baseline sample of twenty-four particleboards was

measured in this way, and the twenty-four profiles are illustrated in Figure 3.1.

Figure 3.1: Vertical Density Profile (VDP) of 24 Particleboards

0.0 0.1 0.2 0.3 0.4 0.5 0.6

3540

4550

5560

65

Depth (inches)

Den

sity

(lbs

ft3)

An approach to understand the profile variation in Phase I is a method proposed

by Jones and Rice (1992) and discussed in Woodall, et. al. (2004). This method is

based on principal components where each profile is represented as an n × 1 vectorof responses and mutually orthogonal linear combinations of the responses are found

which explain as much variation as possible. Jones and Rice (1992) then recommended

24

plotting the profiles with the largest and smallest principal component scores to give

a visual interpretation of what each principal component represents. An example of

this method using the VDP data can be found in Woodall, et. al. (2004). They show

that the first two principal components correspond to the level and the flatness of

the profiles and explain 84% and 10.77% of the variation, respectively. We strongly

recommend the use of such plots.

Young, Winistorfer, and Wang (1999) introduced a statistical method to monitor

VDP data. With their method, one summarizes the density measurements into three

average density measurements: one near the core and one near each face. The three

averages are the quality characteristics that are subsequently monitored using a stan-

dard multivariate T 2 control chart. With this method one basically summarizes each

nonlinear profile into only three numbers with a corresponding loss of information.

An alternative approach without such a considerable loss of information is to

model the profiles themselves parametrically. The nonlinear function we use to model

profile i is a “bathtub” function given by

f(xij,β) =

a1(xij − c)b1 + d xj > ca2(−xij + c)b2 + d xj ≤ c

i = 1, . . . ,m; j = 1, . . . , n, (3.14)

where β = (a1, a2, b1, b2, c, d)′. One advantage of this nonlinear model is the inter-

pretability of the model parameters. For example, a1, a2, b1, and b2 determine the

“flatness”, c is the center, and d is the bottom, or the “level” of the curve. Differing

values of a1 and a2 or different values of b1 and b2 allow for an asymmetric curve about

the center c. Other parameterizations of this model are possible. However, reparam-

eterizing can effect the outcome of the Phase I analysis since different parameters

can produce different T 2 statistics. A strategy one should take is to parameterize

the model so that parameters of interest can be estimated and monitored. For this

example, the proposed parameterization addresses the shape, center, and symmetry

25

of the profiles. Figure 3.2 contains the “bathtub” function fit to board 1 from the

VDP data.

Figure 3.2: “Bathtub” Function Fit to Board 1

0.0 0.1 0.2 0.3 0.4 0.5 0.6

4045

5055

6065

Depth (inches)

Den

sity

(lbs

ft3)

The profile of board 1 is well-modeled by this parametric fit (R2 > 0.9999). For

each of the twenty-four boards in the baseline sample we fit the nonlinear model

in equation (3.14), and calculated the T 2P,i, T2D,i, T

2IPP,i, and T

2MV E,i statistics based

on the β̂i values. Parameter estimates for each of the twenty-four boards and the

corresponding T 2 statistics are given in Table 3.1. We plot the six parameter estimates

for each of the twenty-four boards in Figure 3.3. The plots of a1 and b1 expose a

potential outlier in board 15. The plots of a2 and b2 reveal potential outliers in

boards 4, 18, and 24. Boards 4 and 15 also appear as potential outliers in the plot of

d.

26

Table 3.1: Estimated Parameter Values and T 2 Statistics for the VDP Data

Board â1 â2 b̂1 b̂2 ĉ d̂ T2P;i T

2D;i T

2IPP;i T

2MVE;i

1 6560 3259 5.63 4.40 45.98 0.29 2.65 1.91 2261.01 6.002 470 291 3.01 2.74 42.08 0.32 7.56 5.27 4310.54 6.973 1812 2871 3.99 5.02 47.66 0.34 5.83 7.17 11255.64 8.644 6171 15009 4.25 7.39 46.63 0.39 12.21 17.28 4084.67 1131.815 4963 2251 5.14 4.20 43.43 0.30 1.65 2.27 1031.75 2.886 4556 3758 5.28 4.72 40.13 0.30 8.49 13.03 20105.39 9.837 5542 3815 5.25 5.00 44.15 0.31 2.15 3.49 211.32 3.588 3664 2979 4.89 4.41 44.06 0.30 0.79 0.97 129.64 2.699 28041 8872 7.58 4.95 43.22 0.26 4.62 7.10 3137.89 385.0310 1640 1207 4.17 3.39 41.84 0.28 4.30 5.05 5882.99 4.6111 3492 1031 5.82 3.17 46.06 0.25 8.66 8.95 2964.00 10.0012 915 750 3.45 3.52 44.37 0.32 1.80 1.99 334.45 2.2213 989 1392 3.58 4.05 45.47 0.32 3.42 4.42 1830.18 5.1814 1474 620 4.82 3.29 42.52 0.27 3.28 4.50 3218.19 7.0415 129068 5420 12.40 3.33 45.90 0.15 21.45 22.18 2048.70 17018.9116 10166 3822 5.83 4.86 44.19 0.30 3.83 5.60 266.80 12.9317 1483 603 4.07 3.26 44.83 0.30 2.30 2.53 663.30 2.3618 31156 31069 7.70 5.94 46.46 0.27 14.55 19.75 3113.48 8221.0019 418 198 3.22 2.67 42.84 0.30 4.58 3.90 1915.40 5.1620 3207 4741 4.88 5.02 44.45 0.30 5.34 5.59 23.82 34.0021 672 773 3.37 3.37 44.46 0.31 2.64 3.42 471.33 2.7922 3520 1807 5.10 4.01 45.52 0.29 1.71 1.37 1324.44 1.7323 1979 845 4.24 3.66 45.53 0.32 4.45 4.85 1843.91 7.3824 6095 26778 5.41 6.67 44.46 0.31 9.75 10.55 416.01 6676.21

We simulated UCLs for all four of the T 2 statistics to achieve an overall probability

of a signal equal to 0.05 for m = 24 boards. In our simulations, we sampled from

a multivariate normal distribution of dimension six, mean vector zero, and variance-

covariance matrix I, since the in-control performance of the methods does not depend

on the assumed in-control parameter vector or the variance-covariance matrix. We

repeated our simulation 200,000 times for each T 2 statistic, giving a standard error

for the estimated control limits less than 0.0005. The four UCL values are 14.72,

23.33, 28.22, and 65.37, for the T 2P , T2D, T

2IPP , and T

2MV E control charts, respectively.

27

Figure 3.3: Nonlinear Regression Parameter Estimates a1, a2, b1, b2, c, and d byBoard for the VDP Data

5 10 15 20

040

000

1000

00

Board

a 1

5 10 15 20

010

000

2500

0

Board

a 2

5 10 15 20

46

810

12

Board

b 1

5 10 15 20

34

56

7

Board

b 2

5 10 15 20

4042

4446

Board

c

5 10 15 20

0.15

0.25

0.35

Board

d

In Section 3.1.3 we gave theoretical UCLs for the T 2P , T2D, and T

2IPP control charts.

For m = 24 boards and p = 6 parameters, the theoretical UCLs are 14.71, 11.85,

and 20.63 for the T 2P , T2D, and T

2IPP control charts, respectively. The exact marginal

distribution of the T 2P,i statistic is known, thus the theoretical and simulated UCL

values are very similar. On the other hand, the exact marginal distribution of the

T 2D,i statistic is not known, so we used instead the correction given by Mason and

Young (2002, pp. 26-27). The large difference between the simulated and theoretical

UCL for the T 2D control chart shows that the approximate marginal distribution is

28

inadequate. The UCL of the T 2IPP control chart was computed based on the marginal

asymptotic distribution of the T 2IPP,i statistic because the small-sample distribution

is unknown. For our VDP example, the number of samples in the baseline dataset

is only m = 24 boards. The theoretical UCL values become more exact as m gets

larger.

In Phase I analysis, we are interested in identifying “outlying” or out-of-control

boards or a shift in the process which might affect the estimation of in-control pa-

rameters. We compared the four T 2 control charts for assessing process stability and

identifying outlying profiles. In Figure 3.4 we illustrate all four T 2 control charts for

the VDP data.

Figure 3.4: The T 2 Control Charts for the VDP Data. (a) The T 2P control chart basedon the pooled sample covariance matrix, (b) T 2D control chart based on the successivedifferences estimator, (c) T 2IPP control chart based on the intra-profile pooling method,and (d) T 2MV E control chart based on the minimum volume ellipsoid, with UCL valuesof 14.72, 23.33, 28.22, and 65.37, respectively.

5 10 15 20

05

1015

2025

(a)

Board

5 10 15 20

05

1015

2025

(b)

Board

5 10 15 20

050

0015

000

(c)

Board

5 10 15 20

050

0010

000

(d)

Board

29

The T 2P control chart based on the pooled sample variance-covariance matrix esti-

mator indicates that board 15 has the only out-of-control profile, although the profile

for board 18 is borderline. The T 2D chart based on the successive differences estimator

does not produce an out-of-control signal. Note that the T 2D,i statistic accentuates

the same outlying observations of the T 2P chart, but has a larger UCL. As discussed in

Sullivan and Woodall (1996), the T 2P control chart has greater power to detect isolated

outlying observations than the T 2D control chart based on the successive differences

variance-covariance matrix estimator, however the T 2D chart is better for detecting a

sustained shift in the mean vector. For this dataset, there is no apparent sustained

shift in the regression parameter vector.

In the T 2IPP control chart based on the intra-profile pooling variance-covariance

matrix estimator, all the T 2IPP,i statistics are above the UCL except for board 20,

indicating that most of the profiles are significantly different from each other in the

statistical sense. Recall that SIPP in the T2IPP,i statistic is the average of the m

within-profile variance-covariance matrices of the β̂i vectors. For this dataset, the

within-profile variability is much smaller than the between-profile variability, causing

the T 2IPP,i statistics to be very large. The use of this method is appropriate only if

there is no expected common cause variation between profiles. We expect that in

most applications there will be some common cause variation between profiles.

The T 2MV E control chart based on the MVE estimator indicates that boards 4,

9, 15, 18, and 24 have outlying profiles. The most pronounced outlier is board 15,

which the T 2P chart also indicated as the most severe outlier. As shown by Vargas

(2003), the T 2MV E control chart is very powerful in detecting multivariate outliers.

Investigating the table of parameter estimates for these boards, given in Table 3.1,

it seems reasonable that the boards 15 and 18 are outliers, with boards 4, 9, and 24

worthy of further investigation.

30

As discussed in Section 3.1.5, an alternative approach to modeling the profiles

with a parametric curve is to employ nonparametric smoothing techniques to model

the profiles. Walker and Wright (2002) employed spline smoothing with 16 degrees of

freedom to model the twenty-four boards of the VDP data. We replicated their spline

fits to each profile. After obtaining the spline fits to each profile, ẏij, i = 1, . . . , m; j =

1, . . . , n, the average spline, ỹj, is calculated. For example, the spline fit to board 1

and the average spline are illustrated in Figure 3.5.

Figure 3.5: Spline Fit of Board 1 (Above) and Average Spline (Below) for the VDPData

0.0 0.1 0.2 0.3 0.4 0.5 0.6

4045

5055

6065

Depth (inches)

Den

sity

(lbs

ft3)

The spline fit with 16 degrees of freedom provides a concise summary of the shape

of the profile from board 1. The average spline fit is systematically lower than the

spline fit to board 1. In order to determine which boards are in-control we calculated

31

dissimilarity metrics as given in Section 3.1.5. Since the metrics Mi2 and Mi3 differ

only by a constant, it is not helpful to consider both metrics simultaneously. Instead

we calculate the metrics Mi1, Mi3, Mi4, and Mi5, and then employ an I-chart based

on the moving range to establish control limits, as suggested by researchers at Boeing

(1998). We plot each metric versus i with associated control limits to obtain control

charts. The four charts are given in Figure 3.6.

Figure 3.6: Control Charts on Metrics: (a) Mi1, the Maximum Deviation, (b) Mi3,the Maximum Absolute Deviation, (c) Mi4, the Sum of Squared Differences, and (d)Mi5, the Mean Absolute Deviation for the VDP Data

5 10 15 20

−10

−50

510

(a)

Board

Max

imum

Dev

iatio

n

5 10 15 20

−50

510

(b)

Board

Max

imum

Abs

olut

e De

viatio

n

5 10 15 20

−200

00

2000

6000

(c)

Board

Sum

of S

quar

ed D

iffer

ence

s

5 10 15 20

−20

24

(d)

Board

Mea

n Ab

solu

te D

evia

tion

The charts based on metrics Mi1 and Mi4 both give the same conclusion, that all

the profiles of the boards are in-control. This is not surprising since Mi4 is the absolute

32

value of Mi1, but both are given for illustrative purposes. The most extreme value

of the metrics came from board 14, with values of M14,1 = −5.79 and M14,4 = 5.79 .This value represents the maximum (absolute) deviation of the spline fit to board 14

from the average spline fit.

Similarly, the charts based on metrics Mi3 and Mi5 both give the same conclusion,

that the profile for board 6 is out-of-control. Referring to Figure 3.1, board 6 is

the one with the profile that is consistently lower than all other boards. The next

most extreme value of the two metrics is that of board 3, although it does not give

an out-of-control signal. Again, referring to Figure 3.1, board 3 is the one with the

profile that is consistently higher than all the other boards. It is apparent that these

two metrics measure how consistently different each profile is from the average profile

across the depth values, whereas metrics Mi1 and Mi4 measure the greatest extent to

which a profile is from the average at any particular depth value. It is important to

note that the results for the control charts on the metrics (Figure 3.6) do not show

the same results as the control charts based on the regression estimators in Figure

3.4. If the profile can be adequately represented by a parametric model, then this, in

general, will lead to more effective charts.

In addition to monitoring the regression parameter vectors of the profiles in a

Phase I analysis, we should monitor the variation about the profiles to check for

stability. As mentioned in Section 2.4, we recommend using the methods of Wludyka

and Nelson (1997) to monitor the variance σ2. Use of their method is appropriate

when the error terms within a profile are independent. In our VDP example, however,

the within-profile density measurements are spatially correlated. A more appropriate

control chart in this case to monitor the process variance σ2 is a topic for further

research.

33

3.3 Autocorrelation

Engineering applications that give rise to nonlinear profile data may lead to autocor-

related error terms. A common source of autocorrelated errors is the spatial or serial

manner in which data are collected. The VDP data, for example, is spatially corre-

lated because the density measurements are taken at close intervals along the vertical

depth of the particleboard. On the other hand, some nonlinear profiles may have

independent error terms. One example of this is typical dose-response data where

several doses of a particular drug are administered to different subjects and their

responses are measured. The subsequent error terms in the nonlinear dose-response

curve are typically assumed to be independent.

When the error terms are autocorrelated due to either serial, spatial or any other

effects, the correlation structure should be taken into account in the analysis. Failure

to do so might yield misleading results in some cases, particularly with the control

chart to monitor σ2. In the example of Section 3.2, we estimated parameters of a

nonlinear regression model for each board. For our nonlinear model we assumed that

the errors ²ij are i.i.d. For the VDP data, it may be reasonable to assume that the

²ij are correlated. If this is the case, perhaps an alternative approach would be to

employ either nonlinear mixed model methods or generalized estimating equations

(GEE) methodology. Both methods can be used to estimate the mean function, or

profile, while accounting for autocorrelation in the error structure. A more detailed

treatment of these methods can be found in Schabenberger and Pierce (2002) and

Hardin and Hilbe (2003). In the context of analyzing nonlinear profiles for Phase I

applications, this approach is a topic that requires further investigation.

34

3.4 Discussion

In Phase I, we are interested in identifying outlying observations as well as identifying

step or ramp shifts in the mean vector over time. As shown by Vargas (2003), the

robust variance-covariance matrix and mean vector estimators employed in the T 2MV E,i

statistic are very powerful in detecting multivariate outliers, but are not powerful in

detecting a step shift. However, the opposite is true of the T 2D,i statistic. As shown

by Sullivan and Woodall (1996), the T 2D chart is powerful in detecting a step shift,

but not powerful in detecting multivariate outliers. One possible alternative is to

employ both the T 2D and T2MV E charts simultaneously, the former chart sensitive to

step shifts and the latter sensitive to outliers. However, in examining both charts

simultaneously, one must be cautious of inflating the false alarm probability. This

approach is also a topic for research, as discussed in Chapter 6.

We have not given a detailed treatment of the nonparametric approaches to moni-

toring profiles discussed in Section 3.1.5. Rather, we have only described some meth-

ods that have been proposed and then illustrated their use with the VDP data. Some

issues that need to be addressed, for example, are the best nonparametric estimation

technique for a given scenario, the best metrics to apply, the strengths and weaknesses

of each metric, and the distributional properties of the metrics in order to establish

valid control limits. It is our hope that the present work will generate interest in

investigating these and other unresolved issues.

The field of profile monitoring using control charts has potential to extend statis-

tical process control to a wide variety of engineering and pharmaceutical applications.

With the increasing ease and efficiency in which processes and products can be mea-

sured, there is a need for statistical methodology to be developed which can accom-

modate the growing needs of industry. We have encountered a number of engineering

applications in which a response curve is needed to assess quality. In some cases, the

35

shape of the response curve can be well-represented by a parametric nonlinear regres-

sion function. In this paper we have developed control chart methodology to monitor

such nonlinear profiles for Phase I applications. When a profile cannot be easily de-

scribed by a parametric function, nonparametric methods may be applied. Our VDP

example shows, however, that the parametric and nonparametric approaches do not

always lead to the same conclusions regarding outlying profiles.

36

Chapter 4

Distribution of the T 2 StatisticBased on the SuccessiveDifferences Covariance MatrixEstimator

4.1 The T 2 Statistic

In a Phase I analysis, we begin with an HDS consisting of m independent vectors

of dimension p observed over time, where p is the number of quality characteristics

that are being measured, and p < m. We make a standard assumption that when

the process is in-control the observation vectors, xi, i = 1, . . . , m, are independent

and identically distributed multivariate normal random vectors with common mean

vector and covariance matrix, i.e.,

xi ∼ Np(µ,Σ).

For example, in the context of nonlinear profile monitoring discussed in Chapter 3,

each xi is equal to β̂i, i = 1, . . . , m. It is useful to define the m× p HDS matrix X as

X =

x′1x′2...

x′m

.

37

The Hotelling’s T 2 statistic measures the Mahalanobis distance of the correspond-

ing vector from the sample mean vector. The general form of the statistic is

T 2i = (xi − x̄)′ S−1 (xi − x̄) ,

where x̄ = 1m

∑mi=1 xi and S is some estimator of Σ.

A common choice for S is the sample variance-covariance estimator given by

SP =1

m− 1m∑

i=1

(xi − x̄) (xi − x̄)′ .

The T 2 statistics based on SP are then

T 2P,i = (xi − x̄)′ S−1P (xi − x̄) , i = 1, 2, . . . , m (4.1)

As shown in Chapter 3, the exact distribution of T 2P,i is proportional to a beta distri-

bution, i.e.,

T 2P,im

(m− 1)2 ∼ BETA(

p

2,m− p− 1

2

); i = 1, . . . , m. (4.2)

An alternative choice of S is one based on SD of Equation (3.8), where β̂i is

replaced with xi. The resulting T2 statistics based on SD are given by

T 2D,i = (xi − x̄)′ S−1D (xi − x̄) . (4.3)

As noted in Sullivan and Woodall (1996), Holmes and Mergen (1993) incorrectly

specify the Phase I UCL of a T 2 chart based on T 2D,i statistics by applying con-

trol limits based on a Phase II analysis. Sullivan and Woodall (1996) proposed an

approximate distribution for T 2D,i as

T 2D,im

(m− 1)2 ∼ BETA(

p

2,f − p− 1

2

), (4.4)

where f = 2(m−1)2

3m−4 . Mason and Young (2002, pp. 26-27) suggested an adjustment to

this approximation, replacing each m in Equation (4.4) with