University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange

Masters Theses Graduate School

5-2004

An Applied Statistical Reliability Analysis of the Internal Bond of Medium Density Fiberboard

David Joesph Edwards University of Tennessee, Knoxville

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Recommended Citation Edwards, David Joesph, "An Applied Statistical Reliability Analysis of the Internal Bond of Medium Density Fiberboard. " Master's Thesis, University of Tennessee, 2004. https://trace.tennessee.edu/utk_gradthes/4659

This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council:

I am submitting herewith a thesis written by David Joesph Edwards entitled "An Applied Statistical Reliability Analysis of the Internal Bond of Medium Density Fiberboard." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Statistics.

Frank Guess, Timothy Young, Major Professor

We have read this thesis and recommend its acceptance:

Halima Bensmail

Accepted for the Council: Carolyn R. Hodges

Vice Provost and Dean of the Graduate School

(Original signatures are on file with official studentecor r ds.) To the Graduate Council:

I amsubmitting herewitha thesis written by David Joseph Edwardsentitled "An Applied Statistical Reliability Analysis of the InternalBond of Medium Density Fiberboard." I have examinedthe finalpaper copy of this thesis for form and content and recommend that it be accepted in partial fulfillmentof the requirements for the degree of Master of Science, with a major in Statistics.

Frank Guess, Co-chair

We have read this thesis and recommend its acceptance:

� Halima Bensmail

An Applied Statistical Reliability Analysis of the Internal Bond of Medium Density Fiberboard

Masters Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville

David Joseph Edwards May 2004 Dedication

This masters thesis is dedicated to my wife:

Elizabeth,

who has provided constant love and support

throughoutthis thesis' researchand writing process.

Without her compassion, understanding, and sparkfor life,

progress toward the goal of finishingthis thesis

would not have been easy.

She is truly a gift from God.

To my familywho has always been there for me and

who have always believed I could accomplish anything.

To my friendswho makeme laughand remind me that

lifeis an adventure to be enjoyed.

To our pet Snoopy for being a part ofmy support structure.

Her listening ear andlove for attention has made lifeless stressful

andmore enjoyable.

11 Acknowledgements

The work on this thesis was fundedby Associate ResearchProf essor Timothy

Young fromthe United States Departmentof Agriculture, Special Wood Utilization

ResearchGrants, which is administered by The University of Tennessee Agricultural

Experiment Station andthe Tennessee Forest Products Center.

I would like to thankmy thesis committee co-chairs,Dr. FrankGuess and

Timothy Young, fortheir guidance, persistence,and attention to details. They have helped to provide considerable growthfor future graduate work andmy professional careeras a statistician. In addition, I would like to thankthe other member of my committee, Dr. Halima Bensmail, who has provided needed supportand suggestions to enhancemy understandings.

Also, I appreciate Dr. HamparsumBozdogan and Dr. WilliamSeaver forreading over portionsof this thesis andproviding their helpfulcomments andchanges. Dr.

Bozdogan read andcommented on Chapter 4 on informationcriteria, while Dr. Seaver did Chapter 5 on bootstrapping. I appreciate some additional comments fromDr. Russell

Zaretzkion his reading of Chapter 5 on bootstrapping. Thanks also to the other members of the facultyand staffof the Departmentof Statistics and the Tennessee Forest Products

Center. Working together andnetworking has certainlyaided in the completion of this thesis.

111 Abstract

The forestproducts industryhas seen tremendous growthin recent yearsand has a huge impact on the economies of many countries. For example,in the state of Maine in

1997, the forestproducts industry accounted for 9 billion U.S. dollarsfor that year. In the state of Tennessee, forexample in 2000, this figurewas 22 billion U.S. dollarsfor that year. It has, therefore, become more importantin this industry to focuson producing higherquality products. Statistical reliability methods, amongother techniques, have been employed to help monitor andimprove the quality of forestproducts. With such a largefocus on quality improvement, data is quite plentiful, allowing for more useful analysesand examples.

In this thesis, we demonstrate the usefulnessof statistical reliability tools and apply them to help assess, manage, and improve the internalbond (IB) of medium density fiberboard (MDF). MDF is a highquality engineered wood composite that undergoes destructivetesting during production. Workers test cross sections ofMDF panelsand measure the IB in pounds per squareinches. IB is a key metric of quality since it provides a direct measurement for the strength of MDF, which is important to customers and the manufacturers.

Graphical procedures such as histograms, scatter plots, probability plots, and survivalcurves are explored to help the practitioner gain insights regardingthe distributions of IB and strengths of differentMDF product types. Much information can be revealed froma graphicsapproach.

Thoughuseful, probability plots canbe a subjective way to assess the parametric

lV distribution of a data set. Insightfuldevelopments in informationcriteria, in particular

Akaike's Information Criteria andBozdogan's InformationComplexity Criteria, have made probability plotting more objectiveby assigningnumeric scores to each plot. The plot with the lowest score is deemed the best amongcompeting models. In application to

MDF, we will see that initial intuitions arenot always confirmed. Therefore,information criteria prove to be usefultools for the practitionerseeking more clarityregarding distributional assumptions. We recommend more usage of these helpfulinformation criteria.

Estimating lower percentiles in failuredata analysis canprovide valuable assistance to the practitioner forunderstanding product warranties and their costs. Since data may not be plentiful forthe lower tails, estimation of these percentiles may not be an easy task. Indeed, we stress times to not even try to estimate the lowest percentiles. If samplesare large andparametric assumptions are weakor not available, asymptotic approximationscan be utilized. However, unless the samplesize is sufficientlylarge, such approximations will not be accurate.

Bootstraptechniques provide one solution forthe estimation of lower percentiles when asymptoticapproximations should not be utilized. This computer intensive resamplingscheme provides a method for estimating the true sampling distribution of these percentiles, or anypopulation parameterof interest. This canbe used forvarious parametricmodels or fornonparametric settings, when the parametricmodel mightbe imperfect or misspecified. The empirical bootstrap distributioncan then be used for inferencessuch as determiningstandard errors andconstructing confidenceintervals.

Helpful applications of the bootstrap to the MDF data show this procedure's advantages

V andlimitations in order to aid the practitioner in their decision-making. Graphics can readily warn the practitioner when even certain bootstrap procedures are not advisable.

To be able to say that improvements have been made, we must be able to measure reliability expressed in percentiles that allow for statistical variation. We need to make comparisons of these reliability measures between products and within products before andafter process improvement interventions. Knowing when to trustconfidence intervalsand when not to trust them arecrucial for managers and users ofMDF to make successful decisions.

vi Table of Contents

Chapters Page

1. Introduction ...... 1

2. Literature Review ...... 7 Medium Density Fiberboard...... 8 Reliability ...... 15 Information Criteria ...... 17 Bootstrapping...... 18

3. Exploring Graphically and Statistically the Reliability of Medium Density Fiberboard...... 24 Introduction andMotivation ...... 24 Categorizing Typesof MDF ...... 29 Exploring Graphically andStatistically IB in Typesof MDF...... 31 Summary...... 44

4. Using Helpful InformationCriteria to Improve Objective Evaluation of Probability Plots ...... 45 Introduction andMotivation ...... 4 5 A Brief Summaryof AIC and ICOMP ...... 49 Using AIC andICOMP with Probability Plots to determine the Parametric Distribution of the InternalBond of Medium Density Fiberboard...... 54 Summary...... 59

5. Applying Bootstrap Techniquesfor Estimating Percentiles of the InternalBond of Medium Density Fiberboard...... 61 Introduction andMotivation ...... 61 A Brief Introduction to Bootstrap SamplingMethods andConfidence Intervals...... 69 Methods of Bootstrap Sampling...... 69 BootstrapConfidence Intervals...... 71 Exploring BootstrapMethods andConfidence Intervals for Percentiles of the Internal Bond ofMDF ...... 78

Vll Summaryand Conclusions ...... 98

6. Summaryand Concluding Remarks...... 105

Bibliography...... 114

Vita...... 123

vm List of Tables

Page

Table 3.1. Groupand type numbers for differentMDF products with (A, B, C) where A = density, B = thickness, and C = width where for example Type 1 in Group I represents A = 46 pounds per cubic foot,B = 0.625 inches thickness, and C = 61 inches (or the equivalent metric units) ...... 30

Table 3.2. Descriptive statistics comparison of Types1 and2 ...... 31

Table 3.3. Normalitytest comparisons for Type 1 ...... 38

Table 3.4. Normality testcomparisons for Type2 ...... 38

Table 4.1. AIC and!COMP for Type 1 probability plots ...... 55

Table 4.2. AIC and!COMP forType 2 probability plots ...... 57

Table 4.3. AIC and!COMP forType 3 probability plots ...... 57

Table 4.4. AIC and !COMP forType 5 probability plots ...... 58

Table 5.1. 95% Asymptoticnormal confidence intervals for Type1 MDF ...... 81

Table 5.2. Fully nonparametric 95% bootstrapconfidence intervals forType 1 MDF ...... 81

Table 5.3. Fully parametric 95% bootstrapconfidence intervals forType 1 MDF ...... 85

IX Table 5.4. NBSP 95% bootstrapconfidence intervals for Type 1 MDF ...... 87

Table 5.5. 95% Asymptoticnormal confidence intervals for Type 5 MDF ...... 90

Table 5.6. Fully nonparametric 95% bootstrap confidenceintervals for Type 5 MDF ...... 90

Table 5.7. Fully parametric 95% bootstrap confidenceintervals for Type 5 MDF ...... 94

Table 5.8. NBSP95% bootstrap confidenceintervals for Type 5 MDF ...... 96

X List of Figures

Page

Figure 2.1. A comparisonof common particleboard to MDF...... 9

Figure 3.1. Cross sections of forestproducts fromwidth view...... 25

Figure 3.2. Cross sections of forestproducts from diagonal view...... 25

Figure 3.3. Histogramand Boxplot of Primary Product (Type1) fromJMP ...... 33

Figure 3.4. Histogramand Boxplot of Type2 fromJMP ...... 33

Figure 3.5. Overlay Plot of Types1 and2 fromJMP ...... 34

Figure 3.6. NCSS Weibull probability plots...... 36

Figure 3.7. NCSS Normalprobability plots ...... 36

Figure 3.8. SurvivalPlot ofTypes 1 and2 fromJMP ...... 38

Figure 3.9. SurvivalPlot forTypes 1 and 5 from JMP ...... 40

Figure 3.10. SurvivalPlot for Types 1 and 3 from JMP...... 42

Figure 3.11. Overlay Plot of Types 1 and 3 from JMP ...... 42

Figure 3.12. SummarySurvival Plot forTypes 1, 2, 3, and 5 fromJMP ...... 43

Xl Figure 4.1. Comparing Probability Plots for Type 1 MDF using JMP ...... 46

Figure 4.2. Comparing Probability Plots forType 1 MDF using SAS ...... 46

Figure 4.3. Comparing Probability Plots forType 1 MDF using S-PLUS ...... 46

Figure 4.4. Type 1 probability plots by MATLAB ...... 55

Figure 4.5. Type 2 probability plots by MATLAB ...... 57

Figure 4.6. Type3 probability plots by MATLAB ...... 57

Figure 4. 7. Type5 probability plots by MATLAB ...... 58

Figure 5. 1. Samplingdistribution of percentiles forType 1 MDF under the fully nonparametricbootstrap samplingmethod ...... 82

Figure 5.2. Samplingdistribution of percentiles for Type 1 MDF under the fully parametricsampling method ...... 86

Figure 5.3. Sampling distribution of percentiles forType 1 MDF under the NBSP method ...... 88

Figure 5.4. Sampling distribution of percentiles forType 5 MDF under the fully nonparametric bootstrap samplingmethod ...... 91

Figure 5.5. Samplingdistribution of percentiles forType 5 MDF under the fully parametricbootstrap samplingmethod ...... 95

Figure 5 .6. Samplingdistribution of percentiles forType 5 MDF under the NBSP method ...... 97

XU Chapter 1

Introduction

Meeker andEscobar (2004) stress the importance of reliability formodem products and point out that "manufacturingindustries have gone througha revolution in the use of statistical methods for product quality. Tools forprocess monitoring and experimental design are much more commonly used today to maintain and improve product quality." Consumer expectations and demands for higher quality products are growingalmost as fast as the technology that makes product development possible.

Meeker and Escobar(1 998) indicate "customers expect purchasedproducts to be reliable andsafe" as well as to "performtheir intended function under usual operating conditions, forsome specifiedperiod of time."

In order to narrow the focus here, it should be mentioned that this growth in both technology and customer expectations is certainly familiarto the forest products industry.

This industry hasseen tremendous growthand hasan impact on the economies of countries aroundthe world plus manystates in the United States, e.g., Tennessee, Maine, etc. Forest products are present in furniture, shelving, flooring, structural applications, and unquestionably many others.

According to Williams(200 1) and Young and Winistorfer( 1999), raw materials

th andlabor costs were relatively low and inexpensive during the early20 century. Instead, technology proved to be a major limitation for the enhancement of production.

The quality of finalproducts was of little concernto most forestproducts industries. This has changed drastically with the dawn of the informationage andnewer technologies.

1 Restrictions and regulations have been enforcedthat limit the availability of raw materials, thus driving prices for these materials higher. Stronger interest hasbeen placed on producing more reliable products. Approaches in statistical quality and process managementhave been employed to help improve the quality of forestproducts.

In this thesis, as a case study, statistical reliability ideas and tools are applied to help assess, manageand improve the strength of a particularforest product, Medium

Density Fiberboardor simply MDF. This particular producthas undergone tremendous marketgrowth, and internationaldemands forMDF are steadily increasing.

During the manufacturingprocess ofMDF, the product undergoes extensive monitoring where manyprocess variables such as fiber-matweight, line speed, MDF width andthickness, etc., are collected automatically via sensors. In particular,the real­ time data warehouseused in this thesis stored 2,850 process variables and was obtained froma world-classNorth AmericanMDF manufacturermaking 100 million lineal feetof

MDF per year. A smaller subset of 230 process variableswas used throughoutthis study.

th th The data warehousewe used was fromMarch 19 to September 10 , 2002 containing

1,478 records (data rows). See Young and Guess (2002) foradditional details. Compare, also, Guess, Edwards, Pickrell, andYoung (2003).

Furthermore, the MDF product also undergoes destructive testing at differenttime periods of production in order to determine the strength ofMDF. If this strength complies with the specificationsof quality set forthby the manufacturerand consumer, the productis shipped. Workers performthis testing by samplingcross sections of MDF panels over time. A special measuring device is utilized that pulls the cross section apart andmeasures the strength of internalbond (IB) in pounds per squareinches (psi) until

2 failure. In engineering settings, this is often called tensile strength. Internalbond is a key metric of quality and unfortunatelyis not detenninedvia sensors. It is not automatically available, because it requires humanlabor. This makes reliability improvements in MDF more complicated.

Throughoutthis thesis, differentstatistical tools will be presented foranalyzing andinterpreting reliability data forthe purposes of improving product quality. As just mentioned, the data pertaining to the internalbond of medium density fiberboardwill serve as a usefulexample. It is the intent of this author, not to reprove the well-known theoretical underpinningsof the statistical methodologiesused throughoutthis thesis, but rather to apply them appropriately. This approach will provide a helpful set of tools that canbe used by practitioners seeking to improve thequality and reliability of their respective products.

Chapter 2 of this thesis is a required literature review that serves the reader as brief introductions to the major ideas of the thesis. This chapter also provides a usefulset of literature that not only develops furtherand expounds on the concepts offered,but may lead the reader to other researchinsights that will be infonnative. The literature review beginswith extensive backgroundand work pertainingto the internalbond of medium density fiberboard. It is assumed that the reader ( especially those fromoutside the forest products industry) has had little exposure to MDF, its properties,uses, and current researcharound the world. Therefore,key backgroundwill be presented here in the hopes of creating a more comfortablesense of familiarityfor what is to come in the rest of this thesis. Next, literature review pertaining to statistical reliability methods and studies arepresented in the context of demonstratingthe practicality of analyzingfailure-

3 time and strength data. The followingsegment of the literature review provides an approach of using informationcriteria forthe purposesof selecting the "best" underlying statistical model fordata.

The finalsegment of the literature review summarizesthe computer intensive resamplingmethod known as the bootstrap, its background, and current work utilizing this tool. Again, it is assumed that the reader has had little or modest exposure to the statistical methodologies presented. Greater effortsin Chapter 2 will be made to ensure clear and proper understanding of the backgroundmaterial.

Chapter 3 applies simple statistical tools to aid in the understanding and exploration of the reliability of medium density fiberboard. Further background on MDF will be providedalong with the method used to categorize differenttypes of MDF used in the study. Descriptive statistics and histograms are utilized along with probability plots andsurvival plots (Kaplan-Meier estimates) as a methodology forobtaining more information fromreliability data. This method also allows for greaterease in interpretability. Probability plots help demonstrate how a data set conformsto a particular distribution. Survivalplots arenonparametric plots that canbe utilized when parametricmodels arenot justified.

This thesis demonstrates that graphicalexploration is a helpfulway forthe practitioner to understanddifferences in differentMDF producttypes, e.g., product types of MDF may have differentdensities and thickness. The thesis also explores the sources of variationthat may influenceinternal bond.

Chapter 4 summarizes informationcriteria that are helpful in objectively identifyingthe underlying parametricdistribution of differentproduct types of MDF. It

4 is common to use the models of Weibull, lognormal, and normalin variousreliability settings on the originaldata or the transformeddata. See Meeker and Escobar (1998). In particular, Akaike's Information Criteria (AIC) andfurther work by Bozdogan(2000) with his InformationComplexity Criterion (ICOMP)will be presented. The identificationof a parametricmodel is essential formany statistical tests andallows for better ease in estimating desired population parameterssuch as percentiles.

Recall that in Chapter 3, probability plots will be introduced as a graphical approach foraiding in the identificationof a parametricmodel. In many cases, they can easilyidentif y anunderlying statistical distribution fora particulardata set. However, this method is subjective andthus makes anysuch model determinationmore difficult.

The theorysupporting AIC andICOMP makesprobability plotting more objective by accounting forthe likelihood of the underlying model. They both create a numeric

"score" foreach probabilityplot. The model with the lowest score is picked as the "best" fitof the data.

In Chapter 5, different bootstrap methods are presented forthe purposes of constructingconfidence intervals for model parameters. In particular, interest lies in obtaining confidenceintervals for the extreme lower percentiles forthe internalbond of

MDF. In reliability studies, it is generally of most interest to estimate the lower percentiles. These lower numbers aremore crucialfor estimating percent fallout during warranty, early failuresduring normalusage, as well as percent fallingout of the specificationlimits. The idea behind bootstrapping is to simulate the samplingprocess a specified(usually large) number of times andobtain anempirical bootstrap distribution forthe desired parameter. The bootstrap distribution is then used to acquire

5 characteristicsabout the population parametersuch as bias, standarderror, and confidence intervals. One bootstrap method is completely nonparametric and requires no assumptions about the underlying distribution of the data.

Other methods using the bootstrap do require parametric assumptions. As differentmethods of bootstrap samplingexist, so do different methods forconstructing bootstrap confidenceintervals. In light of many differentpossibilities, these methods will be explored andcompared. Draftrecommendations will be provided regardingthe circumstancesthat dictate which bootstrap methods are most appropriate. Overall, the objective of this portion of the thesis is to illustrate the usefulnessof the bootstrap tool as an alternativeto other statistical methods available such as the normallarge sample approximate confidence intervals and/orthe likelihood-based interval.

The finalchapter of the thesis, Chapter 6, summarizes the entire thesis and also has concluding remarks. Suggestions for possible futurework are referredto and explored in this chapter. The work presented in any thesis is by no means complete and/or definitive. As with many aspects of life, new ideas and technologies become available and work builds upon itself. The bootstrap and its aggressive growth is a classical example of this. The reader should note that the author plans futurework in this field during his career andmore work is forthcoming.

6 Chapter 2

Literature Review

We now begin a brief literature review that describes other research and current issues involving the topics of thisthesis. In particular, the subject matter to be discussed includes:

( 1) the backgroundand uses of medium density fiberboardwith respect to its internal

bond,

(2) reliability data analysis,

(3) using importantinformation criteria such as Akaike's InformationCriteria (AIC)

andInformation Complexity Criterion (ICOMP)to alleviate subjectivity in

probability plotting, and finally

(4) bootstrapping andits applications.

Since readers outside of the forestproducts industrymay have little prior knowledge of medium density fiberboard,substantial emphasisand background arepresented first.

It is assumedthat the reader has some prior knowledgeof statistical methods and applications. Therefore, there will be less focusin this chapter on literature that pertains to researchin reliability analysis, informationcriteria, and bootstrapping. Rather, many of the details forthis will be presented in Chapters 3, 4, and 5 of this thesis.

7 2.1 MEDIUM DENSITY FIBERBOARD

The financialimpact of the forestproducts industry on worldwide economies is overwhelming. For example,in the state of Maine, the forestproducts industry accounted forapproximately 9 billion U.S. dollars per year and in the state of Tennessee in 2000, this figurewas approximately 22 billion U.S. dollars per yearaccording to

English, Jensen, and Menard(2 004). There aremany examples of forest productsbeing used in furniture,cabinets, shelving, flooring, paneling, and molding. One product called medium density fiberboard (MDF),which is a wood composite, will be the focusof this section of the literature review.

Specifically,MDF is a highquality engineered timber product offering superior qualities of consistency of finishand density, freedom from knotsand natural irregularities, as well as having the characteristicsof strength, durability, and uniformity are not always foundin natural timber. MDF is used by home building and furniture manufacturingindustries worldwide and is considered anindustry leader in quality and productivity.

In fact, international demands forthis product are increasing. China is an aggressive growtharea forMDF and produces the largestamount in the world. For example, ten new MDF manufacturing plantsare opening in China. Also, to illustrate the increasing demands forMDF, one such largeproducer ofMDF has a capacity in excess of 100 million lineal feet per year. Suchslandand Woodson ( 1986) and Maloney ( 1993) cover manufacturingpractices of MDF. Figure 2. 1 illustrates a comparison between

MDF and common particleboard. Other illustrative comparisons can be found in Chapter

3 of this thesis.

8 Figure 2.1. A comparison of common particle boardto MDF.

9 A brief literaturesearch on MDF in Web of Science database yielded 160+ references. We then focusedon MDF papers that dealt with the key variableof internal bond(IB ), which yielded a more modest 21 references. IB is a key metric of quality and servesas ameasure of strength forMDF. It is through destructive testing on MDF that

IB informationis obtained. We will briefly describe these referencesand others, which help provide anindication as to why studying MDF and its IB, are important. Manyof the journalarticles listed beloware fromforest products researchersfrom countries as varied as Japan,China, Denmark, Sweden, New Zealand,and the United States of

America. It is importantto note that this literature reviewis not exhaustive andmany more usefuland informative references are available on this topic.

We beginwith Wang,Winistorfer andYoung (2004) who investigate the

"formationcharacteristics of the verticaldensity profileof MD F... . Results of laboratorystudies indicate the verticaldensity profileof MDF is formedfrom a combination of actions that occur bothduring compactionand also afterthe press has reached finalposition." They assertthat methodologies forthe formationof density profilesfor oriented strandboard (OSB) discussed in Wang and Winistorfer (2000a) apply, also, to the formation of density profilesfor MDF. It was determinedthat "high­ density surface layers areeasier to create in MDF thanin OSB."

Widsten, Laine, Tuominen andQvintus-Leino (2003) study IB beingimproved by higher defibrationtemperatures. It was seen that other improvements in MDF properties were also obtained by this approach.

We note that vanHouts, Winistorfer andWang (2003 ) comment, "Acetylation is a treatment knownto reduce the swelling andwater absorptionbehavior of wood." They

10 found m reliability was reasonable foracetylation treated products.

Tsunoda, Watanabe, Fukuda andHagio (2002) "examinedthe resistance of medium density fiberboard treated with zinc borate to fungaland termite attack." They observedthat this treatmentled to no significantloss in m andother MDF properties.

Rials, Kelley and So (2002) used near infrared spectroscopyfor predicting variouscharacteristics of MDF samples. One of the characteristics was m. This would provide a useful process improvementtool formanuf acturing in the future. That is, this prediction method may help alleviate some of the destructive testing that is currently necessaryto obtain m information.

Young and Winistorfer (2001) use simple autocorrelationtime series andprocess improvement techniques on MDF thickness, rather thanm. Wang, Winistorfer, Young and Helton (200 1) study MDF produced in the laboratory. They used a technique called the "step-closure pressing schedule" which helped increase m in analyzed samples.

They also observed, "greatercore density did not result in higher internalbond strength."

"Internal bond (IB) strength ... was closely related with carbonate types andlevel used" according to Park, Riedl,Hsu and Shields (200 1 ). They conducted a study to

"optimize hot pressing time andadhesive content for the manufactureof three-layer medium density fiberboard(MDF) throughthe cure acceleration of phenol-formaldehyde

(PF) adhesives ... ." In particular, they used threecarbonates (propylene carbonate, sodium carbonate, andpotassium carbonate) in their study.

Han, Umemura, Zhang, Honda andKawai (2001) study fiberboard manufactured fromreed straw or fromwheat straw. These MDF's had m ten times higher than particleboard. The thickness swelling, however, of the wheat MDF did not meet industry

11 fiberboardstandards although all other propertieswere met. The IB of both reed and wheat MDF's did meet industry standards.

Also, van Houts, Bhattacharyya andJayaraman (2000) contend that due to "the moisture and temperature gradientsdeveloped during hot pressing of medium density fibreboard (MDF),residual stresses occur within the board as it equilibrates to room conditions." They study the measurement of these residual stresses andshow how these canaffect MDF properties. In particular, theystudied the effectof residual stresses on internalbond.

In vanHouts, Bhattacharyya andJayaraman (2001a) it was shown that a method known as the "Taguchi method of experimental designcan be utilizedto investigate methods for relieving the residual stresses present in medium density fibreboard (MDF)."

The Taguchi method involves subjecting various MDF panelsto varying degreesof heat, moisture, andpressure.

Furthermore, vanHouts, Bhattacharyyaand Jayaraman(2001b) report on the

"Taguchi analysisof the internalbond strength, surface layer tensile modulus, surface layer tensile strength and thickness swell of the treated specimens. These propertieswere measured to indicate whether the treatmentshad anyeffect on panelstrength and dimensional stability." They findthat moisture and heat had little influence on m for 8 mm MDF. Heat increases IB in 17 mm board but moisture had little effect.

Chow, Bao, Youngquist, Rowell, Muehl and Krzysik ( 1996) studied the "effects of fiber acetylation,resin content, and wax content on mechanicalproperties of dry­ process hardboardmade from aspen and pine ... . " The results from the investigation revealed that "Tensile stress parallelto face andinternal bond (IB)were generally higher

12 foruntreated boardsthan foracetylated boards." Increases in resin content increased 1B while increases in waxcontent showed a decrease in IB.

Hsu (1993) employed a self-sealing steampress system forthe effective productionof phenol-formaldehyde-bonded fiberboard. The relationship between the press system andboard properties was the focus of the study. It was determined that 6'the most significant factor affecting.. .i ntemal bond is resin content ... . Althoughmat consolidation time beforesteam in jection affects other boardproperties .. .it has no significant effect on ...intemal bond."

Gomez-Bueso, Westin, Torgilsson, Olesen andSimonson (2000) reporton

"Lignocellulosic fibersof different origins ... acetylated in large batches. The fibers used were of commercial, medium density fiberboard(MDF) pulp quality produced from softwood, beech, waste wood (low quality residue froman intermediate forestcutting) andwheat straw, respectively. Fiber from de-inked, semi-bleached, recycled paper was also included in the study." This composite fiberMDF had greatproperties suchas thickness swelling being decreased around90% andmechanical characteristics were modestly enhanced. They performed what is called in reliability circles an "elephant test." It involved "cyclic testing according to EN 321, (three cycles, each comprising 72 h water immersion, 24 h freezingat -18 degreesC and 72 h drying at 70 degreesC) show that more than 90% of the internal bond, IB, remained afterthe testing. This value can be comparedwith the corresponding value of30-40% obtained for fiberboard's made from unmodified fibers."

Wang,Chen andFann (1 999) investigated "a compression sheardevice foreasy and fast measurementof the bonded shearstrength of wood-basedmaterials to replace

13 the conventional method used to evaluate internalbond strength (IB)." They foundthat measuring strength of MDF or particleboardby the suggested compression shear strength and by the conventional approach of internalbond strengthwere significantlycorrelated.

This provides an alternativeapproach to measuring strengths of materials.

Young and Guess (2002) present modem high technology approaches to managingthe manufacturing ofMDF andrelated data with real time process data feedback. They employ a usefulregression model to predict the MDF strength of internal bond by using knowledgeon more than 230 process variables.

Guess, Edwards, Pickrell andYoung (2003) apply statistical reliability tools to manageand seek improvements in the internalbond ofMDF. As a partof the MDF manufacturingprocess, the product undergoes destructivetesting at various intervals to determinecompliance with customer's specifications. Workers performthese tests over sampled cross sections of the MDF panelto measure the IB in pounds per squareinches until failure. They explore both graphicallyand statistically this "pressure-to-failure"of

MDF.

For other referencesand work that connect to MDF and internalbond among many others, see Park,Riedl, Hsu and Shields ( 1998), Ogawa and Ohkoshi ( 1997), Xu,

Winistorferand Moschler (1996), Yusuf, Imamura, Takahashiand Minato (1995), Xu and Winistorfer(1 995), Hashim, Murphy, Dickinson and Dinwoodie (1994), Labosky,

Yobp, Janowiakand Blankenhom (1993), Chow andZhao (1992), Butterfield,Chapman,

Christie and Dickson (1992), and Rowell, Youngquist, Rowell andHyatt (1991).

For more general andspecific information, see also the followingwebsites: http://web .utk.edu/�tfpc/ and http:// www.spcforwood.com.

14 2.2 RELIABILITY

We now do a brief review of reliability literature. Reliability data analysis along with studying the 1B ofMDF is the core of the rest of this thesis. Currently,the best single source of statistical reliability analysisis Meeker andEscobar (19 98). This

"excellent andcomprehensive book on reliability methods andtheir applications"was listed as number one in the top fiverecommended books forstatisticians by Ziegel (2003) in the September 2003 edition of Amstat News.

According to Meeker andEscobar (1 998), "Reliability is oftendefined as the probabilitythat a system, vehicle, machine, device, andso on will performits intended functionunder operating conditions,for a specifiedperiod of time." They also list some of manyreasons forcollecting reliability data such as "assessing characteristics of materials over a warrantyperiod or over the product's designlife", "predicting product warrantycosts", and"tracking the productin the fieldto provide informationon causes of failureand methods of improving product reliability." Meeker andEscobar ( 1998) provides anexcellent andthorough treatment ofthe Weibull distribution,other reliability functions, andvalidating/exploring graphically these models.

For reliability studies and in particularfor the case of analyzingstrengths of materials, it is common to firstconsider the standardWeibull distribution as the underlyingmodel. According to Cox andOakes (1 984), it was Fisher andTippett (1928) that introduced the Weibull distribution when working with the extreme value distribution. In fact, even Weibull himself analyzedstrengths of materials during the

1930's andfound that the standard normaldistribution did not fithis exampleswell. See

Weibull (1939) as well as Chapter 3 of this thesis formore informationon this famous

15 distribution. Also, see Cox and Oakes (1984) formore on the Weibull distribution as well as the analysisof reliability data in general. Extreme values have been studied by the Russiansalso, independently of Weibull.

Guess, Edwards,Pickrell, andYoung (2003) shows that in the case of analyzing the strength of the internalbond of medium density fiberboard,it was natural to first consider the Weibull distribution. Althoughit sometimes fitparts of the IB data, it was surprisinglynot a valid model forthe total rangeof the internal bond. It was determined that other parametric distributionsfor strengths be investigated andultimately , a nonparametricapproach was needed. For more on specificparametric and/or nonparametric reliabilitymodels, see Meeker andEscobar (1998 ). Hollander and Wolfe

(1973) provide aninsightful and thoroughcoverage ofnonparametricsin general.

Young and Guess (2002) "focuses on how modem data mining canbe integrated with real-time relational databases andcommercial data warehousesto improve reliability in real-time" for forest products. In particular,interest lies in improving reliability in the manufacturingof medium density fiberboard. This improvement is called forsince the

"cost of unacceptable MDF was as large as 5% to 10% of total manufacturingcosts " and

"prevention canresult in annualsavings of millions of U.S ..."

In Walker and Guess (2003), the reliability of the bursting strengths of two designs forpolyethylene terephthalate bottles were compared. It was determined that neither designwas more reliable thanthe other. Furthermore,they stress "( 1) the need of operationalclear de finitions forreliabil ity, (2) the need of graphicalexploratory analysis to discover anomaliesin the data, (3) the value of nonparametric methods, and(4) the problems of using parametric techniques when the assumptions are violated." 16 Urbanik(19 98) presents "multiple load levels" for corrugated fiberboard and

"related them to the probabilityof time to failure." A reliability analysiswas conducted for the logarithm of failuretime data varyingwith load level. The "results were used to

(a) quantifythe performance of two corrugatedfiberboards having significantlydifferent components and(b) show that a safe-load-level test using multiple load levels andcyclic humidity is more sensitive to material strengthdifferences than a dynamicedgewise compression test at standardatm ospheric conditions." Kim, Guess andYoung (2004) discuss the usage of data mining tools in reliability applications. Caution is given forthe use of decision trees in such applicationsand a new tool knownas GUIDEis utilized for the purposes of comparisonto regression techniques.

They present a case study that focuseson predicting the internalbond of medium density fiberboardbased on productspecifications such as density, thickness, andwidth.

Manyother excellent sources of information on reliabilityand its applications exist. Guess, Walker andGallant ( 1992) focuson how differentmeasures of reliability such as means,medians, percentiles, etc. canbe interpretedand used. For a thorough treatment of the underlying theoryof reliability andlife testing, see Barlowand Proschan

(1965, 1974, and 1981). See also andcompare texts by Lawless (2003), O'Connor

(1985), andMann, Schaferand Singpurwalla (1 974).

2.3 INFORMATION CRITERIA

More details andspecifics regarding information criteria will be presentedin

Chapter 4 of this thesis. The focusis not broad andwe present only severalof many excellent sources on informationcriteria. Recall thatprobability plots provide a 17 graphicaldemonstration of how a particulardata set conformsto a specific probability distribution. That is, the data are ordered andthen plotted against the theoretical order statistics forthe desired distribution. If the data set "conforms"to that particular distribution, the points will formroughly a straight line. More informationon probability plotting can be foundin Chapter 6 of Meeker andEscobar (1998).

Manystatistical techniques, such as probability plotting just described, are very subjective andallow fordecisions to be made based on someone's own personal assessment. However, choosing a model based on informationcriteria is much more objective andhelps relieve much of the ambiguitythat is present when looking solely at a probability plot. See the excellent review article ofBozdogan (2000) and the lecture notes ofBozdogan (2001). For additional comments, also compare forexample

Bozdoganand Bearse (2003), Urmanov, Gribok, Bozdogan,Hines andUhrig (2002), and

Bozdogan andHaughton (1998). See these papers and their extensive references plus

ProfessorBozdog an'shelpful website where his lecture notes are available: http://web .utk.edu/---bozdogan/Stat563 2003 .

This approach greatly helps remove the subjectivity in analyzing plots. It allows anobjective score where the model with the lowest score wins as the best. We urge this as an importantadditional tool forpractitioners andengineers in deciding on a parametric model. In our Chapter 4, we will discuss these in greaterdetail.

2.4 BOOTSTRAPPING

According to Efronand Tibshirani(1 993), "the bootstrap is a data-based simulation method forstatistical inference... . The use of the termbootstrap derives from

18 the phrase to pull oneselfup by one 's bootstrap." Efron's bootstrapis a Monte Carlo simulation that requires no parametricassumptions about the underlying population from which the data is drawn. It is a computationally intensive statistical method that can require a largenumber of iterations and hence usually requires the use of the computer.

More recently, Chernick (1999) is anotherexcellent book that provides a thorough and insightfultreatment of manybootstrap methods andtheir applications. A student, practitioner, or others beginning to learnmore about bootstrapping would do well to start here. Chernick(1 999) hasan extensive helpfulbibli ography, also. We follow the approach of Meeker and Escobar(1 998) who use bootstrapping to estimate percentiles while others might consider cross-validationor jackknifing as in Giudici (2003).

Hall (2003) providesa brief and usefulprehistory of the bootstrap. The idea is to further explore past connections with the bootstrap that may not be as well known. That is, "the relationship of bootstraptechniques to certain earlywork on permutationtesting, the jackknifeand cross-validation is well understood. Less known,however, are the connections of the bootstrapto researchon surveysampling forspatial data in the first half of the last century or to work fromthe 1940s to the 1970s on subsampling and resampling."

For further readings on bootstrap methodology, theory, and applications, including some ofEfron's earlier work, see Efron (2003), Efron andTibshirani (19 91),

Efron andGong (1983), and Diaconis and Efron(19 83) among many others.

Meeker and Escobar (19 98) present bootstrapping related directlyto reliability data analysis. Also, see the helpfulinsights and comments on limitations of the bootstrap in Ghosh, Parr, Singhand Babu (1984). In addition, compareParr (1 983, 1985a, and

19 1985b).

A completely nonparametric bootstrap, or Efron's bootstrap is conducted as followsaccording to Martinez andMartinez (2002):

1. Beginning with a random sampledenoted by x, calculate anestimate forsome

parameter, 0.

th 2. Samplewith replacement from x to obtain x *b, where b represents the b

bootstrap replicate.

3. Using x *b, calculate anestimate for 0 .

4. Repeat steps 2 and3 a largenumber of times.

5. Use the distribution of the estimates for0 to obtain desired characteristics such

as standard error, bias, andconfidence intervals.

Other formsof the bootstrap andmethods of implementation have since emerged from Efron's earlierwork. The above nonparametric method, other bootstrap methods, andthe general bootstrap methodology for constructingboot strap confidenceintervals will be described in greaterdetail in this thesis' Chapter5.

Chapter 9 of Meeker and Escobar (1998) provides a useful treatment ofbootstrap methods and applications forreliability data. They present two methods of bootstrap sampling, ( 1) the fullyparametric boot strap that includes parametric samplingfor parametric inference and (2) nonparametricboot strap sampling forparametric inference.

In both cases, it is necessaryto firstdetermine the underlying parametricdistribution of the data. Applications of these methods to the construction of confidence intervals are presented in large detail and in particular,the bootstrap-t method for constructing confidenceintervals is dealt with thoroughly. For anexcellent treatment of the 20 construction of confidenceintervals and other statistical intervals in general, see Hahn and Meeker ( 1991 ).

Pages 217-220 of Chapter 9 in Meeker andEscobar (1 998) reviews the same fully nonparametric bootstrap method as presented in Efron andTibshirani (1993) as it applies to analyzingreliability data. They capture some of the limitations andpresent warningsfor the use of this fully nonparametric bootstrap. Recall, also, Ghosh, Parret al.

(1984).

For example, Meeker andEscobar (1 998) contend that the ''justificationfor the bootstrap is based on large-sampletheory. Even with largesamples, however, there can be difficultiesin the tails of the sample. For the nonparametric bootstrap, there will be a separateboot strap distribution at each time for which there were one or more failuresin the original sample." This would not pose a problem outside the tails of the original data where the bootstrapdistribution will be approximately continuous. However, in the extreme tails of the original data,there may be only a small number of failures or outcomes. In this case, the bootstrap distribution may be anything but continuous. As canbe seen by the examples presented, when the extreme tails'areof interest (as is often the case in reliability studies), the standard fully nonparametric bootstrap methods are not as useful as other bootstrapmethods. Rather the standard bootstrap methods have a place when estimating parameters such as the quartiles(2 5th or 75th percentiles).

Chapter 3 of Chernick(1 999) mentions severaldiff erent methods for the construction of bootstrap confidenceintervals. These include the standardpercentile method, the bias correctedand accelerated percentilemethod, the iterated bootstrap, and the bootstrap-t. The aforementionedmethods aredescribed concisely andtheir ranges of

21 applicability as well as limitations are discussed. The last chapter provides information on the overall limitations of the bootstrap and when the bootstrap fails. These include, but are not limited to, (1) a sample size that is too small and (2) when attempting to estimate extreme values. Also, compare with Efronand Tibshirani (1 993) and Davison andHinkley (1997) as they provide furtherand extensive informationon bootstrap theory, methods, and limitations.

Recall that the bootstrap is a computer intensive statistical method. Martinezand

Martinez(2002) devote Chapter 6 to the bootstrap,including estimation of standarderror, bias, andconfidence in tervals, using MATLAB. Several routines are provided along with examplesto aid the reader in getting started. Other Monte Carlotechniques such as the jackknife are also discussed.

Lunneborg (2000) shows how to construct bootstrap confidenceintervals using

Resampling Stats and/or S-PLUS statistical programmingpackages. This thoroughwork provides step-by-step algorithms forimplementation of many resamplingmethods forthe purposesof analyzing data.

DiCiccio and Efron(1 996) is devoted to bootstrap confidence intervals and provides a usefulsurvey of many such intervals( standard, percentile, bootstrap-t, etc.).

Its focusis to "improve by an order of magnitude upon the accuracy of the standard intervals.. .i n a way that allows routine application even to very complicated problems."

Examplesfor each method are provided and the underlying theory is also presented.

Polansky (1999) shows that bootstrapconfidence intervalsconstructed using percentile methods "have bounds on their finite sample coverage probabilities.

Depending on the functionalof interest and the distribution of the data, these bounds can

22 be quite low." It is said that the "bounds arevalid even formethods that are asymptotically second-order accurate."

Boos (2003) gives his own thoughtson bootstrapping. In particular, he contends that "the real reason the bootstrapwas so path-breakingand has remained so popularis that Efron described it mainlyin terms of creating a 'bootstrapworld,' where the data analystknows everything." In a sense, any population parameterof interest can be estimated simply through simulation. For example,"if the varianceof a complicated parameterestimate in this world is desired, just computer generate B replicate samples

(bootstrapsamples or resamples),compute the estimate foreach resampleand then use the samplevariance of the B estimates as an approximation to the variance." Thus, "In effect thisbootstrap world simulation approachopened up complicatedstatistical methods to anybodywith a computer and a randomnumber generator."

23 Chapter 3

Exploring Graphically and Statistically the Reliability of Medium Density Fiberboard

This chapter is a slightly revised version of a paper by the same namepublished in the International Jo urnalof Reliabilityand Applications in 2003 by Frank M. Guess, David J. Edwards, Timothy M. Pickrell, andTimothy M. Young:

Guess, F. M., Edwards,D. J., Pickrell,T. M., andYoung, T. M. (2003). Exploring Graphically andStatistically the Reliability of Medium Density Fiberboard. In ternational Jo urnalof Reliabilityand Applications, 4(4), 97-110.

My primary contributions to this paper include (1) all of the computer work (charts, graphs, etc.), (2) manyof the interpretationspresented, and(3) portionsof the writing.

3.1 INTRODUCTION AND MOTIVATION

Medium Density Fiberboard(MDF) is used internationally in a host of building needs and furniture construction. It is a superior engineered wood product with great strength, reliability andgrooving ability for uniquedesigns. In addition, MDF offers superior qualities on consistency of finish anddensity, plus freedom fromknots and natural irregularities.

MDF has characteristics of strength, durability and uniformity not always found in natural timber or standard particleboard. It has excellent machinability due to its homogenous consistency and smaller variation in needed characteristics comparedto natural wood. These features makeMDF particularly suited foruse in flooring, paneling, andmanufacturing of furniture,cabinets, andmoldings. It, also, has environmentally friendlyproperties of using wood waste to manufacture useful byproducts. This does not happen as easily with traditional wood products.

Figure 3. 1 and Figure 3.2 demonstrate the marked differencesbetween 24 Figure 3.1. Cross sections of forestproducts fromwidth view. Top is particleboard, middle is natural wood, andbottom is MDF.

Figure3. 2. Cross sectionsof forestproducts from diagonal view. Top is particleboard, middle is natural wood, and bottom is MDF.

25 particleboard, natural timber,and MDF. We have observedthe common usage in writing particleboard or fiberboard( or fibreboard). This is used by most companiesor timber associations, which we sampled. Our sample ranged fromGeorgia-Pacific, Sabah

TimberIndustries Association ofMalaysia, to an Association ofNew ZealandForestry

Companies. In some settings, however, you will see it writtenas two words "particle board." We followedthe more typical industrial usage ofwriting as one word

"particleboard"or "fiberboard."

Suchslandand Woodson (1986) andMaloney (1993) cover manufacturing practices ofMDF. See, also, formore general and specificinformatio n: http:!/web. utk.edu/-tfpc/ and http://www.spcforwood.com.

Young andGuess (2002) present modem high technology approaches to managingthe manufacturingof MDF and relateddata with real time process data feedback. They employ regression prediction of MDF strengths using knowledge on more than 230 process variables.

Here, we are interested in the statistical reliabilityproperties of the strengthto fa ilure as opposedto the time to failure. The strengthto failure data will give us a clear idea ofthe utility of the product. It allows the producer to make assurancesto customers aboutthe usefullife ofthe product. The key measureofMDF 's reliabilityand quality is its internal bond(IB ), which is measured in poundsper square inch (psi) until breaking.

Note that one psi is equivalent to 0.07 kilogramsper square centimeter.

For a numberof reasons, testing the MDF product typesover anyextended time, under a variety ofoperating conditions is not economically feasible. Instead ofan elaborate procedureof designing and implementing lifetests, a "pull apart"destructive

26 approach provides an instantaneousmeasurement of the IB. Destructive "pull apart"test samples are taken periodically to determine IB. Due to the cost and loss of products from destructivetesting, manufacturers obviously want to keep such tests to a minimum.

Another advantage to destructive testingis immediate feedback intothe manufacturing process leadingto rapid process improvement. Also, it can help modifyor stop the manufacturing process; thus, preventing great waste of materials. The cost of unacceptable MDF was as largeas 5% to 10% of total manufacturingcosts, which can result in 10 to 15 million dollars per plant per year. In 2003to 2004, ten such high production plants are anticipated to be built in Asia.

We present new results on different m data regardingthe statistical distributions for various product typesof m. This is important for studying potential warrantyissues, understanding wearing over time, failure of products under misuse, and variation in IB between product types. This is, also, needed within particularproduct typesof MDF.

Our data covers the time period from March19 , 2002 to September 10, 2002.

We explore graphicallyand sta tistically the distributions of the strengths of this material. It would be natural to consider firstthe standard Weibull model for strengthsof materials. Indeed, the researcherWeibull himself first analyzedstrengths of different materials, ranging from cotton to metal. From his data sets, he found the primary available distribution of the normal did not fithis examples well in the 1930's. The alternative parametric model he originally proposed is what we now call the "three parameter''Weibull. See Weibull ( 1939 and 1951).

The originalthree parameterWeibull is oftenreduced andwritten today as a two parameter distribution. Recall this two parameter Weibull density functioncan be written

27 in the following parameterization as

f(x) = ;.,pxP-te<-V> (3.1) where x � 0 ( and f(x) = 0, forx < 0), while the reliability function is

F(x) = e<-V> (3.2) where x � 0 (and F (x) = 1 forx < 0). Another common reason formodeling data with a

Weibull distribution is that it may be suitable foreither increasing, constant (i.e., an exponential) or decreasing hazard functions. For MDF subject to destructive "pull apart" tests, we would conjecture an increasing failurera te. This would lead us to hypothesis, a priori, that the shape parameter p > 1 forany MDF product typethat mightbe Weibull.

See, forexample, the excellent book of Meeker andEscobar (19 98) fora thorough treatment of the Weibull, other reliability functions andvalidating/exploring graphicallythese models. Also, comparetexts by O'Connor (1985), Barlow and

Proschan( 1981 ), etc. Although the Weibull can sometimes fitparts of ourm data forsome categories of MDF, it is surprisinglynot a valid model forthis data forthe total m range. Other parametric distributions of strengths are employed and a nonparametric approach is needed. See Meeker and Escobar( 1998) forreliability parametric/nonparametric models and the insightfulHollan der and Wolfe(1 999) on nonparametrics, in general.

The spirit of thischapter is that of an exploratory analysis via graphs,descriptive statistics, and tests. See the excellent overviewon graphics by Scott (2003) and his references. Section 3.2 covers the typesof MDF and ways these types are determined.

Section 3.3 explores both graphically and statistically particulartypes of MDF, while 28 Section 3 .4 provides concluding comments.

3.2 CATEGORIZING TYPES OF MDF

We begin the analysisby sorting the IB data by three key characteristics:

• density (lbs/ft3)

• thickness (inches)

• and width (inches).

These three characteristics differentiate the MDF's for variousapp lications. Since MDF in this particularstudy wasproduced in continuous length of sheets, length was not a crucialvariable for our purposes. Further, forthe purposeof analysis, the MDF was separated into two main groups:

• Group I: standard density

• Group II: highdensity.

The high densitytype is MDF with densities on the upper end of the scale 47-48 pounds

3 per cubic foot (752.86-768.88 kg/m ). The standard densitytype is the MDF with densities ranging from 45-46 pounds per cubic foot (720.83-736.85 kg/m3).

Within each groupthe m was measured in accordance with classificationby density, thickness andwidth. The typenumbers (with density, thickness, andwidth after each typenumber) are listed in Table 3.1.

Since there were a number of typesin each group, we select the primarytypes, which sold the most, fora more detailed analysis. These were Types 1 and 3 fromGroup

I (standard density) and Types2 and 5 from Group 2 (high density). See Table 3.1 for more details. 29 Table 3.1. Group and typenumbers fordiff erent MDF products with (A, B, C) where A= density, B = thickness, and C = width where forexample Type 1 in Group I represents A = 46 pounds per cubic foot,B = 0.625 inches thickness, and C = 61 inches ( or the eqmva. 1 ent metnc. umts . ) . Group I: standard density Group II: high density Type # Type # 1 (46, 0.625,61) 2 (48,0.75,61) 3 (46, 0.75,49) 5 (4 8,0.62 5,61) 4 (46, 0.75, 61) 9 (4 8,0.75,49) 6 (45,1,61) 10 (48,0.375,61) 7 (46, 0.625,49) 11 (48,0.5,61) 8 (45,1,49) 14 (48, 0.5,49) 12 (46,0.688) 15 (48, 0.625,49) 13 (46, 0.688,49) 16 (47 , 1,61) 17 (45, 1.125,61) 18 (48, 0.4379,49) 20 (46, 0.875,61) 19 (48, 0.375,49) 22 (45,1.12549) 21 (48,0.563,61) 23 (48,0.4379,61) 24 (48, 0.688,61)

30 Another reason behind our splitting into two distinct groupswas that the destructive testing of the MDF is concernedmainly with the m strength. A priori,this is reasonablyhypothesized to be mainly a functionof density. Furthermore, from the nature of the destructivetesting, which involved the cutting of many cross sections from different pieces ofMDF, the lengths and widths were not the major factors effecting strengths. In the next section graphsand statistical tests will demonstrate strikinglythis hypothesis to be true.

3.3 EXPLORING GRAPHICALLY AND STATICALLY IB IN TYPES OF MDF

The initial analysisbegan with the assessment of the underlying distribution of the internal bond strengths, categorizedby the density, thickness and width measurements.

We wantto firstunderstand means, medians, and percentiles of the strengths of MDF.

See, forexample, Guess,Walker, andGallant (1 992) formore on these measures.

Table 3.2 provides a descriptive statistics comparisonof product Types1 and2.

These numbers have been roundedto one decimal place. Note that both the meanand median in Type 1 are 120.2, while for Type 2 the mean and median are close at 180.0 and

179.0. Type 1 has less variationas measured by the IQR and standarddeviation, but

Tabl e 3 ..2 D escnpttve statistics compansono fTypes 1 and2 . Type 1 IB (psi) Type 2 IB (psi) Mean 120.2 180.0 Median 120.2 179.0 Std. Dev. 9.9 12.3 IQR 12.3 17.6 Min 87.2 140.6 Max 164.5 214.5

31 Type 1 has a bigger rangeof 77.3 when comparedto Type 2being 73.9. This bigger rangefor Type 1 canbe understood by its outliers, boxplots, andthe histogramsin

Figures 3.3 and 3.4.

From the histogramin Figure 3.3, we see that the distribution of the primary product, Type 1, is approximately normal. Recall the mean and medianbeing the same.

Figure 3.4 suggests that we explore the reasons behind the weakness of the units in the

140 to 150 psi bins, plus understandthe much better strengths in the higher bins overall, especially for the 190 to 21o+ psi bins, in order to improve the reliability.

Recall the exploratory flavorof Tukey of examining manyviews of the same data. See, also, Scott (2003). Figure 3.5 is an overlay plot that gives anotherlook at the differences and similarities between Types 1 and 2. Notice it can be a little misleading when comparedto the actual raw data or the histogram.The plot shows quite a distinction between the twoproduct types,providing evidence that Type2 is much stronger than

Type 1. That is, heavier products or products requiring more load bearing strength, such as shelving, would make use of Type2 MDF. Type 1, with less strength, would be used more extensively in products not requiringlarge strength, such as picture frames.

Probability plots were used extensively in this analysisbecause they give a clear demonstration of how a particulardata set conformsto a specificcandidate proba bility distribution. The data areordered andthen plotted against the theoretical order statistics for a desired distribution. If the data set "conforms"to that particulardistribution, the points will forma straight line. Simultaneousconfidence bands provide ob jective bounds of deviation from the line or not. Those data points outside the confidencebands are shown to deviate from thecandidate prob ability distribution in question. See

32 • I • . ______,_ I

90 100 110 120 130 140 150 160

Figure3. 3. Histogramand Boxplot of PrimaryProduct (Type 1) fromJMP.

• {)

140 150 160 170 180 190 200 210

Figure 3.4. Histogramand Box plot of Type 2 from JMP.

33 Overlay Plot

"C C � _1 cu C... � C

.9 1 1.1 1.21.31.41.5 1.61.71.81.9 2 2.1 Type

Figure 3.5. Overlay Plot of Types 1 and2 from JMP.

34 Chapter 6of Meeker and Escobar(19 98) for furtherinformation. Normal and Weibull probability plots were producedfor Types 1 and 2 as shown in Figures 3.6 and 3.7.

In Figure 3.6(a) for Type 1, there is cleardeparture from the Weibull in the upper tail, but appears to be following this distribution in the middle and the lower tail. Fitting in the lower tail canbe importantfor estimatingpercent fallout of specification limits.

Figure 3.6(b) forType 2 shows cleardeparture fromthe Weibull distribution overall.

The snake-likemeandering is a systemic patternthat stronglysuggests the Weibull does not fitat all forType 2.

Recall that the histogramof Type 1 as well as the mean and median being the sameprovides some evidence fornorma lity. Figure 3.7(a) shows a normalplot with points that fallmostly within the simultaneousbounds, except forsome outliers. There is some cleardeparture in the tails that may not be following so perfectly a normal distribution. Again, recall the lower tail is important in estimating lower percentiles.

Thus, as shown in Figure 3.6(a), the Weibull may prove to be a better model for estimating these lower percentiles forType 1. However, this is only a conjecture andwe will see in Chapter 4 that such subjectivecon jectures may not always hold.

Figure 3.7(b) shows less departurefrom normality andcertainly ap pearsto be a better fitof the data than thecorresponding Weibull distribution forType 2. In fact, as we will see later, largep-values will not allow for normalityto be rej ected forType 2.

Overall, neither model appearsto be the best, thus; a nonparametric approach may be more appropriate.

As seen, the probability plots have been a very visual and indeed subjective

35 Weibull Probability Plot of lnternal_Bond Weibull Probability Plotof lnternal_Bond

180.0 220.0 0

CD

0 ,, 146.7 193.3 -' -'J i .5 113.3 j 166.7

80.0 140.0 -8.0 -4.7 -1.3 2.0 -6.0 -3.3 -0.7 2.0 Weibull Distribution Weibull Distribution (a) (b) Figure 3.6. NCSS Weibull Probability Plots. (a) Weibull plot forType 1; (b) Weibull plot forType 2.

Nonnal Probability Plotof lntemal_Bond NonnalProbability Plot of lntemal_Bond

220.

155. 200.

-;a' 180.

180.

-1.5 0.0 1.5 3.0 -1.5 0.0 1.5 3.0 Expected Normals Expected Nonnals (a (b) Figure 3.7. NCSS NormalProba bility Plots. (a) Normalplot forType 1; (b) Normalplot forType 2.

36 method forassessing the underlying distribution forthe differentproduct types. In particular,the normaldistribution was determinedto be the reasonable fitcompared to

Weibull forsome producttypes. We do not show all typeshere to save space in the thesis. Therefore, it is natural to ask if the data truly followsa normaldistribution andif this is statistically significantby testing. Tests such as the Shapiro-Wilk, Kolmogorov­

Smimov, andothers exist to help answerthese questions more objectively. These tests will produce differentp-values, asseen in the Tables 3.3 and3.4.

For Type 1, we clearlyre jectnormality using the Shapiro-Wilk andAnderson­

Darlingtest foralpha level of 0.05. The Kolmogorov-Smimov test has bigger p-values, but recall it tends to have low power. Notice that fourdifferent softwarepackages (SAS,

JMP, NCSS, andMinitab) were used forchecking the consistency of the tests statistics andp-values. Table 3.4 with its largerp-values forall three tests shows we can not reject normality for anyreasonable alpha levels. Still we may wantto seek better understandingby other plots. Walker andGuess (2003) stress the need formore nonparametricplots andanalysis, when the parametricmodels may be weakor not the strongest. Nonparametric plots known as Kaplan-Meierestimators, survivalplots, or reliability plots will now be shown for various product types.

Immediately, one should notice the largegap present between Type 1 and2 in the

Kaplan-Meiernonparametric survival plots in Figure 3.8. Recall furtherthat the medians are very different. That is, 120.3 and 179.0 psi forTypes 1 and 2, respectively. Based on this, Type 2 has even more evidence of being significantlystronger than Type 1. A two samplet-test wasconducted with variances assumed unequal. This assumption was based on a test forunequal variances provided by SAS, which yieldeda p-value of

37 T a bl e 3 ..3 N orma r1ty test compansons tior ypeT 1 . Test Statistic I p-value Software / Test Shapiro-Wilk Kolmoeorov-Smirnov Anderson-Darlin2 JMP 0.97947 I <0.0001 NIA NIA SAS 0.97947 I <0.0001 0.0357 I >0.15 0.80213 I 0.0393 Minitab 0.9880 I <0.01 0.035 I >0.15 0.802 I 0.038 NCSS 0.97947 I 0.00002 0.03317 I NIA 0.80213 I 0.03787

T a bl e 3 ..4 N orma rt1 ·�, tes t compansons tior ype T 2 . . Test Statistic I p-value i Software/ Test Shapiro-Wilk Komogorov-Smirnov Anderson-Darling JMP 0.990482 I 0.2514 NIA NIA SAS 0.990482 I 0.2514 0.044945 I >0.15 0.485528 I 0.2311 Minitab 0.9947 / >0.1 0.045 I >0.15 0.485 I 0.224 NCSS 0.99048 I 0.25142 0.0449 /NIA 0.48526 I 0.2268

Survival Plot 1.0 --,---=c:::=------=-----=------, 0.9 2 0.8 0.7 g>o.6 5 :� 0. � 0.4 0.3 0.2 0.1 0 .0 --,----,---,------,-.-----.----�----.-----.------,-----.-,----.----�---+ 80 90 100 120 140 1eo 1ao JOO 220

Figure 3.8. SurvivalPlot of Types 1 and 2 fromJMP.

38 p=0.0004. This is quite a significantresult andallows us to proceed with the stated t-test.

In particular,the t-test gives a small p-value less than0.000 1, which is also highly statistically significantand allows for the conclusion that Types 1 and 2 are significantly differentand thus, Type 2 is significantly strongerthan Type 1.

It is appropriate to take a moment to provide the practitionerunfamiliar with survivalplots with anexplanation of the interpretabilityof these curves. Consider Figure

3.8 showing Types 1 and 2. The survivalplot has the internalbond strengthshown on the horizontal axisand the percentage of product survivingalong the vertical axis. If we are interested in the internalbond strength where 50% of Type 1 MDF is surviving (or equivalently, where 50% have failed), simply find 0.5 on the vertical axis and move horizontally until reaching the survivalcurve forType 1. Reading the horizontalaxis at this point on the curvegives 120.3 psi, which is the medianinternal bond forType 1 and what we would expect to obtain. Other examplesfollow similarly. Chapter3 of Meeker and Escobar(1 998) providesa thoroughand helpful treatment of the construction and interpretationof survivalcurves.

Suppose that interest lies in comparingtwo producttypes of the samethickness, but with a differentdensity. Here, we compareproduct Types 1 and 5. That is, Type 1

3 3 has a smaller density of 46 lbs/ft while Type 5 has a higher density of 48 lbs/ft •

However, they both have a thickness of 0.625 inches. The survivalplot comparingthese two products is shown in Figure 3.9. As with Figure 3.8, notice the large gap separating the two product typesallowing forevidence that Type5 is a much stronger product.

Thus, we areseeing that product typesof a higherdensity appearto be stronger than those at a lower density.

39 Survival Plot ------r:------; 1.0 --,------==--,------1 0.9 � �- 5 0.8 0.7 \.,._ � g>o.6 '\ :� 0.5 � 0.4 \ \ 0.3 0.2 \ 0.1 0.0 ---��-���-----=;,-...... ,-����-�...,....:c...J------1- 80 90 100 120 140 160 180 200 220 ·tnternat_Bond Figure 3.9. Survival Plot for Types 1 and 5 fromJMP.

40 Instead, suppose that interest lies in comparing producttypes of the same density, but with a different thickness. Then, in this case, comparison is between product Types 1

3 and3. Types 1 and 3 both have a density of 46 lbs/ft: , but Type 1 has a thickness of

0.625 inches and Type3 has a thickness of0.75 inches. The survivalplot showing this comparison is shown in Figure 3.10.

Notice that the gap we have been seeing in the plots is no longer present. This provides evidence that there are no differences among these two product types. That is, when density is held constant,thickness does not appearto have any effect on IB.

However, it is importantto verifythis statistically. Figure 3.11 is anoverlay plot of

Types 1 and 3. A two-samplet-test was conducted (again, assumingunequal variances)

and a p-value of 0. 1988 was obtained. Thus, our suspicions are confirmed and it can be

concluded that there are no statistically significant differences between Types 1 and 3 at

particularlevels.

A summary survivalplot showing Types 1, 2, 3, and 5 is shown in Figure 3.12.

From this plot, it is relatively easy to see which product typeshad the higher densityand

which had a lower density. However, it is not as obvious which product types had the

higher or lower thickness making it clearthat density is the main driver in determining

MDF strengthwhereas thickness is not a largecontributor to IB.

One noticeable attribute of the survivalplot shown in Figure 3.12 is that the

survival curves at the same density are crossing each other at some point. The

explanationis quite simple. The significanceof this crossing is that one producthas a

greater strengthat lower pressures whereas the other product will surpass at higher

pressures. For example, Type2 startsout with a greaterstrength than Type 5 at the

41 Survival Plot 1.0 -;-----==--==-==------, 0.9 0.8 0.7 g>o.6 5 :� 0. �0.4 0.3 0.2 0.1 0.0 ------r-����----r------r-�-----r----r--T'--'----�---r-----r----t 80 90 100 110 120 130 140 150 160 I ...... _Bclnd Figure 3 .10. Survival Plot forTypes 1 and 3 fromJMP.

[ Oneway Analysis of Internal Bond By Type 160-- 1 50-- � 140- 0 - m 130-- -'as 120- E - .! 110- ..5 - 100-- . !. . 90-- . . 80------�I- 1 3

Type Figure 3.11. Overlay Plot of Types 1 and 3 from JMP.

42 Survival Plot 1 .0...,.....-c=:::------:,..,,------c------, 0.9 0.8 0.7 g>o.6 5 :� 0. �0.4 0.3 0.2 0.1 TypeI .. 0.0 80 90100 120 140 180 180 200 220 Jnlernal_Bond Figure3. 12. SummarySurvival Plot for Types 1, 2, 3, and5 fromJMP.

43 extreme lower pressures. However, as pressure increases,we see the survivalcurve for

Type 5 crossthat ofType 2 andthus, surpassit in strength at the higher pressures. This crossingof survival plots may be helpfulto note formanufact urers ofMDF when developing new products.

3.4 SUMMARY

In conclusion, we find that exploring graphically andstatistically the MDF's reliability as measured by IB means,medians, and other percentilesreadable from survivalplots are helpfulways forunderstanding each product typebetter. Recall Type 1 had more outliers, which suggests more need forprocess improvements there. Density is a key driver in improving IB average. In fact, it was the key source of variationin IB.

Changesin thickness (or width) do not affect IB as much as changesin the density.

One should be awarethat quality andreliability are more thanjust one number

(not just the mean or median). We need to explore these andother descriptive statistics as well as graphsof the data. Also, be carefulof potential softwarediff erences on some tests, which may be mild or sometimes severe in certain instances. Validation with a different software package than the first softwareanalysis might be advisable. Besides histograms,survival curves are a very helpfuland insightful way to view your data.

These different views may surprise you, suggesting places forreal world process improvements. Compare Deming (1986 and 1993). Future work on estimating C.l.'s on the lower percentiles and other sources ofvariation will be explored later.

44 Chapter 4

Using Helpful Information Criteria to Improve Objective Evaluation of Probability Plots

4.1 INTRODUCTION AND MOTIVATION

In Chapter 3, the reliability of the internalbond (1B) of medium density fiberboard (MDF) was explored graphicallyand statistically comparable to the approach in Meeker andEscobar (1 998). In particular, probability plots andsurvival (reliability function)plots were utilized to allow forgreater ease in obtaining the most information fromthe IB data andfor ease in interpretability. Probability plots were discussed as a method for determiningthe underlying distribution of a particulardata set. Recall that if the data set "conforms" to a distribution,the points on the plot will forma straight line.

A shortcoming of this method, however, is the extreme subjectivity, i.e., forany given probability plot, different peoplemay have conflicting conclusions. Chapter 6 of Meeker and Escobar(1 998) provides examplesof probability plots based on repeated samplesof the samesize. Their graphscan serve as a strongillustration of the subjectivity of plots alone.

Consider the followingcomparison example as shown in Figures 4.1-4.3. For this example, we make use of Type 1 MDF data. Recall that Type 1 is the primaryproduct andis thereforericher in data thanother producttypes. Figure 4. 1 fitsthe normal, lognormal, andWeibull distributions using JMP Statistical Discovery Software. Figures

4.2 and4.3 fitthese same distributions,but use respectively SAS and S-PLUS software packages.

45 • I

2 1.2 j 0 � -1 I -2

Nl � � -5 .3,.....,__+-----r...... -���--r---,--' -6-"-r---....,...... ����----- 80 90 100110120130140150160 80 90 100110120130140150160 J 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2

lntemal_Bond lntemal_Bond log lntemal_Bond (a) (b) (c) Figure 4.1. Comparing Probability Plots for Type 1 MDF using JMP. (a) Normal, (b) Lognormal,and ( c) Weibull probability plots.

Type 11B Type11 B Type11 B

....., ... A.la ' -· ...... �.- ...... ; ...... ; - �c.; ,: ...... � ... �···················! . __- @:· ·············· ········ ... ,., ..·<, .....·• ·············...... ········:. - : ...... ,:...... : -�- l:-f:�-············!�···�·····liii ····· ... - '-T--.-...... ,...,.�.....,...... -r�...... ,...,...... = ��"TTTT".,...,,..,.,,..,....,..,,...... -...... --,.... ·•...... (a) (b) (c) Figure 4.3. ComparingProbability Plots for Type 1 MDF using S-PLUS. (a) Normal,(b) Lognonnal,and ( c) Weibull probabilityplots.

46 Based solely on the plots shown in Figure 4.1-4.3, the choice between the normal, lognormal, andWeibull distributions forthe m of Type 1 MDF may be anythingbut straightfotward. The comparisons of the packages shows that we must concernourselves with comparing one set of probabilityplots withina particularstatistical software package. In addition, we must examine the visual impressions between software packages. In particular, the placement of simultaneousconfidence bands does not appear · to agree forall three softwarepackages. Again, we emphasize, as others, the subjectivity of graphical approaches without informationcriteria. Later in this section we will discuss these very helpfulcriteria.

Hypothetically, one may look at Figure 4.1 andsuggest that the normalmodel provides the best fit. Likewise, a look at Figure 4.2 may suggest that the lognormal model is the best fit. Because of differentscaling in the software packages, the placement of obvious outliers are different, which may also have an affect on which plot is chosen to best represent the data. It should be emphasized, also, that in many cases, probabilityplots arevery useful andcan easily identify anunderlying statistical distribution fora particulardata set. However, it is againfurther stressed that this method is subjective andthus makes model determination difficult.

Fortunately, modem developments have produced more objectiveapproaches for determiningthe best candidatemodel fordata thansubjective probability plots, i.e., informationcriteria or model evaluation criteria. Accordingto Bozdogan(2000), the

"necessity of introducingthe concept of model evaluation has been recognizedas one of the most important technical areas, and the problem is posed on the choice of the best approximating model amonga class of competing models by a suitable model evaluatfon

47 criteria given a data set. Model evaluation criteria are figuresof merit, or performance measures, for competing models." In particular,Bozdogan (2 000) reviews the basic ideas surrounding Akaike(1 973) InformationCriterion or AIC and thenpresents further work based on his InformationComplexity Criterion (ICOMP) which is a new entropic model selection criteria. The theory supporting AIC and ICOMP makesprobability plotting more objective by accounting for the likelihood of the underlying model, which creates a numeric "score" for each probability plot. The model with the lowest score is picked as the "best" fit of the data. See Bozdoganand Bearse (2 003), Bozdoganand

Haughton(1 998), Urmanov, Gribok et al. (2002), Bozdogan(1 990), andBozdogan and

Sclove (1984) who show how informationcriteria plays animportant role in simple and multivariate regressionanalysis, cluster analysis,the detection of influentialobservations, etc.

The spirit of Chapter 4 of this thesis is to provide the reader with the essential backgroundto wisely apply AIC and ICOMP. Also, we show how using these helpful informationcriteria canaid in the importantselection of a better parametricmodel forthe internalbond. Section 4.2 introduces and further develops the ideas behind AIC and

ICOMP as it applies foruse in probability plotting. Section 4.3 uses the IB data on MDF as a brief case study and shows probability plots along with their informationcriteria

"scores" for the normal, lognormal, andWeibull distributions. It is the intent of this section to choose better underlying parametricdistributions for the differentMDF product types previously mentioned, i.e., Types 1, 2, 3, and 5. Section 4.4 provides concluding remarks, plus potential futurework that may be conducted in the use of informationcriteria forquantile modeling.

48 4.2 A BRIEF SUMMARY OF AIC AND ICOMP

Bozdogan(2001) is anexcellent placeto startfor very helpfulbackground

informationon AIC, ICOMP, andtheir manyapplications. In this section, it is reviewed

how these informationcriteria canbe used with probability plotting to prevent the

subjectivity when only using the plots.

Akaike's InformationCriterion (AIC), like other model-selection methods, takes the formof a lack of fitterm (such as minus twice the log likelihood) plus a penalty term.

The penalty termis a "compensation forthe bias in the lack of fitwhen the maximum

likelihood estimators are used" according to Bozdogan(200 1) . AIC has the following

form:

AIC;::-2 log L(O) + 2k (4. 1)

where L(8) is the maximized likelihood functionfor a particularpopulation parameter

IJ either ( scalaror vector valued) and k is the number of parametersin the model. For

2 example,if we consider the normalmodel with the parameters µ anda , then k = 2.

Recall that the model with these lowest informationcriteria score is chosen as the

best fitof the data. In order to better understandwhy we take minus twice the log

likelihood as the lack of fitterm, we takea heuristicapproach of reasoningby extremes

and exponentials, in the spirit of FrankProschan.

Consider anexponential model with failurerate parameterl and a data set with

the sole observationof x = 0. Here we have X rv Exp( l) where the density is

f(x) = k-.tx forx � 0 and l >O(and /(x) =0 for x < 0). Our data hasn = land x = 0.

This means our likelihood is simply L(l) = k-.tx and then plugging in x = 0, we

49 get L(l) = le-.t= l.

Recall that the sample meanhere is 0, andthe failure rate is the reciprocal of the mean. Thus, for this Proschanstyle heuristic, allowing the failurerate to be infinitywe have i = + oo.This yields L(i) = oo andtherefore, -2 logL(i) = -oo.Then, AIC

=-oo+ 2(1) = -oo.

Thus, a model with infinite likelihood will obtain a score of negative infinityand prove to be the "best" underlying model ( or at least tied for "best" model) forthis given data set. ProfessorFrank Proschan would use such extreme heuristics to demonstrate the essentials of importantconcepts and methods in reliability andelsewhere.

In comparisonto AIC, Bozdogan's ICOMP includes firstthe same lack of fit term. However, in contrast, the penalty term is substantiallydifferent. This penalty term takes intoaccount the asymptoticproperties of the maximum likelihood estimators as

1 well as the "complexity" of the inverse Fisher informationmatrix, !J- , of the proposed model.

Basically, rather than twice the number of parameters in the model, I COMP takes on a penalty termthat is viewed as the "degreeof interdependence amongthe components of the model" andhas the goal of providing a "more judicious penalty term thanAIC andother AIC-typecriteria, since counting andpenalizing the number of parametersin the model is necessary but by no meanssufficient" according to Bozdogan

(2000).

Since the complexity termtakes into account the interdependence of the parametersin the model, ICOMP canonly be used formodels with two or more

50 parameters. As with AIC, the model with the lowest ICOMP score is considered the best amongall competing models.

We now present the generalizedformula for ICOMP, which is as follows:

(4.2)

-1 where C1 (9 (0)) is a measure of complexity of the inverse Fisher informationmatrix and is given by:

(4.3) where r is the rank of the inverse Fisher informationmatrix and l•I denotes the determinant. For those interested in the details of the theory underlying AIC and

ICOMP, the reader should turnto Bozdogan(19 87, 1988, and 1996) andBozdogan and

Haughton (1998), amongothers.

We now focuson the use of AIC and ICOMP in association with probability plotting forthe purposesof choosing the best plot that represents the data. Chapter 6 of

Bozdogan(2001) provides extensive informationon quantilemodeling andhow to incorporate AIC and ICOMP into this graphical approach. Recall, first, that in probability plotting, the data are ordered andthen plotted against the theoretical order statistics fora desired distribution. Inorder to make use of the informationcriterion previously discussed, it is necessary to fita regression model through the plotted points.

To do this, we make use of a firstorder approximationof the form:

Y; = Po + P1x; + &; i = 1, 2, ... n (4.4) where Y; represents the ordered data, x1 represents the theoretical quantiles, and &;

51 correspondsto the errorassociated with using the first order approximation. We must also assume that the errorsare approximately normallydistributed with constanterror

2 variance in order to carryout least squares regression. That is, &; rv N(O, u ).

It is without question that anotherregression model other thanthat shown in (4.4) above may be considered for use here. Furthermore, a more complex model that does not assume a constanterror variance can certainlybe utilized andmay prove to be more appropriate. In particular, further research in this area may be conducted later. However, given the usefulnessof this methodology for the practitioner, we choose a rough and quick approximation in the spirit of Frank Wilcoxon in order to aid in the ease of carrying out the use of informationcriteria in quantilemodeling. We employ the derived formulas of AIC andICOMP that will be usefulfor this application as follows:

2 AIC = n In(21l')+ n In(o- ) + n + 2(2) (4.5) where k = 2 since we are estimating Po and p1 for the regressionmodel. Some may argue that k = 3 since we also include the parameter u2 in the likelihood equation.

However, letting k = 2 is the traditional approach in regression analysis, especially in the construction of confidenceintervals. See our resident expert, ProfessorBozdogan, and his papers cited in this chapter for helpfulcomments on this and other insights below.

For more on regressionanalysis in general, see Neter, Kutner, Nachtsheim and

Wasserman (1996) and Montgomery, Peck and Vining (2001). These, among others, providea thorough referenceand would be helpful for the practitioner and those interested in learningmore about the fundamentalsand applications of linear regression models.

52 The derived formulafor ICOMP is:

(4.6)

0 " where -1 2 Lx/ g (Po,P.,u) = i=1 (4.7) 0 n

Without question, one may expectthat since the inverse Fisher informationmatrix shown

1 2 in (4.7) takes into account three parameters, then the form of l- (p0,p1 ,u ) would be a

2 3x3 matrix. However, recall that we estimate u = !.f e/ where &; = Y; - Po - Pix; n ;.1 when we allow the intercept term,Po , to not equal zero. Note that &; = Y; - Pix; when the intercept termis set equal to zero, i.e., Po = 0. Therefore, whether we are dealing with a linearmodel with anintercept term or not (i.e. Po = 0 or not) will have no affect on the form of the inverse Fisher informationmatrix since the intercept term is included in the 2 estimation of u • This is importantsince in some applications ( e.g., engineering andindustrial applications) the intercept termdoes need to be constrained to zero. Therefore, we simply fit the simple regressionmodel that is the most appropriateand calculate the estimated error variance using the residuals of the fitted model. Then, substitute the calculated estimate of the errorvariance into the inverseFisher informationmatrix given in (4.7). For more on the derivation of the formulas forAIC, ICOMP, and the inverse

53 Fisher informationmatrix shown in formulas( 4.5), (4.6), and (4. 7) respectively, see the chapter on thetheory of linearmodels in Bozdogan(200 1). This chapter further illustrates the necessary technical details and shows how (4. 7) is the same forboth a model with or without an intercept term.

Let us next move to show how this can be used in application. In particular,we returnto our data on the IB ofMDF. Types 1, 2, 3, and 5 will be used and the best parametric distributionwill be determinedby scoring AIC and I COMP.

4.3 USING AIC AND ICOMP WITH PROBABILITY PLOTS TO DETERMINE THE PARAMETRIC DISTRIBUTION OF THE INTERNAL BOND OF MEDIUM DENSITY FIBERBOARD

Using the helpfulMATLAB programminglanguage, a routine was constructed to calculate the quantiles, plot them, and score AIC and ICOMP for each of the normal, lognormal, and Weibull distributions. This was done for each of Type 1, 2, 3, and 5

MDF product types as anillustration and foruseful comparison. Recall that product

Types 1 and3 have the same density of 46 lbs/ft3 and a differentthickness of 0.625 and

0.750 inches, respectively. Likewise, productTypes 2 and 5 have the same density of 48 lbs/ft3 anda different thicknessof 0. 750 and 0.625 inches, respectively. Recall Table 3.1.

We beginwith Type 1 MDF. Figure4.4 shows the three probability plots forthe normal, lognormal, and Weibull distribution forType 1 as produced by MATLAB. A close look at these plots clearlyreveals their subjectivity. Two practitioners may not be able to agreeon the best fit. Further,Table 4.1 shows the AIC andICOMP scores for each plot. Based on the minimum values of AIC and!COMP of 1428.6 and 1439.5,

54 0 0

(a) (b) (c) Figure 4.4. Type 1 probability plots by MATLAB. (a) normal, (b) lognonnal, and(c) Weibull probability plots.

T abl e 4.. 1 AIC and ICOMP iior ype T pro 1 a b 1b uy T p t o 1 s. t Distribution AIC ICOMP Normal 1428.6 1439.5 Lognormal 1487.3 1498.1 Weibull 1678.1 1687.5

55 respectively, the normaldistribution appearsto be the best fitof the Type 1 internalbond data. As will be seen furtherin Chapter 5 of this thesis, having knowledge of the parametricdistribution enables the better estimation of population characteristics/parametersof interest. Knowing that Type 1 followsroughly a normal distribution helps in conducting statistical tests (such as the analysisof variance

(ANOVA))where the assumption of normalityis required.

Figure 4.5 shows the probability plots for Type 2 MDF as produced in MATLAB andTable 4.2 gives the AIC and ICOMP scores foreach plot. Given, the minimum values of AIC and ICOMP of 571.12 and 584.11, respectively, choose the lognormal distribution as the best fit for the internalbond of Type 2 MDF. Recall that if

T l"V Lognormal(µ,o-) then Y = log(T) l"V No rmal(µ,o-) . In the case of the lognormal, we definethe parameter µ as the meanof the logarithmof T and o- as the standard deviation of the logarithmof T. That is, the lognormal parametersare the meanand standard deviationof the transformeddata. This distribution appearscommonly in reliability data andfalls into the location-scale familyof distributions. For more informationon the lognormaldistribution, such as formulasfor the expected value, variance,and quantiles, see Meeker and Escobar( 1998).

Figure 4.6 andTable 4.3 give the probabilityplots and the AIC andICOMP scores forType 3 MDF, while Figure 4. 7 andTable 4.4 show the probability plots and

AIC and ICOMP scores forType 5 MDF. Although the Type3 AIC andICOMP scores forthe normaland lognormal were extremely close, the minimum value ''wins," andthus, the normaldistribution is chosen as the best fit for the internalbond of Type3 MDF. For

Type 5, we also will findthat the normaldistribution proves to be the best fitof internal 56 (a) (b) (c) Figure 4.5. Type 2 probability plots by MATLAB. (a) normal,(b ) lognormal, and(c ) Weibull probability plots.

T abl e 4 ..2 AIC and CO I MP for Type 2 probability plots. Distribution AIC ICOMP Normal 614.47 627.24 Lognormal 571.12 584.11 Weibull 933.96 944.57

(a) (b) (c) Figure 4.6. Type 3 probability plots by MATLAB. (a ) normal,(b ) lognormal, and( c ) Weibull probability plots.

T abl e 4 ..3 AIC and ICOMP f4or Type 3 b bpro T I a 1 1ty p ots. Distribution AIC ICOMP I Normal 542.36 553.11 II Lognormal 543.14 553.84 I Weibull 711.85 721.05 I

57 (a) (b) (c) Figure4. 7. Type5 probabilityplots by MATLAB. ( a) normal, (b) lognormal, and( c) Weibull probabilityplots.

1 T a bl e 4 ..4 AIC and ICOMP tior ypeT 5 pro b a b Ttty p 1 ots. Distribution AIC ICOMP Normal 305.54 315.14 Lognormal 342.47 351.57 Weibull 321.08 330.24

58 bond. There wasno conclusive evidence to suggest that the Weibull distributionwas the best underlying distribution forthe top fourMDF product types.

This mightbe surprisingsince the Weibull is oftenthe first choice when studying the strengthsof materials. Indeed, the researcher Dr. Weibull himself firstanalyzed strengths of different materials, rangingfrom cotton to metal. From his data sets, he foundthe primaryavailable distribution of the normaldid not fithis examples well in the

1930's. The alternativeparametric model he originally proposedis what we now call the

"three parameter"Wei bull. CompareWeibull (1939 and 1951).

The internalbond ofMDF is anexample of how importantthe information criteria are to findthe parametric distribution fora data set. Afterall, our data surprises us andour intuition may not always be confirmed. Thesecriteria let the data speak objectively forthe "best" model.

4.4 SUMMARY

It cannot be reiterated enoughthat probability plotting is anextremely useful way to aid in the determinationof the parametricdistribution of a particulardata set. In many cases, when comparingplots of different candidatedistributions, little ambiguityis present andthe choice of distributionis not difficult. However, this method is subjective andwhen ambiguityamong candidate distribution probability plots is present, two or more variant conclusions may be reached.

Akaike's InformationCriterion (AIC) andBozdogan 's InformationComplexity

Criterion (ICOMP)rescues practitioners fromsuch subjectivity. These two formsof information criteria, amongothers, have a lackof fitterm (minus twice the log

59 likelihood) plus a penalty termthat accounts forthe number of parametersin the respectivemodel. Building on the fundamentalideas behind probability plotting, AIC and ICOMP make it much more objective by creating a numeric score foreach plot. The plot with the lowest score is considered to be the best fitof the data set under consideration. This is great to explain to practitioners, who may not have much statistical training.

When applied to the different product types ofMDF, it was discovered that the normal distribution is technically the best fit forTypes 1, 3, and5, while the lognormal distribution is the best fitfor Type 2. Of course forType 3, the lognormalwas extremely second which mightalso be considered in that specificcase . Thus, even though the

Weibull distribution mightbe anintuitive choice when studying the strengths of materials, the actual data analysis of the internalbond ofMDF does not supportthis first intuition. As stated in Chapter 3, data has a way of producingunexpected results in the light of intuition, exploration andtheory .

Note that futurework in the area of informationcriteria might include using anothermodel differentthan this helpfulfirst order approximation assuming normal errorswith constantvariance. Also, informationcriteria are certainlynot limited to applications in probabilityplotting. Other considerations include their use in regression analysis, statistical process control, etc.

60 Chapter 5

Applying Bootstrap Techniques for Estimating Percentiles of the Internal Bond of Medium Density Fiberboard

S.1 INTRODUCTION AND MOTIVATION

In reliabilitystudies, it is generally of high interest to estimate percentiles. In particular,interest usually lies in the estimation of the lower percentiles. These lower numbers arehelpful for warranty analysis, understanding early failures during normal usage, plus improvingthe specification limits. Meekerand Escobar (1 998) observethat the "traditional parametersof a statistical model (e.g., meanand standard deviation) are not of primary interest. Instead, designengineers, reliabilityengineers, managers,and customers areinterested in specificmeasures of product reliability or particular characteristicsof a failure-time distribution( e.g., failure probabilities, quantilesof the life distribution, failure rates)." See, also, Meeker and Escobar(2004).

Nelson (1990) furthermentions that with "lifedata work, one oftenwants to know low percentiles such as the 1% and 10% points, which correspondto earlyfailure. The

50% point is called the medianand is commonly used as the 'typical'life." The first and third quartiles are also useful in studying the lifeof a product. We note that forsome lower percentiles, samples sizes need to be adequately large. If samples are small, the lower percentiles should be avoided andthe quartilesshould be used instead. Recall the medianis the second quartile. Comparethe comments fromPolansky ( 1999) warningto have sufficientsample size for the lower percentiles.

To be able to say that improvements have been made, we must be able to measure

61 reliability expressed in percentilesthat allow forstatistical variation. We need to make comparisonsof these reliability measures between products and within products before and afterprocess improvement interventions. Knowing when to trust confidence intervalsand when not to trust them are crucial formanagers and users of MDF to make successfuldecisions.

Chapter 5 is devoted to the estimation of percentiles of IB and their respective confidenceintervals. In particular,the bootstrap will be presented as a usefulmethod for obtaining the aforementionedestimates and confidenceintervals. We next provide the reader with some usefulbackground infonnation on percentiles, its consistency, and asymptoticdistribution.

Serfling(1 980) definesthe 100 pthpercentile or pthquantile as tP = inf{t : F(t) � p} where F represents the distribution function. That is, the pthquantile is the greatestlower bound of the set of all values, t, such that F(t) is greaterthan or equal to a specifiedvalue of p where 0 � p � 1. In practice, we takethe infimumof the set since it is possible forthe distribution function, F, to yield a set where the minimum does not exist, but the "inf' does. For example,the open interval(0,

1) has an inf of 0, but the min does not exist.

For a sampleof n observations, {t1 ,t2 , ...t n } on F, the sample pth quantile, denoted by i is "definedas the quantile of the sample distribution function", or as P pth p-t (p). Furthennore, it has been shown that i is a consistent estimator of . That is, P tP as the sample size increases, the estimate of the quantile gets closer and closer to the true value. This asymptoticproperty is "such a fundamentalproperty thatthe worthof an 62 inconsistent estimator should be questioned (or at least vigorously investigated)" according to Casellaand Berger (2002). Thus, consistency is certainly a desirable propertyfor anestimator. Stated statistically, for any small & > 0, it followsthat

P(supliP -tP I > &) --+ 0 as the samplesize n approaches infinity. Equivalently, this can be written as P(supjiP -tP I S &) --+ 1 as n approaches infinity. In particular,it should be noted that this convergence rate is exponential.

Serfling(1 980) also thoroughly examinesthe asymptotic distribution of the samplequantile. In particular,under mild requirements (i.e. smoothness of the distribution function), thesample quantiles areasymptotically normal. We state the followingtheorem and corollaryfrom Serfling(1 980) without proof, which for more extensive details, see his book.

Theorem: Assume thatthe leftand right hand derivatives of F exist at tP and that F is continuous at tP. Then, if the lefthand derivative, denoted by F'(tP -), is greaterthan 0, then for t < 0,

J;(i -tP ) limP( P St)= (t) . (5.1) n-+ao Jp(l - p) I F'(tp -)

Furthermore,if the righthand derivative,denoted by F '(tP + ), is greaterthan 0, then for t > 0,

J;(i -t ) lim P( P P St)= (t) . (5.2) n-+ao p(1 - p) / F '( t +) J P Finally, for t = 0, (5.1) and (5.2) canbe simplified as,

lim P(.j°;(iP -tP ) S 0) = (0)= 0.5. (5.3) n-+ao

63 Corollary:Assuming F is differentiableat and F '( ) > 0, then tP tP

(5.4)

where AN stands for"asymptotically normal".

For furtherdetails andan extensive proof of the above theorem andcorollary, see

Serfling( 1980). This is a usefulresult since by possessing asymptotic normality; we can

construct asymptotic normal confidence intervals for the pth quantileof a distribution.

· Chapter 8 of Meeker and Escobar( 1998) provides very helpfulinformation regarding the

construction of such intervals forthe location-scale distributions used commonly in

reliability data analysis. In particular, a normal approximate confidenceinterval for tP is

given by:

(5.5)

�i where p is the standarderror of the estimate andis given by:

which is derived using the delta method and <1>-1 represents the inverse of the cumulative

standardnormal distribution. Vai-(µ),-ra,.(6-), and Cov(µ,6-)are obtained fromthe

variance-covariancematrix or inverse Fisher informationmatrix, 5-t. To compute these

intervalsby handcan, without question, be verytedious andtime consuming.

Fortunately, statistical softwarepackages such as SAS have the capabilities (i.e. PROC

RELIABILITY) to produce these confidenceintervals for desired quantiles.

When the sample size is sufficientlylarge, the asymptoticnormal intervals

64 provide very good approximations. Even though these intervalsare approximations, they areusually good enough for practice. However, forsmall samples,these intervalsmay not provideaccurate approximations. It is in this case that anothermethod, whether it be parametricor nonparametric, is necessary to obtain better confidence intervals fordesired quantiles. Bootstrap methods provide one possibility for better estimation given reasonable samplesizes.

Bootstrappingis a computer intensive statistical method where the basic idea is to simulate the samplingprocess a specified (usually large)number of times and obtain an empirical bootstrap distribution for a desired population parameter. This empirical bootstrapdistribution is then used to acquire characteristicsabout the population parameter. These include, but arenot limited to, the standarderror, .anestimate of bias, andconfidence intervals. Some bootstrap methods are nonparametricand therefore do not require anyparametric assumptions regarding the underlying distribution of a particulardata set. Other methods using the bootstrap are parametric. These methods will be discussed andcompared in detail later on in this chapter.

According to Chernick( 1999), the "bootstrapis a formof a largerclass of methods that resample fromthe originaldata set andthus are called resampling procedures. Some resamplingprocedures similar to the bootstrap go back a long way ... .

However, it was [Bradley] Efronwho unifiedideas andconnected the simple nonparametric bootstrap, which 'resamplesthe data with replacement' with earlier accepted statistical tools for estimating standarderrors such as the jackknifeand the delta method."

Boos (2003)describes the bootstrapas a technique that "has made a fundamental

65 impact on how we carryout statistical inferencein problems without analyticsolutio ns."

Davison andHinkley ( 1997) tell us that the bootstrapis called so since ''to use the data to generate more data seems analogousto a trick used by the fictionalBaron Munchausen, who when he foundhimself at the bottom of a lakegot out by pulling himself up by his bootstraps." They furtherassert the necessity of carefulreasoning andinvestigation of the problem at handdespite the usefulnessof bootstrapmethods . It is contended that

"unless certain basic ideas areunderstood, it is all too easy to produce a solution to the wrong problem, or a bad solution to the rightone . "

Efronand Tibshirani(1 993) is an excellent startingpoint and a way to get acquainted with the fundamentalconcepts andapplications of the bootstrap. Much of their work is written without rigorous technicaldetails in order to focuson ideas rather thanjustification. Those details can be foundin some of their later chapters as well as other works.

DiCiccio and Efron(1 996) is devoted to the constructionof bootstrap confidence intervals. Here, differentmethods arepresented as well as the theoretical underpinnings.

We adopt next the notation of Martinezand Martinez(2002), which is also similar to Efronand Tibshirani(19 93). In general, the basic nonparametricbootstrap procedure

(Efron's bootstrap) canbe summarizedas follows. For a given data set, x = (xi ,x2 , ••• ,xn ) of size n, we estimate a population parameter, say 0, by 0. We then samplewith replacement fromthe original data set to obtain a bootstrap sampleof size n denoted by

*b •b •b . . . x = ( x1 , x2 , •••x n•b ) . Th1s" resamp1· mg with rep 1 acement 1s d one a large num b er o f times and for each bootstrap samplewe calculate the estimate of 0, which is denoted by i)•b

66 where b standsfor the bth bootstrapestimate of a total of B bootstrap replications. The

l empirical bootstrapdistribution of { , is definedand used as an estimate to thetrue distribution of 0. The fundamentalidea behind the bootstrapis that the empirical bootstrap distribution provides an approximationto the theoretical samplingdistribution of the desired population parameteras the samplesize increases. In particular, as n approaches infinity, the bootstrapdistribution becomes more normalfor most cases. The bootstrap has a wide range of applications andhas enjoyed more growth in use in recent years.

However, as with anystatistical method, the bootstrap does have its limitations.

Beran(2003) emphasizes that "Success of the bootstrap.. .i s not universal.

Modificationsto Efron'sdefinition of the bootstrapare needed to makethe idea work for modem procedures that are not classicallyregular."

As also described in Chapter 2 of this thesis, Meeker andEscobar ( 1998) contend that the "j ustificationfor the bootstrapis based on large-sampletheory. Even with large samples,however, there can be difficultiesin the tails of the sample. For the nonparametricbootstrap, there will be a separatebootstrap distribution at each time for which there were one or more failuresin the original sample." This would not pose a problem outside the tails of the original data where the bootstrap distribution will be approximately continuous. However, in the extreme tails of the originaldata, there may be only a small number of failuresor outcomes. In this case, the bootstrapdistribution may be anythingbut continuous. As canbe seen by the examplespresented, when the extreme tails areof interest (as is oftenthe case in reliability studies), thefully

67 nonparametric bootstrapmethods may not prove to be as useful. Rather the standard bootstrap methods have a place when estimating parameterssuch as the quartiles(2 5th'

50th or 75th percentiles).

We will see the above limitation of the bootstrap when applied to the estimation of lower percentiles forthe internalbond of medium density fiberboard. The samplesize also plays a key role in the accuracy of bootstrap methods. Further limitations are described, forexample, in Chernick(1 999) and Ghosh, Parret al. (1984).

In section 5.2 coming, differentmethods forconstructing bootstrapconfidence intervalswill be introduced. These include the standard normal,bootstrap -t, percentile, andbias-corrected percentile intervals. However, for each method of creating bootstrap confidenceintervals, there is also more than one way to create bootstrap samples. In particularwe discuss the completely nonparametricbootstrap, the completely parametric bootstrap, and finallya "nonparametric" bootstrapfor parametric inferenceas described in Meeker and Escobar (1998). Each of these methods forcreating bootstrap samples will be presented along with the above methods forproducing confidenceintervals.

Section 5.3 will apply what was presented in section 5.2 to the MDF data for estimating the lower percentiles of the internalbond. The asymptotic normalintervals will be comparedwith the bootstrap confidenceintervals. Furthermore, the bootstrap confidence intervalswill be comparedamong themselves with respect to samplingand construction method. Section 5 .4 provides a summary and concluding comments with loose recommendations forwhich situations dictate which bootstrap method to use. Also, possible futurework will be presented here.

68 5.2 A BRIEF INTRODUCTION TO BOOTSTRAP SAMPLING METHODS AND BOOTSTRAP CONFIDENCE INTERVALS

5.2.1 Methods of Bootstrap Sampling

Already in section 5.1, we introduced the completely nonparametricbootstrap or

Efron's bootstrap. That is, no assumptions aremade about the underlying parametric distribution of a data set of size n. The desired population parameteris estimated nonparametricallyfrom the initial data. Then, samplingis done with replacement

(usually a largenumber of times). For each sampleof size n obtained, we nonparametricallyestimate the population parameter. These estimates of the desired parameterare used to formthe empirical bootstrapdistribution that will be usefulfor inference. This empirical distributionis a discrete distribution that assigns a probability of l /n to each value of x. This methodof samplingis helpfulsince it has the advantage of no distributional assumptions. When it is not possible or feasibleto make such an

assumption, the completely nonparametricbootstrap sampling method should be

employed. The next two methods of samplingbelow do require a parametric distributional assumption.

The completely parametricbootstrap is described brieflyin Efron andTibshirani

(1993), as well as Chernick(1 999) andMeeker andEscobar (1 998). A parametric

distribution is assumedand the initial data of size n is utilized only to obtain maximum

likelihood estimates of the model parameters. From there, one must simulate a specified

number of samplesof size n fromthe parametricdistribution. The population parameter

is then estimated parametricallyfrom each of the simulated samples, which then helps us

create the desired bootstrapdistribution necessary for inference.

69 Chernick(1 999) interestingly notes that the "parametricform of bootstrapping is equivalent to maximum likelihood. However, in parametric problems, the existing theory on maximum likelihood estimation is adequate andthe bootstrap adds little or nothing to the theory. Consequently, it is uncommon to see the parametric bootstrap used in real problems." However, Efron and Tibshirani (1993) argue that when the fullyparametric bootstrap is used, it "provides more accurate answersthan textbook formulas, and can provide answers in problems where no textbook formulaeexist. ..The parametric bootstrap is usefulin problems where some knowledge about the formof the underlying population is available, and forcomparison to nonparametricanalyses."

Meeker and Escobar(1 998) points out that the parametricbootstrap has a disadvantagein reliability data problems. That is, the complete censoring process must be specified giventhat we are simulating data. This may seem to be unproblematic in simple exampleswhere such specificationis easy. However, this canbe "more difficult forcomplicated systematic or randomcensoring. Oftenthe needed informationmay be unknown." An alternativeto this method requiring parametricassumptions is described next.

Meeker and Escobar (1998) describe andillustrate applications of a

"nonparametric"bootstrap samplingmethod forparametric inference, which we denote as NBSP for nonparametricbootstrap sampling for parametric models. They contend,

"This method is simple to use andgenerally, with moderate to largesamples, provides results that are close to the fullyparametric approach." This samplingscheme does require parametricassumptions. However, rather thansimulating data, we samplewith replacement fromthe original data. For each sampleof size n, maximum likelihood

70 estimates are obtained based on the assumed parametricmodel. Then, the MLEs areused to parametricallyestimate the population parameterof interest.

The distribution of estimates allows us to conduct the desired inferences. See

Chapter 9 of Meeker andEscobar (1 998) formore details on this method of bootstrap sampling. We move now to see how the aforementionedsampling schemes andthe

resulting empirical distributions allow us to constructbootstrap confidence intervals for

population parameters.

5.2.2. Bootstrap Confidence Intervals

Differentalgorithms/methods areavailable forconstructing bootstrap confidence

intervalsfor population parameters. These include, but arecertainly not limited to, the bootstrapstandard confidence interval, bootstrap-t confidenceinterval, bootstrap percentile interval, andbias-corrected bootstrap percentile interval. We describe these

brieflyhere andomit much of the theoretical details here. For those interested in the

theoretical underpinningsand additional topics, manygood books andarticles exist. See,

amongothers, Efronand Tibshirani (1 993), DiCiccio and Efron (1996), Davison and

Hinkley (1997), andPolansky (1 999).

The bootstrap standardconfidence interval is by farthe easiest to implement.

Efron andTibshirani (1993) andothers use the phrase "bootstrap standard confidence

interval"while is it also knownas the normalapproximation bootstrap confidence

interval. These intervalsare basedon thefollowing asymptotic result:

Z = -;::;s:::-0-0 rv N(O,l) . (5.7) se9

71 Then, the standardconfidence interval is given by:

(5.8) where ;;simply the standarddeviation of the bootstrap is estimates of (} and z<012> is the a I 2th quantileof the standard normaldistribution. That is, forexample, z<0·025> =

-1.96.

Although this method is easy to use, (5.7) "is only anapproximation in most problems, and the standardinterval is only anapproximate confidenceinterval, though a very usefulone in anenormous variety ofsituations" according to Efronand Tibshirani

(1993). This intervalcan be used when the asymptotic normalityis valid.

Another usefulmethod is the bootstrap-t confidenceinterva l. For each of B

(some largenumber of choice for B; usually largerthan 1000) bootstrap samples, we compute:

(5.9) where � is the standard error of o•b fora particular bootstrap sample. The difficulty arises in the computation of this estimate standarderror. In manysituations, a nice closed formula does not exist. To remedy this, a possible solution is to bootstrap each bootstrap sample and then take se9 .b to be the standard deviation of the "bootstrapped" bootstrap sample. Basically, one performsa double bootstrap to obtain the desired estimate of the standarderror .

The only problem with this is the amountof computer power required to perform

72 such a largenumber of bootstrapreplications. For example, ifwe want to obtain 2000

bootstrapsamples and boot strapeach of those 50 times to obtain the sampleestimate 's

standarderror, then this requires 100,000 iterations.

After obtaining the B values of z•b, order them andcalculate the a 12th and

1-(a l2)th quantiles of the distribution of z•b values which will be denoted by

i andi (l-ati> respectively. This is findingthe appropriate percentiles of the sampling distribution,which is to be distinguished clearly from the percentiles of the original population data. At this point, the bootstrap-t confidence intervals canbe computed.

Then, a 100( 1- a ) % bootstrap-tconfidence interval is given by:

(5.10)

The bootstrap-t confidence intervalsare second-order accurate ( error goes to zero at a

rate of 1/n), which makesthem a popularchoice in practice.

However, Efron andTibshirani (1 993) warnthat the "bootstrap-tcan give erratic

results, andcan be heavily influencedby a few outlying data points. The percentile based

methods ...are more reliable." Polansky(2000) "investigates two methods forstabilizing

the endpoints of bootstrap-tintervals in the case of small samples. In those cases,this

would be anapproach forothers to use. Two of the percentile-based methods will now

be discussed.

Perhaps one of the most obvious ways to construct a confidenceinterval is to base

it on the quantilesof the bootstrap distribution of estimates. Constructinga bootstrap

confidenceinterval in this manner is known as the standard percentile method. Martinez

andMartinez (2002) and Efronand Tibshirani (199 3) maintain that "this technique has

73 the benefit of being more stable than the bootstrap-t,and it also enjoys better theoretical coverage properties." In particular,this method works well when a monotone transformation, and iJ-<1-a12> respectively. Then, a 100(1- a)% confidenceinterval for 0 is given by:

(5.11)

For example, in order to construct a 95% confidence interval,we simply calculate the 2.5th and97.S th percentiles of the bootstrap distribution forthe parameterof interest.

It is generally recommended that the number of bootstrap replications be equal to or greaterthan 1000 forthis method to produce accurate results.

Though the standardpercentile method is easy to implement, Chernick(1 999) points out that "the percentile method works if exactly 50% of the bootstrap distribution of o· is less than iJ" which may certainly not always be the situation andthat "in the case of small samples, the percentile method does not work well." Furthermore, a two­ sided 100(1- a)% confidenceinterval should have the probability of not covering the true value of a parameter, either above or below, of a I 2 . The standardbootstrap

112 percentile intervalsare first order accurate ( errorgoes to zero at a rate of n- ) which meansthat the errorin getting exactly the desired a /2 probability is anorder of magnitude greater than that of the bootstrap-t intervals which, one can recall, are second

74 order accurate. Fortunately,there aremethods that help improve on the standard

percentile method and one such will be shown next.

The final method for constructing bootstrapconfidence interval that will be

presented here is called the bias-correctedpercentile interval. This method was

introduced in Efron( 1981) and discussed furtherin Efron( 1987). The method is

described there in greater length along with the needed theoretical details. The bias­

corrected percentile method ( or BC) works best when a monotone transformation,

�

bias correctionconstant and r is the constant standard deviation of � .

Assuming, again, that the aforementionedtransformation exists, Efron (1987)

� shows that the transformationleads to the "obvious confidenceinterval ( + rz0) ± rz

for

transformation 0 = g-1 (

automatically from bootstrap calculations,without requiring the statisticianto know the

correcttransformation g."

The bias correctionconstant is defined as the amountof difference between the

medianof the bootstrap distribution of estimates and the estimate, 0 fromthe original

sample. That is, if we takethe bias to be bias = 8 -o:.s, then o:.s = 8 -bias . This is

explainedfurther in Chernick(1 999). Explicitly, we definethe estimate of the bias

correction constant, denotedby z0 , simply as:

(5.12)

75 where <1>-1 represents the inverse cumulativenormal distribution and p• is the cumulative bootstrapdistribution forthe parameterof interest.

Alternatively, in other words, we canexpress (5.12) as the inverse cumulative normaldistribution of the number of bootstrap estimates, o·b' that are less thanthe original sample estimate, 0, divided by the number of bootstrap replicates, B. That is, we rewrite (5.12) as:

" ,. ,. "' 1 #(8• b < 0) - - ( ) (5.13) Zo - "' B

Then, a 100(1 - a)% confidence interval for (} is given by:

(5.14) where a1 and a2 are the new quantitieson which to base the percentile confidence intervalendpoints. Martinezand Martinez (2002) explain that "instead of basing the endpoints of the intervalon the confidencelevel of 1- a, they are adjusted using informationfrom the distribution of bootstrapreplicates." These quantitiesare the lower and upper bias-corrected cut-offperc entages andare defined as:

a - (2z + z ) (5.15) 1- 0 and

(5.16) where <1> is the cumulative standardnormal distribution and z is the a I 2th quantile of the standard normaldistribution. The bias-corrected percentile intervalshave been foundto be second-order accurate, which certainlyimproves on the standardpercentile interval. The method does have the drawback of not being monotone in coverage. That

76 is, if we decreasethe confidencelevel, we do not necessarilyget a shorter intervalthan that obtained at a higher level of confidence. In general, the percentileand bias-corrected percentile methods give more conservativeconfidence intervals than the bootstrap-t.

In summary,Martinez and Martinez (2002) point out that the "bootstrap-tinterval has good coverage probabilities, but does not performwell in practice. The bootstrap percentile intervalis more dependable in most situations, but does not enjoy the good coverage property of the bootstrap-t interval."

Recall, rather, that the percentile intervalpossesses good theoretical coverage

properties, which may not actually hold in practice. The bias-correctedpercentile

intervalhelps to remedythis by being both dependable andhaving good coverage

properties. Efron(2 003) furtherpoints out that even thoughthe bootstrap-t andthe _bias­ correctedintervals are second-order accurate, they are not widely used in application.

Instead, researchers and even seasonedstatisticians "seem all too happy with the standard

intervals"which may certainlybe due to its theoretical simplicity andease in

construction.

Other possibilities for bootstrapconfidence intervals, which will not be described

here, include the iterated or double bootstrap andthe bias-corrected andaccelerated (BCa)

percentile interval, amongothers. The iterated bootstrap requires the user to bootstrap

the B1 bootstrap samples,B 2 times. The price to pay here is an extremely large increase

= = 2 in iterations. For example, if B1 B2 B, then one must go through B iterations, which

can obviously take up a largeamount of computing power. The bias-correctedand

accelerated intervalis built upon the bias-correctedinterval described above. It requires

the calculation of anacceleration constantwhen the standarddeviation of the

77 transformationis not independent of the transformation. This constant,however, can prove to be difficultto determine and the readeris directed to Efron (1987) and Efronand

Tibshirani (1993) formore on the BCa method.

After this degreeof introduction, we move next (in section 5.3) to demonstrate how thesebootstrap sampling methods andconfidence intervals can be applied forthe purposesof estimating percentiles (especially lower percentiles) of the internalbond of medium density fiberboard. It should be emphasized here again that the estimation of lower percentiles is very importantin reliability studies forstrengths of failure. This helps manufacturersgauge process improvements, warranties,proportion of product fallingout of spec, etc. Also, there are cleareconomic advantagesto exploring these lower percentiles.

5.3 EXPLORING BOOTSTRAP METHODS AND CONFIDENCE INTERVALS FOR PERCENTILES OF THE INTERNALBOND OF MDF

In this section, we review the methods of bootstrapsampling and forconstructing confidence intervalsin the context of estimating percentiles forthe internalbond of medium density fiberboard. We also observehow they weighagainst each other. For each method of sampling,the standardnormal, bootstrap-t, standardpercentile, andbias­

st th th correctedpercentile intervals will be constructedand compared for the 1 , 10 , 25 , and

50th percentiles forMDF product Types 1 and 5. These two typeswere chosen to aid in theillustration of the benefits andlimitations of the bootstrap. Type 1 MDF has n=396 observationswhile Type5 MDF has n=74. We will thus be able to see ·how a smaller sample sizecompares to that of a sample that is sufficientlylarge to obtain relatively

78 accurate results. For each method of sampling, B=2000 bootstrap samplesof the same size as the originalsample were created. In manycases, but not always, this should be a sufficientnumber of bootstrapsamples to create the confidence intervals. The asymptoticnormal confidence intervals will also be providedin order to compare with the bootstrapresults.

Furthermore, along with each method of bootstrapsampling, histograms of the empirical bootstrapdistribution will be shown foreach percentile. The fully nonparametricintervals will be shown first, followedrespectively by the fullyparametric intervalsand the NBSP intervals describedby Meeker and Escobar ( 1998). This is important for practitionerswho may not have the luxuryof developing or assuming certainparametric distributions due to their intense time pressures. Also, this protects them more frommisspecified parametricmodels.

MATLAB was utilized as the programof choice forthis author in order to construct these intervals. Certainlyother software packages exist with capabilities of producingbootstrap confidenceintervals. The SPLIDAadd-on for S-PLUS developed by WilliamMeeker plus Resampling Statshave such capabilities.

We begin, as before,with Type I MDF. One should recall fromChapter 4 that through the use of informationcriteria in conjunction with probabilityplotting, it was determinedthat the underlying parametricdistribution forType 1 MDF is better modeled by the normal,than Weibull or lognonnal. This assumption will not be necessaryfor the fullynonparametric bootstrap intervals. However, it will be essential forconstructing the fullyparametric confidence intervals and the NBSP method described by Meeker and

Escobar (1998) that involves nonparametricsampling for the purposesof parametric

79 inference. Again, this particularproduct typehas a large samplesize and this will help alleviate some of the limitations of the bootstrap based on samplesize . Table 5 .1 provides the 95% asymptoticnormal confidence intervals forType 1 MDF, while Table

5.2 shows the fullynonparametric 95% bootstrap confidence intervals. In the tables that follow, LCL stands forlower confidence limit, whileUCL stands forupper confidence limit. The units forthe point estimates and confidence limitsare pounds per square inches (psi) as the reader will recall are the units formeasuring internal bond. This is followed by Figure 5.1, which displays the nonparametricempirical bootstrap sampling distribution foreach of the fourquantil es.

An initial look at the bootstrap samplingdistributions shown in Figure 5.1 shows that the bootstrap distribution becomes narrowerand more peakedas the percentiles increase from 1 to 50, reflectingthe standarderrors being smaller as the numbers get larger. This is what your intuition would expect. It is advantageous to note that based on the histogramsand given the relatively large sample size; the bootstrap distributions for

Type 1 MDF roughlyappear continuous rather thandiscrete (i.e. no holes are present in the histogram). Recall that this problem was described above and does occur frequently with small samplesizes.

The intervals forthe 1st percentile of Type 1 MDF are rather wide. They are, in fact, wider thanthe asymptoticnormal intervals. This, again, is to be expected given the limited amount of data in the extreme lower tail of the IBdata. Users might consider not using them. This is a healthy warning of the dangers of using the bootstrap without thinking!

When we employ the fullynonparametric bootstra p, which sampleswith

80 . Ta bl e 5 ..1 95¾ 0 Asymp t o f1c norma1 con fidence mt erva 1 s 6or T ype 1 MDF p i, = quantile LCL UCL .01 97.2746 95.4023 99.1470 .10 107.5921 106.2795 108.9046 .25 113.5868 112.5093 114.6644 .50 120.2475 119.2749 121.2201

. Ta bl e 5 ..2 F u 11y nonparame t.nc 95¾o b oo ts trap con fid1 ence mt erva 1 s 6or T ype 1 MDF p i, = quantile IntervalType LCL UCL .01 94.4307 Standard 87.2652 100.4228 Bootstrap-t 88.4673 99.1082 Percentile 87.2000 100.6300 Bias-Corrected 87.2000 99.2800 .10 107.6854 Standard 105.6784 109.5216 Bootstrap-t 106.1158 108.7768 Percentile 105.9300 109.7600 Bias-Corrected 105.9008 109.4300 .25 114.3420 Standard 113.4248 115.3752 Bootstrap-t 113.7840 115.0314 Percentile 113.4000 115.4000 Bias-Corrected 112.8000 115.1500 .50 120.2993 Standard 119.1490 121.2510 Bootstrap-t 119.2056 120.7486 Percentile 119.4000 121.6500 Bias-Corrected 119.3000 121.6000

81 (a)

(c) (d) Figure 5.1. Samplingdistribution of percentiles for Type 1 MDF under the full th th tr nonparametricboots trap samplingmeth od. (a) 1 s1, (b) 10 , ( c) 25 , and( d) so .

82 replacement, to obtain a new data set of n=396 observations, it may or may not select any of the few failuresthat occur in theextreme tails. This certainlyproves to be an extreme limitation when our goal is to estimate the 1st percentile. We would expect only 4 out of about 400 to be below the 1st percentile.

On the other hand, the bootstrap is designedto simulate the sampling process and such variabilitypresent, as that in Figure 5.1 (a) is certainlypossib le. Furthermore, these wide bootstrap intervalsmay be providingmore usefulinformation (warningson uncertainty) to the engineer and/orpractitioner regardingthe variabilitypresent in the destructivesampling process. Note that the asymptotic normal intervals, which are theoretical, may be too narrowto capture all the informationdesired about the 1st percentile. I.e., they may be overly optimistic about the standarderrors being smaller.

For example, the asymptotic intervalfor the 1st percentile was [95.40 , 99.15] while the standard intervalwas [8 7.27 , 100.42]. This is quite an obvious difference!

As the percentiles increase andthe "relative"1B data becomes more plentiful,the bootstrap confidenceintervals are more closely matching the asymptotic intervals. For example,the standard bootstrap interval for the 50th percentile was [ 119.15 , 121.25] while the asymptoticinterval was [119.2 7 , 121.22]. Also, it is usefulto acknowledge that the different methods for constructing the bootstrap confidence intervalsyielded very similarresults. Thissupports that Figure 5.1 yields plots reasonablyclose enoughto normalityfor all of these fourintervals to be in agreement.

The reader need only brieflycompare the intervalsin Table5.2 to see this result.

This typeof agreementcan be expected when the samplesize is sufficientlylarge since the bootstrapdistributions will tend usually to be approximatelynormally distributed.

83 Thus, any of the proposed methods forthe construction of bootstrap confidence intervals would prove to be useful here. As will be seen shortly, suchagreement may not occur when the sample size is much smaller.

Table 5.3 and Figure 5.2 display the fullyparametric bootst rap confidence intervals andthe bootstrapsampling distributions for each percentile for Type 1 MDF, respectively. In this situation, the assumption of normalityof Type 1 MDF is required.

Here, the normalparameters were estimated from the original data and then used to simulate samplesof size n=396.

A glanceat the samplingdistributions in Figure 5.2 reveals immediately its differences to the nonparametric samplingdistributions of Figure 5.1. Notice that the distribution of the 1st percentile followsa normaldistribution quitewell in this larger sample case. Due to this, the intervalswe obtain match very closely the asymptotically normalconfidence inter vals. Thus, rather thanproviding much more informationabout the IBpercentiles, they help to confirm the accuracy of the asymptotic intervals. This canbe a useful doublecheck or potential warning, when needed. As with the nonparametric intervals for Type 1, there appearto be little differences between the methods for constructing bootstrap intervals.

Table 5.4 and Figure 5.3 show the confidenceintervals and samplingdistribution, respectively, forType 1 MDF basedon the NBSP samplingmethod described by Meeker andEscobar (1 998). Recall that the samplingwas done fromthe original data with replacement just as was done in the fullynonparametric method. The difference is that we must also assume the normal distribution as was done in the fullyparame tric case.

Then, foreach bootstrap sample,we estimate the normalparameters and usethem to

84 . . T a bl e 5 ..3 F u 111y parame tnc 95o/co boo t s trap con fid1 ence mt erva 1 s fior ype T 1MDF p i, = quantile IntervalType LCL UCL .01 97.2830 Standard 95.3632 99.1280 Bootstrap-t 95.8696 98.5501 Percentile 95.4033 99.1887 Bias-Corrected 95.3767 99.1410 .10 107.5917 Standard 106.2712 108.8809 Bootstrap-t 106.6612 108.5103 Percentile 106.2543 108.8704 Bias-Corrected 106.2201 108.8598 .25 113.5573 Standard 112.4981 114.6588 Bootstrap-t 112.8016 114.3739 Percentile 112.4541 114.6764 I Bias-Corrected 112.5109 114.7343 .50 120.2372 Standard 119.3135 121.1804 Bootstrap-t 119.6273 120.9105 Percentile 119.2881 121.1449 Bias-Corrected 119.2973 121.1502

85 101 I

(a) (b)

1 5

(c) (d) Figure 5.2. Sampling distributionof percentiles forType 1 MDF under the fully th th th parametricbootstrap samplingmethod. (a) 1 s1, (b) 10 , (c) 25 , and (d) 50 .

86 . Ta bl e 5 ..4 NBSP 95'¼o b oo ts t rap con fidence mt erva 1 s fior ype T 1MDF p i, = quantile IntervalType LCL UCL .01 97.3126 Standard 94.8629 99.6282 Bootstrap-t 95.4981 98.9364 Percentile 94.8999 99.6350 Bias-Corrected 94.6364 99.4291 .10 107.6269 Standard 106.0747 109.0774 Bootstrap-t 106.5334 108.6309 Percentile 106.0586 109.1413 Bias-Corrected 105.9957 109.0460 .25 113.6049 Standard 112.4488 114.7080 Bootstrap-t 112.7936 114.3879 Percentile 112.4553 114.6907 Bias-Corrected 112.3902 114.6478 .50 120.2417 Standard 119.2569 121.2381 Bootstrap-t 119.5950 120.9517 Percentile 119.2832 121.2338 Bias-Corrected 119.3141 121.3019

87 (a) (b)

(c) (d) Figure 5.3. Sampling distribution of percentiles forType 1 MDF under the NBSP 51 1 1 th• method. (a) 1 , (b) 10 \ (c) 25 \ and(d) 50

88 parametrically estimate the percentiles of interest.

Therefore,this method proves to be a combination of the two aforementioned

bootstrap methods andis in factthe method of choice forMeeker andEscobar ( 1998) for

analyzingreliability data. As previously mentioned, they argue that a downside of the

fullyparametric bootstrap requires complete knowledge of the censoring mechanisms involved andmust be takeninto consideration when simulating the bootstrapsamples.

Furthermore, the fullynonparametric method canlead to samplingdistributions

that are discrete, especially in the case of smaller sample sizes. Thus, they use the NBSP

method, which does not require knowledge of the censoring mechanisms, andalso does

not givediscrete samplingdistributions with a reasonable samplesize (i.e. n=7 or more).

The samplingdistributions shown in Figure 5.3 are also normalfor each of the

percentiles. Again, we observethat the intervalsare similar to the asymptoticint ervalsas

well as similaramong themselves.

Table 5.5 provides the 95% asymptoticnormal int ervalsfor Type 5 MDF. Table

5.6 andFigure 5.4 display the fullynonp arametricint ervalsand sampling distributions,

respectively, forType 5 MDF percentiles. The samplingdistributions shown provide an

exampleof a limitation of the fullynonp arametricbootstrap. When the sample size is

relatively small, as is the case of Type5 MDF, the samplingdistributions are more

discrete. Figure 5.4(a) and 5.4(b) show this clearly. Furthermore, the sampling

distribution forthe 1 st percentileis very much skewed to the right. As the percentiles

increase,the distribution becomes more symmetric(more normal) andhas a more

continuous appearance. Here, more differences amongthe various confidence intervals

emerge.

89 . Ta bl e 5.. 5 95¾ 0 A sympt o f1c nonna1 con fid1 ence mt erva 1 s f4or ype T MDF 5 p LCL UCL iP = quantile .01 150.4675 144.2243 156.7108 .10 165.3393 160.9623 169.7159 .25 173.9803 170.3872 177.5734 .so 183.5811 180.3380 186.8242

Table5.6. Fun,, nonparametric 95% bootstrap confidencemtervals forType 5 MDF P IntervalType LCL UCL iP = quantile .01 149.3956 Standard 138.6630 155.0410 Bootstrap-t 136.5856 147.2492 Percentile 146.3000 161.1200 Bias-Corrected 146.3000 150.7120 .10 165.3361 Standard 159.3674 169.4326 Bootstrap-t 161.4890 166.6291 Percentile 161.0000 168.9000 Bias-Corrected 157.4000 168.2900 .25 172.8687 Standard 166.6393 177.9606 Bootstrap-t 167.8679 175.4658 Percentile 168.3000 177.80 00 Bias-Corrected 168.3000 177 .6000 .SO 185.2743 Standard 181. 1404 189.9596 Bootstrap-t 182.8249 190.2666 Percentile 178.8000 189.0000 Bias-Corrected 178.5433 188.8567

90 (a) (b)

(c) (d) Figure 5.4. Sampling distribution of percentiles forType 5 MDF under the full th th tl nonparametricbootstrap sampling meth od. ( a) 1 s1, (b) 10 , ( c) 25 , and ( d) so .

91 For example, forthe 1 st percentile, the standardinterval is extremelywide covering a range from 138.66 to 155.04. The percentile interval, in this case, simply cuts the distribution at the 2.5th percentile and 97 .5th percentile. It therefore,does not take the right skewness of the distribution into account and gives us a confidenceinterval of

[ 146.3 , 161.12]. The bias-corrected intervalprovides significant help in this situation by correcting forthe skewness present. Here, we obtain a much smaller interval of [ 146.3 ,

150.71]. By looking at the histogram,the interval appearsto better capturethe information regarding the 1 st percentile forType 5 MDF. The bootstrap-talso appears to help account forthe skewness present giving an intervalof [136.59 , 147.25]. However, we would not trustthe bootstrap-t given thatit does not even contain the point estimate forthe 1 st percentile of 149.4. The intervaltypes forthe other percentiles in Table 5.6 are much more agreeableto each other. Workers andmanagers aregiven warningto not trust only one set of intervalsin such cases. It provides a reality check. The histogram, also, present healthy warningsigns to the user.

Practitioners areadvised that when these histogramsare discrete or appear

"snaggle-toothed" to up the resamplingsize to, say, B=5000. If it no longer has a

"snaggle-toothed" appearance, then the larger resampling size has helped. If it still, however, appears "snaggle-toothed" then practitioners are advised not to use the fully nonparametric approach for constructing bootstrap confidence intervals, at least not for the extreme lower percentiles. Instead, the three quartiles are likely safer or even just the median. Note well the warningof Polansky ( 1999) when the percentiles are very small such as 1 % or 5% to not use bootstrapestimates.

In order to practice what we preach, since the histogramsfor the lower percentiles

92 in Figure 5.4 do appear"snaggle-toothed", the fully nonparametric bootstrap intervals were reconstructedusing B=5000 bootstrap samples. Whatresulted were confidence intervalsvery similarto those in Table 5.6 andhistograms that continued to have a

"snaggle-toothed" appearance. Thus, even though the bias-corrected andbootstrap-t intervalshave helped our situation a little, it is advised that the practitioner not use the lower percentile estimates here. It is very fortunateto have a graphicalwarning.

As more than likely suspected, the fullyparametric intervals types shown in Table

5.7 are all agreeableto each other. Furthermore,they match very closely to the asymptotic normal intervals. Stated before, the usefulness of this would mainly be to assist in confirmingthe asymptoticintervals. Recall,from Chapter 4, that it was determinedthat Type5 MDF followsa normaldistribution. Figure 5.5 shows the samplingdistributions, which arecontinuous andfollow a normaldistribution nicely.

Placing a lot of faithin these parametricintervals may cause anincorrect inference about the Type5 percentiles, especially if the distribution hasbeen misspecified. It is essential to also point out that if the samplesize had been larger, it is possible (andlikely) that the bootstrap samplingdistribution would approacha normaldistribution making lifein the bootstrap world much easier. The NBSPmethod intervals and sampling distributionsare shown in Table 5.8 andFigure 5.6 respectively.

Comments analogousto the parametricintervals for Type 5 MDF described above applyto the NBSP method described by Meeker andEscobar (1 998). That is, the intervalstypes are in greateragreement and the sampling distributions are normally distributed foreach percentile. The plots arevery helpfuldiagnostics.

Samplingusing the NBSP method described by Meeker and Escobar( 1998) may

93 . . T abl e 5 ..7 F u 11yp aramet nc 95o/co oo b s trap t confid 1 ence mt erva 1 s £or Type 5 MDF p i = quantile IntervalType LCL UCL P .01 150.3468 Standard 143.9692 156.5137 Bootstrap-t 145.6272 154.7383 Percentile 143.8976 156.3403 Bias-Corrected 143.2824 156.0068 .10 165.2401 Standard 160.9071 169.5224 I Bootstrap-t 161.9716 168.2961 Percentile 161.0064 169.5044 Bias-Corrected 161.0522 169.5169 .25 173.9953 Standard 170.2529 177.5766 Bootstrap-t 171.2273 176.4632 Percentile 170.4550 177.8767 Bias-Corrected 170.3784 177.7188 .50 183.5092 Standard 180.3048 186.8574 Bootstrap-t 181.2664 185.8846 Percentile 180.2122 186.7233 Bias-Corrected 180.3048 186.8574

94 (a) (b)

(c) (d) Figure 5.5. Sampling distributionof percentiles forType 5 MDF under the fully 51 th th th parametricboo tstrap samplingmethod. (a) 1 , (b) 10 , (c) 25 , and (d) 50 •

95 Ta bl e 5 ..8 NBSP 95o/co b oo ts tr ap confid 1 ence m. t erva I s fior ype T 5MDF p IntervalT ype LCL UCL iP = quantile .01 150.5676 Standard 143.8357 156.6472 Bootstrap-t 145.2324 154.9846 Percentile 144.2618 157.0768 Bias-Corrected 143.5739 156.530 1 .10 165.3426 Standard 160.4306 169.9989 Bootstrap-t 161.6941 168.508 1 Percentile 160.7296 170.0846 Bias-Corrected 160.6547 169.9636 .25 173.9540 Standard 170.0271 177.8024 Bootstrap-t 171.2549 176.5879 Percentile 170.0607 177.7451 Bias-Corrected 169.9757 177.7111 .so 183.5619 Standard 180.3897 186.7724 Bootstrap-t 18 1.1307 185.9261 Percentile 180.2257 186.6669 Bias-Corrected 180. 1838 186.5517

96 (a) (b)

�) �) Figure 5.6. Samplingdistribution of percentiles forType 5 MDF under the NBSP s th 1 th method. (a) 1 1, (b) 10 , (c) 25 \ and(d) 50 •

97 be a more sensible choice when the sample size is small, providedyou have confidence in the underlyingparametric model. This NBSP method of Meeker and Escobar (1998) does require a parametricassumption to build upon, but samplesthe original data with replacement. For reliability data, which is our concern here, thisallo ws forno assumptions to be made on the censoring mechanism in the data, which is required forthe fullyparametric approach.

S.4 SUMMARY AND CONCLUSIONS

This chapter has given the reader an opportunity to explore brieflythe basic ideas surrounding bootstrap methods, the construction of bootstrap confidenceintervals, and how it can be applied to the estimation of percentiles. In particular, we continued our study of the internalbond of particular product typesof medium density fiberboard.

The different meihods described forobtaining bootstrap samples include the fully parametric,fully nonparametric, and a mix between the parametricand nonparametric methods. Along with the different samplingmethods, fourdiff erent typesof bootstrap confidence intervalswere discussed. These include the standardinterval, bootstrap-t interval,percentile, and bias-correctedpercentile interval. For Type 1 and Type5 MDF, bootstrap confidenceintervals for each of the described sampling methods were

st th th th constructed forthe 1 , 10 , 25 , and 50 percentiles. The asymptotic normalintervals were shown to aid in the comparison.

For a sufficiently largesample size, as is the case forType 1 MD�, the bootstrap sampling distributions appearcontinuous (i.e. does not have any holes) and follow roughly a normaldistribution. In this case, it is relatively a matter of preference as to

98 which of the bootstrap intervaltypes is used. They all provide very similarand accurate results as was easilyobserved. This is even the case using the fullynonparametric approach, althoughsome careshould be takenwhen examiningthe 1 st percentile. Also, this is usefulsince no distributional assumptions are required and the worryof misspecificationof the model is alleviated.

Furthermore, with a largesample size, the bootstrap sampling distribution appears continuous, allowing forreliable results. Note the reader canunderstand it being recommendedthat when the samplesize is large, nonparametricsampling is an appropriatesafer choice andcan be usedmore confidently. A largesample size helps to makeup forinformation that is lost when not assuming a parametric distribution. Again, we repeat that anyof the described intervaltypes would be useful in this environment.

They were all approximately the same, which is reassuring forthe practitioner.

It was shown that when the samplesize is sufficientlylarge, the methods for constructing bootstrap confidence intervals were comparableto the asymptotic intervals as would be expected. As the percentilesincreased from 1 to 50, the confidenceintervals becamenarrower, given the largerquantities of observedfailure data. I.e., the standard errorsgrew smaller. This is especially seen with the fullynonparametric case. The

st th th interval forthe 1 percentile is much wider than the intervalsfor the 10 , 25 , and 50th percentiles. However, this result followsnaturally fromthe samplingmethod and the lack of observedfailure data in the extreme lower tail. Althoughthis occurs, the nonparametricbootstrap canprovide accurate results when the samplesize is largeand is recommended when the parametric assumptionsare suspect.

Conversely, when the samplesize is much smaller, as is the case for Type5 MDF,

99 and when samplingis done using the fullynonparametric method, the bootstrap sampling distributions can be anythingbut continuous and may or may not followa normal distribution. Recall to always check the plots. Furthermore,the nonparametricbootstrap does not yield intervalsthat aresimilar. Naturally, this adds complications andrequires other considerations thanthose recommended for the largesample case.

If no distributional assumptionscan be made, it is recommended that the practitioner makeuse of the bootstrap-tintervals or perhaps as a firstchoice the bias­ correctedpercentile intervals withgreat humility. Doing this canstill produceroughly accurate results forthe medianor quartiles usingthe nonparametricmethod when the samplesize is small. These intervals help to alleviate some of the frustrationthat canbe caused by having a samplingdistribution that does not follow,at least roughly,a normal distribution. Furthermore,they areboth second-order accurate intervals. However, we would recommend not using confidenceintervals forthe lower percentiles andinstead resort to anotherapproach. Thus, we should place little faithin the confidence intervals for the 1st or 10th percentiles of Type 5 MDF shown in Table 5.6 . Recall the warningsof

Polansky(1 999) to not even estimate the lower percentiles when the plots appeardiscrete or "snaggle-toothed." Also, note Polansky(2000) andhis helpful insightsinto using kernel smoothing to better estimate lower percentiles in smaller samples.

Ideally, the best answerto estimating lower percentiles realistically is to have a larger sample. Note that the 1st percentile is not robust in anysample less than 100 because by just changingthe minimum, we can makethe 1st percentile virtuallyany number less than the old minimum up to the second order statistic. We wantto stress the non-robustness of the 1st percentilein sampleslike Type 5 with n=7 4. It is recommend

100 that at least a sampleof size 200 or more be used. Further study mightbe done in the future to investigate this in more detail. Next, three alternatives are suggested to get around this difficultyif cost is prohibitive.

First, as an alternative,we need to study the outliers,which canbe classifiedas outliers due to measurement error or due to statistical variation. One might do bootstrapping in a way thattakes into account the few or manyoutliers in anyparticular data set. We leave that fora future study.

Another approach to estimating lower percentiles with a small sample would be to use the multiple regression equation in Young and Guess (2002) for estimating IB for a much larger sample. Then, use that largersample to get more realistic estimates on the

lower percentiles. This would save money andtime but would need to be continuously validated as anappropriate model by actual destructive sampling. Alternatively, engineering judgmentand experiences could be incorporated into a helpful Bayesian approach to get more realistic estimates on the lower percentiles when the data is small.

Chapter 14 of Meeker and Escobar( 1998) provides a thorough treatment of Bayesian methods forreliability data.

Samplingusing the NBSP method described by Meeker and Escobar ( 1998) may

be a more sensible choice when the sample size is small, provided you have confidencein

the underlying parametric model. Recall the informationcriteria discussed in chapter 4

combined with Q-Q plots help with such needed parametricassessments with less data.

By constructing intervals in this manner, thebootstrap sampling distributions appear

continuous androughly follow a normal distribution. In this case, the confidence interval

construction methods produced similar intervals. Althoughrequiring at least

101 approximate parametricassumptions, this method was usefulin constructing intervals for the extreme lower percentiles.

This NBSP method of Meeker andEscobar ( 1998) does require a parametric assumption to build upon, but samplesthe original data with replacement. For reliability data, which is our concernhere, this allows for no assumptions to be made on the censoring mechanismin the data, which is required forthe fullyparametric approach.

For our data here, we had no censoring.

As a brief aside, let us recall the dangerswhen a user may misspecify a model mentioned previously in the thesis. The distributions of 1B appear continuous and followingnormal distributions approximately,but not perfectly. If it was perfectly normal,one can do exact confidenceintervals. See, forexample, Lawless (1982). Our approach with the nonparametricbootstrap protects the user fromassuming a perfect normal distributionand still applies then, whereas the exact procedure would not be completely exact. Also, the reader will note that the bootstrap-t would approximate very well those exact confidenceintervals in the perfectlynormal world. This is a nice feature and extra validation in practice. When, however, exact procedures are not available we canstill do bootstrapping.

The bootstrap samplingdistributions fromthe importantdiagnostic graphsfor the

NBSP method appear normal. They should always be checkedto prevent the misuse of bootstrapinappropriately. There is never getting somethingfor nothing. The different bootstrap intervalsproduce relatively similarresults andthe choice of interval foruse can be based on preferencebecause of the normalityof the samplingdistributions. It is, then, recommended that when a small samplesize cannotbe avoided and when one has

102 confidencein parametric assumptions, that one should makeuse firstof the NBSP method from Meeker andEscobar (1 998).

The fullyparametric bootst rap is useful forverifying classical results using the familiartextbook formulas. Otherwise,there is no significantadvantage forusing the parametricbootst rap over the commonly known classical formulas,except fordouble­ checking. Recall previous related comments.

Overall, samplesize is the key player in our gameof bootstrapping. A large sample size allows formore promising results when no distributional assumptionsare made. Smaller samplesizes giveway to needed limitations. Chernick( 1999) tells us that

"the main concernin small samples is that with only a few values to select from, the bootstrap samplewill under represent the truevaria bility as observationsare frequently repeated andthe bootstrap samplesthemselves repeat." This does not meanthat the bootstrap should not be used with small samplesizes. Rather, much greater care should be takenwhen analyzingthe results andtheir accuracy. It has been recommended that in the case of constructingconfidence intervals, that more than 1000 bootstrap samples should be generated. This number canbe and should be increased even more when the samplesize is small.

It is standard practice to create bootstrap samplesthe samesize as the original data being sampled from, as we have done in our MDF examples. Thepractitioner, however, may find it useful to create bootstrap samples as large as the data with the most observations. For illustration, since Type 1 MDF had n=396 observations,we resampled from this data to create bootstrap samples of size n=396. However, forType 5 MDF with n=7 4 observations,we would recommenda futurepractitioner resample from this data

103 and create bootstrapsamples also of size n=396. By creating bootstrap samplesin this manner, one is able to control the overall samplingvariation and focusinstead on other sources of variation that are of greaterinterest to the practitioner. This was done forthe fullynonparametric bootstrapand a percentile intervalwas constructed forthe 1st percentile. The results yielded [146.3 , 148.6] which, as expected, had a smaller length thanthat previously shown. Comparewith Table 5.6 above. Even thoughwhat we have done is very standardcurr ently, we recommend this alternative highly for practitioners.

We thank Seaver (2004) forthese helpfulinsights.

It is the hopes of this author that the reader will take away a general knowledge of the bootstrap and find it to be a usefuland helpful tool foranalyzing data (reliability data, in particular). The common and practical use of the computer andease of implementing the bootstrap algorithms makeit a good candidate for conducting statistical inference. Possible futurework with respect to bootstrapping and MDF includes observingdifferences in percentiles over time and shift. Efron(2 003) remarks, "These days statisticiansare being asked to analyzemuch more complicated problems ... . I believe, or maybe just hope, that a powerfulcombination of Bayesianand frequentist methodology will emerge to deal withthis deluge of data andthat computer-intensive methods like the bootstrap will facilitatethe combination."

104 Chapter 6

Summary and Concluding Remarks

It has been the purposeof this thesis to introduceand illustrate usefulmethods for analyzingrelia bility data. The discussions in the previous chapters have been concise for the purpose of creating a setting conducive forthe practitioner and to keep the thesis from being too long. However, if interested, the reader is encouraged to referto the appropriate cited sources, among othersthat these authors cite, to obtain extra details. By presenting the subject in this manner, it is hopefulthat the practitioner will be able to easily understand and implement these methods without having to sift through excessive theoretical discussions. Applications of these methods in the forestproducts industry were used throughout.

The forestproducts industry was anappropriate choice fordemonstration since it has seen tremendous growthin recent years and impacts the economies of many countries. Recall the state of Tennessee has anannual impact of all forestproducts on the order of $22 billion per year, comparedto Maine of around $9 billion per year.

With government regulations being enforced that limit the amountof available raw materials; it has become more importantto focuson environmental issues and producing higherquality products. Statistical reliability and quality control methods, to name a few, have been employed more to monitor the quality of forestproducts. With significantresearch and applied efforts devoted to this area, data is plentiful. This certainly allows formany helpful examples and case studies to illustratespecific statistical methods. In particular,this thesis focused onapp lying reliability methods, 105 informationcriteria, and bootstrapping to better understand the strength to failureof the internal bond (IB) of medium density fiberboard(MDF). Recall that MDF is a high quality engineered timber product. m, measured in pounds per square inch (psi), is one of the key metricsof quality that is obtained during destructive testing. Basically, mis one measurement forthe strength of MDF.

Chapter 2 reviewed currentliterature with a largefocus on MDF, IB research, and its improvement. Recall as anexample that Wang, Chen et al. (1999) investigated "a compression sheardevice foreasy and fast measurement of the bonded shearstrength of wood-based materials to replace the conventional method used to evaluate internalbond strength (IB)." They foundthat measuring strengthof MDF or particleboardby the suggested compression shearstrength andby the conventional approach of internalbond strength were significantlycorrelated. This provides an alternativeapproach to measuring strengths of materials. Mentioning again the above referencehelps to

illustrate the importance of understanding the strength of the 1B. Other helpful resources regardingm canbe found in Chapter 2. Additionally, Chapter 2 delves into the statistical literature pertaining to

reliability, informationcriteria, and bootstrapping. The cited referencesare helpful in

obtaining more thorough discussions of the topics covered in this thesis. Reliability data analysis,along with MDF, is the recurrenttheme of this thesis. As previously mentioned, reliability data refersto survivalor failuretime data andthe analysisof this data is an

importanttopic forindustry andgovernment. Manygreat books are devoted to this topic,

including the classic Meeker andEscobar (1 998).

In Chapter 3, exploratory data analysistechniques were utilized to examinethe m

106 ofMDF andto drawuseful initial conclusions. It, also, helped motivate the need for

Chapter 4 using informationcriteria. Histograms andscatter plots were shown firstin

Chapter 3 to gain preliminaryinsight on the distributionsof the MDF producttypes with respect to m. In particular,histograms for Types 1 and2 revealed a symmetric distribution, which certainly canprovide us with useful informationregarding variability in the data plus insight on the underlyingparametric distribution. Recall additional comments in Chapter3. The scatter plot of each typeintuitively tells us that Type2 is a strongerproduct thanType 1. At-test confirmed this withp<0. 0001. Thus, Type2 would more likely be used in shelving, forexample, thanType 1.

The use andhelpfulness of probability plots to characterizethe underlying distribution of the 1B for different product typeswere also described. The points on the plot will forman approximate straight line if the data set "conforms"to a particular distribution. Thus, these plots, in manyinstances, can correctly identify the parametric distribution. Meeker andEscobar (1 998) illustrate instances, however,where these plots may not correctlyidentify the parametricdistri bution. This misspecification is largely basedon samplesize andsub jective visualization. Quite obviously, then, these plots do have subjectivity to them andone must be awareof this. Probability plots areuseful tools,but the results producedshould not always be takento be concrete anddefinitive.

When the parametricmodel assumption is weakor absent, nonparametricplots known as theKaplan-Meier estimators, survival plots,or reliability plots provide another usefulway to explore reliability data. These plots canprovide useful insight andshow surprisingresults fromthe data. For the different MDF producttypes, the survivalplots clearlyindicated that Type2 is stronger thanType 1 MDF.

107 What is interesting to note here is that Typ·e 2 has a higher density of 48 lbs/ft3

3 thanType 1 with a density of 46 lbs/ft • When, producttypes of the same density but different thickness were compared, little to no differences was evident fromthe plots.

Therefore, density appearsto be a key driver in IB variability. This informationwas determinedbased on the plots rather thana particular statistical test. It is remarkable the amount of informationthat canbe extracted from the data based on graphical procedures!

Often,though, we do wish to knowthe parametric distribution with more statistical assurance. More exact inferencescan followfrom havingthis parametric model, when valid.

Probability plots were discussed in Chapter 3 and abovein an exploratory data analysis context. They were revealed as a helpfulyet subjective tool that can provide the practitioner with anappropriate startingpoint fordetermining the parametricdistribution of a particulardata set. Rather than dispensing with these plots, Chapter 4 presented tools that make probability plots more objective and allow the choice of parametric distribution to be much clearer. This is accomplished through the use of information criteria that assign a numeric score to each plot. The distribution characterized by the probability plot with the lowest score is considered to be the best amongcompeting models. This is a greathelp to practitioners.

The particularinformation criteria discussed in Chapter 4 include Akaike's

Information Criteria (AIC) andBozdogan 's InformationComplexity Criterion (ICOMP).

These criteria have a lack of fitterm that accounts forthe likelihood of the proposed model as well as a termthat accounts forthe number of parameters/complexity in the model. A first order approximate model was fit throughthe points on the probability plot

108 in order to score them using AIC and I COMP. The backgroundis provided forthese two criteria as well as forhow the probabilityplot scores were obtained. However, more referencesare provided to point the reader to more extensive information.

An exampleusing MDF product Types 1, 2, 3, and 5 was shown to illustrate the usefulnessof AIC and I COMP. In particular,probability plots forthe normal,lognormal, andWeibull distributionswere constructed andshown along with their corresponding

AIC andICOMP scores. For example,Type 1 MDF probability plots had ICOMP scores of 1439.5, 1498.1, and 1687.5 for the normal,lognormal, andWeibull distributions,

respectively. It is quite easy to see that the lowest score of 1439.5 correspondsto the

normaldistribution. Therefore, we determinedthat amongthe competing models, the

normaldistribution providedthe best fit for Type 1 MDF. Recall that this was suspected

based on the exploratorygraphics developed in Chapter 3. However, informationcriteria

makethis assumptionmore plausible. Furthermore,Type 2 was determined to followthe

lognormaldistribution while Types3 and 5 followa normaldistribution.

Aside fromknowing the parametricdistribution, percentiles are oftenof key

importancein reliability studies. Reliability engineers are frequently interested in the

time at which 10% (or even 1% ) of a particularproduct will fail. Specifically, percentiles

canhelp in understandingproduct warranties and their costs. The meanor standard

deviation of failuretime data is not usually of concernand would not provideas much

informationas the percentiles in this context. Recall, again, Meeker andEscobar (1998).

One of the difficultiesin estimating the percentiles(in particular,lower

percentiles) is that the data may not be plentifulin thelower tail of thedistribution.

Therefore, if the parametricdistribution is available and is considered strong, then

109 estimating the percentiles parametricallycan provide accurate results.

If the parametricdistribution is not available, asymptotic approximations can be utilized. In particular,the samplepercentiles have been shown to be asymptotically normal. However, unless the samplesize is sufficientlylarge, these asymptotic intervals will not necessarilyyield accurate results. It then becomes importantto search for anothermethod forestimating these desired characteristics. Bootstrappingmethods were presented in Chapter 5 as a usefulmethod forconstructing confidence intervals for percentiles. Histogramsprovide a diagnosticto warnof the misuse of this method in some cases. As always, the practitioner/user must check procedures forappropriateness of use.

Chapter 5 presents several methods for resampling the data along with different methods forconstructing bootstrap confidenceintervals. Bootstrap sampling methods described included nonparametric,parametric, and mixed (i.e. nonparametricsampling forparametric inference or NBSP). The confidence interval methodsincluded the standard,bootstrap-t, percentile, and bias-correctedpercentile intervals. Sufficient background informationwas provided in order for the practitioner to implement the methods presented.

An application of these methods to the MDF data was shown also in Chapter 5.

th th Bootstrapconfidence intervalsfor the 1st, 10 , 25 , and 50th percentiles were constructed forTypes 1 and 5 MDF. These two product types were chosenas they would aid in illustration and providea usefulcontrast. Type 1 is a rather largesample size of n=396 m observationswhile Type 5 is much smaller with n=74 m observations. It was shown that when the samplesize is sufficientlylarge, the methods for

110 constructingbootstrap confidence intervals all producedsimilar results andwere comparable to the asymptoticintervals as would be expected. As the percentiles

increased from 1 to 50, the confidenceintervals became narrower, giventhe larger quantitiesof observed failure data. I.e., the standarderrors grew smaller. This is especially seen with the fullynonparametric case. The interval for the 1st percentile is

th th th much wider than the intervals forthe 10 , 25 , and50 percentiles. However, this result

followsnaturally from the samplingmethod and the lack of observed failuredata in the

extreme lower tail. Althoughthis occurs, the nonparametric bootstrapcan provide accurate results when the samplesize is large and is recommended when the parametric

assumptions are weak.

When the sample size is small, the nonparametricbootstrap does not yield intervals that aresimilar . Inthis case, the bootstrapsampling distributions canbe discrete

and standardmethods are not usually recommended. To partlyremedy this, it is loosely recommended that the bootstrap-t or the bias-correctedpercentile methods be utilized.

Doing this canstill produce roughlyaccurate results forthe medianor quartilesusing the nonparametric method when the sample size is small. These intervalshelp to alleviate

some of the frustration that canbe caused by having a samplingdistribution that does not

follow,at least roughly, a normaldistribution. Furthermore,they are both second-order

accurate intervals. However, we would recommend not using confidence intervals forthe

lower percentiles andinstead resortto anotherapproach such as imputation and/or

Bayesianmethods.

It is thus recommended, but not always, that when the samplesize is small, that

the practitioner makesuse of the NBSP method described thoroughlyby Meeker and

111 Escobar(19 98). The plots cangive warningwhen not to use this approach. As described, this method does require parametricassumptions, but samplesthe data as done in the fullynonparametric case. By constructing intervalsin this manner,the bootstrap sampling distributionsappear continuous and roughlyfollow a normaldistribution. In this case, the confidence intervalconstruction methods produced similarintervals.

Although requiring at least approximate parametricassumptions, this method was useful in constructing intervals forthe extremelower percentiles.

As alreadymentioned several times before, to be able to say that improvements have been made, we must be able to measure reliability expressed in percentiles that allow forstatistical variation. We need to makecomparisons of these reliability measures betweenproducts andwithin products beforeand after processimprovement interventions. Knowing when to trust confidenceintervals and when not to trust them are crucial for managersand users ofMDF to makesuccessful decisions.

Throughall the explanations,illustrations, and summaries, it is hopefulthat the reader will come away with some insight on practically analyzing reliabilitydata and will be able to implement the statistical methods presented into their own research. At this point, it is common to wonder what might come next. With regardsto the 1B ofMDF, futurework on studying other sources of variationpresent is a possibility.

In general, other work is helpfulregarding information criteria andbootstrapping.

In Chapter 4, the model fitto the probability plots was a simple first order approximation that assumed normal errors with constantvariance. It is possible and likely that another model would provide a better fit. Certainly,it is also feasibleto assume that the error variationis not constant andrather, instead, varieswith the data. We leave that work to

112 anothertime.

With respect to bootstrapping,the methods presented in Chapter 5 are not exhaustive. Other methods for constructingintervals certainly exist andthese may be

explored in greaterdetail. Furthermore, more percentiles may be estimated forthe MDF data to comparethem over time, by shift,month, etc. Bootstrapping is receiving much more attention in recent years thanit ever has. It is likely that bootstrap methods will be

further developed, refined,and built upon to better suit the practitioner's needs. See

comments in Chapter 5 about Type5 MDF foralternative approaches for estimating

lower percentiles as possible futurework.

In anycase, more researchis always possible andlikely to appearin futureworks.

Other possibilities thanthose listed above arecertain to be explored. Reliability data

analysishas rich theoretical foundationsas can easilybe seen in excellent works such as

Barlowand Proschan (1975) andMeeker andEscobar (19 98). However, the theoryleads

to manyapplications that areof interest to engineers, researchers, andother practitioners

in industry, government, academia, etc. It is for these applications that this thesis is

written.

While focusingon the forestproducts industryand application to MDF, the

methods discussed andillustrated flow easilyinto a plethora of other worlds. We close,

appropriately, with words fromMeeker andEscobar (1 998) who assertthat "reliability is

being viewed asthe productfe ature that hasthe potential to provide animportant

competitive edge. A currentindustry concernis in developing better processes to move

rapidly fromproduct conceptualization to a cost-effectivehighly reliable product. A

reputation forunreliability candoom a product, if not the manufacturing company."

113 Bibliography

114 Akaike, H. (1973 ). Information Theoryand an Extension of the Maximum Likelihood Principle. Second international symposium on informationtheory, Budapest, Academiai Kiado.

Barlow, R. E. andProschan, F. (1965). Mathematical Theoryof Reliability.Wiley, New York, NY.

Barlow, R. E. andProschan, F. (1974). Statistical Theoryof Reliabilityand Life Testing: ProbabilityMo dels. Holt Rinehart andWinston, New York, NY.

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122 Vita

David J. Edwards is a graduateresearch assistant in statistics forthe Tennessee

Forest Products Center at the University of Tennessee. He received a B.S. in

Mathematics with a minor in Statistics fromVirginia Polytechnic Institute and State

University andis currently pursuing an M.S. in Statistics fromthe University of

Tennessee with plansto graduatein May 2004. David received two Hatcher Scholarships in Mathematics while attending Virginia Tech. He is a member of Golden Key

International Honor Society, Phi Beta Kappa National Honor Society, Pi Mu Epsilon

National Mathematics Honor Fraternity, andAlpha Phi Omega National Service

Fraternity. David was a speaker at the 57'h Annual Forest Products Society conferencein

Seattle, WA in June 2003. He presented furtherwork at the Massachusetts Institute of

Technology'sInternational Conference on InformationQuality in November 2003. He servedas a Statistics Instructorfor the Ronald McNair Post-Baccalaureate Achievement

Programduring summer 2003. Afterobtaining a Master's degree,David will pursue a

Ph.D. in Statistics fromVirginia Tech and intends to remain in an academic setting in

order to teach andconduct research.

123