Introduction to Robust Estimation and Hypothesis Testing

689 Pages·2012·5.09 MB·English

by  Rand Wilcox

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Introduction to Robust Estimation
and Hypothesis Testing
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Introduction to Robust
Estimation and Hypothesis
Testing
3rd Edition
Rand Wilcox
AMSTERDAM•BOSTON•HEIDELBERG•LONDON
NEWYORK•OXFORD•PARIS•SANDIEGO
SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO
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WILCOX preface-9780123869838 2011/10/25 1:23 Page xix #1
Preface
Thisbookfocusesonthepracticalaspectsofmodernandrobuststatisticalmethods.The
increasedaccuracyandpowerofmodernmethods,versusconventionalapproachestothe
analysisofvariance(ANOVA)andregression,isremarkable.Throughacombinationof
theoreticaldevelopments,improvedandmoreflexiblestatisticalmethods,andthepowerof
thecomputer,itisnowpossibletoaddressproblemswithstandardmethodsthatseemed
insurmountableonlyafewyearsago.
Themostcommonapproachwhencomparingtwoormoregroupsistocomparemeans,
assumingthatobservationshavenormaldistributions.Whencomparingindependentgroups,
itisfurtherassumedthatdistributionshaveacommonvariance.Conventionalwisdomisthat
thesestandardANOVAmethodsarerobusttoviolationsofassumptions.Thisviewisbasedin
largepartonstudies,publishedbeforetheyear1960,showingthatifgroupsdonotdiffer
(meaningthattheyhaveidenticaldistributions),thengoodcontrolovertheprobabilityofa
typeIerrorisachieved.However,ifgroupsdiffer,hundredsofmorerecentjournalarticles
havedescribedseriouspracticalproblemswithstandardtechniquesandhowtheseproblems
mightbeaddressed.Oneconcernisthatthesamplemeancanhavearelativelylargestandard
errorunderslightdeparturesfromnormality.Thisinturncanmeanlowpower.Another
problemisthatprobabilitycoverage,basedonconventionalmethodsforconstructing
confidenceintervals,canbesubstantiallydifferentfromthenominallevel,andundesirable
powerpropertiesariseaswell.Inparticular,powercangodownasthedifferencebetweenthe
meansgetslarge.Theresultisthatimportantdifferencesbetweengroupsareoftenmissed,
andthemagnitudeofthedifferenceispoorlycharacterized.Putanotherway,groupsprobably
differwhennullhypothesesarerejectedwithstandardmethods,butinmanysituations,
standardmethodsaretheleastlikelytofindadifference,andtheyofferapoorsummaryof
howgroupsdifferandthemagnitudeofthedifference.Yetanotherfundamentalconcernis
thatthepopulationmeanandvariancearenotrobust,roughlymeanthatunderarbitrarily
smallshiftsfromnormality,theirvaluescanbesubstantiallyalteredandpotentiallymislead.
Thus,evenwitharbitrarilylargesamplesizes,thesamplemeanandvariancemightprovide
anunsatisfactorysummaryofthedata.
Whendealingwithregression,thesituationisevenworse.Thatis,thereareevenmoreways
inwhichanalyses,basedonconventionalassumptions,canbemisleading.Thevery
xix
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Preface
foundationofstandardregressionmethods,namelyestimationviatheleastsquaresprinciple,
leadstopracticalproblems,asdoviolationsofotherstandardassumptions.Forexample,if
theerrorterminthestandardlinearmodelhasanormaldistribution,butisheteroscedastic,
theleastsquaresestimatorcanbehighlyinefficient,andtheconventionalconfidenceinterval
fortheregressionparameterscanbeextremelyinaccurate.
In1960,itwasunclearhowtoformallydevelopsolutionstothemanyproblemsthathave
beenidentified.ItwasthetheoryofrobustnessdevelopedbyP.HuberandF.Hampelthat
pavedtheroadforfindingpracticalsolutions.Today,therearemanyasymptoticallycorrect
waysofsubstantiallyimprovingonstandardANOVAandregressionmethods.Thatis,they
convergetothecorrectanswerasthesamplesizesgetlarge,butsimulationstudieshave
shownthatwhensamplesizesaresmall,notallmethodsshouldbeused.Moreover,formany
methods,itremainsunclearhowlargethesamplesizesmustbebeforereasonablyaccurate
resultsareobtained.Oneofthegoalsofthisbookistoidentifythosemethodsthatperform
wellinsimulationstudies,aswellasthosethatdonot.
Thisbookdoesnotprovideanencyclopedicdescriptionofalltherobustmethodsthatmight
beused.Althoughsomemethodsareexcludedbecausetheyperformpoorlyrelativetoothers,
manymethodshavenotbeenexaminedinsimulationstudies,sotheirpracticalvalueremains
unknown.Indeed,therearesomanymethods,amassiveeffortisneededtoevaluatethem.
Moreover,somemethodsaredifficulttostudywithcurrentcomputertechnology.Thatis,they
requiresomuchexecutiontimethatsimulationsremainimpractical.Ofcourse,thismight
changeinthenearfuture,butwhatisneedednowisadescriptionofmodernrobustmethods
thathavepracticalvalueinappliedwork.
Althoughthegoalistofocusontheappliedaspectsofrobustmethods,itisimportantto
discussthefoundationsofmodernmethods,sothisisdoneinChapters2and3,andtosome
extentinChapter4.Onegeneralpointisthatmodernmethodshaveasolidmathematical
foundation.Anothergoalistoimpartthegeneralflavorandaimsofrobustmethods.Thisis
importantbecausemisconceptionsarerampant.Forexample,someindividualsfirmlybelieve
thatoneofthegoalsofmodernrobustmethodsistofindbetterwaysofestimatingµ,the
populationmean.Fromarobustpointofview,thisgoalisnotremotelyrelevant,anditis
importanttounderstandwhy.Anothermisconceptionisthatrobustmethodsonlyperform
wellwhendistributionsaresymmetric.Infact,boththeoryandsimulationsindicatethat
robustmethodsofferanadvantageoverstandardmethodswhendistributionsareskewed.
Apracticalconcernisapplyingthemethodsdescribedinthisbook.Manyofthe
recommendedmethodshavebeendevelopedinonlythelastfewyearsandarenotavailable
instandardstatisticalpackagesforthecomputer.Todealwiththisproblem,easy-to-useR
functionsaresupplied.(Withthisthirdedition,S-PLUSfunctionsarenolongersupported.)
TheycanbeobtainedasindicatedinSection1.8ofChapter1.WithoneRcommand,allof
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WILCOX preface-9780123869838 2011/10/25 1:23 Page xxi #3
Preface
thefunctionsdescribedinthisbookbecomeapartofyourversionofR.Illustrations,using
thesefunctions,areincluded.
Thebookassumesthatthereaderhashadanintroductorystatisticscourse.Thatis,allthatis
requiredissomeknowledgeaboutthebasicsofANOVA,hypothesistesting,andregression.
Thefoundationsofrobustmethods,describedinChapter2,arewrittenatarelatively
nontechnicallevel,buttheexpositionismuchmoretechnicalthantherestofthebook,andit
mightbetootechnicalforsomereaders.ItisrecommendedthatChapter2bereadoratleast
skimmed,butthosewhoarewillingtoacceptcertainresultscanskiptoChapter3.Oneofthe
mainpointsinChapter2isthattherobustmeasuresoflocationandscalethatareusedarenot
arbitrary,butwerechosentosatisfyspecificcriteria.Moreover,thesecriteriaeliminatefrom
considerationthepopulationmean,variance,andtheusualcorrelationcoefficient.
Fromanappliedpointofview,Chapters4–11,whichincludemethodsforaddressing
commonproblemsinANOVAandregression,formtheheartofthebook.Technicaldetails
arekepttoaminimum.Thegoalistoprovideasimpledescriptionofthebestmethods
available,basedontheoreticalandsimulationstudies,andtoprovideadviceonwhich
methodstouse.Usually,nosinglemethoddominatesallothers,onereasonbeingthatthere
aremultiplecriteriaforjudgingaparticulartechnique.Accordingly,therelativemeritsofthe
variousmethodsarediscussed.Althoughnosinglemethoddominates,standardmethodsare
typicallytheleastsatisfactory,andmanyalternativemethodscanbeeliminated.
xxi
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CHAPTER 1
Introduction
Introductorystatisticscoursesdescribemethodsforcomputingconfidenceintervalsand
testinghypothesesaboutmeansandregressionparametersbasedontheassumptionthat
observationsarerandomlysampledfromnormaldistributions.Whencomparingindependent
groups,standardmethodsalsoassumethatgroupshaveacommonvariance,evenwhenthe
meansareunequal,andasimilarhomogeneityofvarianceassumptionismadewhentesting
hypothesesaboutregressionparameters.Currently,thesemethodsformthebackboneofmost
appliedresearch.Thereis,however,aseriouspracticalproblem:Manyjournalarticleshave
illustratedthatthesestandardmethodscanbehighlyunsatisfactory.Oftentheresultisapoor
understandingofhowgroupsdifferandthemagnitudeofthedifference.Powercanbe
relativelylowcomparedtorecentlydevelopedmethods,leastsquaresregressioncanyielda
highlymisleadingsummaryofhowtwoormorerandomvariablesarerelatedascantheusual
correlationcoefficient,theprobabilitycoverageofstandardmethodsforcomputing
confidenceintervalscandiffersubstantiallyfromthenominalvalue,andtheusualsample
variancecangiveadistortedviewoftheamountofdispersionamongapopulationof
participants.Eventhepopulationmean,ifitcouldbedeterminedexactly,cangiveadistorted
viewofwhatthetypicalparticipantislike.
Althoughtheproblemsjustdescribedarewellknowninthestatisticsliterature,many
textbookswrittenfornonstatisticiansstillclaimthatstandardtechniquesarecompletely
satisfactory.Consequently,itisimportanttoreviewtheproblemsthatcanariseandwhythese
problemsweremissedforsomanyyears.Aswillbecomeevident,severalpiecesof
misinformationhavebecomepartofstatisticalfolkloreresultinginafalsesenseofsecurity
whenusingstandardstatisticaltechniques.
1.1 Problems with Assuming Normality
Tobegin,distributionsarenevernormal.Forsomethisseemsobvious,hardlyworth
mentioning,butanaphorismgivenbyCrame´r(1946)andattributedtothemathematician
Poincare´ remainsrelevant:“Everyonebelievesinthe[normal]lawoferrors,the
experimentersbecausetheythinkitisamathematicaltheorem,themathematiciansbecause
theythinkitisanexperimentalfact.”Granted,thenormaldistributionisthemostimportant
IntroductiontoRobustEstimationandHypothesisTesting.DOI:10.1016/B978-0-12-386983-8.00001-9
Copyright©2012ElsevierInc.Allrightsreserved. 1
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2 IntroductiontoRobustEstimationandHypothesisTesting
distributioninallaspectsofstatistics.Butintermsofapproximatingthedistributionofany
continuousdistribution,itcanfailtothepointthatpracticalproblemsarise,aswillbecome
evidentatnumerouspointsinthisbook.Tobelieveinthenormaldistributionimpliesthat
onlytwonumbersarerequiredtotelluseverythingabouttheprobabilitiesassociatedwitha
randomvariable:thepopulationmeanµandpopulationvarianceσ2.Moreover,assuming
normalityimpliesthatdistributionsmustbesymmetric.
Ofcourse,nonnormalityisnot,byitself,adisaster.Perhapsanormaldistributionprovidesa
goodapproximationofmostdistributionsthatariseinpractice,andthereisthecentrallimit
theorem,whichtellsusthatunderrandomsampling,asthesamplesizegetslarge,thelimiting
distributionofthesamplemeanisnormal.Unfortunately,evenwhenanormaldistribution
providesagoodapproximationtotheactualdistributionbeingstudied(asmeasuredbythe
Kolmogorovdistancefunctiondescribedlater)practicalproblemsarise.Also,empirical
investigationsindicatethatdeparturesfromnormality,thathavepracticalimportance,are
rathercommoninappliedwork(e.g.,Hill&Dixon,1982;Micceri,1989;Wilcox,2009a).
Evenoveracenturyago,KarlPearsonandotherresearcherswereconcernedaboutthe
assumptionthatobservationsfollowanormaldistribution(e.g.,Hand,1998,p.649).In
particular,distributionscanbehighlyskewed,theycanhaveheavytails(tailsthatarethicker
thananormaldistribution),andrandomsamplesoftenhaveoutliers(unusuallylargeorsmall
valuesamongasampleofobservations).Outliersandheavy-taileddistributionsareserious
practicalproblemsbecausetheyinflatethestandarderrorofthesamplemean,sopowercan
berelativelylowwhencomparinggroups.Modernrobustmethodsprovideaneffectiveway
ofdealingwiththisproblem.Fisher(1922),forexample,wasawarethatthesamplemean
couldbeinefficientunderslightdeparturesfromnormality.
Aclassicwayofillustratingtheeffectsofslightdeparturesfromnormalityiswiththe
contaminated ormixednormaldistribution(Tukey,1960).Let X beastandardnormal
randomvariablehavingdistribution(cid:56)(x)= P(X ≤x).Thenforanyconstant K >0,
(cid:56)(x/K)isanormaldistributionwithstandarddeviation K.Let(cid:15) beanyconstant,0≤(cid:15) ≤1.
Thecontaminatednormaldistributionis
H(x)=(1−(cid:15))(cid:56)(x)+(cid:15)(cid:56)(x/K), (1.1)
whichhasmean0andvariance1−(cid:15)+(cid:15)K2.(Stigler,1973,findsthattheuseofthe
contaminatednormaldatesbackatleasttoNewcomb,1896.)Inotherwords,thecontaminated
normalarisesbysamplingfromastandardnormaldistributionwithprobability1−(cid:15);
otherwise,samplingisfromanormaldistributionwithmean0andstandarddeviation K.
Toprovideamoreconcreteexample,considerthepopulationofalladults,andsupposethat
10%ofalladultsareatleast70yearsold.Ofcourse,individualsatleast70yearsoldmight
haveadifferentdistributionfromtherestofthepopulation.Forinstance,individualsunder
theageof70mighthaveastandardnormaldistribution,butindividualsatleast70yearsold
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Chapter1 Introduction 3
mighthaveanormaldistributionwithmean0andstandarddeviation10.Then,theentire
populationofadultshasacontaminatednormaldistributionwith(cid:15) =.1and K =10.In
symbols,theresultingdistributionis
H(x)=0.9(cid:56)(x)+0.1(cid:56)(x/10), (1.2)
whichhasmean0andvariance10.9.Moreover,Eq.(1.2)isnotanormaldistribution,
verificationofwhichisleftasanexercise.
To illustrate problems that arise under slight departures from normality, we first examine
Eq. (1.2) more closely. Figure 1.1 shows the standard normal and the contaminated normal
probability density function corresponding to Eq. (1.2). Notice that the tails of the
contaminated normal are above the tails of the normal, so the contaminated normal is said
to have heavy tails. It might seem that the normal distribution provides a good
approximation of the contaminated normal, but there is an important difference. The
standard normal has variance 1, but the contaminated normal has variance 10.9. The reason
for the seemingly large difference between the variances is that σ2 is very sensitive to the
tails of a distribution. In essence, a small proportion of the population of participants can
have an inordinately large effect on its value. Put another way, even when the variance is
0.4
Normal curve
0.3
0.2
0.1
0.0
−3 −2 −1 0 1 2 3
x
Figure1.1: Normalandcontaminatednormaldistributions.
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4 IntroductiontoRobustEstimationandHypothesisTesting
known, if sampling is from the contaminated normal, the length of the standard confidence
interval for the population mean, µ, will be over three times longer than it would be when
sampling from the standard normal distribution instead. What is important from a practical
point of view is that there are location estimators other than the sample mean that have
standard errors that are substantially less affected by heavy tailed distributions. By “measure
of location,” it is meant that some measure intended to represent the typical participant or
object, the two best-known examples being the mean and the median. (A more formal
definition is given in Chapter 2.) Some of these measures have relatively short confidence
intervals when distributions have a heavy tail, yet the length of the confidence interval
remains reasonably short when sampling from a normal distribution instead. Put another
way, there are methods for testing hypotheses that have good power under normality, but
that continue to have good power when distributions are nonnormal, in contrast to methods
based on means. For example, when sampling from the contaminated normal given by
Eq. (1.2), both Welch’s and Student’s method for comparing the means of two independent
groups have power approximately 0.278 when testing at the 0.05 level with equal sample
sizes of 25 and when the difference between the means is 1. In contrast, several other
methods, described in Chapter 5, have power exceeding 0.7.
Inanattempttosalvagethesamplemean,itmightbearguedthatinsomesensethe
contaminatednormalrepresentsanextremedeparturefromnormality.Theextremequantiles
ofthetwodistributionsdodiffersubstantially,butbasedonvariousmeasuresofthedifference
betweentwodistributions,theyareverysimilarassuggestedbyFigure1.1.Forexample,the
Kolmogorovdistancebetweenanytwodistributions, F andG,isthemaximumvalueof
(cid:49)(x)=|F(x)−G(x)|,
themaximumbeingtakenoverallpossiblevaluesof x.(Ifthemaximumdoesnotexist,the
supremumorleastupperboundisused.)Ifdistributionsareidentical,theKolmogorov
distanceis0,anditsmaximumpossiblevalueis1,asisevident.Nowconsiderthe
Kolmogorovdistancebetweenthecontaminatednormaldistribution, H(x),givenby(1.2),
andthestandardnormaldistribution,(cid:56)(x).Itcanbeseenthat(cid:49)(x)doesnotexceed0.04for
any x.Thatis,basedonaKolmogorovdistancefunction,thetwodistributionsaresimilar.
Severalalternativemethodsareoftenusedtomeasurethedifferencebetweendistributions.
(SomeofthesearediscussedbyHuberandRonchetti,2009.)Thechoiceamongthese
measuresisofinterestwhendealingwiththeoreticalissues,buttheseissuesgobeyondthe
scopeofthisbook.Sufficeittosaythatthedifferencebetweenthenormalandcontaminated
normalisagainsmall.Gleason(1993)discussesthedifferencebetweenthenormaland
contaminatednormalfromadifferentperspectiveandalsoconcludesthatthedifferenceis
small.
Evenifitcouldbeconcludedthatthecontaminatednormalrepresentsalargedeparture
fromnormality,concernsoverthesamplemeanwouldpersist,forreasonsalreadygiven.
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