ALDA Motivation

Applied LongitudinalData Analysis

ModelingChangeandEventOccurrence

Judith D. Singer

JohnB. Willett

OXFORDUNIVERSITY PRESS

2003

OXFORDUNWEKSITY PRESS

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Library ofCongressCataloging-in-PublicationDataSinger,JudithD.

Appliedlongitudinaldataanalysis:modelingchangeand eventoccurrence/byJudith 0. SingerandJohnS.Willett.

p. cm.Includesbibliogntphicalreferencesand index.

ISBN 0-19-5152964I. Longitudinal methods. 2- Socialsciences—Kesearch.

1. Willett, JohnB, It. Title.1-162.S4775520020O1,4’2—dc2l 2002007055

987654S21

Printed in the United Statesof Americaon acid-freepaper

Contents

PART I

1 A Frameworkfor InvestigatingChangeoverTime 3

1.1 WhenMight You StudyChangeoverTime? 4

1.2 DistinguishingBetweenTwo Typesof Questionsabout Change 7

1.3 ThreeImportantFeaturesof a Studyof Change 9

2 Exploring LongitudinalData on Change 16

2.1 Creatinga LongitudinalData Set 17

2.2 DescriptiveAnalysisofIndividualChangeoverTime 23

23 ExploringD~[ferencesin ChangeacrossPeople 33

2.4 ImprovingthePrecisionandReliability of OLS-EstimatedRatesof Change:

Lessonsfor ResearchDesign 41

3 Introducing the Multilevel Model for Change 453.1 WhatIs thePurposeof theMultilevelModelfor Change? 46

3.2 TheLevel-i Submodelfor IndividualChange 49

3.3 TheLevel-2Sulrmoddfor SystematicInterindividuoiDifferencesin Change 57

3.4 Fitting theMultilevel Modelfor Changeto Data 63

3.5 ExaminingEstimatedFixedEffects 68

3.6 ExaminingEstimatedVarianceComponents 72

4 Doing DataAnalysis with the Multilevel Model for Change 75

4,1 Example’ Changesin AdolescentAlcohol Use 76

4.2 The CompositeSpecificationof theMultilevel Modelfor Change 80

4.3 MethodsofEstimation,Revisited 85

4.4 First Steps:Fitting Two UnconditionalMultilevelModelsfor Change 92

xviii Contents

4.5 PracticalData AnalyticStrategiesfor ModelBuilding 104

4.6 ComparingModels UsingDevianceStatistics 116

4.7 UsingWaldStatisticsto Test CompositeHypothesesAboutFixedEffects 122

4,8 Evaluatingthe Tenability of a Model’sAssumptions 127

4.9 Model-Based(EmpiricalBayes)Estimatesof theIndividualGrowth Parameters 132

5 TreatingTIME More Flexibly 138

5.1 V&siably SpacedMeasurementOccasions 139

5.2 1Q~ryingNumbersofMeasurementOccasions 146

5.3 Time-VaryingPredictors 159

5.4 RecenteringtheEffectofTIME 181

6 ModelingDiscontinuousandNonlinearChange 1896.1 DiscontinuousIndividualChange 190

6.2 UsingTransformationsto ModelNonlinearIndividualChange 208

6.3 RepresentingIndividual ChangeUsinga PolynomialFunction ofTIME 213

6.4 Truly Nonlinear Trajectories 223

7 Examiningthe Multilevel Model’s Error CovarianceStructure 243

7.1 The “Standard” Specificationof theMultilevel Modelfor Change 243

7,2 Using theCompositeModelto UnderstandAssumptionsaboutthe

Error CovarianceMatrix 246

7.3 Postulatingan AlternativeError CovarianceStructure 256

S Modeling ChangeUsingCovarianceStructureAnalysis 266

8,1 The General CovarianceStructureModel 266

& 2 TheBasicsofLatent GrowthModeling 280

8,3 Cross-DomainAnalysisof Change 295

8.4 Extensionsof LatentGrowthModeling 299

PART U

9 A Frameworkfor InvestigatingEventOccurrence 305

9.1 ShouldYou Cmducta SurvivalAnalysis?The “Whether” and “When” Test 3069.2 Framinga ResearchQuestionAboutEventOccurrence 309

93 Censoring.’How CompleteAretheData on EventOccurrence? 315

10 DescribingDiscrete-TimeEventOccurrenceData 325

10.1 TheL#fisTable 326

10.2 A Frameworkfor CharacterizingtheDistribution ofDiscrete-Time

EventOccurrenceData 330

10.3 DevelopingIntuition AboutHazardFunctions,SurvivorFunctions,

andMedian Lifetimes 339

Contents nix

10.4 QuantifyingtheEffectsof SamplingVariation 348

10.5 A Simpleand UsefulStrategyfor ConstructingtheLife Table 351

11 Fitting Basic Discrete-TimeHazardModels 35711.1 Thwarda StatisticalModelforDiscrete-TimeHazard 358

11.2 AFormal RepresentationofthePopulationDiscrete-TimeHazardModel 36911.3 Fitting aDiscrete-TimeHazardModelto Data 378

11.4 InterpretingParameterEstimates 38611.5 DisplayingFittedHazard andSurvivorFunctions 391

11.6 ComparingModels UsingDevianceStatisticsandInformation Criteria 397

11.7 StatisticalInferenceUsingAsymptoticStandardErrors 402

12 Extendingthe Discrete-TimeHazardModel 407

12.1 AlternativeSpecificationsfor the “Main Effectof TIME” 40812.2 Using theComplementaryLog-LogLink to Specifya Discrete-Time

HazardModel 419

12.3 Time-VaryingPredictors 426

12.4 TheLinearAdditivity Assumption:UncoveringViolationsand

SimpleSolutions 443

12.5 TheProportionality Assumption:UncoveringViolations and

SimpleSolutions 451

12.6 TheNo UnobservedHeterogeneityAssumption:No SimpleSolution 461

12.7 ResidualAnalysis 463

13 DescribingContinuous-TimeEventOccurrenceData 468

13.1 A Frameworkfor CharacterizingtheDistribution ofContinuous-Time

EventData 469

13.2 GroupedMethodsfor EstimatingContinuous-TimeSurvivor

andHazardFunctions 475

113 TheKaplan-MeierMethodofEstimatingtheContinuous-Time

SurvivorFunction 48313.4 The CumulativeHazardFunction 488

13.5 Kernel-SmoothedEstimatesof theHazardFunction 494

13.6 Developingan Intuition about Continuous-TimeSurvivor~

CumulativeHazard, andKernel-SmoothedHazardFunctions 497

14 Fitting Cox RegressionModels 50314.1 Toward a StatisticalModelfor Continuous-TimeHazard 503

14.2 Fitting theCox RegressionModel to Data 51614.3 InterpretingtheResultsofFitting theCoxRegressionModel to Data 523

14.4 Nonparairtetric StrategiesforDisplayingtheResultsofModelFitting 535

xx Contents

15 Extendingthe CoxRegressionModel 543

15.1 Time-VaryingPredictors 54-4

15.2 NonproportionalHazardsModelsvia Stratification 556

15.3 NonproportionalHazardsModels via Interactionswith Time 562

15.4 RegressionDiagnostics 570

15.5 CompetingRisks 586

15.6 LateEntryinto theRiskSet 595

Notes 607

References 613

Index 627

1

A Frameworkfor InvestigatingChangeoverTime

Changeis inevitable.Changeis constant.—BenjaminDisraeli

Changeis pervasivein everyday life. Infants crawl andwalk, childrenlearn to readandwrite, the elderly becomefrail andforgetful. Beyondthesenaturalchanges,targetedinterventionscanalsocausechange:cho-lesterol levels may declinewith new medication;test scoresmight riseafter coaching.By measuringandchartingchangeslike these—bothnat-uralistic andexperimentallyinduced—weuncoverthe temporalnatureof development.

The investigationof changehasfascinatedempirical researchersforgenerations.Yet it is only since the 1980s,when methodologistsdevel-opeda classof appropriatestatistical models—knownvariouslyas indi-vidualgrowthmodels,randomcoefficientmodels,multilevelmodels,mixedmodels,and hierarchical linear models—thatresearchershave beenable to studychangewell. Until then,the technicalliteratureon the measurementofchange was awash with broken promises,erroneous half-truths, andname-calling.The l960sand 1970swereespeciallyrancorous,with mostmethodologistsoffering little hope, insistingthat researchersshouldnoteven attempt to measurechange becauseit could not be done well(Bereiter,1963; Linn & Slinde,1977). For instance,in their paper,“Howshouldwe measurechange?Or shouldwe?,” CronbachandFurby (1970)tried to endthe debateforever, advisingresearchersinterestedin thestudyof changeto “frame their questionsin otherways.”

Todaywe know that it is possibleto measurechange,andto do it well,if yo-u have longitudinaldata (Rogosa,Brandt, & Zimowski, 1982; Willett,1989). Cross-sectionaldata—soeasyto collect andsowidely available—will notsuffice.In this chapter,we describewhy longitudinaldataare nec-essaryfor studyingchange.We begin,in section1.1,by introducingthree

3

4 Applied LongitudinalDataAnalysis

longitudinalstudiesof change.In section1.2, wedistinguishbetweenthetwo typesof questiontheseexamplesaddress,questionsabout: (1) within-individual change—Howdoes eachperson changeover time?—and (2)interindividual differencesin change—Whatpredicts differences amongpeoplein their changes?This distinction providesan appealingheuris-tic for framing researchquestionsand underpinsthe statisticalmodelswe ultimately present.We conclude,in section1.3, by identifying threerequisitemethodologicalfeaturesof anystudyof change:the availability of(1) multiplewavesof data; (2) asubstantivelymeaningfulmetricfor time;and(3) anoutcomethat changessystematically.

1.1 When Might You StudyChangeoverTune?

Many studies lend themselvesto the measurementof change.Theresearchdesigncanbe experimentalor observational.Datacan be col-lectedprospectivelyor retrospectively.Time canbemeasuredin avarietyof units—months,years,semesters,sessions,andso on. The datacollec-tion schedulecan be fixed (everyonehas the sameperiodicity) or flex-ible (eachpersonhasauniqueschedule).Becausethe phrases“growthmodels”and“growth curve analysis”havebecomesynonymouswith themeasurementof change,many people assumethat outcomesmust“grow” or increaseover time. Yet the statisticalmodelsthat wewill specifycare little aboutthe direction (or eventhe functional form) of change.They lend themselvesequally well to outcomesthat decreaseover time(e.g., weight loss amongdieters) or exhibit complex trajectories(in-cluding plateausand reversals),as we illustrate in the following threeexamples.

1.1.1 Changesin Antisocial Behaviorduring Adolescence

Adolescenceisaperiodof greatexperimentationwhenyoungsterstry outnew identities and explore new behaviors.Although most teenagersremainpsychologicallyhealthy,someexperiencedifficulty andmanifestantisocialbehaviors,includingaggressiveexternalizingbehaviorsanddepres-sive internalizing behaviors.For decades,psychologistshave postulatedavariety of theoriesaboutwhy some adolescentsdevelopproblemsandothersdo not,but lackingappropriatestatisticalmethods,thesesupposi-tionswent untested.Recentadvancesin statisticalmethodshaveallowedempirical exploration of developmentaltrajectoriesandassessmentoftheir predictabilitybaseduponearlychildhoodsignsandsymptoms.

A Frameworkfor InvestigatingChangeover Time 5

Coie,Terry, Lenox,Lochman,andHyman (1995) designedan ingen-ious study to investigate longitudinal patternsby capitalizing on datagatheredroutinely by the Durham, North Carolina, public schools.Aspartof asystemwidescreeningprogram,everythird gradercompletesabatteryof sociometricinstrumentsdesignedto identify classmateswhoare overly aggressive(who start fights,hit children,or say meanthings)or extremelyrejected(whoareliked by few peersanddislikedby many).To investigatethe link betweentheseearly assessmentsandlaterantiso-cial behavioraltrajectories,the researcherstrackeda randomsampleof407 children,stratifiedby their third-gradepeerratings.Whentheywerein sixth, eighth, andtenthgrade,thesechildrencompletedabatteryofinstruments,including the Child AssessmentSchedule(GAS), a semi-structuredinterview that assesseslevelsof antisocialbehavior.Combin-ing datasetsallowedtheresearchersto examinethesechildren’spatternsof changebetweensixth andtenthgradeand thepredictabilityof thesepatternson the basisof the earlier peerratings.

Becauseof well-known genderdifferencesin antisocialbehavior,theresearchersconductedseparatebut parallelanalysesby gendetFor sim-plicity here,we focuson boys. Nonaggressiveboys—regardlessof theirpeer rejection ratings—consistentlydisplayedfew antisocial behaviorsbetweensixth andtenthgrades.For them, the researcherswere unableto rejectthe null hypothesisof no systematicchangeover time. Aggres-sive nonrejectedboyswereindistinguishablefrom this groupwith respectto patternsof externalizingbehavior,but their sixth-gradelevelsof inter-nalizing behaviorwere temporarily elevated(declining linearly to thenonaggressiveboys’ level by tenthgrade).Boyswhowerebothaggressiveand rejectedin third gradefollowed avery different trajectory.Althoughtheywere indistinguishablefrom the nonaggressiveboys in their sixth-grade levels of either outcome,over time theyexperiencedsignificantlinear increasesin both.Theresearchersconcludedthatadolescentboyswho will ultimately manifestincreasinglevels of antisocialbehaviorcanbe identified asearlyas third gradeon the basisof peeraggressionandrejection ratings.

1.1.2 Individual Differencesin ReadingTrajectories

Some children learn to read more rapidly than others. Yet despitedecadesof research,specialistsstill do not fully understandwhy. Educa-tors and pediatriciansoffer two major competing theoriesfor theseinterindividual differences:(1) the lag hypothesis,which assumesthatevery child can becomeaproficientreader—childrendiffer only in therate at which they acquire skills; and (2) the deficit hypothesis,which

6 AppliedLongitudinalDataAnalysis

assumesthat some children will never read well becausethey lack acrucialskill. If the lag hypothesiswere true, all childrenwould eventuallybecomeproficient; we needonly follow them for sufficient time to seetheir masteryIf the deficit hypothesiswere true, somechildren wouldneverbecomeproficientno matter how long theywere followed—theysimply lack the skills to do so.

Francis, Shaywitz, Stuebing,Shaywitz,andFletcher (1996) evaluatedthe evidence for and against these competinghypothesesby follow-ing 363 six-year-olds until age 16. Each year, children completedtheWoodcock-JohnsonPsycho-educationalTest Battery, a well-establishedmeasureof readingability; every other year, they also completedtheWechslerIntelligenceScale for Children (WISC). By comparingthird-grade reading scoresto expectationsbasedupon concomitantWTSCscores,the researchersidentified threedistinctgroupsof children: 301“normal readers”;28 “discrepant readers,”whose readingscoresweremuch different than their WISC scoreswould suggest;and 34 “lowachievers,”whosereadingscores,while not discrepantfrom their WISCscores,were far belownormal.

Drawing from arich theoreticaltraditionthatanticipatescomplextra-jectories of development,the researchersexaminedthe tenability ofseveralalternativenonlineargrowth models.Basedupona combinationof graphicalexploration andstatisticaltesting, theyselectedamodelinwhich readingability increasesnonlinearly over time, eventuallyreach-ing an asymptote—themaximum readinglevel the child could beexpectedto attain (if testing continued indefinitely). Examining thefitted trajectories,the researchersfoundthat the two groupsof disabledreaderswere indistinguishablestatistically,but thatbothdifferedsignifi-cantly from the normalreadersin their eventualplateau.Theyestimatedthatthe averagechild in the normalgroupwould attain areadinglevel30 points higher than that of the avengechild in either the discrepantor low-achievinggroup (a largedifferencegiven the standarddeviationof 12). The researchersconcludedthat their datawere moreconsistentwith thedeficithypothesis—thatsomechildrenwill neverattainmastery—thanwith the lag hypothesis.

1.1.3 Efficacyof Short-TermAnxiety-ProvokingPsychotherapy

Manypsychiatristsfind that short-termanxiety-provokingpsychotherapy(STAPP) can ameliorate psychological distress. A methodologicalstrength of the associatedliterature is its consistentuse of a well-developedinstrument:the SymptomCheckList (SCL-90),developedby


Derogatis(1994).A methodologicalweaknessis its relianceon two-wavedesigns:onewaveof datapretreatmentandasecondwaveposttreatment.Researchersconcludethat the treatmentis effectivewhen the decreasein SCL-90scoresamongSTAPPpatientsislower thanthedecreaseamongindividuals in a comparisongroup.

Svartberg, Seltzer, Stiles, and Thoo (1995) adopted a differentapproachto studyingSTAPP’sefficacy.Insteadof collectingjust twowavesof data, the researchersexamined“the course,rate and correlatesofsymptomimprovementas measuredwith the SCL-90 during and afterSTAPP” (p. 242). A sampleof 15 patientsreceived approximately20weekly STAPP sessions.During the study, eachpatientcompletedtheSCL-90 up to seven times: once or twice at referral (before therapybegan),onceat mid-therapy,onceat termination,andthree timesaftertherapyended (after 6, 12, and24 months). Suspectingthat STAPP’seffectivenesswouldvarywith the patients’abilitiesto control their emo-tional andmotivationalimpulses(known as ego rigidity), two independ-ent psychiatristsreviewed the patients’ intake files and assignedegorigidity ratings.

Plotting eachpatient’sSCL-90 dataover time, the researchersidenti-fied two distinct temporalpatterns,oneduring treatmentandanotherafter treatment.Betweenintakeandtreatmenttermination (an averageof 8.5 months later), most patientsexperiencedrelatively steeplineardeclinesin SCL-90 scores—anaveragedecreaseof 0.060symptomspermonth (from an initial meanof 0.93). During the two yearsafter treat-ment,the rate of linear declinein symptomswas far lower—only 0.005permonth—althoughstill distinguishablefrom 0. In addition to signifi-cantdifferencesamongindividuals in their ratesof decline beforeandafter treatmenttermination, ego rigidity was associatedwith rates ofsymptomdeclineduring therapy(but not after). The researcherscon-cludedthat: (1) STAPPcandecreasesymptomsof distressduring therapy;(2) gains achievedduring STAPP therapycan be maintained;but (3)majorgains aJler STAPPtherapyendsarerare.

1.2 DistinguishingBetweenTwo Typesof QuestionsaboutChange

Fromasubstantivepoint of view, eachof thesestudiesposesauniquesetof researchquestionsaboutits own specific outcomes(antisocialbehav-ior, readinglevels, and SCL-90 scores) and its own specific predictors(peerratings,disability group,andegorigidity ratings).From astatisti-cal point of view, however,eachposesan identicalpair of questions:(1)


How doesthe outcomechangeover time? and (2) Can we predict dif-ferencesin thesechanges?From this perspective,Coie and colleagues(1995) are asking: (1) How does each adolescent’slevel of antisocialbehaviorchangefrom sixth throughtenthgrade?;and(2) Canwe predictdifferencesin thesechangesaccordingto third gradepeerratings?Sim-ilarly, Francis andcolleagues(1996) are asking: (1) How does readingability changebetweenages6 and 16?; and (2) Can we predict differ-encesin thesechangesaccordingto thepresenceor absenceof areadingdisability?

Thesetwo kindsof questionform thecoreof everystudyaboutchange.The first questionis descriptiveandasksusto characterizeeachperson’spatternof changeover time. Is individual changelinear?Nonlinear?Isit consistentover time or doesit fluctuate?The secondquestionis rela-tional andasksusto examinethe associationbetweenpredictorsandthepatternsof change.Do differenttypesof peopleexperiencedifferentpat-ternsof change?Which predictorsareassociatedwithwhich patterns?Insubsequentchapters,we use thesetwo questionsto providethe concep-tual foundationfor our analysisof change,leadingnaturallyto thespec-ificationof apairofstatisticalmodels—oneperquestion.To developyourintuition aboutthe questionsandhow theymapontosubsequentstudiesof change,herewe simply emphasizetheir sequentialandhierarchicalnature.

In the first stageof ananalysisof change,knownas level-I, we askaboutwithin-individual changeover time. Here, we characterizethe individualpatternof changesothat wecandescribeeachperson’sindividualgyowth

trajectory—theway his or heroutcomevaluesriseandfall over time.Doesthis child’s readingskill grow rapidly, so that shebegins to understandcomplextext by fourth or fifth grade?Doesanotherchild’s readingskillstartout lower andgrow moreslowly? The goalof a level-i analysisis todescribethe shapeof eachperson’sindividual growth trajectory.

In the secondstageof an analysisof change,known as level-Z we askabout interindividualdifferencesin change.Here, we assesswhetherdiffer-entpeoplemanifestdifferentpatternsof withinindividualchangeandaskwhat predictsthesedifferences.We askwhetherit is possibleto predict,on the basisof third-gradepeerratings,which boys will remain psycho-logically healthyduring adolescenceandwhichwill becomeincreasinglyantisocial?Canego rigidity ratingspredictwhich patientswill respondmostrapidly to psychotherapy?The goalof a level-2 analysisis to detectheterogeneityin change acrossindividuals and to determinethe rela-

tionship betweenpredictors andthe shapeof each person’sindividualgrowth trajectory.

In subsequentchapters,we mapthesetwo researchquestionsonto a

A Frameworkfor InvestigatingChangeoverTime 9

pair of statistical models: (1) a level-i model, describing within-individual changeover time; and (2) a level-2 model,relatingpredictorsto any interindividual differencesin change.Ultimately, we considerthesetwo modelsto be a linked pair” andrefer to themjointly as themultilevelmodelfor change.But for now, we askonly that you learn to dis-tinguish the two types of questions.Doing so helpsclarify why researchstudiesof changemustpossesscertain methodologicalfeatures,a topicto which we nowturn.

1.3 ThreeImportantFeaturesof aStudyof Change

Not everylongitudinal studyis amenableto the analysisof change.Thestudiesintroducedin section1.1 sharethreemethodologicalfeaturesthatmakethem particularlywell suitedto this task.Theyeachhave:

• Threeor morewavesof data• An outcomewhosevalueschangesystematicallyover time• A sensiblemetric for clocking time

We commenton eachof thesefeaturesof researchdesignbelow.

i.3.i Multiple Wavesof Data

To model change,you needlongitudinal data that describehow eachpersonin the samplechangesover time.We beginwith this apparenttau-tology becausetoo manyempiricalresearchersseemwilling to leapfromcross-sectionaldata that describedifferencesamongindividualsof dif-ferent agesto making generalizationsabout changeover time. Manydevelopmentalpsychologists,for example, analyzecross-sectionaldatasetscomposedof childrenof differing ages,concludingthatoutcomedif-ferencesbetweenagegroups-inmeasuressuchas antisocialbehavior—reflect real change over time. Although change is a compellingexplanationof this situation—it might even be the true explanation—cross-sectionaldatacan never confirm this possibility becauseequallyvalid competingexplanationsabound.Even in a sampledrawn from asingle school, a random sample of older children may differ from arandomsampleofyoungerchildren in importantways: thegroupsbeganschoolin differentyears,theyexperienceddifferent curricula and lifeevents,and if datacollection continuesfor a sufficient period of time,the older sample omits age-mateswho dropped out of school. Anyobserveddifferencesin outcomesbetweengrade-separatedcohortsmaybedueto theseexplanationsandnot to systematicindividual change.In


statistical terms,cross-sectionalstudiesconfound ageand cohort effects(andageandhistory effects)and areprone to selectionbias.

Studiesthat collect two wavesof dataare only marginallybetter. Fordecades,researcherserroneouslybelievedthat two-wavestudiesweresuf-ficient for studyingchangebecausetheynarrowlyconceptualizedchangeas an increment the simple difference betweenscoresassessedon two

measurementoccasions(seeWillett, 1989).Thislimited perspectiveviewschangeas the acquisition(or loss) of the focal increment:a “chunk” ofachievement,attitude,symptoms,skill, or whatever.But thereare tworeasonsan increment’ssize cannotdescribethe processofchange.First, itcannottell usaboutthe shapeof eachperson’sindividual growth trajec-tory, thefocus of our level-I question.Did all the changeoccurimmedi-ately after the first assessment?Was progresssteadyor delayed?Second,it cannotdistinguishtrue changefrom measurementerror. If measure-menterror renderspretestscorestoo low andposttestscorestoo high,you might concludeerroneouslythat scoresincreaseover time whenalongertemporalview wouldsuggestthe opposite.In statisticalterms,two-wavesstudiescannotdescribeindividual trajectoriesof changeandtheyconfoundtrue changewith measurementerror (seeRogosa,Brandt,&Zimowski, 1982).

Onceyou recognizethe needfor multiple wavesof data, the obviousquestionis, How many wavesare enough?Are three sufficient? Four?Shouldyougathermore?Notice thatCoie’sstudyof antisocialbehaviorincludedjustthreewaves,while Svartberg’sSTAPPstudyincludedatleastsix andFrancis’sreadingstudyincludedup to ten, In general,morewavesarealwaysbetter,within costandlogisticalconstraints.Detaileddiscussionof thisdesignissuerequiresclearunderstandingof the statisticalmodelspresentedin this book.Sofor now, we simplynotethatmorewavesallowyou to positmore elaboratestatisticalmodels. If your datasethas onlythree waves, you must fit simpler modelswith stricter assumptions—usuallyassumingthatindividualgrowthis linearovertime (asCoieandcol-leaguesdid in their studyof antisocialbehavior).Additional wavesallowyoutopositmoreflexible modelswith lessrestrictiveassumptions;youcanassumethat individual growth is nonlinear(as in the readingstudy) orlinearin chunks(asin the STAPPstudy). In chapters2—5,we assumethatindividual growthis linear over time. In chapter6, we extendthesebasicideasto situationsin which level-i growth is discontinuousor nonlinear.

1.3.2 A SensibleMetric for Time

Time is the fundamentalpredictorin everystudyof change;it mustbemeasuredreliably and validly in a sensiblemetric. In our examples,


readingscoresare associatedwith particular ages,antisocialbehavior isassociatedwith particular grades,andSCL-90 scoresare associatedwithparticular monthssinceintake.Choiceof atime metric affectsseveralinter-relateddecisionsaboutthe numberandspacingof datacollectionwaves.Eachof these,in turn, involvesconsiderationof costs,substantiveneeds,andstatisticalbenefits.Onceagain, becausediscussionof theseissuesrequiresthestatisticalmodelsthatwe haveyetto develop,we do not delveinto specificshere.Insteadwe discussgeneralprinciples.

Ouroverarchingpoint is thatthereis no singleanswerto the seeminglysimplequestionaboutthemostsensiblemetric for time.Youshouldadoptwhateverscalemakes most sensefor your outcomesandyour researchquestion.Coieandcolleaguesusedgradebecausetheyexpectedantisocialbehaviorto dependmoreonthis “social” measureof time thanon chrono-logical age. In contrast,Francisandcolleaguesusedage becauseeachreadingscore was basedon the child’s ageat testing.Of course,theseresearchersalsohadtheoptionof analyzingtheir datausinggradeas thetime metric; indeed,theypresenttablesin this metric.Yet when it cameto dataanalysis,theyusedthe child’s ageat testingso as to increasetheprecisionwith which theymeasuredeachchild’s growth trajectory.

Many studiespossessseveralplausiblemetrics for time. Suppose,forexample,your interestfocuseson the longevityof automobiles.Most ofuswouldinitially assesstime usingthe vehicle’sagl7—thenumberof weeks(ormonths)sincepurchase(ormanufacture).And for manyautomotiveoutcomes-particularlythose that assessappearancequalities like rustandseatwear—thischoiceseemsappropriate.But for other outcomes,othermetricsmaybe better.Whenmodelingthedepthof tire treads,youmight measuretime in miles, reasoningthat tire weardependsmore onactualuse,not yearson the road.The tiresof a one-year-oldcar thathasbeendriven 50,000miles will likely bemore worn than thoseof a two-year-oldcar thathasbeendriven only 20,000miles. Similarly,whenmod-eling the health of the starter/igniter,you might measuretime in trips,reasoningthat thestarteris usedonly onceeachdrive. The conditionofthestartersin two carsof identicalageandmileagemaydiffer if onecaris driven infrequentlyfor long distancesandthe other is driven severaltimesdaily for short hops.So, too,whenmodelingthe life of theengine,you might measuretime in oil changes,reasoningthat lubricationis mostimportantin determiningenginewear.

Our point is simple:chooseametric for time thatreflects thecadenceyou expectto be mostuseful for your outcome.Psychotherapystudiescan clock time in weeksor numberofsessions.Classroomstudiescanclocktime in gradeor age. Studiesof parentingbehaviorcan clock time usingparentalage or child age. The only constraintis that, like time itself, the

12 Appfled LongitudinalDataAnalysis

temporal variable can changeonly monotonically—in other words, itcannotreversedirection. This means,for example,thatwhen studyingchild outcomes,you coulduseheight,but notweight, as agaugeof time.

Havingchosenametric for time, you havegreatflexibility concerningthe spacingof the wavesof datacollection.Thegoalis to collectsufficientdata to providea reasonableview of eachindividual’sgrowth trajectory.Equally spacedwaveshaveacertainappeal,in that theyoffer balanceandsymmetry.But thereis nothing sacrosanctaboutequal spacing.If youexpectrapid nonlinearchangeduring sometime periods,you shouldcollectmoredataat thosetimes.If you expectlittle changeduringotherperiods, spacethose measurementsfurther apart. So in their STAPPstudy, Svartbergand colleagues(1995) spacedtheir early waves moreclosely together—atapproximately0, 4, 8, and12 months—becausetheyexpectedgreaterchangeduring therapy.Their laterwaveswerefurtherapart—at18 and30 months-becausetheyexpectedfewer changes.

A relatedissue is whethereveryoneshould sharethe samedatacol-lection schedule—inotherwords, whethereveryoneneedsan identicaldistribution of waves.If everyoneis assessedon an identicalschedule—whetherthe wavesareequallyor unequallyspaced—wesaythatthe dataset is time-structured.If datacollection schedulesvary acrossindividuals,wesaythedatasetis time-unstructured.Individual growth modelingis flex-ible enough to handleboth possibilities.For simplicity, we begin withtime-structureddatasets(in chapters2, 3, and4). In chapter5, weshowhow the samemultilevel modelfor changecanbeusedto analyzetime-unstructureddatasets.

Finally, theresultantdatasetneednot be balanced;in otherwords,eachperson neednot have the samenumber of waves. Most longitudinalstudiesexperiencesomeattrition. In Coie andcolleagues’(1995) studyof antisocialbehavior, 219 children hadthreewaves, 118 had two, and70 hadone. In Francisandcolleagues’ (1996) readingstudy, the totalnumberof assessmentsperchildvariedbetweensixandnine.While non-randomattrition can be problematicfor drawing inferences,individualgrowth modeling does not require balanceddata. Each individual’sempiricalgrowth recordcancontainauniquenumberof wavescollectedat uniqueoccasionsof measurement—indeed,as we will seein chapter5, someindividualscanevencontributefewer than threewaves!

1.3.3 A ContinuousOutcomeThat ChangesSystematicallyOverTime

Statisticalmodelscare little about the substantivemeaningof the indi-vidual outcomes.Thesamemodelscanchartchangesin standardizedtest


scores,self-assessments,physiologicalmeasurements,or observerratings.This flexibility allows individual growth modelsto beusedacrossdiversedisciplines,from the social and behavioralsciencesto the physical andnaturalsciences.The contentof measurementis asubstantive,not statis-tical, decision.

Howto measurea givenconstruct,however,is a statisticaldecision,andnot all variablesare equally suitable. Individual growth modelsaredesignedfor continuousoutcomeswhosevalues change systematicallyover time.’ This focus allows us to representindividual growth trajecto-riesusingmeaningfulparametricforms (an ideawe introducein chapter2). Of course,it must makeconceptualandtheoreticalsensefor theoutcometo follow suchatrajectory.Francisandcolleagues(1996) invokedevelopmentaltheory to arguethat readingability will follow a logistictrajectoryas more complexskills arelayereduponbasicbuilding blocksandchildrenheadtowardanupperasymptote.Svartbergandcolleagues(1995) invoke psychiatric theory to argue that patients’ trajectoriesofsymptomatologywill differ when they are in therapyandafter therapyends.

Continuousoutcomessupport all the usual manipulationsof arith-metic: addition, subtraction,multiplication, and division. Differencesbetweenpairsof scores,equidistantlyspacedalong the scale,haveiden-tical meanings.Scoresderivedfrom standardizedinstrumentsdevelopedby testing companies-including the Woodcock Johnson Psycho-educationalTestBattery—usuallydisplay theseproperties.So, too, doarithmetic scoresderived from most public-domain instruments, likeHodges’s Child AssessmentSchedule and Derogatis’s SCL-90. Evenhomegrowninstrumentscan producescoreswith the requisitemeasure-mentpropertiesas long as theyinclude alargeenoughnumberof items,eachscoredusinga largeenoughnumberof responsecategories.

Ofcourse,youroutcomesmustalsopossessdecentpsychometricprop-erties. Using well-known or carefully piloted instrumentscan ensureacceptablestandardsof validity andprecision.But longitudinal researchimposesthreeadditionalrequirementsbecausethe metric, validity, andprecisionof the outcomemustalsobe preservedacrosstime.

Whenwe saythat the metric in which the outcomeis measuredmustbe preservedacrosstime, we meanthat the outcomescoresmust beequatableovertime—agiven valueof theoutcomeon anyoccasionmustrepresentthe same “amount” of the outcomeon every occasion.Out-comeequatabilityis easiestto ensurewhenyou usethe identicalinstru-ment for measurementrepeatedlyover time, as did Coie andcolleagues(1995) in their studyof antisocialbehaviorandSvartvergandcolleagues(1995) in their studyof STAPR Establishingoutcomeequatabilitywhen


the measuresdiffer over time—like the WoodcockJohnsontestbatteryused by Francis and colleagues(1996)—requiresmore effort. If theinstrumenthasbeendevelopedby atestingorganization,you canusuallyfind supportfor equatabilityovertimein the testingmanuals.Francisandcolleagues(1996) note that:

The Rasch-scaledscorereportedfor the reading-clusterscoreis a trans-formationof the numbercorrectfor eachsubtestthat yields ascorewithintervalscalepropertiesandaconstantmetric.The transformationis suchthata scoreof 500 correspondsto the averageperformancelevel of fifthgraders. Its interval scale and constantmetric propertiesmake theRasch-scaledscoreidealfor longitudinal studiesof individualgrowth.(p. 6)

If outcomemeasuresarenotequatableovertime, thelongitudinalequiv-alenceof the scoremeaningscannotbe assumed,renderingthe scoresuselessfor measuringchange.

Note thatmeasurescannotbemadeequatablesimplyby standardizingtheir scoreson eachoccasionto acommonstandarddeviation.Althoughoccasion-by-occasionstandardizationappearspersuasive—itseemsto letyou talk aboutchildrenwho are “I (standarddeviation)unit” abovethemeanatage 10 and“1.2 units” abovethemeanatage11, say—the“units”from which thesescoresare derived (i.e., the underlying age-specificstandarddeviationsusedin the standardizationprocess)are themselvesunlikely to havehadeither the samesize or the samemeaning.

Second,your outcomesmustbe equallyvalid acrossall measurementoccasions.If you suspectthat cross-wavevalidity might becompromised,you shouldreplacethe measurebeforedatacollectionbegins.Sometimes,asin the psychotherapystudy,it is easyto arguethatvalidity ismaintainedover time becausethe respondentshavegoodreasonto answerhonestlyon successiveoccasions.But in otherstudies,suchasCoieandcolleagues’(1996) antisocial behavior study, instrumentvalidity over time may bemoredifficult to assertbecauseyoung children maynot understandallthe questionsabout antisocial behavior included in the measureandolderchildrenmaybe lesslikely to answerhonestly.Take the time to becautiousevenwhenusing instrumentsthatappearvalid on the surface.In hislandmarkpaperon dilemmasin themeasurementof change,Lord(1963) arguedthat,just becauseameasurementwas valid on oneocca-sion,it wouldnot necessarilyremainso on all subsequentoccasionsevenwhenadministeredto the sameindividualsunderthe sameconditions.He arguedthat amultiplication testmay be avalid measureof mathe-maticalskill amongyoungchildren,but becomesameasureof memoryamongteenagers.

Third, you should try to preserveyour outcome’sprecisionover time,


although precisionneednot be identical on everyoccasion.Within thelogistical constraintsimposedby datacollection, the goal is to minimizeerrors introduced by instrumentadministration. An instrument that is“reliable enough”in a cross-sectionalstudy—perhapswith a reliability of.8 or .9—will no doubt be sufficient for a studyof change.So, too, themeasurementerror variance can vary across occasionsbecausethemethodswe introducecaneasilyaccommodateheteroscedasticerrorvari-

ation. Although the reliability of changemeasurementdependsdirectlyon outcomereliability, the precisionwith which you estimateindividualchangedependsmore on the numberand spacingof the wavesof datacollection. In fact, by carefully choosing and placing the occasionsofmeasurement,you can usuallyoffset the deleteriouseffects of measure-ment error in the outcome.

2

Exploring Longitudinal Dataon Change

Changeis the nurseryof music,joy, life, andEternity.—JohnDonne

Wise researchersconductdescriptiveexploratoryanalysesof their databefore fitting statistical models. As when working with cross-sectionaldata, exploratoryanalysesof longitudinal data can reveal generalpat-terns,provideinsightintofunctional form, andidentify individualswhosedatado not conform to the generalpattern.The exploratoryanalysespresentedin this chapterare basedon numericalandgraphicalstrate-giesalreadyfamiliar from cross-sectionalwork. Owing to the natureoflongitudinaldata,however,theyareinevitably morecomplexin this newsetting.For example,beforeyou conductevenasingle analysisof longi-tudinal data,you mustconfrontaseeminglyinnocuousdecisionthathasseriousramifications: how to storeyour longitudinal dataefficiently. Insection2.1, we introduce two differentdataorganizationsfor longitudi-nal data—the“person-level” format and the “person-period”format—and arguein favor of the latter.

We devotethe restof this chapterto describingexploratoryanalysesthat canhelpyou learnhow differentindividualsin your samplechangeover time. These analysesserve two purposes: to identify importantfeaturesof your dataandto prepareyou for subsequentmodel-basedanalyses.In section2.2, we addressthe within-personquestion—Howdoeseachpersonchangeover time?—byexploringandsummarizingemØri-cal growth records, which list eachindividual’s outcomevaluesover time.In section 2.3, we addressthe between-personquestion—Howdoes indi-vidual change differ acrosspeople?—byexploring whether differentpeoplechangein similar or differentways. In section2.4, we show howto ascertaindescriptivelywhetherobserveddifferencesin changeacrosspeople(interindividual differencesin change) are associatedwith individual

1t5

Exploring LongitudinalDataon Change 17

characteristics.Thesebetween-personexplorationscanhelpidentify vari-ablesthatmayultimately prove to be importantpredictorsof change.Weconclude,in section 2.5, by examiningthe reliability andprecision ofexploratoryestimatesof changeandcommentingon their implicationsfor the designof longitudinalstudies.

2.1 CreatingaLongitudinalDataSet

Your first stepis to organizeyour longitudinal datain aformat suitablefor analysis.In cross-sectionalwork, data-setorganizationis so straight-forwardas to not warrantexplicit attention—allyouneedis a “standard”dataset in which eachindividual hashis or her own record. In longitu-dinal work, data-setorganizationis lessstraightforwardbecauseyou canusetwo verydifferentarrangements:

• A person-level data set, in which each person has one recordandmultiple variablescontain the datafrom eachmeasurementoccasion

• A person-period data set, in which each person has multiplerecords—onefor eachmeasurementoccasion

A person-leveldataset has as many recordsas thereare peoplein thesample.As you collect additionalwaves, the file gainsnewvariables,notnewcases.A person-perioddatasethasmanymorerecords—onefor eachperson-periodcombination.As you collect additionalwavesof data,thefile gainsnew records,but no new variables.

All statisticalsoftwarepackagescan easilyconverta longitudinal dataset from oneformat to the other. The website associatedwith our bookpresentsillustrative codefor implementingthe conversionin avariety ofstatistical packages.If you are using SAS, for example, Singer (1998,2001) providessimplecodefor theconversion.In STATA, the “reshape”commandcan be used. The ability to move from one format to theother meansthat you can enter,andclean,your datausingwhicheverformat is most convenient.But as we show below, when it comes todataanalysis—eitherexploratoryor inferential—youneedto haveyourdata in a person-periodformat becausethis most naturally supportsmeaningfulanalysesof changeover time.

We illustrate the difference betweenthe two formats in figure 2.1,which presentsfive waves of datafrom the National Youth Survey (N\’S;Raudenbush& Chan,1992).Eachyear,whenparticipantswere agesII,12, 13, 14, and 15, they filled out a nine-item instrument designedto assesstheir tolerance of deviantbehavior. Using a four-point scale

“Person-Level”dataset

ID TOLJI TOLI2 TOLI3 TOLI4 TOLlS MALE EXPOSURE9 2.23 1.79 1.9 2.12 2.66 0 1.54

45 1.12 1.45 1.45 1.45 1.99 1 1,16268 1.45 1.34 1.99 1.79 1.34 1 0.9314 1.22 1.22 1.55 1.12 1.12 0 0,81442 1.45 1.99 1.45 1.67 1.9 0 1,13514 1.34 1.67 2.23 2.12 2.44 1 0.9569 1.79 1.9 1.9 1.99 1.99 0 1.99624 1.12 1.12 1.22 1.12 1.22 1 0.98723 1.22 1.34 1.12 1 1.12 0 0.81918 1 1 1.22 1.99 1.22 0 1.21949 1.99 1.55 1.12 1.45 1.55 1 0.93978 1.22 1.34 2.12 3.46 3.32 1 1.59

1105 1.34 1.9 1.99 1.9 2.12 1 1.381542 1.22 1.22 1.99 1.79 2.12 0 1.441552 1 1.12 2.23 1.55 1.55 0 1.041653 1.11 1.11 1.34 1.55 2.12 0 1.25

‘Person-Period”datasetID AGE TOL MALE EXPOSURE

9 11 2.23 0 1.549 12 1.79 0 1.549 13 1.9 0 1.549 14 2.12 0 1.549 15 2.66 0 1.54

45 11 1.12 1 1.1645 12 1.45 1 1.1645 13 1.45 1 1.1645 14 1.45 1 1.1645 15 1.99 1 1.16

1653 1.11 0 1.251653 12 1.11 0 1.251653 13 1.34 0 1.251653 14 1.55 0 1.251653 15 2.12 0 1.25

Figure 2.1. Conversionof aperson-leveldatasetinto aperson-perioddatasetfor selectedparticipantsin the tolerancestudy.


(1 = very wrong, 2 = wrong, 3 = a little bit wrong, 4 = notwrong at all),

they indicatedwhetherit waswrong for someonetheir ageto: (a) cheaton tests, (b) purposelydestroypropertyof others,(c) usemarijuana, (d)stealsomethingworth lessthan five dollars, (e) hit or threatensomeonewithout reason, (f) use alcohol, (g) break into a building or vehicle tosteal, (h) sell hard drugs,or (i) steal somethingworth more than fiftydollars. At eachoccasion,the outcome,TOL, is computedas the respon-dent’s averageacrossthe nine responses.Figure 2.1 also includes twopotential predictorsof changein tolerance:MALE, representingrespon-dent gender, and EXPOSURE,assessingthe respondent’sself-reportedexposureto deviantbehaviorat age11.To obtainvaluesof this latterpre-dictor, participantsestimatedthe proportion of their close friendswhowere involved in eachof the samenine activities on anotherfour-pointscale(rangingfrom 0 = none, to 4 = all). Like TOL, eachrespondent’svalue of EXPOSUREis the averageof his or her nine responses.Figure2.1 presentsdatafor a randomsampleof 16 participantsfrom the largerNYS dataset. Although the exploratorymethodsof this chapterapply indata sets of all sizes,we havekept this examplepurposefully small toenhancemanageabilityand clarity. In later chapters,we apply the samemethodsto larger datasets.

2.1.1 The Person-LevelDataSet

Many peopleinitially storelongitudinal dataasaperson-leveldataset (alsoknown as the multivariateformat) , probablybecauseit mostresemblesthefamiliar cross-sectionaldata-setformat. The top panel of figure 2.1 dis-

plays the NYS data using this arrangement.The hallmark feature of aperson-leveldataset is that eachpersonhas only one row (or “record”)of data,regardlessof the numberof wavesof datacollection.A 16-persondata set has 16 records;a 20,000-persondata set has 20,000. Repeatedmeasurementsof eachoutcomeappearasadditionalvariables(hencethealternate“multivariate” label for theformat). In the person-leveldatasetof figure 2.1, the five valuesof toleranceappearin columns2 through 6(TOLI1, TOL12,... TOL15).Suffixesattachedto column headingsiden-

tify the measurementoccasion (here, respondent’sage) andadditionalvariables—here,MALEand EXPOSURE—appearin additional columns.

The primary advantageof aperson-leveldatasetis the easewith whichyou can examinevisually eachperson’sempiricalgrowth record, his or her

temporally sequencedoutcomevalues. Eachperson’sempirical growthrecordappearscompactlyin asinglerow makingit is easyto assessquicklythe way he or she is changingover time. In examining the top panel offigure 2.1, for example, notice that changediffers considerablyacross


Table 2.1: Estimatedbivariatecorrelationsamongtolerancescoresassessedon fivemeasurementoccasions(n = 16)

TOLII TOLI2 TOLI3 TOLI4 TOLJ5

TOLI1 1.00TOL.12 0,66 1.00TOLI3 0.06 0.25 1.00TOL14 0.14 0.21 0.59 1.00TOLJ5 0.26 0.39 0.57 0,83 1.00

adolescents.Although mostbecomemore tolerantof deviantbehaviorover time (e.g., subjects514 and 1653), many remain relatively stable(e.g., subjects569 and624), noneof the 16 becomesmuchlesstolerant(althoughsubject949 declinesfor awhile beforeincreasing).

Despitethe easewith which you can examineeachperson’sempiricalgrowth record visually, the person-leveldataset hasfour disadvantagesthat render it apoor choicefor mostlongitudinal analyses:(1) it leadsnaturally to noninformativesummaries;(2) it omits an explicit “time”variable; (3) it is inefficient, or useless,when thenumberandspacingof waves varies acrossindividuals;and (4) it cannot easily handle thepresenceof time-varying predictors.Below, we explain these difficul-ties; in section2.1.2,we demonstratehow eachis addressedby a con-versionto aperson-perioddataset.

First, let us begin by examining the five separatetolerancevariablesin the person-leveldatasetof figure 2.1 andaskinghowyoumightanalyzetheselongitudinaldata.For mostresearchers,the instinctiveresponseisto examinewave-to-wave relationshipsamong TOL1J through TOL15

using bivariate correlation analyses (as shown in table 2.1) or com-panion bivariate plots. Unfortunately, summarizingthe bivariate rela-tionshipsbetweenwavestells us little aboutchangeover time, for eitherindividualsor groups.What, for example,doesthe weak but generallypositive correlation between successiveassessmentsof TOLERANCEtell us?For anypair of measures,say TOLl I and TOLI2, we know thatadolescentswho were more tolerant of deviantbehavior at one wavetendto be moretolerantat the next.Thisindicatesthatthe rank orderofadolescentsremainsrelativelystableacrossoccasions.But it doesnot tellushoweachpersonchangesover time; it doesnot eventell usabout thedirectionof change.If everyone’sscoredeclinedby onepointbetweenage11 andage 12, but the rank ordering was preserved,the correlationbetweenwaveswould be positive (at +1)! Temptingthoughit is to infera direct link betweenthe wave-to-wavecorrelationsand change,it is a

Exploring Longitudinal Dataon Change 21

futile exercise.Evenwith asmall dataset—herejust five wavesof datafor16 people—wave-to-wavecorrelations and plots tell us nothing aboutchangeover time.

Second,the person-leveldatasethasno explicit numericvariableiden-tifying the occasionsof measurement.Informationabout “time” appearsin the variable names,not in the data, and is thereforeunavailableforstatistical analysis.Within the actual person-leveldata set of figure 2.1,for example, information on when these TOLERANCEmeasureswereassessed—thenumericvalues11, 12, 13, 14, and 15—appearsnowhere.Without including thesevaluesin the dataset,we cannotaddresswithin-person questionsabout the relationship between the outcome and“time.”

Third, the person-levelformat is inefficient if either the number,or spacing,of wavesvaries acrossindividuals. The person-levelformat isbest suited to researchdesignswith fixed occasionsof measurement—eachpersonhasthe samenumberof wavescollected on the sameexactschedule.The person-leveldataset of figure 2.1 is compactbecausetheNYS usedsuch adesign—eachadolescentwasassessedon the samefiveannual measurementoccasions(at ages11, 12, 13, 14, and 15). Manylongitudinal data sets do not share this structure. For example, if wereconceptualized“time” as the adolescent’sspecificage (say, in months)at eachmeasurementoccasion,we would need to expandthe person-level dataset in someway. Wewould needeither five additional columnsto recordtherespondent’spreciseageon eachmeasurementoccasion(e.g.,variableswith nameslike AGEII, AGE12,AGED, AGEI4, andAGE15)orevenmore additional columns to record the respondent’stoleranceof

deviant behavioron eachof the many unique measurementoccasions(e.g., variableswith nameslike TOLJJ.1, TOLIJ.2,. . . TOLI5.ll). Thislatterapproachis particularlyimpractical.Not only wouldwe add55 vari-ables to the data set, we would havemissing values in the cells corre-

sponding to each month not used by a particular individual. In theextreme,if eachpersonin the dataset hashis or her own uniquedatacollection schedule—aswould be the casewere AGErecordedin days—the person-levelformat becomescompletely unworkable.Hundredsofcolumnswould beneededandmostof the dataentrieswouldbe missing!

Finally, person-leveldata sets become unwieldy when the valuesofpredictors can vary over time. The two predictors in this data set aretime-invariant—thevaluesof MALE and EXPOSUREremain the sameoneveryoccasion.Thisallowsusto useasinglevariableto recordthe valuesof each. If the data set contained time-varying predictors—predictorswhosevaluesvaryover time—wewould needan additionalsetof columnsfor each—oneper measurementoccasion.If, for example,exposureto


deviantbehaviorweremeasuredeachyear,wewould needfouradditional

columns.While the datacouldcertainlyberecordedin this way, this leadsto the samedisadvantagesfor time-varying predictors as we havejustdescribedfor time-varyingoutcomes.

Taken together, thesedisadvantagesrender the person-levelformat,so familiar in cross-sectionalresearch,ill suited to longitudinal work.Althoughwewill return to the multivariateformat in chapter8, whenweintroducea covariancestructureanalysisapproachto modelingchange(known as latent growth modeling), for now we suggestthat longitudinaldataanalysisis facilitated—andmademore meaningful—if you use the

“person-period”format for your data.

2.1,2 The Person-PeriodDataSet

In a person-perioddata set, also known as univariate format, each indi-vidual hasmultiple records,onefor eachperiod in which he or shewasobserved.The bottompanelof figure 2.1 presentsillustrative entriesfortheNYS data.Both panelspresentidentical information; theydiffer onlyin structure.The person-perioddata set arrayseachperson’sempirical

growth record vertically, not horizontally. Person-perioddatasetsthere-fore havefewer columnsthan person-leveldatasets (here, five insteadof eight), but many more rows (here, 80 insteadof 16). Even for thissmall example,the person-perioddataset has so many rows that figure

2.1 displaysonly a small subset.All person-perioddatasetscontainfour typesof variables:(1) asubject

identifier; (2) a time indicator; (3) outcomevariable(s);and (4) predic-

tor variable(s).TheID number,which identifies the participantthateachrecorddescribes,typically appearsin the first column. Time-invariantbydefinition, IDs areidenticalacrosseachperson’smultiple records.Includ-ing an ID numberismore than goodrecordkeeping;it is anintegralpartof the analysis.Without an ID, you cannotsort the datasetinto person-specific subsets(a first step in examining individual changetrajectories

in section2.2).The secondcolumn in the person-perioddata set typically displaysa

timeindicator—usuallylabeledAGE, WAVE,or TIME—which identifies thespecificoccasionof measurementthat the recorddescribes.For the NYSdata, the secondcolumn of the person-perioddata set identifies therespondent’sAGE (in years)on eachmeasurementoccasion.A dedicatedtime variableis afundamentalfeatureof everyperson-perioddataset;itis whatrenderstheformat amenableto recording longitudinal datafromawide rangeof researchdesigns.Youcaneasilyconstructaperson-perioddataset even if eachparticipanthasa unique data collection schedule

ExploringLongitudinal Dataon Change 23

(aswould be the caseif we clocked time usingeachadolescent’spreciseageon the dateofinterview). The newAGEvariablewould simply recordeachadolescent’sage on that particular date (e.g., 11.24, 12.32, 13.73,14.11, 15.40 for one case;11.10, 12.32, 13.59, 14.21, 15.69 for the next,etc.). A dedicatedTIME variable also allows person-perioddatasets to

accommodateresearchdesignsin which the number of measurementoccasionsdiffers acrosspeople.Eachpersonsimply hasas many recordsashe or shehaswavesof datain the design.Someonewith threewaveswill havethreerecords;someonewith 20 will have20.

Eachoutcomein a person-perioddataset—here,just TOL—is repre-

sentedby asinglevariable(hencethe alternate“univariate” label for theformat) whosevalues representthat person’sscore on each occasion.In figure 2.1, every adolescenthas five records,one per occasion,eachcontaininghis or her toleranceof deviantbehaviorat the ageindicated.

Every predictor—whether time-varying or time-invariant—is alsorepresentedby a single variable. A person-perioddata set can includeasmany predictorsof either type as you would like. The person-perioddataset in figure 2.1 includes two time-invariantpredictors,MALE andEXPOSURE.The former is time-invariant;the latter is time-invariantonlybecauseof the wayit wasconstructed(asexposureto deviantbehavioratonepoint in time, age11).Time-invariantpredictorshaveidenticalvaluesacrosseach person’s multiple records; time-varying predictors havepotentially differing values.We defer discussionof time-varying predic-tors to section5.3.For now, we simply note how easyit is to include themin aperson-perioddataset.

We hope that this discussionconvincesyou of the utility of storinglongitudinal data in a person-periodformat. Although person-period

data sets are typically longer than their person-levelcousins, the easewith which they can accommodateany data collection schedule,anynumber of outcomes,and any combination of time-invariantand time-varying predictorsoutweigh the cost of increasedsize.

2.2 DescriptiveAnalysis of Individual Changeover Time

Having createdaperson-perioddataset,you arenow poisedto conductexploratory analysesthat describehow individuals in the dataset changeover time. Descriptive analysescan revealthe natureand idiosyncrasiesof each person’stemporal pattern of growth, addressingthe question:How doeseachpersonchangeover time? In section 2.2.1,we presentasimple graphical strategy; in section 2.2.2, we summarizethe observedtrendsby superimposingrudimentaryfitted trajectories.

24 AppliedLongitudinal DataAnalysis

2.2.1 Empirical GrowthPlots

The simplestway of visualizing how apersonchangesover time is toexaminean empiricalgrowthplot, a temporallysequencedgraphof his orher empirical growth record. You can easily obtain empirical growthplots from anymajor statisticalpackage:sort the person-perioddatasetby subject identifier (II?) andseparatelyplot eachperson’soutcomevs.time (e.g., TOLvs.AGE). Becauseit is difficult to discernsimilaritiesanddifferencesamongindividuals if eachpagecontainsonly asingle plot,we recommendthat you cluster sets of plots in smaller numbersofpanels.

Figure2.2presentsempiricalgrowthplots for the 16 adolescentsin theNYS study.To facilitate comparisonandinterpretation,we useidenticalaxesacrosspanels.We emphasizethis seeminglyminor point becausemany statistical packageshave the annoying habit of automaticallyexpanding(or coutracting)scalesto fill out apageor plot area,Whenthis happens,individualswho changeonly modestlyacquire seeminglysteeptrajectoriesbecausethe verticalaxisexpandsto covertheir limitedoutcomerange;individualswho changedramaticallyacquireseeminglyshallow trajectoriesbecausethe vertical axis shrinks to accommodatetheir wide outcomerange. If your axes vary inadvertently,you maydraw erroneousconclusionsaboutany similarities and differencesinindividual change.

Empirical growthplots canrevealagreatdeal abouthoweachpersonchangesover time. You can evaluatechange in both absoluteterms(againstthe outcome’soverall scale) andin relative terms(in compari-son to other samplemembers).Who is increasing?Who is decreasing?Who is increasingthe most?The least?Doesanyoneincreaseandthendecrease(or vice versa)?Inspectionof figure 2.2 suggeststhat toleranceof deviantbehaviorgenerallyincreaseswith age (only subjects314, 624,723,and949 do not fit this trend).But we alsoseethatmostadolescentsremainin the lower portionof theoutcomescale—hereshownin its fullextensionfrom I to 4—suggestingthat tolerancefor deviantbehaviorneverreachesalarming proportions(except,perhaps,for subject978).

Shouldyou examineeverypossibleempiricalgrowth plot if your dataset islarge,includingperhapsthousandsof cases?We do notsuggestthatyou sacrificea reamof paperin the nameof dataanalysis.Instead,youcan randomly selectasubsampleof individuals (perhapsstratified intogroupsdefinedby the valuesof importantpredictors)to conducttheseexploratoryanalyses.All statisticalpackagescan generatethe randomnumbersnecessaryfor suchsubsampleselection;in fact, this is howweselectedthese16 individualsfrom the NYS sample.


TOE IDE Tot TOE4 809 4~ 1045 4 ¶0264 4 80314

0L~ 0L~~Ii 12 13 84 15 11 12 13 14 15 II 12 13 14 85 II 12 13 14 15

AGE AGE AGE AGE

TOE TOE TOE IDE

10442 4 10514 4 10569 4 10624

31 3121 2~ * 2 . 4 2~

:Lli :L~:E.. L~11 12 13 14 15 11 12 13 34 15 II 12 13 84 15 II 12 83 14 15

AGE AGE AGE AGE

TOE TOE TOE TOE80723 10918 4j 10949 1 i0978

__ :L±±:L±0L._II 12 83 14 15 11 12 13 14 15 11 82 13 14 85 II 12 83 14 15

AGE AGE AGE AGE

TOE TOE TOE TOE

801105 101542 4~ 801552 801653

~ 4.421 2~

O~7~1:2 13 ~7T~7s ~r-iT-1,AGE AGE AGE AGE

Figure 2.2. Exploring how individuals changeover time. Empirical growth plots for 16participantsin the tolerancestudy.

2.2.2 Using aTrajectoryto SummarizeEachPerson’sEmpiricalGrowth Record

It is easyto imagine summarizingthe plot of eachperson’sempiricalgrowth recordusingsome type of smoothtrajectory.Although we often


begin by drawing freehandtrajectories,we strongly recommendthatyou also apply two standardizedapproaches.With the nonparametricapproach,you let the “data speakfor themselves”by smoothingacrosstemporal idiosyncrasieswithout imposing a specific functional form.With the parametric approach,you select a commonfunctional formfor the trajectories—astraight line, aquadraticor someother curve—andthenfit aseparateregressionmodel to eachperson’sdata,yieldingafitted trajectory.

The fundamentaladvantageof thenonparametricapproachis that itrequiresno assumptions.Theparametricapproachrequiresassumptionsbut, in return,providesnumericsummariesof the trajectories(e.g.,esti-matedinterceptsandslopes)suitablefor furtherexploration.We find ithelpful to begin nonparametrically—asthesesummariesoften informthe parametricanalysis.

SmoothingtheEmpirical GrowthTrajectaryNonpara.metrically

Nonparametrictrajectoriessummarizeeachperson’spatternof changeover time graphicallywithout committingto aspecific functional form.All major statisticalpackagesprovide several options for assumption-free smoothing, including the use of splines,besssmoothers,kernelsmoothers,and moving averages.Choice of a particular smoothingalgorithm is primarily a matter of convenience;all are adequatefor theexploratory purposeswe intend here.

Figure 2.3 plots the NYS empirical growth recordsandsuperimposesa smoothnonparainetrictrajectory (obtainedusing the “curve” optionin Harvard Graphics).When examiningsmoothedtrajectorieslike these,focuson their elevation,shape,andtilt. Wheredo the scoreshover—atthe low, medium,or high endof the scale?Doeseveryonechangeovertime or do somepeopleremainthe same?What is the overall pattern of

change?Is it linear or curvilinear; smooth or steplike? Do the trajecto-ries havean inflection point or plateau?Is the rate of changesteeporshallow?Is this rate of changesimilar or different acrosspeople?Thetrajectoriesin figure 2.3 reinforce our preliminary conclusionsaboutthe nature of individual changein the tolerance of deviant behavior.Most adolescentsexperienceagentleincreasebetweenages11 and15,exceptfor subject978,who registersa dramaticleapafterage 13.

After examining the nonparametrictrajectoriesindividually, stareatthe entiresettogetheras agroup. Group-levelanalysiscan helpinformdecisionsthat you will soon need to make about a functional formfor the trajectory. In our example,severaladolescentsappearto havelinear trajectories(subjects514, 569, 624, and 723) while othershave

8094

TOE 80454

TOE 80268

AGE AGE AGE

TOE4 80442

11 12 13 14 15

AGE

4TOE

2

AGE

4TOE ID 514 lOt

H 4~69

11 12 13 84 15 11 12 13 14 15

AGE AGETOE

4TOE

10723 7 80918

~ 0L~II 12 13 84 15 11 12 U 14 IS

AGE AGE

ID 1542 ID 8552TOE4

TOE

- -

11 12 13 14 15 11 12 13 14 15

AGE AGE

TOE801105

~121ms

AGE

TOE

4 80314

2~

11 12 83 14 IS

AGE

TOE4 80624

AGETOE

4 80978xAGE

TOE4 101653

AGE

Figure 2.3. Smooth nonparametricsummariesof how individuals changeover time.Smooth nonparametric trajectoriessuperimposedon empirical growth plots forparticipantsin the tolerancestudy.

3-

2~

1-

TOE

3.


curvilinearonesthateitheraccelerate(9, 45, 978, and1653) or riseandfall aroundacentralpeakor trough (268, 314, 918, 949, 1552).

Smoothing the Empirical Growth Trajectory UsingOILS Regression

We can also summarizeeach person’s growth trajectory by fitting aseparateparametric model to each person’s data. Although manymethodsof modelfitting arepossible,we find thatordinaryleastsquares(OLS) regressionisusuallyadequatefor exploratorypurposes.Of course,fitting person-specificregressionmodels,one individual at a time, ishardly the most efficient use of longitudinal data; that’s why we needthe multilevel modelfor changethatwewill soonintroduce.But becausethe “fitting of little OLS regressionmodels” approachis intuitive andeasyto implementin aperson-perioddataset,we find that it connectsempirical researcherswith their datain adirect andintimateway.

To fit anexploratoryOLS regressionmodelto eachperson’sdata,youmustfirst selectaspecificfunctionalformfor thatmodel.Not only is thisdecision crucial during exploratory analysis, it becomeseven moreimportantduring formal model fitting. Ideall); substantivetheory andpast researchwill guide your choice. But when you observeonly arestrictedportionof the life span—aswe do here—orwhenyouhaveonlythree or four wavesof data,modelselectioncanbe difficult.

Two factorsfurthercomplicatethe choiceof a functional form. First,exploratoryanalysesoftensuggestthatdifferentpeoplerequiredifferentfunctions—changemight appearlinear for some,curvilinearfor others.We observethispattern,to someextent,in figure2.3.Yet thesimplificationthat comesfrom adoptingacommonfunctional form acrosseveryoneinthedataset is socompellingthatits advantagestotally outweigh its disad-vantages.Adopting a commonfunctional form acrosseveryonein thesampleallowsyouto distinguishpeopleeasilyusingthesamesetofnumeri-cal summariesderivedfrom their fitted trajectories.This processis espe-cially simple if you adoptalinear changemodel, aswe do here;you canthencompareindividualsusingjusttheestimatedinterceptsandslopesoftheir fitted trajectories.Second,measurementerrormakesit difficult todiscernwhethercompellingpatternsin theempiricalgrowthrecordreallyreflecttrue changeor aresimplydueto randomfluctuation.Remember,eachobservedscoreis justafallible operationalizationof an underlyingtrue

.score—dependingupon the sign of the error, the observedscorecanbeinappropriatelyhighor low. The empiricalgrowthrecordsdo not presenta person’strue patternof changeover time; they presentthe fallibleobservedreflectionof thatchange.Someof whatwe seein the empiricalgrowthrecordsandplotsis nothingmorethanmeasurementerror.


Thesecomplicationsarguefor parsimonywhenselectingafunctional

form for exploratoryanalysis,driving you to adoptthe simplesttrajectorythat cando thejob. Often the bestchoice is simply astraight line. In thisexample,we adoptedalinear individual changetrendbecauseit providesa decent description of the trajectories for these 16 adolescents.Inmaking this decision,of course,we assumeimplicitly that any deviations

from linearity in figure 2.3 result from eitherthe presenceof outliers ormeasurementerror. Useof an individual linear changemodel simplifiesour discussionenormouslyand has pedagogic advantagesas well. We

devote chapter 6 to a discussionof models for discontinuous andnonlinearchange.

Having selectedan appropriateparametricform for summarizingthe

empirical growth records,you obtain fitted trajectoriesusingathree-stepprocess:

1. Estimateawithin-personregressionmodelfor eachpersonin thedataset.With a linearchangemodel,simply regressthe outcome(here TOL) on somerepresentationof time (here,AGE) in theperson-perioddataset.Be sureto conducta separateanalysisforeachperson(i.e., conductthe regressionanalyses“by ID”).

2. Collect summarystatisticsfrom all the within-person regressionmodelsinto a separatedataset.For a linear-changemodel, eachperson’sestimatedinterceptand slopesummarizetheir growthtrajectory; the R2andresidualvariancestatisticssummarizetheirgoodnessof fit.

3. Superimposeeachperson’sfitted regressionline on a plot of hisor her empirical growth record. For each person,plot selectedpredictedvaluesandjoin them togethersmoothly.

We now apply this three-stepprocessto the NYS data. -,

We begin by fitting a separatelinear changemodel to eachperson’sempiricalgrowth record.Although wecanregressTOL on AGEdirectly, weinsteadregressTOLon (AGE— 11) years,providingacenteredversionofAGE.

Centeringthe temporalpredictor is optional, butdoing soimproves theinterpretabilityof the intercept.HadwenotcenteredAGE,the fitted inter-ceptwould estimatethe adolescent’stoleranceof deviantbehaviorat age0—anagebeyondthe rangeof thesedataandhardlyone atwhich a child

canreportan attitude.Subtracting11 yearsfrom eachvalueof AGEmovestheorigin of the plot sothateachinterceptnow estimatesthe adolescent’stoleranceof deviantbehaviorat themorereasonableageof 11 years.

CenteringAGE has no effect on the interpretationof eachperson’sslope:it still estimateshis or herannualrateof change.Adolescentswithpositiveslopesgrowmore tolerant of deviantbehavioras theyage; thosewith the largestslopesbecomemore tolerant the most rapidly. Adoles-

30 Applied Longitudinal DataAnalysis

Table 2.2: Resultsof fitting separatewithin-personexploratoryOLS regressionmodelsfor TOLERANCEasafunction of linear time

ID

Initial s

Estimate

tatus

se

Rateof c

Estimate

hange

se

.

Residualvariance R2

MALE EXPOSURE

0009 1.90 0.25 0.12 0,10 0.11 0.31 0 1.540045 1.14 0.13 0.17 0.05 0.03 0.77 II 1.160268 1.54 0.26 0.02 0.11 0.11 0.02 1 0.900314 1.31 0.15 —0.03 0.06 0.04 0.07 0 0.810442 1.58 0.21 0.06 0.09 0.07 0.14 0 1.130514 1.43 0.14 0.27 0.06 0_os 0.88 1 0,900569 1.82 0.03 0.05 0.01 0.00 0.88 0 1.990624 1.12 0.04 0.02 0.02 0.00 0.33 1 0.980723 1.27 0.08 —0.05 0.04 0.01 0.45 0 0.810918 1.00 0.30 0.14 0.13 0.15 0.31 0 1.210949 1.73 0.24 —0.10 0.10 0.10 0.25 1 0.930978 1.03 0.32 0.63 0.13 0.17 0.89 1 1,591105 1.54 0.15 0.16 0.06 0.04 0.68 1 1.381542 1.19 0.18 0.24 0.07 0.05 0.78 0 1.441552 1.18 0.37 0.15 0.15 0.23 0.25 0 1.041653 0.95 0.14 0.25 0.06 0.03 0.86 0 1.25

centswith negative slopes grow less tolerant of deviant behavior over

time; thosewith the mostnegativeslopesbecomeless tolerantthe mostrapidly. Becausethe fitted slopesestimatethe annualrate of changeinthe outcome,theyare theparameterof centralinterestin an exploratoryanalysisof change.

Table 2.2 presentsthe results of fitting 16 linear-changeOLS regres-sion modelsto the NYSdata.ThetabledisplaysOLS-estimatedinterceptsand slopesfor eachpersonalongwith associatedstandarderrors, resid-ual variance,and R2 statistics.Figure 2.4presentsastem-and-leafdisplayof each summarystatistic. Notice that both the fitted interceptsandslopes vary considerably, reflecting the heterogeneity in trajectories

observedin figure 2.3. Although most adolescentshavelittle tolerancefor deviantbehavioratage 11, some—likesubjects9 and569—aremoretolerant.Notice, too, that many adolescentsregisterlittle changeovertime. Comparingtheestimatedslopesto their associatedstandarderrors,we find that the slopesfor nine people (subjects9, 268, 314, 442, 624,723, 918, 949, and1552) are indistinguishablefrom 0. Threehavemod-erate increases(514, 1542, and 1653) and one extremecase (978)increasesthreetimesfasterthanhis closestpeer.

Figure2.5 superimposeseachadolescent’sfitted OLS trajectoryon hisor her empirical growth plot. All major statisticalpackagescangenerate


0.6~0.50.40.30.2 4 5 70.1 2 4 5 6 7022560 3 5

-0.1 0

Residualvariance R2 statistic

Figure 2.4. Observedvariationin :~~fitted OLS trajectories.Stemand 1 Ia 0 11leafdisplaysfor fitted initial .0hi 5 7status,fitted rateof change, .0 Ia 00133344residualvariance,andR~statisticresultingfrom fitting separateOLS regressionmodelsto thetolerancedata.

such plots. For example,becausethe estimatedinterceptandslope forsubject514 are 1.43 and0.27, the fitted valuesat ages11 and15 are: 1.43(computedas1.43+ 0.27(11—11))and2.51 (computedas1.43+ 0.27(15— 11)). To preventextrapolation beyond the temporal limits of the data,

we plot this trajectory only betweenages11 and 15.Comparingthe exploratory OLS-fitted trajectorieswith the observed

datapointsallowsusto evaluatehowwell the chosenlinearchangemodelfits eachperson’sgrowth record. For someadolescents(such as569 and624), the linear changemodel fits well—their observedandfitted valuesnearly coincide. A linear changetrajectory may also be reasonableformany othersamplemembers(including subjects45, 314, 442, 514, 723,949, 1105, and 1542) if we are correct in regarding the observeddevia-

tions from thefitted trajectoryasrandom error. For five adolescents(sub-

jects 9, 268, 918, 978, and 1552), observedand fitted values are moredisparate.Inspectionof their empiricalgrowth recordssuggeststhattheirchangemaywarrantacurvilinearmodel.

Table2.2 presentstwo simplewaysof quantifying the quality of fit foreachperson:an individual R2 statistic andan individual estimatedresid-ual variance.Even in this small sample,notice the strikingvariability in

Fitted rateof change

3

Fitted initial status

1.9 01.8 21.7 31.61.5 4 4 81.4 31.3 11.2 71,1 2 4 89103

0.9 5

0.80.7:0.60.50.40.30.20.1

0

6889788

5

11355427


11 12 13 14 15

AGE

90514 ID 569

AGE AGETot

II) 918 ID 949

3.~

AGE

ID 1542

Tot

10268 10394

21

0.111 12 13 14 15

AGE

Tot

ID 624

2~

1~---

0•.—11 12 13 14 15

AGETOE

4 90978

AGE

?~ 2’

12 13 140

15 11 12 13 14 150-

11 12 13 140,

15 11 12 13 94 15

AGE AGE AGE AGE

Figure 2.5. OLSsummariesof how individualschangeovertime. Fitted OLS trajectories

superimposedon empiricalgrowthplotsfor participantsin the tolerancestudy.

the individual R2 statistics.They rangefrom alow of 2% for subject268(whose trajectory is essentiallyflat andwhose dataarewidely scattered)to highsof 88% for subjects514and569 (whoseempiricalgrowth recordsshowremarkablelinearity in change)and89% for subject978 (who has

Tot Tot91345 4

Tot

9139

2Lc~

11 ________________

oL~.II 12 13 14 15

AGE AGE

Tot47 10442

AGETOE

ID 723

0~11 12

4 TOt

0~.

13 14 15 11 12 13 14 15

AGE AGElrlt

101105 4

3~

ID 1552


the most rapid rate of growth). The individual estimatedresidualvari-

ancesmirror this variability (asyou might expect, given that they arean

elementin the computationof the R2 statistic). Skewedby definition (asapparentin figure 2.4), they rangefrom a low near0 for subjects569 and624 (whosedataarepredictednearlyperfectly) to highsof 0.17and0.23for subjects978 and 1552 (who eachhavean extremeobservation).Weconclude that the quality of exploratory model fit variessubstantiallyfrom personto person;the linear changetrajectoryworks well for somesamplemembersandpoorly for others.

By now you may be questioningthe wisdom of using OLS regressionmethodsto conductevenexploratory analysesof thesedata.OLS regres-

sion methodsassumeindependenceand homoscedasticityof residuals.Yet theseassumptionsare unlikely to hold in longitudinal datawhere

residualstend to be autocorrelatedandheteroscedasticover timewithinperson. Despite this concern, OLS estimatescan be very useful for

exploratorypurposes.Although they are lessefficient when the assump-tion of residual independenceis violated (i.e., their samplingvarianceis too high), they still provide unbiasedestimatesof the interceptandslope of the individual change(Willett, 1989). In other words, these

exploratory estimates of the key features of the individual changetrajectory—eachperson’s intercept and slope—will be on target, if alittle noisy.

2.3 Exploring Differences in ChangeacrossPeople

Having summarizedhow each individual changesover time, we nowexaminesimilaritiesanddifferencesin thesechangesacrosspeople.Doeseveryonechangein the sameway? Or do the trajectoriesof changediffersubstantiallyacrosspeople?Questionslike thesefocuson the assessmentof intthndividual d~ffrrencesin change.

2.3.1 Examining the Entire SetofSmoothTrajectories

The simplestway of exploring interindividual differencesin changeis toplot, on asinglegraph,the entire setof smoothedindividual trajectories.

Theleft panelof figure 2.6 presentssuchadisplayfor theNYS datausingthe nonparametricsmoother; the right panel presentsa similar display

using OLS regressionmethods.In both, we omit the observeddata todecreaseclutter.


Figure 2.6. Examining the collection of smooth nonparametricand OLS tn~Jectoriesacrossparticipantsin the tolerancestudy. PanelA presentsthe collection of smoothnonparametrictrajectories;PanelB presentsthe conectionof fitted OLS trajectories.Both panelsalsopresentan average change tra~ectary for theentire group.

Eachpanelin figure2.6 alsoincludesanew summary:an average changetrajectory for the entire group. Depictedin bold, this summaryhelpsuscompareindividual changewith group change.Computing an averagechangetrajectory is a simple two-step process.First, sort the person-

period dataset by time (here, AGE), and separatelyestimate the meanoutcome (here, TOLERANCE) for each occasion of measurement.Second,plot thesetime-specific meansand apply the samesmoothingalgorithm, nonparametricor parametric,used to obtain the individual

trajectories.Both panelsin figure 2.6 suggestthat, on average,the changein tol-

eranceof deviantbehaviorbetweenages11 and15 is positivebut modest,risingby oneto two-tenthsof apoint peryear (on this 1 to 4 scale).Thissuggeststhat asadolescentsmature,theygraduallytoleratemore deviant

behavior.Note that eventhe nonparametricallysmoothedaveragetra-jectory seemsapproximatelylinear. (The slight curvatureor discontinu-ity betweenages12 and 13 disappearsif we set asidethe extremecase,

subject978.) Both panelsalso suggestsubstantialinterindividual hetero-geneity in change.For someadolescents,toleranceincreasesmoderatelywith age;for others,it remainsstable;for some,it declines.This hetero-geneitycreatesa “fanning out” of trajectoriesas increasingageengen-dersgreaterdiversity in tolerance.Notice that the OLS regressionpanelis somewhateasierto interpret becauseof its greaterstructure.

13 13

AGE AGE


Although the averagechange trajectory is a valuable summary,weinject anoteof caution: the shapeof the averagechangetrajectorymaynot mimic the shapeof the individual trajectoriesfrom which it derives.We seethis disconcertingbehaviorin figure 2.6, wherethe nonparamet-rically smoothedtrajectoriesmanifestvariouscurvilinearshapesbut theaveragetrajectoryis nearlylineaL Thismeansthatyoushouldneverinferthe shapeof the individual changetrajectoriesfrom the shapeof theiraverage.As we explainin section6.4,theonly kind of trajectoryforwhichthe “averageof the curves”is identicalto the “curve of the averages”isonewhosemathematicalrepresentationis linear in theparameters(Keats,1983).All polynomials—includinglinear,quadratic,andcubictrajectories—are linear in the parameters;their averagetrajectorywill alwaysbe apolynomial of the sameorderas the individual trajectories.The averageof a set of straight lines will be a straight line; the averageof a set ofquadraticswill be aquadratic.But many other commoncurvesdo notsharethis property. The averageof a set of logistic curves,for example,is usually a smoothed-out step function. This meansthat you must ex-erciseextremecautionwhenexamininganaveragegrowth trajectory.Wedisplay the averagesimply for comparison,not to learn anything about

underlying shapesof the individual trajectories.

2.3.2 Using the Resultsof Model Fitting to FrameQuestionsaboutChange

Adopting a parametric model for individual change allows us to re-

expressgeneric questionsaboutinterindividual differencesin “change” asspecific questions about the behavior of parametersin the individualmodels.If we haveselectedour parametricmodelwisely, little informa-tion is lost andgreatsimplification is achieved.If you adopta linear indi-vidual change model, for instance, you are implicitly agreeing tosummarizeeachperson’sgrowth usingjust two parameterestimates:(1)the fitted intercept;and(2) the fitted slope.For the NYS data,variationin fitted interceptsacrossadolescentssummarizesobservedinterindivid-ual differencesin toleranceat age 11. If theseinterceptsdescribefittedvaluesat the first waveof datacollection, as they do here,we saythat theyestimatesomeone’s“initial status.”Variation in the fitted slopesdescribesobservedinterindividual differencesin the ratesat which tolerancefordeviantbehaviorchangesover time.

Greaterspecificity andsimplification accruesif we reframe generalquestionsabout interindividual heterogeneityin changein terms of keyparametersof the individual changetrajectory.Rather than asking “Doindividuals differ in their changes,andif so, how?” we cannow ask “Do


individuals differ in their intercepts?in their slopes?”To learnabouttheobservedaverage patternof change,we examinethe sampleaveragesofthe fitted interceptsandslopes; thesetell us about the averageinitialstatusand the averageannualrateof changein the sampleasawhole.To learnaboutthe observedindividual d~fferences in change,we examinethe sample variances and standard deviations of the interceptsandslopes;thesetell us aboutthe observedvariability in initial statusandratesofchangein the sample.And to learn about the observedrelationshipbetweeninitial statusandthe rateof change,we canexaminethe samplecovariance or correlation betweeninterceptsandslopes.

Formal answersto thesequestionsrequire the multilevel model forchangeof chapter3. Butwe can presagethiswork by conductingsimpledescriptiveanalysesof the estimatedinterceptsandslopes.In addition toplotting their distribution (as in figure 2.4), we can examinestandarddescriptivestatistics(meansand standarddeviations)andbivariatesum-maries (correlation coefficients) obtained using the data set thatdescribesthe separatefitted regressionresultsin table 2.2.

We find it helpful to examinethreespecificquantities,the:

• Samplemeansof the estimatedinterceptsand slopes.The level-i OLS-estimatedinterceptsandslopesare unbiasedestimatesof initialstatusandrate of changefor eachperson.Their samplemeansarethereforeunbiasedestimatesof thekeyfeaturesof theaverageobservedchangetrajectory.

• Samplevariances (or standard deviations) of the estimated interceptsand slopes. These measuresquantify the amount of observedinterindividualheterogeneityin change.

• Samplecorrelation between the estimated intercepts and slopes. Thiscorrelationsummarizesthe associationbetweenfitted initial statusandfitted rateof changeandanswersthe question:Are observedinitial statusandrateof changerelated?

Resultsof theseanalysesfor the NYS dataappearin table2.3.Across thissample,we find an averageestimatedinterceptof 1.36 and

an averageestimatedslope of 0.13. We therefore conclude that theaverageadolescentin this samplehasanobservedtolerancelevel of 1.36atage11 andthat thisincreasesby anestimated0.13pointsperyear. Themagnitudeof the samplestandarddeviations (in comparisonto theirmeans)suggeststhatadolescentsarescatteredwidely aroundboth theseaverages.This tells us that the adolescentsdiffer considerablyin theirfitted initial statusandfitted ratesof change.Finally, thecorrelationcoef-ficient of —0.45 indicatesa negativerelationshipbetweenfitted initialstatusandfitted rate of change,suggestingthatadolescentswith greater


Table 2.3: Descriptivestatisticsfor theindividual growthparametersobtainedby fitting separatewithin-personOLSregressionmodelsfor TOLERANCEas afunction of lineartime (n= 16)

tnitial status(intercept)

Rateof change(slope)

Mean 1.36 0.13Standarddeviation 0.30 0.17Bivariatecorrelation —0.45

initial tolerance tend to become more tolerant less rapidly over time(althoughwe mustbe cautiousin our interpretationbecauseof negativebias introducedby the presenceof measurementerror).

2.3.3 Exploring the RelationshipbetweenChangeandTime-InvariantPredictors

Evaluatingthe impactof predictorshelpsyou uncoversystematicpatternsin the individual changetrajectoriescorrespondingto interindividualvariation in personalcharacteristics.For the NYS data,we considertwotime-invariant predictors: MALE and EXPOSURE.Asking whether theobservedtolerancetrajectoriesdiffer by genderallows us to explorewhetherboys (orgirls) areinitially moretolerantof deviantbehaviorandwhether they tend to have different annual rates of change.Askingwhether the observedtolerancetrajectories differ by early exposureto

deviantbehavior(at ageii) allows usto explorewhetherachild’s fittedinitial level of toleranceis associatedwith earlyexposureandwhetherthefitted rateof changein toleranceis relatedaswell. All of thesequestionsfocus on systematicinterindividual differencesin change.

GraphicallyExaminingGroups ofSmoothedIndividualGrowth Trajectories

Plotsof smoothedindividual growth trajectories,displayedseparatelyforgroups distinguished by important predictor values, are valuableexploratorytools. If a predictor is categorical,display constructionisstraightforward.If apredictor is continuous,you can temporarilycate-gorize its values.For example, we split EXPOSUREat its median (1.145)for the purposesof display.For numericanalysis,of course,we continueto use its continuousrepresentation.

Figure 2.7 presentssmoothedOLS individual growth trajectoriesseparatelyby gender(upperpair of panels)andexposure(lower pair of

4-

3-

TOL

Males

11 12 13 14 15

TOL4—

Low exposure

—

4TOL

11 12 13 14 15

TOL

011 12 13 14 15

AGE AGE

Figure 2.7. Identifying potential predictors of change by examining OLS fittedtrajectoriesseparatelyby levels of selectedpredictors.Fitted OLS trajectoriesfor thetolerancedatadisplayedseparatelyby gender(upperpanel)andexposure(lower panel).

AGE

2

1

AGE

4

3

High exposure

3

2

1 1

011 12 13 14 15

38


panels).The bold trajectory in eachpaneldepictsthe averagetrajectoryfor the subgroup.Whenyou examineplotslike these,lookfor systematicpatterns:Do the observedtrajectoriesdiffer acrossgroups?Do observeddifferencesappearmore in the interceptsor in the slopes?Are somegroups’ observedtrajectoriesmoreheterogeneousthanothers’?Settingasidesubject978, who hadextremelyrapidgrowth,we find little differ-ence in the distribution of fitted trajectoriesby gender.Each group’saverageobservedtrajectoryis similar in intercept,slope,andscatter.Wealso find little difference in fitted initial status by exposure, but we do

discernadifferencein the fitted rateof change.Evendiscountingsubject978, those with greater initial exposure to deviant behavior seemtobecometolerantmore rapidly asthey age.

TheRelationshipbetweenOL&EstimatedTrajectoriesand SubstantivePredictors

Just as we describedthe distribution of fitted interceptsand slopesinsection 2.3, we can alsouse them asobjectsof further exploratory analy-

sis. To investigatewhether fitted trajectoriesvary systematicallywith pre-dictors,wecantreat the estimatedinterceptsandslopes asoutcomes and

explorethe relationshipbetweenthemandpredictors.For the NYS data,theseanalysesexplorewhether the initial toleranceof deviantbehavioror the annualrate of changein toleranceis observedto differ by: (1)genderor (2) early exposureto deviantbehavior.

Because these analysesare exploratory—soon to be replaced inchapter3 by the fitting of amultilevel model for change—werestrictour-selvesto the simplestof approaches:the useof bivariateplotsandsamplecorrelations.Figure 2.8 plots the fitted interceptsandslopesversusthetwo predictors: MALE and EXPOSURE.Accompanyingeach plot is asamplecorrelation coefficient. All signspoint to little or no genderdif-ferential in either fitted initial statusor rate of change.But with respectto EXPOSURE,it doesappearthat adolescentswith greaterearly expo-sureto deviantbehaviorbecomemoretolerantat a fasterratethan peerswho were lessexposed.

Despite their utility for descriptiveandexploratory analyses,OLS esti-matedinterceptsand slopesare hardly the final word in the analysisofchange.Estimatesare not true values—theyare imperfect measuresofeachperson’strue initial statusandtrue rateof change.They havebiasesthatoperatein known directions;for example,their samplevariancesareinflated by the presenceof measurementerror in the outcome.Thismeansthat the variance in the true rate of changewill necessarilybesmaller than the varianceof the fitted slope becausepart of the latter’s


Fitted initial status Fittedinitial status2.5~ 2.5-~

1.5~1

051 ~ O5~________ rO.19

MALE EXPOSURE

o s Fittedrateof change Fittedrateof change

~1 . o~8

j

$ O.4~

01 . 0

r=0.23 • r=0.44

0 1 0 1 2MALE EXPOSURE

Figure 2.8. Examining the relationshipbetweenOLS parameterestimates(for initialstatusandratesof change)andpotentialpredictors.FittedOLS interceptsandslopesforthe tolerancedataplottedvs. two predictors:MALEandEXPOSURE.

variability is error variation. So, too, the samplecorrelationbetweenthefitted interceptandslope is negativelybiased(it underestimatesthe pop-ulation correlation)becausethemeasurementerror in fitted initial statusis embedded,with oppositesign, in the fitted rateof change.

Thesebiasessuggestthatyoushouldusethe descriptiveanalysesof thischapterfor exploratorypurposesonly. Theycan helpyou get your feetwet andin touch with your data.Although it is technicallypossible toimprove theseestimates—forexample,we can deflate the samplevari-

ancesof OLS estimatesandwe cancorrectthecorrelation coefficientformeasurementerror (Willett, 1989)—wedo not recommendexpendingthis extraeffort. The needfor ad hoc correctionshasbeen effectively


replacedby the widespreadavailability of computer software for fitting

the multilevel model for changedirectly.

2.4 Improving the Precision andReliability ofOLS-EstimatedRatesof Change:LessonsforResearchDesign

Before introducing the multilevel model for change, let us examine

anotherfeatureof the within-personexploratoryOLS trajectoriesintro-ducedin this chapter:the precisionandreliability of the estimatedratesof change.We do so not becausewe will be using theseestimatesforfurtheranalysis,but becauseit allowsus to commenton—in aparticu-larly simplearena—somefundamentalprinciplesof longitudinaldesign.As you would hope, thesesamebasicprinciplesalsoapplydirectly to themore complexmodelswe will soonintroduce.

Statisticiansassessthe precisionof aparameterestimatein termsof its

samplingvariation,a measureof the variability thatwould be found acrossinfinite resamplings from the same population. The most commonmeasureof samplingvariability is an estimate’sstandarderror, the squarerootof its estimatedsamplingvariance.Precisionandstandarderrorhavean inverserelationship; the smallerthe standarderror, the moreprecisethe estimate.Table 2.2 revealsgreatvariability in the standarderrorsofthe individual slopeestimatesfor the NI’S data.Forsome,the estimatedrate of changeis very precise (e.g., subjects569 and624); for others,itis not (e.g., subject1552),

Understandingwhy the individualslopeestimatesvaryinprecisionpro-videsimportantinsightsinto howyoucanimprovelongitudinalstudiesofchange.Standardresultsfrom mathematicalstatisticstell usthat the pre-cisionof an OLS-estimatedrateof changedependsupon an individual’s:(1) residualvariance,theverticaldeviationsofobservedvaluesaroundthefitted line; and (2) numberandspacingof the wavesof longitudinal data.If individual ihas Twavesof data,gatheredattimest

0, ~ AT, thesam-

pling varianceof the OLS-estimatedrateof changeis1:

( Sampling variance a2of the OLS rateof change~= = r~ (21)

for individual i ) ~(t~ —i)2j”l

where a~representsthe residualvariancefor the ith individual and GSST~representshis or her correctedsum of squaresfor TIME, the sum ofsquareddeviationsof the time valuesaroundthe averagetime, t~.


Equation 2.1 suggeststwo ways of increasing the precision of OLSestimatedratesof change: (1) decreasethe residualvariance (becauseit appearsin the numerator);or (2) increasevariability in measurementtimes (becausethe correctedsumsof squaresfor time appearsin thedenominator). Of course, the magnitude of the residual variance islargely outsideyour control; strictlyspeaking,you cannotdirectly modifyits value. But becauseat least some of the residualvarianceis nothingmore than measurementerror, you can improve precision by usingoutcomemeasureswith betterpsychometricproperties.

Greaterimprovementsin precisionaccrueif you work to increasethecorrectedsum of squaresfor time by modifying your researchdesign.Inspectionof equation2.1 indicatesthat the greaterthe variability in thetiming of measurement,the more precise the assessmentof change.Therearetwo simplewaysof achievingincreasedvariability in the timingofmeasurement:(1) redistributethe timing of theplannedmeasurementoccasionsto be further away from their average;and (2) increasethenumberof waves. Both strategiesyield substantialpayoffsbecauseit isthe squareddeviationsof the measurementtimesabouttheir averageinthe denominatorof equation2.1. A changeassimple asaddinganotherwave of data to your researchdesign,far afield from the central set ofobservations,can reap dramatic improvements in the precision withwhich changecanbe measured.

We can reachsimilar conclusionsby examining the reliability of theOLS estimatedratesof change.Even thoughwebelieve thatprecisionisabettercriterionforjudging measurementquality, wehavethreereasonsfor also examining reliability. First, the issueof reliability so dominatesthe literature on the measurementof changethat it may be unwise toavoid all discussion.Second,it is useful to definereliability explicitly soas to distinguish it mathematicallyfrom precision.Third, eventhoughreliability andprecisionaredifferentcriteriafor evaluatingmeasurementquality, they do, in this case, lead to similar recommendationsaboutresearchdesign.

Unlike precision,which describeshowwell anindividualslopeestimatemeasuresthat person’s true rate of change,reliability describeshowmuchthe rateof changevariesacrosspeople.Precisionhasmeaningforthe individual; reliability hasmeaningfor the group.Reliability is definedin terms of interindividual variation: it is the proportion of a measure’sobservedvariancethat is true variance.When testdevelopersclaim thata testhasa reliability of .90 in apopulation, they meanthat 90% of theperson-to-personvariation in observedscoresacrossthe population is

variability in true scores.Reliability of changeis definedsimilarly. The populationreliability of


the OLS slopeis the proportion of populationvariancein observedrateof changethat is variancein true rateof change(seeRogosaet al., 1982;Willett, 1988, 1989). If reliability is high,a largeportion of the interindi-vidual differencesin observedrate of changewill be differencesin truerate of change.Were we to rank everyonein the population on theirobservedchanges,we would then be pretty confident that the rankingsreflect the rankorderof the truechanges.If reliability is low, the rankingson observedchangemight not reflect the trueunderlyingrankingsat all.

Improvementsin precisiongenerallyleadto improvementsin reliability—when you measureindividual changemore accurately,you can betterdistinguishindividuals on the basisof thesechanges.But asagroup-levelparameter,reliability’s magnitudeis also affectedby the amountof vari-ability in true changein the population. If everyonehas an identicalvalue of true rateof change,you will be unableto effectively distinguishamongpeopleevenif their observedratesof changeare precise,soreli-ability will be zero. This meansthatyou can simultaneouslyenjoy excel-lent individual precision for the rate of change and poor reliabilityfor detecting interindividual differences in change;you can measureeveryone’s change well, but be unable to distinguish people becauseeveryone’schangesare identical. For a constantlevel of measurementprecision, aspopulation heterogeneityin truechangeincreases,sodoesreliability.

The disadvantageof reliability as a gaugeof measurementquality is

that it confoundsthe effectof within-personprecisionwith the effect ofbetween-personheterogeneityin truechange.When individual precisionis poor or when interindividual heterogeneityin true changeis small, reli-ability tendsto 0. Whenprecision is high or when heterogeneityin true

changeis large,reliability tendsto 1. This meansthat reliability doesnottell you uniquelyabouteither precisionor heterogeneityin true change;instead,it tells you aboutboth simultaneously,impairing its value as anindicator of measurementquality.

We can confirm theseinadequaciesalgebraically,albeit under a pairof limiting assumptions: (1) that the longitudinal data are fully bal-anced—everyone in the population is observed on the same set ofoccasions,t

1, t

2t~ and (2) that each person’sresidualsare drawn

identically andindependentlyfrom acommondistribution with variancea~.The population reliability of the OLS estimateof individual rate ofchangeis then:

Reliability of the OLS rateof change= alruesb,P, 2 (2.2)

+


where aL5~6is the populationvarianceof the true rateof changeandCSSTis the correctedsum-of-squares-time,now commonacrossindivid-uals (Willett, 1988).BecauseC~ti,e.v#,appearsin both the numeratoranddenominator,it playsacentralrole in determiningreliability. If everyoneis growing at the sametrue rate, all true growth trajectorieswill be par-allel and therewill be no variability in the true rate of changeacrosspeople.When this happens,both aL5~~.and the reliability of change

will be 0, no matter how precisely the individual changeis measured.Ironically, this meansthat the OLS slopecan be a verypreciseyet com-

pletely unreliablemeasureof change.If thereare largedifferencesin thetrue rate of changeacrosspeople,the true growth trajectorieswill criss-crossconsiderably.When this happens,cL~.will be large,dominatingboth numeratorand denominator,and the reliability of the OLS slopewill tend to 1, regardlessof its precision. This meansthat the OLS slopecanbe an impreciseyet reliablemeasureof change.Theconclusion:youcanbe fooled about the quality of your changemeasurementif you usereliability asyour sole criterion.

We canalso useequation2.2 to reinforceour earlierconclusionsaboutlongitudinal researchdesign.First, for agivenlevel of interindividual dif-ferencein true changein the population,the reliability of the OLSslopedependssolely on the residual variance. Once again, the better the

quality of your outcome measurement, the better the reliability withwhich changecanbe measuredbecauseat leastpart of the residualvari-anceis simply measurementerror. Second,reliability can be improvedthrough design,by manipulating the number and spacingof the meas-urementoccasions.Anything thatyou cando to increasecorrectedsum-of-squarestime, CSST,will help. As you add wavesof dataor move theexistingwavesfurther awayfrom the centerof the datacollection period,the reliability with which changecan be measuredwill improve.

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