PerformingrepeatedmeasuresanalysisGraemeL.Hickey

@graemeleehickey www.glhickey.com graeme.hickey@liverpool.ac.uk

Conflictsofinterest

• None• AssistantEditor(StatisticalConsultant)forEJCTSandICVTS

Whatare“repeatedmeasures”data

A

BD A

BD A

BD

“Condition”:chocolatecake “Condition”:lemoncake “Condition”:cheesecake

Measurement:tastescore Measurement:tastescore Measurement:tastescore

Samepeoplescoreeachcondition

Whatare“repeatedmeasures”data

A

BD A

BD A

BD

Measurement:systolicBP Measurement:systolicBP Measurement: systolicBP

SamepeopleprovideBPateveryfollow-upappointment

Whydoweneedspecialmethodology?

• Dataarenotindependent:repeatedobservationsonthesameindividualwillbemoresimilartoeachotherthantoobservationsonotherindividuals

• Guidelines forreportingmortalityandmorbidityaftercardiacvalveinterventionsalsoproposetheuseoflongitudinaldataanalysisforrepeatedmeasurementdata

Simplestcase:2measurementtimes

A

BD A

BD

Measurement:AVgradient Measurement:AVgradient

pre-surgery post-surgery

Suitablemethods: pairedt-testorWilcoxonsigned-ranktest

Whatifwehavetreatmentgroups?

AB

D

Measurementtaken Measurementtaken

beforetreatment aftertreatment

AB

D

EF

H EF

H

Placeb

oActive

treatm

ent

Question:ifpatientsarerandomisedtotreatmentarms,howcanwetestwhetheractivetreatment ismoreeffectivethanplacebo?

Methods: shoulderpainexample

Source:Vickers&Altman.BMJ.2001;323:1123–4.

Placebo(n =27)

Acupuncture(n =25)

Differencebetweenmeans

(95%CI)

P

Follow-up 62.3(17.9) 79.6(17.1) 17.3(7.5to27.1)

Moregeneralscenario

• Werecordmeasurementsofeachpatient>2times• Two(ormoretreatmentgroups)

Designconsiderations

• Balancedversus unbalanced• Balanced follow-up(e.g.baseline,1-hr,2-hr,8-hr,16-hr,24-hr)• Unbalanced (e.g.patientAvisitstheirphysicianondays1,4,6,9,12,andpatientBvisitsonlyondays5,9,and15)

• Missingdata• E.g.patientfailstoattendscheduled follow-upappointment

Hownot toproceed

• Multipletestingissues• Noaccountofsamepatientsbeingmeasured⇒successiveobservationslikelycorrelated• Visualization+reportingissues

Source:Matthewsetal.BMJ.1990;300:230–5.

Dataformat/collection

WideformatSubject Jan01 Aug30 Dec08

A 120 113 115

B 94 94 110

C 140 145 160

D 100 101 100

LongformatSubject Date BP(mmHg)

A Jan01 120

A Aug30 113

A Dec08 115

B Jan01 94

B Aug30 94

B Dec08 110⠇ ⠇ ⠇

D Aug30 101

D Dec08 100

Goodforbalanceddatasets

Goodforunbalanceddatasets

Firststep(always!):visualizethedata

Source:Gueorguieva &Krystal.ArchGenPsychiatry.2004;61:310–317.

Meanprofileplot

Source:Matthewsetal.BMJ.1990;300:230–5.

IndividualpanelplotsIndividualplotsgrouped

bytreatment

Analysisoptions

• Repeatedmeasuresanalysisofvariance(RM-ANOVA)• Linearmixedmodels(LMMs)• Summarystatistics/data-reductiontechniques• Multivariateanalysisofvariance(MANOVA)• Generalizedleastsquares(GLS)• Generalizedestimatingequations• Non-linearmixedeffectsmodels• EmpiricalBayesmethods• …

RM-ANOVATotal

variation

Between-subjectsvariation

Within-subjectsvariation

Treatment

Errorduetosubjectswithin

treatment

Time Treatment*Time Error

Testfor: treatmenteffecttimeeffectinteractioneffect

Sphericity

• RM-ANOVAdependsontheusualassumptionsforANOVA…• … andtheassumptionofsphericity

SDT2– T1 ≅ SDT3– T1 ≅ SDT3– T2 ≅ …

• Restrictiveforlongitudinaldata⇒measurementstakencloselytogetherareoftenmorecorrelatedthanthosetakenatlargertimeintervals

• TestforsphericityusingMauchly’stest

Tomorrow(14:15– 15:45):Checkingmodelassumptionswithregressiondiagnostics

Whensphericityisviolated

• Ifsphericityisviolated,thentypeIerrorsareinflatedandinteractiontermeffectsbiased– thatisserious• Mauchly’stestmaynotrejectsphericityifthesamplesizeissmall,evenifthevariancesarevastlydifferent

Correctionproposal:1. Calculatetheepsilonstatistic

i. Greenhouse-Geisserii. Huynh-Feldt

2. MultiplytheF-statisticdegreesoffreedombyepsilon

Linearmixedmodels

• Generalizeslinearregressiontoaccountforcorrelationinrepeatedmeasureswithinsubjects• Alsodescribedasrandomeffectsmodels,mixedeffectsmodels,randomgrowthmodels,multi-levelmodels,hierarchicalmodels,…

Outcome

Time

𝑦"# = 𝛽& + 𝛽(𝑡"# + 𝜀"#

Fixed effects regression line

Time

Outcome

𝑦"# = 𝛽&" + 𝛽(𝑡"# + 𝜀"#

Fixed effects regression line + within-subject intercepts

Time

Outcome

Within-subjects fixed effects regression lines

𝑦"# = 𝛽&" + 𝛽("𝑡"# + 𝜀"#

Time

Outcome

Linearmixedmodels

• Acompromiseisthemodel

𝑌"# = 𝛽& + 𝑏&" + 𝛽( + 𝑏(" 𝑡"# + 𝜀"#

• 𝑏&", 𝑏(" arecalledsubject-specificrandomintercepts:interceptandsloperespectively,distributedN2(0,Σ)

• Observationswithin-subjectsaremorecorrelatedthanobservationsbetween-subjects• Canbeadjustedforother(possiblytime-varying)covariatesandbaselinemeasurements

Summarystatistics

• Atwo-stageapproach:1. Reducetherepeatedmeasurementsforeachsubjecttoasinglevalue2. Applyroutinestatisticalmethodsonthesesummaryvaluestocompare

treatments,e.g.usingindependentsamplest-test,ANOVA,Mann-WhitneyU-test,…

• Benefits• Easytodo,andconceptuallyeasytounderstand• Canbeusedtocontrastdifferentfeaturesofthedata• Encouragesresearcherstothinkaboutthefeaturesofthedatamostimportanttotheminadvance

• Choiceofsummarystatisticdependsonthedata

T0 T1 T3 T4

Outcome

ymax

T2

T0 T1 T3 T4

Outcome

T2T0 T1 T3 T4

Outcome

ypreT2

ypost - ypre

T0 T1 T3 T4T2

Outcome

Ifthedatadisplaya‘peakedcurve’trend…

Areaunderthecurve Maximummeasurement

TimetoreachmaximumMeanfollow-up– baseline

Ifthedatadisplaya‘growthcurve’trend…

Changescore Finalvalue

Timetoacertain%increase/decreaseSlope

T0 T1 T3 T4

Outcome

T2

ychange

T0 T1 T3 T4

Outcome

T2

yfinal

T0 T1 T3 T4

Outcome

T2

slope

T0 T1 T3 T4T2

Outcome

Missingdata

Method Canithandlemissingdata? Canithandleunbalanceddata?

RM-ANOVA

No– typically excludepatientswith1ormissingvalue

No

LMM Yes– fordatathatismissing(completely)atrandom Yes

Summarystatistics

Dependsonthechoiceofsummary statistic

Dependsonthechoiceofsummary statistic

Software

• Allmethodsimplementedinstandardstatisticalsoftware

• Summarystatisticsusuallyrequire‘manual’calculation,butcanbedoneeasilyinMicrosoftExcelorprogrammedinastatisticssoftwarepackage

Thankyouforlistening…anyquestions?

Slidesavailable(shortly)from:www.glhickey.com

StatisticalPrimerarticletobepublishedsoon!