PerformingrepeatedmeasuresanalysisGraemeL.Hickey
@graemeleehickey www.glhickey.com [email protected]
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!