2009 JSM - Meta Time to Event Data

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Meta-Analysis of Time-to-Event Survival Curves in Drug-Eluting Stent Data

Hsini (Terry) Liao*, PhD Yun Lu, MSc

Hong Wang, MScBoston Scientific Corporation

*Contact: [email protected]

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OutlinesOutlines

•• MotivationMotivation•• MetaMeta--Analysis OverviewAnalysis Overview•• Application to Survival CurvesApplication to Survival Curves•• Case Study and SimulationCase Study and Simulation•• Summary and Future WorkSummary and Future Work•• ReferencesReferences

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MotivationMotivation

•• MetaMeta--analysis provides a structure of consolidating analysis provides a structure of consolidating the outcomes from several studies and deriving the outcomes from several studies and deriving statistical inference of the outcomesstatistical inference of the outcomes

•• MetaMeta--analysis of timeanalysis of time--toto--event data is less common event data is less common than metathan meta--analysis of binary or continuous dataanalysis of binary or continuous data

•• Fixed effect vs. random effects modelsFixed effect vs. random effects models•• PatientPatient--level vs. studylevel vs. study--level datalevel data•• Hazard ratio (HR) vs. KaplanHazard ratio (HR) vs. Kaplan--Meier (KM) curveMeier (KM) curve•• Different followDifferent follow--up schedulesup schedules

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Motivation Motivation (Cont(Cont’’d)d)

Sutton, A.J., Higgins, J.P. Sutton, A.J., Higgins, J.P. ““Recent Developments in MetaRecent Developments in Meta--AnalysisAnalysis””, Stat in Med. 2008; 27:625, Stat in Med. 2008; 27:625--650650

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MetaMeta--Analysis OverviewAnalysis Overview

•• A systematic review of literature to measure the A systematic review of literature to measure the effect sizeeffect size

•• Single study/effectSingle study/effect•• Many studies/narrative reviewMany studies/narrative review•• Effect magnitude/adequate precisionEffect magnitude/adequate precision•• Combine the effects to give overall mean effectCombine the effects to give overall mean effect

•• Effect size: event rate, OR, RR, HR, etc.Effect size: event rate, OR, RR, HR, etc.•• Sample size/standard error to assign weightSample size/standard error to assign weight

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(Current) Application to(Current) Application toSurvival CurvesSurvival Curves

•• Extract data from KM curvesExtract data from KM curves•• Estimate Estimate ln(HRln(HRijij) and ) and var[ln(HRvar[ln(HRijij)] for each )] for each

studystudy•• The HR is a summary of the difference The HR is a summary of the difference

between two KM curvesbetween two KM curves•• Consider timeConsider time--toto--event and censoring, event and censoring,

otherwise HR=RRotherwise HR=RR•• Variety of scenarios (e.g. CI)Variety of scenarios (e.g. CI)•• MS Excel spreadsheet computationMS Excel spreadsheet computation

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HR MatrixHR Matrix

Study 1

Study 2

Study K

Time 1 Time 2 Time JTime 3

HR11 HR12 HR13

HRKJ

HR1J

HRK1 . . .

.

.

.

HR21

HRK2

.

.

.

. . .

HR22 HR23

.

.

.

.

.

.

HRK3

HR2J

.

.

.

. . .

. . .

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Application to Survival Curves Application to Survival Curves (Cont(Cont’’d)d)•• Formal definition of the log hazard ratioFormal definition of the log hazard ratio•• Reported number of observed events and Reported number of observed events and

number of expected events:number of expected events:

)//

ln()ln(ijij

ijijij ECOC

ETOTHR =

ijijij ECET

HR 11)]var[ln( +=

For each studyFor each study i and each time j, i and each time j,

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Application to Survival Curves Application to Survival Curves (Cont(Cont’’d)d)

∑

∑

=

=⋅ = K

i ij

K

i ij

ij

j

HR

HRHR

HR

1

1

)]var[ln(1

)]var[ln()ln(

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⎣

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i ijj HR

HR

For each studyFor each study i=1,i=1,……,K and each time point j, ,K and each time point j,

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Most ScenariosMost Scenarios

•• Reported the initial #patient at risk and Reported the initial #patient at risk and observed event count for each time point, observed event count for each time point, or equivalent informationor equivalent information

•• Take censoring into account if applicableTake censoring into account if applicable•• Able to estimate expected eventsAble to estimate expected events

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Method 1: ProposedMethod 1: Proposed(Overall Observed and Expected Events)(Overall Observed and Expected Events)

•• For each study i, compute the sum of observed For each study i, compute the sum of observed events (events (OTOTii.., , OCOCii..) and the sum of expected events ) and the sum of expected events ((ETETii.., , ECECii..) across all time points ) across all time points

•• Use the formal definition of the log hazard ratioUse the formal definition of the log hazard ratio

)//ln()ln(

⋅⋅

⋅⋅⋅ =

ii

iii ECOC

ETOTHR

⋅⋅⋅ +=

iii ECET

HR 11)]var[ln(

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Method 2: Method 2: ParmarParmar(Observed and Expected Events)(Observed and Expected Events)

•• For each study i, compute For each study i, compute log(HRlog(HR) and associated ) and associated variance for each time point with observed events variance for each time point with observed events ((OTOTijij, , OCOCijij) and expected events () and expected events (ETETijij, , ECECijij))

•• Calculate the weighted mean of Calculate the weighted mean of log(HRlog(HR) across time ) across time points for each study ipoints for each study i

)//

ln()ln(ijij

ijijij ECOC

ETOTHR =

ijijij ECET

HR 11)]var[ln( +=

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Method 3: Williamson Method 3: Williamson (Observed Events and #Patient(Observed Events and #Patient--AtAt--Risk)Risk)

•• For each study i, compute For each study i, compute log(HRlog(HR) and associated ) and associated variance for each time point with the observed variance for each time point with the observed events (events (OTOTijij, , OCOCijij) and #patient) and #patient--atat--risk (risk (NTNTijij, , NCNCijij))

•• Calculate the weighted mean of Calculate the weighted mean of log(HRlog(HR) across ) across time points for each study itime points for each study i

)//

ln()ln(ijij

ijijij NCOC

NTOTHR =

ijijijijij NCOCNTOT

HR 1111)]var[ln( −+−=

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Case Study: Stent DataCase Study: Stent Data

•• Study outcome: Target Vessel Revascularization (TVR)Study outcome: Target Vessel Revascularization (TVR)•• PostPost--hoc analysis set: diabetic vs. nonhoc analysis set: diabetic vs. non--diabetic patients with diabetic patients with

drugdrug--eluting stenteluting stent•• Propensity score adjustment (Propensity score adjustment (““likelike--toto--likelike””): 1): 1--toto--1 match1 match•• Data: total 1,554 DES patients over 7 studiesData: total 1,554 DES patients over 7 studies

•• Study 1: (n=308) 5 yearsStudy 1: (n=308) 5 years•• Study 2: (n=352) 4 yearsStudy 2: (n=352) 4 years•• Study 3: (n= 76) 5 years Study 3: (n= 76) 5 years •• Study 4: (n=436) 3 yearsStudy 4: (n=436) 3 years•• Study 5: (n= 84) 2 yearsStudy 5: (n= 84) 2 years•• Study 6: (n=186) 2 yearsStudy 6: (n=186) 2 years•• Study 7: (n=112) 2 yearsStudy 7: (n=112) 2 years

•• The hazard ratio estimate of study outcomes from patient level The hazard ratio estimate of study outcomes from patient level data is compared with that from the study level data to assess data is compared with that from the study level data to assess the treatment effect in DES patients (Diabetic vs. Nonthe treatment effect in DES patients (Diabetic vs. Non--Diabetic).Diabetic).

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HR MatrixHR Matrix(Calculation Using Formal Definition)(Calculation Using Formal Definition)

NANANANANANA1.061.061.401.40Study 5Study 5



NANANANA2.182.181.441.442.232.23Study 4Study 4

1.071.071.031.030.280.282.06x102.06x106 6 **1.331.33Study 3Study 3

NANA0.110.111.391.392.482.480.910.91Study 2Study 2

0.740.746.326.321.071.071.271.271.991.99Study 1Study 1

Year 5Year 5Year 4Year 4Year 3Year 3Year 2Year 2Year 1Year 1

NA = Not Available* Due to zero event in non-diabetic arm, using tiny number (10-6) instead of zero to make formula work.

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Comparison of Estimates for Comparison of Estimates for Overall HROverall HR

1.33 [1.02, 1.75]1.33 [1.02, 1.75]1.31 [1.04, 1.67]1.31 [1.04, 1.67]Method 3 Method 3 (Williamson)(Williamson)

1.33 [0.98, 1.82]1.33 [0.98, 1.82]1.32 [1.03, 1.67]1.32 [1.03, 1.67]Cox Model in Cox Model in IPDIPD

1.46 [0.90, 2.37]1.46 [0.90, 2.37]1.31 [1.03, 1.66]1.31 [1.03, 1.66]Method 2 Method 2 ((ParmarParmar) )

1.34 [0.99, 1.81]1.34 [0.99, 1.81]1.31 [1.03, 1.67]1.31 [1.03, 1.67]Method1 Method1 (Proposed)(Proposed)

Random Effects ModelRandom Effects ModelHR [95% CI]HR [95% CI]

Fixed Effect ModelFixed Effect ModelHR [95% CI]HR [95% CI]

IPD = Individual Patient Data

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SimulationSimulation

•• Simulation of KM curves in terms of Simulation of KM curves in terms of numbers of patient at risk, censoring and numbers of patient at risk, censoring and events for 5 yearsevents for 5 years

•• MetaMeta--analysis of 10 studiesanalysis of 10 studies•• TwoTwo--arm with initial sample size ratio 1:1arm with initial sample size ratio 1:1•• Event rates are centered at 13%, 16%, Event rates are centered at 13%, 16%,

20%, 23%, and 24% for treatment arm 20%, 23%, and 24% for treatment arm based on pooling historical databased on pooling historical data

•• Generated 1,000 timesGenerated 1,000 times

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Simulation (ContSimulation (Cont’’d)d)

•• Varying the first year event rate difference Varying the first year event rate difference ranging from 0.2% to 4% (treatment effect)ranging from 0.2% to 4% (treatment effect)

•• Varying heterogeneity in terms of standard Varying heterogeneity in terms of standard deviation of event rates of 2%, 3%, 4%, 5%deviation of event rates of 2%, 3%, 4%, 5%

•• Calculated the coverage probability defined as Calculated the coverage probability defined as the percentage of 95% the percentage of 95% CIsCIs that contain the that contain the true underlying value of the log HR over true underlying value of the log HR over 1,000 simulated runs1,000 simulated runs

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STD = 2%

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.20 1.22 1.24 1.26 1.28 1.30 1.32True Hazard Ratio

Estim

ated

Haz

ard

Rat

io

Proposed Parmar Williamson

STD = 3%

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80


Estim

ated

Haz

ard

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STD = 4%

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80


Estim

ated

Haz

ard

Rat

io


STD = 5%

1.00

1.10

1.20

1.30

1.40

1.50

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1.70

1.80


Estim

ated

Haz

ard

Rat

io


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STD = 2%

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100


% C

over

age


STD = 3%

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STD = 4%

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Result of SimulationResult of Simulation

•• Small heterogeneitySmall heterogeneity-- For small or large treatment effect, all For small or large treatment effect, all

three methods perform wellthree methods perform well•• Increasing heterogeneityIncreasing heterogeneity

-- Bias increasesBias increases-- Coverage decreases. Some values Coverage decreases. Some values

decrease dramaticallydecrease dramatically-- For large treatment effect, coverage For large treatment effect, coverage

decreases less for proposed methoddecreases less for proposed method

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Summary: MetaSummary: Meta--Analysis of Analysis of KaplanKaplan--Meier CurvesMeier Curves

•• Reported KaplanReported Kaplan--Meier curves with Meier curves with #patient at risk at baseline#patient at risk at baseline

•• Read off survival probabilities at each Read off survival probabilities at each time pointtime point

•• Estimate the minimum and the Estimate the minimum and the maximum followmaximum follow--up timeup time

•• Assumption for distribution of censored Assumption for distribution of censored subjects: Missing at random (uniform)? subjects: Missing at random (uniform)?

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HR Matrix Different FollowHR Matrix Different Follow--UpUp

Study 1

Study 2

Study K

Time 1 Time 2 Time JTime 3

HR11 HR12 HR13

HRKJ

HR1J

HRK1 . . .

.

.

.

HR21

. . . . . .

. . .

.

.

.

. . .

HR22 HR23

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Future WorkFuture Work

•• Missing value imputation for the HR matrixMissing value imputation for the HR matrix•• Constant Constant HRsHRs over timeover time•• Test of equality for all nonTest of equality for all non--missing missing HRHRijij over over

time (within each study)time (within each study)•• Inclusion/exclusion criteriaInclusion/exclusion criteria•• Distribution of censoringDistribution of censoring•• Summary KM curve of many KM curves Summary KM curve of many KM curves

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ReferencesReferences

•• ParmarParmar, M.K.B., , M.K.B., TorriTorri, V. and Stewart, L. , V. and Stewart, L. ““Extracting Extracting Summary Statistics to Perform MetaSummary Statistics to Perform Meta--Analyses to the Analyses to the Published Literature for Survival EndpointsPublished Literature for Survival Endpoints””, Stat in Med. , Stat in Med. 1998; 17:28151998; 17:2815--28342834

•• Tierney, J.F., Stewart, L.A., Tierney, J.F., Stewart, L.A., GhersiGhersi, D., Burdett, S. and , D., Burdett, S. and SydesSydes, M.R. , M.R. ““Practical Methods for Incorporating Summary Practical Methods for Incorporating Summary TimeTime--toto--Event Data into MetaEvent Data into Meta--AnalysisAnalysis””, Trials 2007; 8:16, Trials 2007; 8:16

•• ArendsArends, L.R., , L.R., HuninkHunink, M.G.M. and , M.G.M. and StijnenStijnen, T. , T. ““MetaMeta--Analysis Analysis of Summary Survival Dataof Summary Survival Data””, Stat in Med. 2008; 27:4381, Stat in Med. 2008; 27:4381--43964396

•• Williamson, P.R., Smith, C.T., Hutton, J.L. and Williamson, P.R., Smith, C.T., Hutton, J.L. and MarsonMarson, A.G. , A.G. ““Aggregate Data MetaAggregate Data Meta--Analysis with TimeAnalysis with Time--toto--Event Event OutcomesOutcomes””, Stat in Med. 2002; 21:3337, Stat in Med. 2002; 21:3337--33513351

•• Sutton, A.J., Higgins, J.P. Sutton, A.J., Higgins, J.P. ““Recent Developments in MetaRecent Developments in Meta--AnalysisAnalysis””, Stat in Med. 2008; 27:625, Stat in Med. 2008; 27:625--650650

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2009 JSM - Meta Time to Event Data