This document presents a meta-analysis of time-to-event survival curves in drug-eluting stent data, aimed at consolidating study outcomes and deriving statistical inferences. It discusses various methodologies, such as the log hazard ratio and different models for calculating hazard ratios, while also presenting a case study comparing diabetic and non-diabetic patients. The analysis covers challenges like patient-level versus study-level data, and future work involves addressing missing values and improving methodologies for survival data.

1. Meta-Analysis of Time-to-Event Survival
Curves in Drug-Eluting Stent Data
Hsini (Terry) Liao*, PhD
Yun Lu, MSc
Hong Wang, MSc
Boston Scientific Corporation
*Contact: terry.liao@bsci.com
JSM 2009 1

2. Outlines
• Motivation
• Meta-Analysis Overview
• Application to Survival Curves
• Case Study and Simulation
• Summary and Future Work
• References
JSM 2009 2

3. Motivation
• Meta-analysis provides a structure of consolidating
the outcomes from several studies and deriving
statistical inference of the outcomes
• Meta-analysis of time-to-event data is less common
than meta-analysis of binary or continuous data
• Fixed effect vs. random effects models
• Patient-level vs. study-level data
• Hazard ratio (HR) vs. Kaplan-Meier (KM) curve
• Different follow-up schedules
JSM 2009 3

4. Motivation (Cont’d)
Sutton, A.J., Higgins, J.P. “Recent Developments in Meta-Analysis”, Stat in Med. 2008; 27:625-650
Meta- Analysis” 27:625-
JSM 2009 4

5. Meta-Analysis Overview
• A systematic review of literature to measure the
effect size
• Single study/effect
• Many studies/narrative review
• Effect magnitude/adequate precision
• Combine the effects to give overall mean effect
• Effect size: event rate, OR, RR, HR, etc.
• Sample size/standard error to assign weight
JSM 2009 5

6. (Current) Application to
Survival Curves
• Extract data from KM curves
• Estimate ln(HRij) and var[ln(HRij)] for each
study
• The HR is a summary of the difference
between two KM curves
• Consider time-to-event and censoring,
otherwise HR=RR
• Variety of scenarios (e.g. CI)
• MS Excel spreadsheet computation
JSM 2009 6

7. HR Matrix
Study 1 HR11 HR12 HR13 . . . HR1J
HR21 HR22 HR23 . . . HR2J
Study 2
. . . . .
. . . . .
. . . . .
Study K HRK1 HRK2 HRK3 . . . HRKJ
Time 1 Time 2 Time 3 . . . Time J
JSM 2009 7

8. Application to Survival Curves
(Cont’d)
• Formal definition of the log hazard ratio
• Reported number of observed events and
number of expected events:
For each study i and each time j,
OTij / ETij
ln( HRij ) = ln( )
OCij / ECij
1 1
var[ln(HRij )] = +
ETij ECij
JSM 2009 8

9. Application to Survival Curves
(Cont’d)
For each study i=1,…,K and each time point j,
K
∑ var[ln(HR )]
ln( HRij )
ij
ln( HR⋅ j ) = i =1
K
∑i =1
1
var[ln(HRij )]
−1
⎡ K ⎤
var[ln(HR⋅ j )] = ⎢
⎢ ∑
⎢ i =1
1 ⎥
var[ln(HRij )] ⎥
⎥
⎣ ⎦
JSM 2009 9

10. Most Scenarios
• Reported the initial #patient at risk and
observed event count for each time point,
or equivalent information
• Take censoring into account if applicable
• Able to estimate expected events
JSM 2009 10

11. Method 1: Proposed
(Overall Observed and Expected Events)
• For each study i, compute the sum of observed
events (OTi., OCi.) and the sum of expected events
(ETi., ECi.) across all time points
• Use the formal definition of the log hazard ratio
OTi⋅ / ETi⋅
ln( HRi⋅ ) = ln( )
OCi⋅ / ECi⋅
1 1
var[ln(HRi⋅ )] = +
ETi⋅ ECi⋅
JSM 2009 11

12. Method 2: Parmar
(Observed and Expected Events)
• For each study i, compute log(HR) and associated
variance for each time point with observed events
(OTij, OCij) and expected events (ETij, ECij)
• Calculate the weighted mean of log(HR) across time
points for each study i
OTij / ETij
ln( HRij ) = ln( )
OCij / ECij
1 1
var[ln(HRij )] = +
ETij ECij
JSM 2009 12

13. Method 3: Williamson
(Observed Events and #Patient-At-Risk)
• For each study i, compute log(HR) and associated
variance for each time point with the observed
events (OTij, OCij) and #patient-at-risk (NTij, NCij)
• Calculate the weighted mean of log(HR) across
time points for each study i
OTij / NTij
ln( HRij ) = ln( )
OCij / NCij
1 1 1 1
var[ln(HRij )] = − + −
OTij NTij OCij NCij
JSM 2009 13

14. Case Study: Stent Data
• Study outcome: Target Vessel Revascularization (TVR)
• Post-hoc analysis set: diabetic vs. non-diabetic patients with
drug-eluting stent
• Propensity score adjustment (“like-to-like”): 1-to-1 match
• Data: total 1,554 DES patients over 7 studies
• Study 1: (n=308) 5 years
• Study 2: (n=352) 4 years
• Study 3: (n= 76) 5 years
• Study 4: (n=436) 3 years
• Study 5: (n= 84) 2 years
• Study 6: (n=186) 2 years
• Study 7: (n=112) 2 years
• The hazard ratio estimate of study outcomes from patient level
data is compared with that from the study level data to assess
the treatment effect in DES patients (Diabetic vs. Non-
Diabetic).
JSM 2009 14

15. HR Matrix
(Calculation Using Formal Definition)
Year 1 Year 2 Year 3 Year 4 Year 5
Study 1 1.99 1.27 1.07 6.32 0.74
Study 2 0.91 2.48 1.39 0.11 NA
Study 3 1.33 2.06x106 * 0.28 1.03 1.07
Study 4 2.23 1.44 2.18 NA NA
Study 5 1.40 1.06 NA NA NA
Study 6 1.24 3.90 NA NA NA
Study 7 1.20 1.00 NA NA NA
NA = Not Available
* Due to zero event in non-diabetic arm, using tiny number (10-6) instead of zero to make formula work. 15
JSM 2009

16. Comparison of Estimates for
Overall HR
Fixed Effect Model Random Effects Model
HR [95% CI] HR [95% CI]
Method1
1.31 [1.03, 1.67] 1.34 [0.99, 1.81]
(Proposed)
Method 2
1.31 [1.03, 1.66] 1.46 [0.90, 2.37]
(Parmar)
Method 3
1.31 [1.04, 1.67] 1.33 [1.02, 1.75]
(Williamson)
Cox Model in
1.32 [1.03, 1.67] 1.33 [0.98, 1.82]
IPD
JSM 2009
IPD = Individual Patient Data 16

17. Simulation
• Simulation of KM curves in terms of
numbers of patient at risk, censoring and
events for 5 years
• Meta-analysis of 10 studies
• Two-arm with initial sample size ratio 1:1
• Event rates are centered at 13%, 16%,
20%, 23%, and 24% for treatment arm
based on pooling historical data
• Generated 1,000 times
JSM 2009 17

18. Simulation (Cont’d)
• Varying the first year event rate difference
ranging from 0.2% to 4% (treatment effect)
• Varying heterogeneity in terms of standard
deviation of event rates of 2%, 3%, 4%, 5%
• Calculated the coverage probability defined as
the percentage of 95% CIs that contain the
true underlying value of the log HR over
1,000 simulated runs
JSM 2009 18

19. STD = 2% STD = 3%
1.80 1.80
1.70 1.70
Estimated Hazard Ratio
Estimated Hazard Ratio
1.60 1.60
1.50 1.50
1.40 1.40
1.30 1.30
1.20 1.20
1.10 1.10
1.00 1.00
1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.20 1.22 1.24 1.26 1.28 1.30 1.32
True Hazard Ratio True Hazard Ratio
Proposed Parmar Williamson Proposed Parmar Williamson
STD = 4% STD = 5%
1.80 1.80
1.70 1.70
Estimated Hazard Ratio
Estimated Hazard Ratio
1.60 1.60
1.50 1.50
1.40 1.40
1.30 1.30
1.20 1.20
1.10 1.10
1.00 1.00
1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.20 1.22 1.24 1.26 1.28 1.30 1.32
True Hazard Ratio True Hazard Ratio
JSM 2009 Proposed Parmar Williamson Proposed Parmar Williamson 19

20. STD = 2% STD = 3%
100 100
90 90
80 80
70 70
% Coverage
% Coverage
60 60
50 50
40 40
30 30
20 20
10 10
0 0
1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.20 1.22 1.24 1.26 1.28 1.30 1.32
True Hazard Ratio True Hazard Ratio
Proposed Parmar Williamson Proposed Parmar Williamson
STD = 4% STD = 5%
100 100
90 90
80 80
70 70
% Coverage
% Coverage
60 60
50 50
40 40
30 30
20 20
10 10
0 0
1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.20 1.22 1.24 1.26 1.28 1.30 1.32
True Hazard Ratio True Hazard Ratio
JSM 2009 Proposed Parmar Williamson Proposed Parmar Williamson 20

21. Result of Simulation
• Small heterogeneity
- For small or large treatment effect, all
three methods perform well
• Increasing heterogeneity
- Bias increases
- Coverage decreases. Some values
decrease dramatically
- For large treatment effect, coverage
decreases less for proposed method
JSM 2009 21

22. Summary: Meta-Analysis of
Kaplan-Meier Curves
• Reported Kaplan-Meier curves with
#patient at risk at baseline
• Read off survival probabilities at each
time point
• Estimate the minimum and the
maximum follow-up time
• Assumption for distribution of censored
subjects: Missing at random (uniform)?
JSM 2009 22

23. HR Matrix Different Follow-Up
Study 1 HR11 HR12 HR13 HR1J
HR21 HR22 HR23
Study 2
. .
. . . . . . . .
. .
Study K HRK1 . . . . . . HRKJ
Time 1 Time 2 Time 3 . . . Time J
JSM 2009 23

24. Future Work
• Missing value imputation for the HR matrix
• Constant HRs over time
• Test of equality for all non-missing HRij over
time (within each study)
• Inclusion/exclusion criteria
• Distribution of censoring
• Summary KM curve of many KM curves
JSM 2009 24

25. References
• Parmar, M.K.B., Torri, V. and Stewart, L. “Extracting
Summary Statistics to Perform Meta-Analyses to the
Published Literature for Survival Endpoints”, Stat in Med.
1998; 17:2815-2834
• Tierney, J.F., Stewart, L.A., Ghersi, D., Burdett, S. and
Sydes, M.R. “Practical Methods for Incorporating Summary
Time-to-Event Data into Meta-Analysis”, Trials 2007; 8:16
• Arends, L.R., Hunink, M.G.M. and Stijnen, T. “Meta-Analysis
of Summary Survival Data”, Stat in Med. 2008; 27:4381-4396
• Williamson, P.R., Smith, C.T., Hutton, J.L. and Marson, A.G.
“Aggregate Data Meta-Analysis with Time-to-Event
Outcomes”, Stat in Med. 2002; 21:3337-3351
• Sutton, A.J., Higgins, J.P. “Recent Developments in Meta-
Analysis”, Stat in Med. 2008; 27:625-650
JSM 2009 25