Where HRi,j are the hazard ratios estimated at time i in trial j through the use of the log-rank observed minus expected number of deaths and its variance.
S1 Supporting Information Methodological details Approach 1: Area between the two pooled survival curves Stewart-Parmar: Stewart and Parmar methodology allows one to estimate the pooled survival curve for the experimental arm SExp,pooled(t) using the pooled hazard ratio (inverse variance weighted average) and the naive Kaplan-Meier survival curve in the control group SControl,KM(t). The pooled survival curve for the control arm SControl,pooled(t) is simply set equal to SControl,KM(t): SControl,pooled(t) SControl,KM (t)
(S1)
SExp,pooled(t) SControl,KM (t)
(S2)
HR pooled
Peto: In the Peto method, survival probabilities are estimated at predetermined time intervals i (actuarial method) based on survival probability pi of the whole population and a pooled hazard ratio HRpooled,i which can vary between time periods. p i exp(-
Di ) PPi
(S3)
Where Di is the number of deaths during period i and PPi the total number of person-periods at period i. One person-period is equivalent to one person-year when the time interval chosen is one year. N
ln( HR pooled,i )
ln( HRi , j )
Var ln( HR ) j 1
i, j
(S4)
N
1 j 1 Var ln( HRi , j )
Where HRi,j are the hazard ratios estimated at time i in trial j through the use of the log-rank observed minus expected number of deaths and its variance. Survival probabilities at each period i in the control arm (pControl,i) and in the experimental arm (pExp,i) are estimated as follow:
p Control,i p i 0,5 p i (p i 1) ln HR pooled,i
p Exp p i 0,5 p i (p i 1) ln HR pooled,i
(S5) (S6)
1
The survival probability at time t in each arm is the product of the probabilities across ni
ni
SControl,pooled(t) p Control,i
periods:
S Exp,pooled(t) p Exp,i
and
i 1
(S7)
i 1
Approach 2: Pooling differences in restricted mean survival time The method to estimate the overall difference in restricted mean survival time (rmstD) aggregating the rmstDj estimated in each of the N trials is developed below with a fixed-effect model:
rmstD j
N
rmstD
Var rmstD j 1
j
N
1 j 1 Var rmstD j
N 1 Var rmstD j 1 Var rmstD j
and
1
(S8)
With difference in restricted mean survival time in each trial: t*
t*
rmstD j SExp,j (t).dt SControl,j (t).dt ˆ Exp ,t * , j ˆ Control,t * , j 0
(S9)
0
ˆ Exp ,t , j and ˆ Control,t , j are the restricted mean survival times estimated in trial j, respectively *
*
for the experimental (Exp) and for the control arm. Var[rmstDj] in each trial j can be estimated based on the following formula (Karrison T. Control Clin Trials 1997;18:151-67):
rmstDj ˆ Exp ,t* , j ˆ Control,t* , j Var rmstDj Var ˆ Exp ,t* , j Var ˆ Control,t* , j
(S10)
Where the variances of mean survival time are estimated (Klein JP, Moeschberger ML. 1997. Survival Analysis: techniques for censored and truncated data, 1st ed. New York: SpringerVerlag):
Var ˆ arm,T , j
2
t* d i, j Sarm, j (t).dt Yi , j (Yi , j d i , j ) i 1 ti , j Dj
where arm = Exp or Control
(S11)
In each trial j: Dj is the number of distinct event times (t1,j< t2,j < …< tD,j), Yi,j is the number of individuals who are at risk at time ti,j and di,j is the number of events at time ti,j. To calculate the overall rmstD (equation (S8)), Sarm,j(t) has then to be replaced by its estimator depending on the survival analysis method chosen (Kaplan-Meier and exponential in our empirical application) in the equations (S9) and (S11)
2
Pooled Kaplan-Meier: Replacing Sarm,j(t) by the Kaplan-Meier estimator Sˆarm, j (t ) , it leads to: rmstD j ˆ Exp ,t* , j ˆ Control,t* , j
Var ˆ arm,T , j
where ai , j
D j 1
mj mj
1 Y i 1
D j 1
Sˆ l i
i, j
Dj
Dj
i 1
i 1
Sˆ Exp , j (ti , j ) (ti 1, j ti , j ) SˆControl, j (ti , j ) (ti 1, j ti , j ) (S9-bis)
ai , j ²
(S11-bis)
(Yi , j d i , j ) D
arm, j
(tl , j ) (tl 1, j tl , j ) and m j d l , j l 1
Pooled Exponential .t Replacing Sarm,j(t) by the exponential estimator Sˆarm, j (t ) e arm, j , it leads to:
t*
rmstDj e
Exp , j .t
t*
.dt e
0
and Var ˆ arm,T , j
Control, j .t
.dt
1 e
Exp , j
0
e arm, j .ti e arm, j .t arm, j i 1 D
Exp , j .t *
*
1 e
Control, j .t *
Control, j
(S9-ter)
2
di , j Yi , j (Yi , j di , j )
(S11-ter)
3
