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Sample Size Re-estimation of Event-Driven Trialswith Composite Events and Delayed Data
Ascertainment
Brittany Schwefel, PhD 1 Thomas D. Cook, PhD 2
1AbbVie
2Department of Biostatistics and Medical InformaticsUniversity of Wisconsin-Madison
May 16, 2016
Introduction
▶ Clinical trials with a time-to-event primary endpoint frequentlyhave an event-driven sample size, i.e. number of expectedprimary events determines subject enrollment and length offollow-up
1
Primary Endpoint of interest
▶ Survival time to a single endpoint e.g. death▶ Survival time to a composite outcome
▶ More than one event type of interest - primary eventconstituents
▶ Events may be recurrent, or of multiple types▶ E.g. first cardiovascular event - MI, stroke, CV death▶ Interested in time to first event in this set of primary event
constituents▶ Note: events may be reported to the database out of order
(non-monotone event reporting)
2
Introduction
▶ When designing a trial, event-rate estimated from historicaldata e.g. literature, Phase 2 trials
▶ During the course of a trial, if event rate is higher or lowerthan expected, enrollment and/or length of follow-up mayneed to be adjusted
▶ Use observed data at an interim time point
3
Observed Event Rate and Projected Number of Events
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
Survival Estimate Interim Analysis Time = 2
time
S(t
)
Reported KM
0 1 2 3 4
020
040
060
080
010
00
Study Time
Num
ber
of e
vent
s
Reported EventsReported KM
Projected Number of Composite Events based on Interim Analysis
▶ E.g. 4 year trial, re-evaluate at year 2
▶ Enroll n = 1580, target number of events = 1000
▶ Based on observed projection, will fall short of target ⇒increase enrollment and/or follow-up time
4
Event reporting in clinical trials
Subject hasevent
Event reportedto database
-Delay
Issue with using observed data for event projection - not allevents may be reported to the database at interim time point
5
Event reporting in clinical trials
Subject hasevent
Event reportedto database
-Delay
Issue with using observed data for event projection - not allevents may be reported to the database at interim time point
6
Event reporting in clinical trials
Subject hasevent
Event reportedto database
-Delay
Issue with using observed data for event projection - not allevents may be reported to the database at interim time point
6
Delay Illustration and Notation
t11 V(t11)
t21 V(t21)
t31 t32 V(t31) V(t32)
t41 t42 V(t42) V(t41)
First observed event
Subject N∗i (Ci ) N∗∗
i (Ci ) NR (Ci )Reported Unreported TotalEvents Events Events
1 1 0 12 0 1 13 1 1 24 1 1 2
tij j th event for subject iCi Administrative censoring time for subject iV (tij) reporting time of the event at tijDij V (tij)− tij = delay time corresponding to the event at tijNR
i (t) count of (possibly recurrent) primary event constituents in sub-ject i at time less than or equal to t = sum of reported andunreported events = N∗
i (t) + N∗∗i (t)
Non-Monotone Event Reporting: ti1 < ti2 =⇒ V (ti1) < V (ti2)
7
Observed Event Rate and Projected Number of Eventscompared with Actual Number of Events
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
Survival Estimate Interim Analysis Time = 2
time
S(t
)
Composite KMReported KM
0 1 2 3 4
020
040
060
080
010
00
Study TimeN
umbe
r of
eve
nts
True EventsReported EventsReported KM
Projected Number of Composite Events based on Interim Analysis
▶ Events may be reported with delays up to 1.0 years
▶ True event rate higher than observed due to unreported at interim timepoint
▶ Observed: Must increase enrollment and/or follow-up time
▶ Truth: On target
8
Objective: Consistently estimate the survivalfunction of the time to primary outcome (either asingle event or composite outcome) in the contextof delayed event ascertainment and non-monotone
event reporting.
9
Inverse Probability of Censoring Weighted (IPCW)Estimator
Correct for unreported events at interim time point due to delayedreporting to the database
▶ IPCW estimator proposed by Robins and Rotnitzky (1992)
▶ Weight contribution of observed events to the survivalestimate by the probability of not being censored
▶ How many events should we have seen had events not beenreported with delay?
▶ Event with large weight ⇒ expect to see more events at timepoint but don’t due to censoring
▶ Note: Kaplan-Meier estimator can be written as an IPCWestimator
10
Inverse Probability of Censoring Weighted (IPCW)Estimator
Correct for unreported events at interim time point due to delayedreporting to the database
▶ IPCW estimator proposed by Robins and Rotnitzky (1992)
▶ Weight contribution of observed events to the survivalestimate by the probability of not being censored
▶ How many events should we have seen had events not beenreported with delay?
▶ Event with large weight ⇒ expect to see more events at timepoint but don’t due to censoring
▶ Note: Kaplan-Meier estimator can be written as an IPCWestimator
10
Inverse Probability of Censoring Weighted (IPCW)Estimator
Correct for unreported events at interim time point due to delayedreporting to the database
▶ IPCW estimator proposed by Robins and Rotnitzky (1992)
▶ Weight contribution of observed events to the survivalestimate by the probability of not being censored
▶ How many events should we have seen had events not beenreported with delay?
▶ Event with large weight ⇒ expect to see more events at timepoint but don’t due to censoring
▶ Note: Kaplan-Meier estimator can be written as an IPCWestimator
10
Example: Kaplan-Meier Estimator as IPCW estimator
▶ Observe Yi =min(Ti ,Ci ), ∆i = I (Ti ≤ Ci )
▶ KM estimator: S(t) =t∏
s=0
(1−
∑i ∆i I (Yi = s)∑i I (Yi ≥ s)
)
▶ Let G (t) = P(C ≥ t)
▶ Gn(t) = KM estimate reversing the roles of censoring and
event times = Gn(t) =t∏
s=0
(1−
∑i (1−∆i )I (Yi = s)∑
i I (Yi ≥ s)
)Write KM estimator as IPCW estimator:
S(t) = 1− 1
n
n∑i=1
∆i I (Yi ≤ t)
Gn(t)
= S(t)
11
Example: Kaplan-Meier Estimator as IPCW estimator
▶ Observe Yi =min(Ti ,Ci ), ∆i = I (Ti ≤ Ci )
▶ KM estimator: S(t) =t∏
s=0
(1−
∑i ∆i I (Yi = s)∑i I (Yi ≥ s)
)▶ Let G (t) = P(C ≥ t)
▶ Gn(t) = KM estimate reversing the roles of censoring and
event times = Gn(t) =t∏
s=0
(1−
∑i (1−∆i )I (Yi = s)∑
i I (Yi ≥ s)
)
Write KM estimator as IPCW estimator:
S(t) = 1− 1
n
n∑i=1
∆i I (Yi ≤ t)
Gn(t)
= S(t)
11
Example: Kaplan-Meier Estimator as IPCW estimator
▶ Observe Yi =min(Ti ,Ci ), ∆i = I (Ti ≤ Ci )
▶ KM estimator: S(t) =t∏
s=0
(1−
∑i ∆i I (Yi = s)∑i I (Yi ≥ s)
)▶ Let G (t) = P(C ≥ t)
▶ Gn(t) = KM estimate reversing the roles of censoring and
event times = Gn(t) =t∏
s=0
(1−
∑i (1−∆i )I (Yi = s)∑
i I (Yi ≥ s)
)Write KM estimator as IPCW estimator:
S(t) = 1− 1
n
n∑i=1
∆i I (Yi ≤ t)
Gn(t)
= S(t)
11
Proposal: Weight contribution of first observedevent by probability of not being censored whenevent is reported with delay i.e. censoring time is
greater than event time plus delay time
12
Estimator
▶ Upweight based on probability of observing event that hasbeen reported with delay
Delay distribution H(d) = P(D ≤ d)
Censoring distribution (with delay) G∗(s) = P(C ≥ s + D)
=
∫H(c − s)dG (c)
Denominator estimate G∗n (s) =
n∑i=1
H(ci − s)dGn(ci )
▶ Numerator (monotone event reporting): First observed eventtime per subject
SMON(t) = 1− 1
n
n∑i=1
∫ t
0
I{N∗i (s−) = 0}dN∗
i (s)
G ∗n (s)
.
13
Estimator
▶ Upweight based on probability of observing event that hasbeen reported with delay
Delay distribution H(d) = P(D ≤ d)
Censoring distribution (with delay) G∗(s) = P(C ≥ s + D)
=
∫H(c − s)dG (c)
Denominator estimate G∗n (s) =
n∑i=1
H(ci − s)dGn(ci )
▶ Numerator (monotone event reporting): First observed eventtime per subject
SMON(t) = 1− 1
n
n∑i=1
∫ t
0
I{N∗i (s−) = 0}dN∗
i (s)
G ∗n (s)
.
13
Estimator
▶ Upweight based on probability of observing event that hasbeen reported with delay
Delay distribution H(d) = P(D ≤ d)
Censoring distribution (with delay) G∗(s) = P(C ≥ s + D)
=
∫H(c − s)dG (c)
Denominator estimate G∗n (s) =
n∑i=1
H(ci − s)dGn(ci )
▶ Numerator (monotone event reporting): First observed eventtime per subject
SMON(t) = 1− 1
n
n∑i=1
∫ t
0
I{N∗i (s−) = 0}dN∗
i (s)
G ∗n (s)
.
13
Estimator
▶ Upweight based on probability of observing event that hasbeen reported with delay
Delay distribution H(d) = P(D ≤ d)
Censoring distribution (with delay) G∗(s) = P(C ≥ s + D)
=
∫H(c − s)dG (c)
Denominator estimate G∗n (s) =
n∑i=1
H(ci − s)dGn(ci )
▶ Numerator (monotone event reporting): First observed eventtime per subject
SMON(t) = 1− 1
n
n∑i=1
∫ t
0
I{N∗i (s−) = 0}dN∗
i (s)
G ∗n (s)
.
13
Estimator
▶ Upweight based on probability of observing event that hasbeen reported with delay
Delay distribution H(d) = P(D ≤ d)
Censoring distribution (with delay) G∗(s) = P(C ≥ s + D)
=
∫H(c − s)dG (c)
Denominator estimate G∗n (s) =
n∑i=1
H(ci − s)dGn(ci )
▶ Numerator (monotone event reporting): First observed eventtime per subject
SMON(t) = 1− 1
n
n∑i=1
∫ t
0
I{N∗i (s−) = 0}dN∗
i (s)
G ∗n (s)
.
13
Issue with Non-Monotone Reporting
t11 V(t11)
t21 V(t21)
t31 t32 V(t31) V(t32)
t41 t42 V(t42) V(t41)
First observed event
▶ Can’t weight each observed event by probability that event is censoreddue to delay
▶ First observed event =⇒ first true event▶ Solution: Also weight each first observed event by probability that there
are no unreported events prior to it
▶ wi (s) = P(N∗∗i (s−) = 0 | Ci , {N∗(u), u < Ci})
SNM(t) = 1− 1
n
n∑i=1
∫ t
0
wi (s)I{N∗i (s−) = 0}dN∗
i (s)
G ∗(s).
14
Trial planning
▶ Event-driven trials are designed presuming some event rate(e.g. based on event rate from phase 2 trial)
▶ Reassessment of sample-size and followup time based onsurvival estimates at interim analysis
▶ Simulated example: trial enrollment 3.5 years, followup for 6months after enrollment of last subject, target number ofevents = 1000, exponential event rate µ = .5⇒ n = 1580
▶ Events may be reported out of order
15
Trial running on time, long delays
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
Survival Estimate Interim Analysis Time = 2
time
S(t
)
Composite KMReported KMMon IPCWNonmon IPCW
0 1 2 3 4
020
040
060
080
010
00
Study Time
Num
ber
of e
vent
s
True EventsReported EventsReported KMMon IPCWNonmon IPCW
Projected Number of Composite Events based on Interim Analysis
Based on interim projection using non-monotone IPCW estimator,trial on target for completion at year 4 ⇒ no adjustment toenrollment or follow-up time needed
16
Event rates higher than planned
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
Survival Estimate Interim Analysis Time = 2
time
S(t
)
Composite KMReported KMMon IPCWNonmon IPCW
0 1 2 3 4
020
040
060
080
010
00
Study Time
Num
ber
of e
vent
s
True EventsReported EventsReported KMMon IPCWNonmon IPCW
Projected Number of Composite Events based on Interim Analysis
Based on interim projection using non-monotone IPCW estimator,trial on target for completion around 3 years, 5 months ⇒ canshorten length of follow-up in trial
17
Conclusion
▶ Correct estimation of the survival function at the interimmonitoring time of event-driven trials is crucial to adequatelyplanning the future of the trial.
▶ If there are long delays between event occurrence andreporting to the database, this should be taken into accountin the survival estimate and event projection
▶ Proposed solution: Correct for delayed event ascertainmentusing IPCW estimator
▶ Note: May also need to adjust for non-monotone eventreporting
▶ Can use similar approach if event reporting is delayed,non-monotone, and event adjudication is required butincomplete at time of interim monitoring
18
Conclusion
▶ Correct estimation of the survival function at the interimmonitoring time of event-driven trials is crucial to adequatelyplanning the future of the trial.
▶ If there are long delays between event occurrence andreporting to the database, this should be taken into accountin the survival estimate and event projection
▶ Proposed solution: Correct for delayed event ascertainmentusing IPCW estimator
▶ Note: May also need to adjust for non-monotone eventreporting
▶ Can use similar approach if event reporting is delayed,non-monotone, and event adjudication is required butincomplete at time of interim monitoring
18
Conclusion
▶ Correct estimation of the survival function at the interimmonitoring time of event-driven trials is crucial to adequatelyplanning the future of the trial.
▶ If there are long delays between event occurrence andreporting to the database, this should be taken into accountin the survival estimate and event projection
▶ Proposed solution: Correct for delayed event ascertainmentusing IPCW estimator
▶ Note: May also need to adjust for non-monotone eventreporting
▶ Can use similar approach if event reporting is delayed,non-monotone, and event adjudication is required butincomplete at time of interim monitoring
18
Questions?
19