sample size re-estimation of event-driven trials with ... - sample size... · with composite events...

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

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


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


▶ 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

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




s=0

(1−

∑i ∆i I (Yi = s)∑i I (Yi ≥ s)

)▶ Let G (t) = P(C ≥ 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




s=0

(1−

∑i ∆i I (Yi = s)∑i I (Yi ≥ s)

)▶ Let G (t) = P(C ≥ 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

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


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


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


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


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

Questions?

19

sample size re-estimation of event-driven trials with ... - sample size... · with composite events...

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