Using multiple imputation
and delta adjustment to
implement sensitivity
analyses for time-to-event
data.
Michael O’Kelly, Quintiles
Ilya Lipkovich, Quintiles
Copyright © 2013 Quintiles
Acknowledgements
• DIA Scientific Working Group (SWG) for Missing Data
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This presentation stems from work with Bohdana Ratitch (inVentiv Health).
The authors of these slides are members of the SWG.
Chair: Craig Mallinckrodt, Eli Lilly.
James Roger and Mouna Akacha, speakers at this session, are also members.
Great downloadable SAS macros for control-based imputation and other MNAR
approaches available SWG webpage at www.missingdata.org.uk.
> SWG members have growing interest in discrete endpoints with missing data.
• Gary Koch (University of North Carolina)
> regular advice;
> in press, with Zhao and others: describes the approach used in this presentation.
• Taylor and others (2002) showed how to implement multiple imputation for
time-to-event outcomes.
• Michael Hughes (Harvard School of Public Health) kindly shared the example
time-to-event data.
2
ACTG 175: HIV study*
• Subjects randomized to four antiretroviral regimes in equal proportions.
• Primary event analysed: 50% decline in CD4 count or death.
• Study start Dec1991; enrolment ended Oct1992; follow-up until end Nov1994
> max follow-up just 4 years.
• For this presentation, we examine two treatment arms
> zidovudine
> zidovudine+didanosine.
* Lu and Tsiatis (2008); Hammer et al. (1996);
the analyses are by O’Kelly and are not the responsibility of the authors of the cited papers.
3
ACTG 175: HIV study*
Zidovudine+
Zidovudine Didanosine
Enrolled
619
613
Event: 50% decline in CD4
182
98
Censored
437
515
Completed study
313
384
Other reasons
124
131
4
ACTG 175: HIV study*
Zidovudine+
Zidovudine Didanosine
Enrolled
619
613
Event: 50% decline in CD4
182
98
Censored
437
515
Completed study
313
384
Other reasons
124
131
5
ACTG 175: HIV study
6
Kaplan-Meier analysis
Logrank statistic
46.12
Standard error
7.726
p-value
<0.0001
Assumes censoring at random (CAR).
(CAR is analogous to missing at random)
7
How robust is this result?
• How robust is this result to the assumption of CAR?
• One way to assess this: tipping point analysis.
• Tipping point for continuous variable:
> Add unfavourable quantity δ to efficacy score when imputed for experimental arm;
> Make δ more extreme until the p-value from the primary analysis is no longer
significant – the “tipping point”.
> Was the “tipping point” δ clinically plausible for subjects who withdrew early?
> If not, the primary result may be judged robust to the missing-at-random
assumption.
8
Tipping point for time to event,
Kaplan-Meier (KM) analysis
• Impute time of event using some hazard worse by δ than that estimated by
Kaplan-Meier.
• Make δ more extreme until the p-value from the primary analysis is no longer
significant – the “tipping point”.
> Was the “tipping point” δ clinically plausible for subjects who withdrew early?
> If not, the primary result may be judged robust to the CAR assumption.
• Note unstatistical terminology in following slides:
• “p(no event)” = p(T>t)
• “p(event)” = p(T<=t)
9
How make p(event) worse than KM
in a statistically principled way?
• Inversion method
• Case 1: assuming CAR
> p(event) = 1- p(no event)
10
How make p(event) worse than KM
in a statistically principled way?
• Inversion method
• Case 1: assuming CAR
This is missing. To impute, first
calculate prob(no event)
associated with time of censoring.
Interpolate between events, if
necessary.
> p(event) = 1- p(no event)
11
How make p(event) worse than KM
in a statistically principled way?
• Inversion method
• Case 1: assuming CAR
This is missing. To impute, first
calculate prob(no event)
associated with time of censoring.
Interpolate between events, if
necessary.
> p(event) = 1- p(no event)
> Imputed p(event|T>t) = U (1-p(no event), 1)
12
ACTG 175: HIV study
13
ACTG 175: HIV study
Impute event for this censored subject.
14
ACTG 175: HIV study
1 – U[1-p(no event), 1]
15
ACTG 175: HIV study
Imputed
time of
event,
case 1
1 – U(1-p(no event), 1)
16
ACTG 175: HIV study
Imputed
time of
event,
case 2
1 – U(1-p(no event), 1)
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ACTG 175: HIV study
Case 3:
imputation
results in
censoring
1 – U(1-p(no event), 1)
18
How make p(event) worse than KM
in a statistically principled way?
• Inversion method
• Case 2: assuming CAR + some δ.
> p(event) = 1- p(no event)
19
How make p(event) worse than KM
in a statistically principled way?
• Inversion method
This is missing. To impute, first
calculate prob(no event)
• Case 1: assuming CAR + some δ.
associated with time of censoring.
Interpolate between events, if
necessary.
> p(event) = 1- p(no event)
20
How make p(event) worse than KM
in a statistically principled way?
• Inversion method
This is missing. To impute, first
calculate prob(no event)
• Case 1: assuming CAR + some δ.
associated with time of censoring.
Interpolate between events, if
necessary.
> p(event) = 1- p(no event)
> Imputed p(event|T>t) = U (1-p(no event)δ, 1)
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ACTG 175: HIV study
p(no event)
22
ACTG 175: HIV study
reference line for p(no event) δ, δ = 2
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ACTG 175: HIV study
Imputed
time of
event,
δ=2
1 – U(1-p(no event)δ, 1)
24
ACTG 175: HIV study
Imputed
time of
event, no
δ
1 – U(1-p(no event), 1)
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ACTG 175: HIV study
Imputed
time of
event,
δ=2
1 – U(1-p(no event)δ, 1)
Imputed event times
tend to be shorter as
δ increases
26
ACTG 175: HIV study
Imputed
time of
event,
δ=2
1 – U(1-p(no event)δ, 1)
Note: this is just
single imputation!
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How to use multiple imputation
here?
• Bootstrap original data set.
• Calculate p(no event)δ associated with time of censoring, using the bootstrap
KM estimates of p(no event).
• Use inversion to find corresponding time on original data set.
28
ACTG 175: HIV study
Bootstrapped data set #1
Bootstrapped data set #2
Bootstrapped data set #3
Bootstrapped data set #4
29
ACTG 175: HIV study
Bootstrap
approximates
variability of
draws from
posterior
distribution
needed for MI
p(no event)
= 0.958
p(no event)
= 0.947
p(no event)
= 0.952
p(no event)
= 0.950
30
ACTG 175: HIV study
Imputations
include
variability from
U() and from the
1 – U(1-p(no event), 1)
differences in
bootstrapped
data sets
31
ACTG 175: HIV study
Imputed
p(no event)
is applied to
the original
data set
32
ACTG 175: HIV study
1 – U(1-p(no event)δ, 1)
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ACTG 175: HIV study
Imputed
p(no event)
is applied to
the original
data set, with δ
applied
34
ACTG 175: HIV study
Sample imputations
with and without δ
might look like this...
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ACTG 175: HIV study
1 – U(1-p(no event), 1)
36
ACTG 175: HIV study
1 – U(1-p(no event)δ, 1)
37
Result of tipping point analysis for
HIV study
δ
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Logrank Standard
statistic* error+
5.58
1.031
5.17
1.057
4.82
1.102
4.50
1.090
4.06
1.121
3.68
1.108
3.53
1.131
3.27
1.211
2.90
1.130
2.54
1.233
2.35
1.199
2.17
1.183
1.99
1.210
p-value
<0.0001
<0.0001
<0.0001
<0.0001
0.0003
0.0010
0.0019
0.0076
0.0105
0.0413
0.0516
0.0674
0.1019
*chi-squared statistic transformed to normal using Wilson-Hilferty transformation
+transformed statistic has variance = 1; standard error includes between-imputation variability
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What if primary analysis is Cox
prop’l hazards or parametric?
• Implementation of MI version of Cox proportional hazards is similar to that of
KM.
• Other implementations of MI for time-to-event analysis in progress by
Lipkovich and Ratitch:
> logistic regression (suggested by Carpenter and Kenward (2013));
> piecewise exponential.
• SAS macros for all four approaches planned to be available at DIA SWG web
page at www.missingdata.org.uk.
> tasks undertaken as part of DIA SWG “New Tools” subgroup.
• The above methods can also be used to implement “control based
imputation” for missing time to event outcomes.
39
References
• Carpenter J and Kenward M (2013) Multiple imputation and its application.
Chichester: Wiley.
• Hammer S, Katzenstein D, Hughes M, Gundaker H, Schooley R, Haubrich R,
Henry W, Lederman M, Phair J, Niu M, Hirsch M, and Merigan T, for the Aids
Clinical Trials Group Study 175 Study Team (1996). A trial comparing
nucleoside monotherapy with combination therapy in HIV-infected adults with
CD4 counts from 200 to 500 per cubic millimeter. The New England Journal
of Medicine 335 1081-1089.
• Lu X, Tsiatis, A (2008) Improving the efficiency of the log-rank test using
auxiliary covariates, Biometrika 95 679-694.
• Taylor J, Murray S, Hsu C-H (2002) Survival estimation and testing via
multiple imputation. Statistics and probability letters 6 77-91.
• Zhao Y, Herring A, Zhou H, Ali M, Koch G (submitted) A multiple imputation
method for sensitivity analyses of time-to-event data with possibly informative
censoring.
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Questions?
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