1. To change this title, go to Notes Master
4/16/2017
Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data.
Michael O’Kelly, Quintiles
Ilya Lipkovich, Quintiles

2. Acknowledgements DIA Scientific Working Group (SWG) for Missing Data
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
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.

3. 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.

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* Zidovudine Zidovudine+ Didanosine Enrolled 619
613
Event: 50% decline in CD4
182 98
Censored
437
515
Completed study
313
384
Other reasons
124
131

6. ACTG 175: HIV study

7. 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)

8. 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.

9. 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)

10. How make p(event) worse than KM in a statistically principled way?
Inversion method
Case 1: assuming CAR
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
p(event) = 1- p(no event)
This is missing. To impute, first calculate prob(no event) associated with time of censoring.
Interpolate between events, if necessary.

12. How make p(event) worse than KM in a statistically principled way?
Inversion method
Case 1: assuming CAR
p(event) = 1- p(no event)
Imputed p(event|T>t) = U (1-p(no event), 1)
This is missing. To impute, first calculate prob(no event) associated with time of censoring.
Interpolate between events, if necessary.

13. ACTG 175: HIV study

14. ACTG 175: HIV study
Impute event for this censored subject.

15. ACTG 175: HIV study
1 – U[1-p(no event), 1]

16. Imputed time of event, case 1
ACTG 175: HIV study
Imputed time of event, case 1
1 – U(1-p(no event), 1)

17. Imputed time of event, case 2
ACTG 175: HIV study
Imputed time of event, case 2
1 – U(1-p(no event), 1)

18. Case 3: imputation results in censoring
ACTG 175: HIV study
Case 3: imputation results in censoring
1 – U(1-p(no event), 1)

19. 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)

20. How make p(event) worse than KM in a statistically principled way?
Inversion method
Case 1: assuming CAR + some δ.
p(event) = 1- p(no event)
This is missing. To impute, first calculate prob(no event) associated with time of censoring.
Interpolate between events, if necessary.

21. How make p(event) worse than KM in a statistically principled way?
Inversion method
Case 1: assuming CAR + some δ.
p(event) = 1- p(no event)
Imputed p(event|T>t) = U (1-p(no event)δ, 1)
This is missing. To impute, first calculate prob(no event) associated with time of censoring.
Interpolate between events, if necessary.

22. ACTG 175: HIV study
p(no event)

23. reference line for p(no event) δ, δ = 2
ACTG 175: HIV study
reference line for p(no event) δ, δ = 2

24. ACTG 175: HIV study Imputed time of event, δ = 2
1 – U(1-p(no event)δ, 1)

25. Imputed time of event, no δ
ACTG 175: HIV study
Imputed time of event, no δ
1 – U(1-p(no event), 1)

26. ACTG 175: HIV study Imputed time of event, δ = 2 Imputed event times
1 – U(1-p(no event)δ, 1)
Imputed event times
tend to be shorter as
δ increases

27. ACTG 175: HIV study Imputed time of event, δ = 2 Note: this is just
1 – U(1-p(no event)δ, 1)
Note: this is just
single imputation!

28. 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.

29. ACTG 175: HIV study Bootstrapped data set #1 Bootstrapped data set #2

30. ACTG 175: HIV study p(no event) = 0.958 p(no event) = 0.947
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

31. ACTG 175: HIV study Imputations include variability from
U() and from the
differences in
bootstrapped
data sets
1 – U(1-p(no event), 1)

32. ACTG 175: HIV study Imputed p(no event) is applied to the original
data set

33. ACTG 175: HIV study
1 – U(1-p(no event)δ, 1)

34. ACTG 175: HIV study Imputed p(no event) is applied to the original
data set, with δ
applied

35. ACTG 175: HIV study
Sample imputations with and without δ might look like this...

36. ACTG 175: HIV study
1 – U(1-p(no event), 1)

37. ACTG 175: HIV study
1 – U(1-p(no event)δ, 1)

38. Result of tipping point analysis for HIV studyδ
Logrank statistic*
Standard error+
p-value 1
5.58
1.031
<0.0001
1.5
5.17
1.057 2
4.82
1.102
2.5
4.50
1.090 3
4.06
1.121
0.0003
3.5
3.68
1.108
0.0010 4
3.53
1.131
0.0019
4.5
3.27
1.211
0.0076 5
2.90
1.130
0.0105
5.5
2.54
1.233
0.0413 6
2.35
1.199
0.0516
6.5
2.17
1.183
0.0674 7
1.99
1.210
0.1019
*chi-squared statistic transformed to normal using Wilson-Hilferty transformation
+transformed statistic has variance = 1; standard error includes between-imputation variability

39. 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
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.

40. 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
Lu X, Tsiatis, A (2008) Improving the efficiency of the log-rank test using auxiliary covariates, Biometrika
Taylor J, Murray S, Hsu C-H (2002) Survival estimation and testing via multiple imputation. Statistics and probability letters
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.

41. Questions?