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Copyright © 2013 Quintiles 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 Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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Page 1: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

Copyright © 2013 Quintiles

Using multiple imputation and delta

adjustment to implement sensitivity analyses for time-to-

event data.Michael O’Kelly, Quintiles

Ilya Lipkovich, Quintiles

Page 2: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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 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.

Page 3: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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.

Page 4: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study*

ZidovudineZidovudine+Didanosine

Enrolled 619 613

Event: 50% decline in CD4 182 98

Censored 437 515

Completed study 313 384

Other reasons 124 131

Page 5: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study*

ZidovudineZidovudine+Didanosine

Enrolled 619 613

Event: 50% decline in CD4 182 98

Censored 437 515

Completed study 313 384

Other reasons 124 131

Page 6: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

Page 7: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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)

Page 8: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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.

Page 9: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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)

Page 10: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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How make p(event) worse than KM in a statistically principled way?

• Inversion method

• Case 1: assuming CAR

> p(event) = 1- p(no event)

Page 11: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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.

Page 12: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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.

Page 13: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

Page 14: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

Impute event for this censored subject.

Page 15: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U[1-p(no event), 1]

Page 16: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U(1-p(no event), 1)

Imputed time of event, case 1

Page 17: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U(1-p(no event), 1)

Imputed time of event, case 2

Page 18: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U(1-p(no event), 1)

Case 3: imputation results in censoring

Page 19: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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)

Page 20: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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.

Page 21: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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.

Page 22: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

p(no event)

Page 23: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

reference line for p(no event) δ, δ = 2

Page 24: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U(1-p(no event)δ, 1)

Imputed time of event, δ = 2

Page 25: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U(1-p(no event), 1)

Imputed time of

event, no δ

Page 26: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U(1-p(no event)δ, 1)

Imputed time of event, δ = 2

Imputed event times tend to be shorter asδ increases

Page 27: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U(1-p(no event)δ, 1)

Imputed time of event, δ = 2

Note: this is justsingle imputation!

Page 28: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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.

Page 29: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

Bootstrapped data set #1

Bootstrapped data set #3

Bootstrapped data set #2

Bootstrapped data set #4

Page 30: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

p(no event) = 0.958

p(no event) = 0.947

p(no event) = 0.950

p(no event) = 0.952

Bootstrapapproximatesvariability of draws from posterior distributionneeded for MI

Page 31: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U(1-p(no event), 1)

Imputations include variability from U() and from the differences in bootstrappeddata sets

Page 32: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

Imputed p(no event)is applied to the original data set

Page 33: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U(1-p(no event)δ, 1)

Page 34: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

Imputed p(no event)is applied to the original data set, with δapplied

Page 35: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

Sample imputations with and without δ might look like this...

Page 36: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U(1-p(no event), 1)

Page 37: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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ACTG 175: HIV study

1 – U(1-p(no event)δ, 1)

Page 38: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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Result of tipping point analysis for HIV study

δLogrank statistic*

Standard error+ p-value

1 5.58 1.031 <0.00011.5 5.17 1.057 <0.00012 4.82 1.102 <0.00012.5 4.50 1.090 <0.00013 4.06 1.121 0.00033.5 3.68 1.108 0.00104 3.53 1.131 0.00194.5 3.27 1.211 0.00765 2.90 1.130 0.01055.5 2.54 1.233 0.04136 2.35 1.199 0.05166.5 2.17 1.183 0.06747 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

Page 39: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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.

Page 40: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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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.

Page 41: Copyright © 2013 Quintiles Using multiple imputation and delta adjustment to implement sensitivity analyses for time-to-event data. Michael O’Kelly, Quintiles

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Questions?