Statistical Methods for Clinical Trials with Treatment Switching

Transcription

1 Statistical Methods for Clinical Trials with Treatment Switching Fang-I Chu Prof. Yuedong Wang Department of Statistics and Applied Probability University of California Santa Barbara July 28, 2014 Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

2 Outline Introduction Problem: Treatment Switching Existing Methods Proposed Model: AFT model with Frailty Model Estimation Simulation Study Result and Discussion Summary Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

3 Treatment Switching In clinical trials, patients are assigned randomly to either control or treatment group. Treatment switching occurs when patients fail to comply with assigned regime. Drop-in: control group to treatment group. Drop-out: treatment group to control group. Fail to account for drop-in or drop-out case will lead to biased conclusion about evaluation of treatment effect. We focus mainly on drop-in cases. Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

4 Treatment Switching Figure : Treatment switching: drop-in Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

5 Existing Methods Intention-To-Treat and Per- Protocal[Fergusson et al.(2002)fergusson, Aaron, Guyatt, and Herbert] Adjusted Hazard Ratio Method [Law and Kaldor(1996)] Accelerated Failure Time Model (Counterfactual Time approach) Casual model form [Robins and Tsiatis(1991)], [Branson and Whitehead(2002)] Joint Parametric model [Walker et al.(2004)walker, White, and Babiker] With Switching Effect [Shao et al.(2005)shao, Chang, and Chow] Semi-Parametric Semicompeting Risks Transition Approach [Zeng et al.(2012)zeng, Chen, Chen, Ibrahim, and research group] Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

6 Notations: (1). Randomization regime R, 1 as treatment and 0 as control (2). Progression status indicator U, 1 as progression before failure and 0 as no progression before failure. (3). Switching status indicator V, 1 as switch and 0 as no switch. (4). T D,T U, and T G denote time to failure, time to disease progression and time from disease progression to failure, respectively. (5). h D (t R, X, U = 0), h U (t R, X, U = 1) and h G (t R, Z, V, U = 1, T U ) are the conditional hazard functions of T D, T U, and T G, respectively. (6). S D (t R, X, U = 0), S U ((t R, X, U = 1, ω) and S G (t R, Z, V, U = 1, T U, ω) are survival function of T D, T U, and T G, respectively. (7). h 0 (t), h 1 (t), h 2 (t), S 0 (t), S 1 (t) and S 2 (t) are unknown baseline hazard and survival functions. (8). X represents baseline covariates; Z reflects covariates collected at baseline and at progression. Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

7 Existing Method: Semi-parametric semicompeting risks transition model Strategy: [Zeng et al.(2012)zeng, Chen, Chen, Ibrahim, and research group] proposed model which contains four components (1). model to determine the progression status U: logit{pr(u = 1 R, X )} = α 0 + α 1 R + α 2 X (2). hazard function for T D : h D (t R, X, U = 0) = h 0 (t)exp [β 00 R + β 01 X ] (3). hazard function for T U : h U (t R, X, U = 1) = h 1 (t)exp [β 10 R + β 11 X ] (4). hazard function for T G : h G (t R, Z, V, U = 1, T U ) = h 2 (t)exp [ β 20 R + β 21 V (1 R) + β T 2 (Z T, T U ) T ] Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

8 Semi-parametric semicompeting risks transition model Figure : Illustration of the model Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

9 Existing method: Semi-parametric semicompeting risks transition model Advantage: (1). Model the time to the intermediate event and time from the intermediate event to failure. (2). Account for heterogeneity at baseline and the heterogeneity at treatment switching. Limitation: The hazards of two individuals at time t is related by a proportionality constant that does not depend on t, which may be too restrictive for some applications. Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

10 Proposed Model: AFT model with frailty Assumption Hazard of encountering event of death and progression depends on time. Frailty term accounts for random factor from subjects. Given frailty term, the survival functions for time to progression and time of progression to failure are independent to each other. (by definition of shared frailty) Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

11 Proposed Model: AFT model with frailty Consider T U, and T G in log-linear form with a frailty term, ω, and T D in the same form without a frailty term as log T D = µ D + θ 00 R + θ 01 X + σ D ɛ D, log T U = µ U + θ 10 R + θ 11 X + ( ω) + σ U ɛ U, log T G = µ G + θ 20 R + θ 21 V (1 R) + θ T 2 (Z T, T U ) T + θ 3 ( ω) + σ G ɛ G. where µ D, µ U, µ G, σ D, σ U and σ G are unknown location and scale parameters, while ɛ D, ɛ U and ɛ G have some specific probability distribution. Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

12 Proposed Model: AFT model with frailty Model Form S D (t R, X, U = 0) = S 0 (exp [β 00 R + β 01 X] t) S U,G (t u, t g R, X, U = 1, ω) = S U (t u R, X, U = 1, ω)s G (t g R, V, U = 1, ω) = S 1 (exp [β 10 R + β 11 X + ω] t u ) S 2 (exp[β 20 R + β 21 V (1 R) + β22(z T T, T U ) T + β 3 ω]t g ) Note: where θ 00 = β 00, θ 01 = β 01, θ 10 = β 10, θ 11 = β 11, θ 20 = β 20, θ 21 = β 21, θ 2 = β 2 and θ 3 = β 3. Given frailty ω, AFT model of T D, T U and T G are assumed as above. The progression status is determined by logit model: logit{pr(u = 1 R, X)} = α 0 + α 1 R + α 2 X Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

13 Proposed Model: AFT model with frailty Advantage and Limitation Advantage: Accommodate the situation that the hazard depends on time. The relation between time to progression and time of progression to death is captured by (1) conditional survival function (2) frailty term. Limitation: Require parametric model assumption. Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

14 Estimation: Likelihood function The likelihood are constructed through four categorized subgroups: 1. No progression event is observed, failure occurs before progression. L 1 (θ) = h D (t X, R i )S D (t X, R) P(U = 0 R, X)f X (x R)P(R) 2. Progression event is observed, failure occurs after progression. L 2 (θ) = h U (t u R, X, U = 1, ω) h G (t g R,, U = 1, ω) S U,G (t u, t g R, X, ω) P(U = 1 R, X)f X (x R)P(R) Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

15 Estimation: Likelihood function 3. Progression event is observed, failure event is censored. L 3 (θ) = h U (t u R, X, U = 1, ω) S U,G (t u, t g R, X, ω) P(U = 1 R, X)f X (x R)P(R) 4. No progression event is observed, failure event is censored L 4 (θ) = [S D (t X, R)P(U = 0 R, X) + S U (t u R, X, U = 1, ω) P(U = 1 R, X)]f X (x R)P(R) Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

16 Estimation: Likelihood function Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

17 Simulation Study All survival times are assumed to follow exponential distribution. Frailty ω follows standard normal. α 0 = 1.6, α 1 = 1.8, α 2 = 1 and α 3 = 0.1 determine progression status. Let β 00 = 1, β 01 = 1, β 02 = 0.2, β 10 = 0.5, β 11 = 1, β 12 = 0, β 20 = 0.3, β 21 = 0.5, β 22 = 0.6, β 23 = 0.5, β 24 = 0.5, β 25 = 0.4 and β 3 = 0.2. The duration time is set to be τ = 3. The censoring time is generated from a uniform distribution on (1, 7); the scheme is assumed to take the minimum of two. A special case with β 3 = 1 that we do not estimate β 3 is also considered. The simulation study is conduct for sample sizes of n = 400 and n = The results from 100 replicates for general model is summarized. The setting is similar to those in [Zeng et al.(2012)zeng, Chen, Chen, Ibrahim, and research group]. Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

18 Simulation Study: AFT with frailty for n = 400 n = 400 True EST Bias SD ESE MSE CP(%) σ Susceptibility Model α α α α Survival model of no-progression population β β β Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

19 Simulation Study: AFT with frailty for n = 400 cont d Disease progression model of progression population β β β Gap time model of progression population β β β β β β β Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

20 Simulation Study: AFT with frailty for n = 1000 Table : Scenario: frailty follows standard normal for large sample. n = 1000 True EST Bias SD ESE MSE CP(%) σ Susceptibility Model α α α α Survival model of no-progression population β β β Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

21 Simulation Study: AFT with frailty for n = 1000 cont d Table : Scenario: frailty follows standard normal for large sample. Disease progression model of progression population β β β Gap time model of progression population β β β β β β β Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

22 Comparison of MSE with existing model Disease progression model of progression population AFT Semin MSE(β 10 ) MSE(β 11 ) MSE(β 12 ) Gap time model of progression population MSE(β 20 ) MSE(β 21 ) MSE(β 22 ) MSE(β 23 ) MSE(β 24 ) MSE(β 25 ) Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

23 Comparison of MSE with existing model Susceptibility Model AFT Semin MSE(α 0 ) MSE(α 1 ) MSE(α 2 ) MSE(α 3 ) Survival model of no-progression population MSE(β 00 ) MSE(β 01 ) MSE(β 02 ) AFT n MSE(σ 2 ) MSE(β 3 ) Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

24 Result and Discussion Simulation work is implemented in R and SAS. As sample size increases, the MSE of all estimators in proposed model appear to be as good as the ones in existing methods, while proposed model accommodate the situation of time-dependent hazard and the variation among subjects. The scaling parameter, β 3, for frailty term in function for T G, models the different post-progression impact on survival time among subjects The casual effect of treatment is obtainable by inserting obtained estimates in the derived expression of survival function for control and treatment arm. Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

25 Summary The use of AFT model rather than hazard model accommodate the situation that the hazard depends on time. Proposed AFT models with frailty captures the relation between time to progression and time from progression to death through (1) conditional survival function (2) frailty term All estimators appear to be unbiased and efficient in simulation study. Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

26 M. Branson and J. Whitehead. Estimating a treatment effect in survival studies in which patients switch treatment. Statistics in Medicine, 21: , D. Fergusson, S. D. Aaron, G. Guyatt, and P. Herbert. Post-randomisation exclusions: the intention o treat principle and excluding patients from analysis. British Medical Journal, 325: , M. G. Law and J. M. Kaldor. Survival analyses of randomized clinical trials adjusted for patients who switch treatments. Statistics in Medicine, 15: , J. M. Robins and A. A. Tsiatis. Correcting for non-compliance in randomized trials using rank preserving structural failure time models. Communications in Statistics-Theory and Methods, 20: , Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25

27 J. Shao, M. Chang, and S. C. Chow. Statistical inference for cancer trails with treatment switching. Statistics in Medicine, 24: , A. S. Walker, I. R. White, and A. G. Babiker. Parametric randomization-based methods for correcting for treatment changes in the assessment of the causal effect of treatment. Statistics in Medicine, 23: , D. Zeng, Q. Chen, M. H. Chen, J. G. Ibrahim, and Amgen research group. Estimating treatment effects with treatment switching via semi competing risks models: an application to a colorectal cancer study. Biometrika, 99(1): , Fang-I Chu Prof. Yuedong Wang (UCSB) Stat. Meth. for Treatment Switching July 28, / 25