1. Subgroup analysis on time-to-event: a Bayesian approach
August 2019
Duy Ngo, Richard Baumgartner, Joseph Heyse, Shahrul Mt-Isa, Jie Chen, Dai Feng & Patrick Schnell (Ohio State University)

2. Overview
Identifying patient subgroups with favorable benefit from treatment is of interest to regulators and health technology assessment agencies worldwide
Evaluate Bayesian credible subgroups in a setting of survival analysis to identify the baseline covariate profiles of benefiting patients
Treatment effect for benefiting subgroup estimated using two summaries
log hazard ratio
restricted mean survival time (RMST)
Application of Bayesian credible subgroups in this setting offers several advantages
Avoids practice of testing for interaction
Addresses multiplicity
Does not require pre-specification of subgroups and work directly with the covariate space
Naturally makes statistical inferences from the full posterior distribution.
Methods applied to a case study of prostate carcinoma and simulated large clinical dataset.
PROPRIETARY ICONS HERE

3. Outline Background and Motivation
Bayesian Credible Subgroups for Survival Endpoints
Simulation Study
The Prostate Cancer Dataset
The Simulated Dataset from a Large Clinical Trial
Discussion and Conclusion
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4. Background and Motivation
Subgroup analysis for survival endpoints from a simulated data.
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5. Concepts
Goal: Finding subgroups of population that benefiting the treatment, i.e. estimate B.
B: unknown true benefit
D: evidence of benefit
S \ D: insufficient evidence (uncertainty region)
𝑆 𝐶 : no benefit.
PROPRIETARY ICONS HERE

6. Existing methods Existing methods to estimate B:
Testing for treatment-covariate interaction. Limitation: not appropriate for subgroup identification.
Tree-based methods. Limitation: instability.
Main challenges of subgroup analysis:
Lack of power to detect the overall main effect difference in response between treatment groups,
Multiplicity arising from simultaneous inference on all subpopulation members (predictive covariate points),
Post hoc analysis (unplanned analyses or data dredging),
Interpretation and conclusions (inference for each individual separately).
Schnell (2016) proposed Bayesian credible subgroup for continuous endpoints with several advantages:
Control multiplicity,
Easily make statistical inferences from the full posterior distribution.
PROPRIETARY ICONS HERE

7. Notation 𝑇≥0 denotes the response variable (failure or survival time).
Suppose that the observed time-to-event data consist 𝑛 independent subjects of ( 𝑌 𝑖 , 𝑥 𝑖 , 𝑧 𝑖 , 𝜅 𝑖 , 𝜃 𝑖 ) where
𝑌 𝑖 =min⁡( 𝑇 𝑖 , 𝐶 𝑖 ) and 𝐶 𝑖 is a random variable for censoring (right censoring),
𝑥 𝑖 and 𝑧 𝑖 be px1 and qx1 vectors of prognostic and predictive covariates, respectively,
𝜅 𝑖 is a censoring indicator, i.e. 𝜅 𝑖 =1 for 𝑇 𝑖 ≤ 𝐶 𝑖 and 0 otherwise,
𝜃 𝑖 ={0, 1} is a treatment indicator.
PROPRIETARY ICONS HERE

8. Personalized Treatment Effects
Average treatment effect (ATE) is the average over the entire population of the individual treatment effects.
Personalized treatment effect (PTE) is the treatment effect for a patient given their baseline characteristics.
Optimal treatment regimes find a set of decision rules to provide optimal treatment for a given patient. They are related to the subgroup analysis, but since their focus is on prediction for a single individual, they are not concerned with the multiplicity issues.
PROPRIETARY ICONS HERE

9. A Log of Hazard Ratio as a PTE
In a two-arms study with censoring, a traditional Cox regression model has the form:
𝜆 𝑡 𝑥, 𝑧, 𝜃 = 𝜆 0 𝑡 exp 𝑥 ′ 𝛽+𝜃 𝑧 ′ 𝛾 ,
where 𝜆 0 (𝑡) is a baseline hazard, 𝛽 and 𝛾 are regression coefficients.
The PTE for a patient with covariate 𝑧 is
Δ 𝐻 = 𝜆 𝑡 𝑥, 𝑧, 𝜃=1 𝜆 𝑡 𝑥, 𝑧, 𝜃=0 = exp 𝑧 ′ 𝛾 < 𝛿 𝐻 ,
where 𝛿 is a predetermined clinical significance.
Alternatively, the log of hazard ratio as a PTE is log Δ 𝐻 = 𝑧 ′ 𝛾< log 𝛿 𝐻 .
PROPRIETARY ICONS HERE

10. A Difference in Restricted Mean Survival Time (RMST)
The RMST is the area under a survival curve 𝑆(𝑡) between 𝑡=0 and 𝑡= 𝜈:
𝜓= 𝑜 𝜈 𝑆 𝑡 𝑑𝑡 .
The difference in RMST between two arms up to time point 𝜈 is
Δ 𝑅𝑑 = 𝜓 𝜃=1 − 𝜓 𝜃=0 = 𝑜 𝜈 [𝑆 𝑡| 𝜃=1 −𝑆 𝑡 𝜃=0)] 𝑑𝑡 ,
The PTE for a patient 𝑧 is Δ 𝑅𝑑 > 𝛿 𝑅𝑑 where 𝛿 𝑅𝑑 is a predetermined threshold of clinical significance.
We use conventional Cox proportional hazard model to estimate the two survival functions on the grid of subgroup-defining covariates in order to compute Δ 𝑅𝑑 .
PROPRIETARY ICONS HERE

11. Construct Bayesian Credible Subgroup
The two-step regression-classification procedure:
Step 1: Define a model, fit a regression and obtain the joint posterior of coefficients of predictive covariates (interacting with treatment choice),
Step 2: Computing the bounds and obtain a pair 𝐷, 𝑆 where 𝐷 is an exclusive credible subgroup and 𝑆 is an inclusive credible subgroup.
PROPRIETARY ICONS HERE

12. 𝜆 𝑡 𝑥 𝑖 , 𝑧 𝑖 , 𝜃 𝑖 = 𝜆 0 𝑡 exp 𝑥 𝑖 ′ 𝛽+ 𝜃 𝑖 𝑧 𝑖 ′ 𝛾 ,
Simulation Study
Suppose that the hazard function for 𝑖 𝑡ℎ subject is
𝜆 𝑡 𝑥 𝑖 , 𝑧 𝑖 , 𝜃 𝑖 = 𝜆 0 𝑡 exp 𝑥 𝑖 ′ 𝛽+ 𝜃 𝑖 𝑧 𝑖 ′ 𝛾 ,
where 𝑥= 𝑥 1 , 𝑥 2 ′ and 𝑧= 1, 𝑧 1 , 𝑧 2 ′.
Let 𝑥 1 = 𝑧 1 ={0, 1}, and 𝑥 2 = 𝑧 2 uniformly distributed on interval (-3, 3).
Let 𝜃= 0,1 , and 𝜆 0 𝑡 =𝜆𝑝 𝜆𝑡 𝑝−1 is a Weibull baseline hazard with 𝜆=0.05 and 𝑝=1.1.
Perform diagnostic test for credible subgroup for sample size 𝑛={50, 100, 500, 1000}, credible level at 0.8, and different settings for 𝛽 and 𝛾:
The prognostic features are with no or small effect 𝛽= 0, 0 , set 𝛾= 0,0,0 and (1,−1,3).
The prognostic features have moderate effect 𝛽= 0.2, 0.2 , set 𝛾= 1, 1, 1 .
The prognostic features have higher effect 𝛽= 1, −2 , set 𝛾= 1, 0.1, 1 .
Each scenario, we simulate 1000 datasets, and for each dataset, we use 1000 posterior draw kept after 500 burn- in iteration when we perform Bayesian method for Cox model. .
PROPRIETARY ICONS HERE

13. Simulation Study We use 𝛿 𝐻 =1 and 𝛿 𝑅 =0.
We report the performance of Bayesian credible subgroup in the following criteria:
Total coverage: the frequency with which 𝐷⊂𝐵⊂𝑆 under a fixed value 𝛾.
Pair size: proportion of the population included in the uncertainty region, i.e. 𝑃 𝑧∈𝑆∖𝐷 𝐷, 𝑆) with uniform measure on 𝑧.
Specificity and Sensitivity of D: how well the credible subgroup align with the benefiting group.
PROPRIETARY ICONS HERE

14. Simulation Results 𝛿 𝐻 =1, 𝛿 𝑅 =0, and credible level 0.8.
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15. Simulation Results 𝛿 𝐻 =1, 𝛿 𝑅 =0, and credible level 0.8
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16. The Prostate Cancer Dataset
The prostate cancer dataset has been analyzed in literature for exploratory subgroup analysis.
Ballarini et al. (2018) proposed a multiple regression model with a Lasso-type penalty to estimate benefiting subgroups.
The dataset includes 475 patients who were randomly assigned either to a combination of placebo and the lowest does level of diethyl stilbestrol (control group) or the higher doses (treatment group).
The interest covariates are: existence of bone metastasis (bm), disease stage either 3 or 4 (stage), performance (pf), history of cardiovascular events (hx), age and weight (wt).
Denote rx as treatment indicator, and include the two interactions bm:rx and age:rx in the model.
PROPRIETARY ICONS HERE

17. Result
A log of hazard ratio as a PTE:
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18. Result
A difference in RMST as a PTE:
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19. The Simulated Dataset Motivated by a Large Clinical Trial.
The simulated dataset is motivated by a large clinical trial reported in Scirica et.al. (2012)
A simulated dataset pertains to patients of whom 8898 were assigned to treatment and 8881 were assigned to placebo.
There are 5 variables to consider: age at entry (years), baseline weight (kilograms), history of hyperlipidemia, smoking status and prior coronary revascularization.
For each treatment group, we randomly selected 20% of subjects and added a Gaussian noise with zero mean and standard deviation of 1 and 5 for continuous covariates age and baseline weight, respectively.
The primary efficacy endpoint is the time of first cardiovascular death, myocardial infarction or stroke. The median follow–up was 2.5 years (IQR years).
Goal: search for benefiting subgroups without prespecified subgroups of interest.
PROPRIETARY ICONS HERE

20. Baseline characteristics
Continuous variable: Median (IQR). Categorical variable: percentage.
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21. Result
A log of hazard ratio as a PTE
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22. Result
A RMST differences
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23. Discussion
We introduced a Bayesian credible subgroup for time-to-event data by using a log of hazard ratio and the difference in RMST as PTEs.
Limitations: parametric model, model selection and missing covariates.
Research in subgroup analysis is mainly focused on assessment of benefit.
It is desirable to assess both benefit and risk in subgroup analysis.
Potential approach: Extend Bayesian credible subgroup method for multiple treatments and multiple endpoints.
Multiple endpoints can include both benefit and risk.
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24. Thank you
Questions
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25. BACK UP
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26. Bayesian Credible Subgroups
Let 𝑍 be a covariate space, a goal of Bayesian credible subgroups searches the covariate points 𝑧 such that
𝐵 𝐻 = 𝑧∈𝑍: Δ 𝐻 𝑧 < 𝛿 𝐻 ,
for HR case, and
𝐵 𝑅𝑣 = 𝑧∈𝑍: Δ 𝑅𝑣 𝑧 > 𝛿 𝑅
for a RMST difference case.
In Bayesian framework, common estimators are
𝐵 𝐻, 𝛼 = 𝑧∈𝑍: 𝑃(Δ 𝐻 𝑧 < 𝛿 𝐻 ) | 𝐷𝑎𝑡𝑎)>1−𝛼
𝐵 𝑅𝑣, 𝛼 = 𝑧∈𝑍: P(Δ 𝑅𝑣 𝑧 > 𝛿 𝑅 𝐷𝑎𝑡𝑎 >1−𝛼
PROPRIETARY ICONS HERE

27. Construct Bayesian Credible Subgroups
Step 1: Define the model, fit a regression and obtain the marginal posterior of 𝛾
Model: 𝜆 𝑖 𝑡 𝑥 𝑖 , 𝑧 𝑖 , 𝜃 𝑖 = 𝜆 0 𝑡 exp 𝑥 𝑖 ′ 𝛽+ 𝜃 𝑖 𝑧 𝑖 ′ 𝛾
Priors: We specify a joint priors for all unknown parameters (𝛽, 𝛾, 𝜆 0 ) as followings:
𝑃 𝛽, 𝛾, 𝜆 0 =𝑃 𝛽, 𝛾 𝑃( 𝛾 0 ),
𝛽, 𝛾 ∼𝑁 𝜇 0 , Σ 0 where 𝑁 𝜇 0 , Σ 0 is he multivariate normal distribution with 𝑝+𝑞 ×1 mean vector 𝜇 0 and a 𝑝+𝑞 ×(𝑝+𝑞) covariance matrix Σ 0 ,
We choose a nonparametric gamma process prior on the baseline hazard 𝜆 0 .
Obtain the posterior of 𝛾 by using Gibbs sampling (from an R package spBayesSurv)
Obtain the posterior for PTEs Δ.
PROPRIETARY ICONS HERE

28. Construct Bayesian Credible Subgroups
Step 2: Compute the bounds and obtain a pair (𝐷, 𝑆)
The simultaneous credible bands for Δ(z) on a covariate space 𝑍 is
Δ 𝑧 ∈ Δ 𝑧 ± 𝑊 𝛼 𝑉𝑎𝑟(Δ(𝑧)) ,
where 𝑊 𝛼 is the 1 −𝛼 quantile of the distribution of 𝑊= sup 𝑧∈𝑍 Δ 𝑧 − Δ 𝑧 2 𝑉𝑎𝑟(Δ(𝑧)) and Δ z is the posterior mean of Δ 𝑧 .
In a case of Δ 𝑧 ≡ Δ 𝐻 (𝑧), the exclusive credible subgroup 𝐷= 𝑧∈𝑍: Δ 𝑧 + 𝑊 𝛼 𝑉𝑎𝑟 Δ 𝑧 , and the inclusive credible subgroup S= 𝑧∈𝑍: Δ 𝑧 − 𝑊 𝛼 𝑉𝑎𝑟 Δ 𝑧
PROPRIETARY ICONS HERE

29. Simulate Time-to-event Data
Suppose that the hazard function of the 𝑖 𝑡ℎ individual is 𝜆 𝑖 𝑡 𝑥 𝑖 , 𝑧 𝑖 , 𝜃 𝑖 = 𝜆 0 𝑡 exp 𝑥 𝑖 ′ 𝛽+ 𝜃 𝑖 𝑧 𝑖 ′ 𝛾 .
We assume a Weibull baseline hazard, i.e. 𝜆 0 𝑡 =𝜆𝑝 𝜆𝑡 𝜈 −1 where 𝜆 and 𝜈 are the scale and shape parameters, respectively.
If 𝑈 is uniformly distributed on [0, 1], the survival time 𝑇 𝑖 =− log 𝑈 𝜆 exp 𝑥 𝑖 ′ 𝛽+ 𝜃 𝑖 𝑧 𝑖 ′ 𝛾 𝜈 .
Suppose that 𝐶 𝑖 ∼𝐸𝑥𝑝(𝑏) is censoring time. Due to censoring, we observe 𝑌 𝑖 = min ( 𝑇 𝑖 , 𝐶 𝑖 ) with a censoring indicator 𝜅 𝑖 .
PROPRIETARY ICONS HERE

30. Simulation study under nonproportional hazard assumption
When the PH assumption is violated, the HR may not accurate represent PTEs and RMST is an alternative approach.
We simulated two groups with different hazard rates. The treatment group ( 𝜃 𝑖 =1) had a constant exponential hazard with rate 𝜆 0 𝑡 =0.01. The control group had a piecewise exponential hazard with rate 𝜆 0 𝑡 =0.01 for 0≤𝑡< 𝑡 𝑐 , and 𝜆 0 𝑡 =0.1 for 𝑡 𝑐 ≤t.
We use similar settings for the prognostic and predictive covariates in simulation study under PH assumption.
We consider 𝛽= 0.7, 0.7 and 𝛾= 0.5, −0.5, −0.5 .
We chose 𝛿 𝑅𝑑 =0 and 𝑡 𝐶 =30. The RMSTs were computed at the change point 𝑡 𝐶 up to 𝑡 𝐶 +50.
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31. Simulation study under nonproportional hazard assumption
Average summary statistics for 80% credible subgroup pairs under nonproportional hazard assumption
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