1. Bayesian-based decision making in early oncology clinical trials
2017 Duke-Industry Statistics Symposium
Cassie Dong, Ph.D., Study lead statistician/lead for phase ii Bayesian Initiatives
senior manager, amgen

2. Acknowledgement
Amgen Hematology/Oncology TA Initiatives – Bayesian Phase I sub-team:
Dr. Haijun Ma for providing background material on Phase I designs.
Amgen Hematology/Oncology TA Initiatives – Bayesian Phase II sub-team members:
Michael Wolf, Chunlei Ke, Qi Jiang, Kejia Wang, Tian Dai, Xuena Wang, Qui Tran for their contribution/suggestion on the Phase II project.

3. Outline Overview Phase I Designs Phase II Designs
Bayesian Simulation Studies on Interim Futility
Discussion
Reference

4. Overview
Early phase studies are critical to increase a success rate of Phase III studies.
The aims of Phase I studies
First experimentation of a new drug / clinical procedure in human subjects
Find a safe, yet potentially effective, dose for future Phase II experimentation
Seek the highest possible dose subject to toxicity constraints, known as the maximum tolerated dose (MTD)
The aims of Phase II studies:
Evaluate the efficacy of the treatment at the recommended dose determined during the Phase I study;
Go/No Go decision for future Phase III studies (stop the trial due to futility or toxicity): screening out ineffective treatments

5. Overview Some common parts in Phase I/II.
Small sample size: Phase I (6 ~ 50), Phase II (30 ~ 200)
Usually use binary endpoint: Phase I (experience the DLT event or not), Phase II (responder or not).
Interim data are necessary or helpful for decision making during the trial:
Phase I: Based on the interim data, which dose should be given to the next patient/cohort, re-escalate/escalate/no change/stop?
Phase II: Based on the interim data, should we stop for futility/efficacy/toxicity or continue the trial?
Both Phase I/II can be single arm trials.
Both Phase I/II studies can be conducted under the Bayesian framework.

6. Phase I designs: 3+3
The aim of the Phase I studies is to find the MTD or a recommended dose for Phase II studies.
The traditional 3+3 is a rule-base design: a rigid, pre-specified design
Enroll 3 pts at the current does K
0/3 DLT:
Go to the next higher dose K+1
1/3 DLT:
3 more pts at the current dose K
≥ 2/3 DLT:
De-escalate to K -1
1/6 DLT:
Go to K+1
2/6 DLT:
MTD is K-1
> 2/6 DLT:
De-escalate K-1

7. Phase I designs: 3+3
3+3 and other rule based dose finding design are easy to implement by non-statisticians.
But it’s known that such designs have major limitations:
Low probability of finding the true MTD
High variability of the toxicity rate (wider confidence interval)
Not fully utilize the data (focus on the data at the current dose level) in addition to the small sample size …

8. Phase I designs: Continual Reassessment method (CRM)
First Bayesian model-based design introduced by O’Quigley et al., 1990.
Pre-specify the target toxicity level for Pr(DLT|x), say 33%. Then the MTD is the x with Pr(DLT|x) closest to 33% under the dose-toxicity curve.
Assume Pr(DLT|x,a) is a function of parameter a
With a non-informative prior of a
Treat 1 pt at x closest to the current MTD
Given the observed the data (1 or 0), update the Posterior Pr: Pr(a|data);
Estimate the Pr(DLT|x,a) at all dose level x and the posterior mean of a.
Choose the dose with Pr(DLT) closest to the target level, say 33%.
Target Pr(DLT) = 0.33
MTD
Modified from H. J . Ma, slides of “Bayesian Designs for Oncology Phase I Dose Escalation Trials”

9. Phase I designs: Escalation with overdose control (EWOC)
In CRM, we usually choose the next dose as the mean of the PostPr(MTD|data)
EWOC has a similar process of CRM, but use a modified decision making to control the overdose probability:
choose a lower percentile, say 25th percentile from the PostPr as the next dose to have a expect % of pts treated at doses > MTD is less than θL

10. Phase I designs: Interval Toxicity
Instead of a fixed target toxicity probability, now the acceptable values of θx =Pr(DLT|x) is an interval, say 𝟎.𝟐,𝟎.𝟑𝟓
Now, give the observed data and the current dose x, we can calculate
Under-dosing: p1 = PostPr(θx < 0.2 | data)
Target-dosing: p2 = PostPr(0.2<θx < 0.35 | data)
Over-dosing: p3 = PostPr(θx >0.35 | data)
Decision Making is based on the max(p1, p2, p3):
If p1 is the max, then escalate the dose to x+1
If p2 is the max, then remain at the current dose x
If p3 is the max, then de-escalate to x - 1

11. Phase II Study: frequentist vs. Bayesian
Phase II studies provide an initial assessment of the treatment effect at the dose determined at Phase I.
It’s common to use interim data for a Go/No Go decision.
Is there convincing evidence in favor of the null or alternative hypothesis? – best addressed using estimation, p-values or Bayesian posterior probabilities
Is the trial likely to show convincing evidence in favor of the alternative hypothesis if additional data are collected? (efficacy) – best addressed using conditional power or Bayesian predictive probabilities
Whether a trial is likely to reach a definitive conclusion by the end of the study? (futility) – as futility is defined as a trial being unlikely to achieve its objective, is inherently a prediction problem, thus is best addressed using predictive probability rather than posterior probability

12. Phase II Study: frequentist vs. Bayesian
Phase II studies provide an initial assessment of the treatment effect at the dose determined at Phase I.
It’s common to use interim data for Go/No Go decision.
Conventional Frequentist interim analysis:
Simon’s two-stage designs for Futility (1989): Optimal/MinMax (phase 2a)
Interim analysis using Group sequential designs (more sample size: Phase 2b)
Bayesian’s sequential stopping rule to decision making:
Posterior Probability : pi = p(Ha| yobs)
Predictive Probability:
PP = E{ I[ Pr( θT> θS |Ynew, Yobs) > 0.95 ] | Yobs}
PP = E{ I(Reject H0 | Ynew, Yobs) > 0.95 ] | Yobs}

13. Phase II Study: frequentist vs. Bayesian
Simon’s Two Stage Designs (1989): A commonly used design for single-arm cancer trials allowing early stop for futility with one interim analysis.
Decision Rule: let y1 and t2 be the # of responders at the interim (N1)and the final stage (N1 + N2)
Claim Futility if: y1 ≤ B1 or (y1 > B1) & (y2 ≤ B2)
Claim Efficacy if: y1 > B1 & y2 > B2
Determination of Stopping Boundary (B1, N1) and (B2, N2): α, β criteria
Claim Efficacy Given Not Efficacious Pr (Y1 > B1 and (Y1+Y2) > B | θT = θC) = α
Claim Efficacy Given Efficacious Pr (Y1 > B1 and (Y1+Y2) > B | θT*) = 1- β
Additional Criteria:
Optimal: min E((N1+N2) | H0) with E((N1+N2) | H0) = N1 + N2 × Pr(B1 < Y1 ≤ N1| H0)
MinMax: min (N1+N2)

14. Phase II Study: frequentist vs. Bayesian
Bayesian Interim Monitoring
Decision Making: update the posterior probability of H1 with the current data:
Data Y ~ Bin(θT, N), θT ~ Beta(at,bt), θc ~ Beta(ac,bc),
Claim Futility if: Pr(θT > θC|Data) ≤ λL at the interim stage or Pr(θT > θC|Data) ≤ λ at the final stage
Claim Efficacy if: Pr(θT > θC|Data) > λL at the interim stage and Pr(θT > θC|Data) > λ at the final stage
Determination of Stopping Boundar (λL, λ) given Nmax: (via simulations)
Under H0: Pr(Claim Efficacy | θ = θC) ≤ α
Under H1: Pr(Claim Efficacy | θ = θT*) ≥ 1 - β

15. Phase II Study: frequentist vs. Bayesian
Flow Chart
Start the Trial
Meet the Futility Bound (PostP ≤ λL):
NO-GO
Not Meet the Futility Bound (PostP > λL): GO
Not Meet Efficacy Bound (PostP ≤ λ):
Meet the Efficacy Bound (PostP > λ):
Claim Efficacy
Start the Trial
Meet the Futility Bound (Y1≤ B1):
NO-GO
Not Meet the Futility Bound (Y1 > B1): GO
Meet the futility Bound (Y2 ≤ B2):
Not Meet the Futility Bound (Y2 > B2):
Claim Efficacy
One Interim at N1
Update the Posterior Pr
N = Nmax
Simon Two-Stage
Bayesian Sequential Monitor

16. Phase II Study: simulation studies for futility analysis
H0: θT ≤ θC vs. H1: θT > θC
Desired α = β = 0.10, θC = 0.2, target θT = 0.4
Design parameters for Simon Two-stage design:
Simon MinMax
Simon Optimal
Nmax 36 37
Interim Sample Size
N1= 19
N1 = 17
Stopping Boundary
B1 = 3, B2 = 10
E(Nmax)
28.3
26.0
Pr(early stop | θT = 0.2)
0.46
0.55
Actual α
0.086
0.095

17. Phase II Study: simulation studies for futility analysis
H0: θT ≤ θC vs. H1: θT > θC
Desired α = β = 0.10, θC = 0.2, target θT = 0.4
Design parameters for Bayesian: no easy solution. Needs to conduct simulation to determine the stopping boundary of λL(interim futility), λ(final efficacy).
Basic idea: for a range of λL and λ, find the values such that:
Generate data under θT = θC = 0.2, the empirical rejection rate of H0 (claim efficacy) should be close to α = 0.10 (mimic Type I error)
Generate data under θT = 0.4, the empirical rejection rate of H0 (claim efficacy) should be close to 1- β = 0.90 (mimic Power)

18. Phase II Study: simulation studies for futility analysis
Determination of the stopping boundaries: With one interim look at N1 = 10, choose λL=0.3 and λ= 0.76: please note, λL has little impact on α λL λ
θT = 0.2
θT = 0.4
E(N)
Early.no.go
final.go
early.no.go
0.1
0.76
33.3
0.1035
0.1005
35.9
0.0051
0.9128
0.765
33.2
0.1058
0.0854
35.8
0.0078
0.9057
0.77
33.1
0.1107
0.0895
0.0073
0.9088
0.2
0.755
0.1099
0.1255
0.0077
0.9279
0.1131
0.0917
0.0071
0.9143
0.1103
0.0875
0.0053
0.9104
0.3
26.3
0.3718
0.1183
34.8
0.0447
0.9029
26.2
0.3761
0.0905
0.0451
0.8889
26.1
0.3822
0.0833
0.0472
0.8849

19. Phase II Study: simulation studies for futility analysis
Interim Decision: λL=0.3 and λ= 0.76, N1 = 10, Nmax = 36, α = β = 0.10

20. Simulation Results (Design Stage)
E(N) and OC curve: λL=0.3 and λ= 0.76, N1 = 10, Nmax = 36, α = β = 0.10

21. Phase II Study: simulation studies for futility analysis
With only N1=10, Bayesian Posterior Probability-based interim futility analysis behaves similar as Simon two-stage design (N1 = 17 or 19) in our simulation studies.
What about more frequent interim looks using Bayesian PP? λL=0.3 and λ= 0.76:
Interim Looks
θT = 0.2
θT = 0.4
E(N)
Early.no.go
Final.go
10, 36
26.2
0.3761
0.0905
34.8
0.0451
0.889
10, 20, 36
25.7
0.4122
0.0953
10, 20, 30, 36
25.3
0.4675
0.0894
10, 15, 20, 25, 30, 36
23.8
0.5187
0.0878

22. Phase II Study: simulation studies for futility analysis
Multiple Interim Looks: Pr(No GO): Bayesian PP (λL=0.3 and λ= 0.76)

23. Phase II Study: simulation studies for futility analysis
Multiple Interim Looks: E(N) and Pr(Claim Efficacy): Bayesian PP (λL=0.3 and λ= 0.76)

24. Discussions In both Phase I/II studies, Bayesian methods:
provide flexible decision making framework utilizing the observed data and the prior information:
Phase I: data from all dose
Phase II: all observed data
utilize similar decision matrix to update the parameter:
Phase I: Posterior Probability of θ = Pr (DLT|dose)
Phase II: Posterior Probability of θ = Pr (Ha)
sequentially monitor the trial (the cohort size can be as small as 1) to achieve a target bound:
Phase I: choose a MTD close to the target toxicity level, say θmax = 0.33
Phase II: Futility / efficacy stopping boundaries.

25. Discussions But the computational can be intensive:
Phase I: complicated computation of the model-based methods.
Phase II: simulation studies to determine the design parameter can be intense in order to achieve a desirable type I and type II error levels.
Without closed form of the posterior probability, MCMC simulations can take lots of time.
Based on our simulation study on futility, Bayesian PP behaves similarly as the Simon two-stage, but providing the chance to stop early (with smaller N) when the drug is not efficacious.
When the drug is efficacious, all methods produce similar results with continuation to Nmax.

26. Reference
Thall and Simon (1994), “Practical Bayesian Guidelines for Phase IIB Clinical Trials”, Biometrics 50, 337 – 349
J Jack Lee, Diane D Liu (2008). A predictive probability design for phase II cancer clinical trials, Clinical Trials, Vol 5, Issue 2, pp  
Saville Ben (2015). “The Utility of Bayesian Predictive Probabilities for Interim Monitoring of Clinical Trials”. KOL Lecture Series.
Thall, Simon and Estey (1995), “Bayesian Sequential Monitoring Designs for Single-arm Clinical Trials with Multiple Outcomes”, Statistics in Medicine 14, 357 – 379
Thall and Yuan (2017), “Bayesian Designs for Phase I-II Clinical Trials”, Short Course, Joint Statistical Meetings.

27. Thank You!