1. Cure Survival Data in Oncology Study
Shih-Yuan (Connie) Lee, Ph.D.
Takeda
JSM 2018 meeting

2. Generate the Random Variable
Outline
Introduction
Generate the Random Variable
Parametric Method
Non-Parametric Method
Application
Sample Size Calculation
Event Projection
Summary

3. Introduction
For some caners, the cure of the disease is a possibility especially in the earlier line of treatment (i.e. Hodgkin Lymphoma)
Viviani S, Zinzani PL, Rambaldi A, Brusamolino E, Levis A, Bonfante V, Vitolo U, Pulsoni A, Liberati AM, Specchia G, Valagussa P, Rossi A, Zaja F, Pogliani EM, Pregno P, Gotti M, Gallamini A, Rota Scalabrini D, Bonadonna G, Gianni AM; Michelangelo Foundation; Gruppo Italiano di Terapie Innovative nei Linfomi; Intergruppo Italiano Linfomi. ABVD versus BEACOPP for Hodgkin's lymphoma when high-dose salvage is planned. N Engl J Med Jul 21;365(3): doi: /NEJMoa
Events of first progression were defined as progression while receiving treatment, a lack of complete remission at the end of treatment, relapse, or death from any cause. The probability of event-free survival is shown in Panel B. Events were defined as progression while receiving treatment, a lack of complete remission at the end of treatment, relapse, death from any cause, discontinuation of treatment owing to life-threatening toxic effects, and secondary leukemia.

4. Proper vs. Improper Survival Curve
S(t): Survival Function
H(t): Cumulative Hazard Function
Proper Random Variable
H(∞)= ∞, S(∞)=exp(-H(∞))=0
Improper Random Variable
H(∞)< ∞ (bounded), S(∞)=exp(-H(∞))>0

5. Generate the Random Variable
Parametric Method

6. Formulation for Cure Survival Function
H(∞) is bounded (<∞) in cure setting
and EXP(-H(∞)) is the cure rate
An arbitrary proper CDF (F(t)=1-G(t))
G(t) arbitrary proper survival
→ ∞

7. Generate Random Variable
Inverse transform
Generate Random Variable U ~ Uniform(0, 1)
Find T such that CDF of T, F(T) = U (i.e. T=F-1(U))
Since 1-U also ~Uniform (0,1), this implies the Survival of T, S(T)=U, (i.e. T=S-1(U))
Since we can write the Cure Survival Function as
Where G(T) is the proper survival function (i.e. Weibull Survival Function)
-H(∞) will be bounded (i.e. a constant) with exp(-H(∞)) is the Cure Rate

8. Example: Weibull Cure Random Variable
Using the Inverse Transformation
Need to be >0
If <=0, then T is ∞(i.e. return a large number)

9. Adding Treatment Effect Under the Proportional Hazard (PH) Assumption
Under the proportional hazard (PH) assumption, the treatment effect θ can be applied to the survival function

10. HR=0.75 (Active vs. Control)
G(t): Weibull Shape=1
2 year survival=0.8 for the control arm
Cure rate=50% for the control arm

11. Generate the Random Variable
Non-parametric Method

12. Non-parametric Method Based on the Discrete Hazard Function
The probability of event occurred between t1 and t2 given the subject is event free at t1 is
Time
Survival
Number Failed
Number Left
0.0000
1.0000 20
0.9500 1 19
2.0000
0.9000 2 18
3.0000
0.8500 3 17
5.0000 * . 16
6.0000
0.7969 4 15
7.0000
0.7438 5 14
8.0000 13
9.0000 12 11
0.6761 6 10
For example:
The subject is had event at time 1 is: (1-0.95)/1=0.05
The subject is event free at time 1. The chance of getting an event at time 2 is:
( )/0.95=0.053

13. Graphical Illustration
Porb. of getting an event at time 1
Event free at time 1.
Porb. of getting an event at time 2
Event at time1
(Stop)
Bernoulli (p=0.05) 1
Bernoulli (p=0.053)
Event at time 2
(Stop) 1
Censored at time 1 (Continue)
Censored at time 2 …
Continue till event or the last F/U time
This could be computational intensive depending on the time interval and sample size evaluated

14. Method Illustrated Using the Simulation

15. Sample Size Calculation
Application
Sample Size Calculation

16. Number of Events Required
Does not depend on the distribution assumption

17. Sample Size Estimation Example
2 year PFS (Active: 81% vs. Control: 73%)
HR=0.67 (assuming an emergent plateau in the PFS event rate after 2 year)
Alpha: 1-sided 0.025
Power: 90%
Study duration: 60 months for the primary analysis.
36 months of accrual, a 5% annual dropout rate and 24 months of F/U after last patient in

19. Sample Size Calculation
Plug into the Software
Typically on the exponential distribution assumption
Number of event=262
Sample size=790
Simulation Approach
Generate R.V. based on the protocol assumption (Assuming cure after 2 years)
Number of event~260
~ 1240 patients were required

20. Event Projection for Cure Survival Curve Non-Parametric Method
Application
Event Projection for Cure Survival Curve Non-Parametric Method

21. Graphic Illustration of Event Projection Simulation
Observed Event Time
Future Survival Time
Lost to follow up
Ongoing at the data cut off
0 (FPI)
Current Data Cutoff
Calendar Time

22. Event Projection for Cure Survival Data
Typical exponential distribution with memory-less property will not hold
Parametric method such as Piecewise Exponential Distribution with very low rate at the tail is an option
Non-parametric method using the discrete hazard function based on the observed K-M curve

23. Non-parametric Method Based on the Discrete Hazard Function
The probability of event occurred between t1 and t2 given the subject is event free at t2 is
Time
Survival
Number Failed
Number Left
0.0000
1.0000 20
0.9500 1 19
2.0000
0.9000 2 18
3.0000
0.8500 3 17
5.0000 * . 16
6.0000
0.7969 4 15
7.0000
0.7438 5 14
8.0000 13
9.0000 12 11
0.6761 6 10
For example:
The subject is event free at time 6. The chance of getting an event at time7 is:
( )/0.7969=0.0666

24. Non-parametric Method Based on the Discrete Hazard Function
Time 12 24 36 48
At Risk
1334
1227
1137
584
110

25. Summary
Introduce two methods to simulate cure survival data which can easily incorporate the treatment effect under PH or non-PH assumption
For a study with potential curable disease, we should not use the traditional software (under exponential distribution assumption) for sample size estimation.
Project future event number using traditional exponential distribution is not appropriate. A pure non-parametric approach could be an option based on the observed K-M survival curve

26. Acknowledgement
Dr. Alex Tsodikov and Dr. Jeremy Taylor, Department of Biostatistics, University of Michigan
Takeda Cancer Immunotherapy Working Group

27. Estimate Sample Size using East
Estimated Sample Size using EAST: N=790

28. Sample Size Calculation
Naïve Approach
Plug into the formula: # of event required: ~262
N is the total sample size (with 1:1 Randomization Ratio , 0.5*N is sample size in each arm)
Assuming 5% annual dropout rate, the cumulative dropout by the end of year 2 is ~10% (1-0.95*0.95)
(1-0.1)*(0.5*N*(1-0.81)+0.5*N*(1-0.73))=262
→ N=1266
Simulation Approach
Generate R.V. based on the protocol assumption
~ 1240 patients were required in about 60 months