1. Use of Piecewise Weighted Log-Rank Test for Trials with Delayed Effect
Boguang Zhen, Ph.D.
Therapeutics Evaluation Branch
Division of Biostatistics
CBER/FDA
Disclaimer: My contributions are an informal communication and represent my own best judgment. These comments do not bind or obligate FDA

2. Outline
Background
Concerns
Evaluations/Discussions
Concluding Remark

3. Delayed Effect (DE)
In chemotherapy, study drugs act directly on cancer
In therapeutic cancer vaccine therapy considerable time might be needed for cancer vaccines to induce immunity
similar to many other immunotherapies,
The impact of the so delayed effect (or lag time) should be considered

4. Sipuleucel-T case: 6 months~ DE
NEJM: 363 (5): 418, 2010

5. Issues on Delayed Effect
For time-to-event endpoint
Regular log-rank test commonly used
Under non-proportional hazards setting
Ignore it or increase sample size
Lose power or increase cost/duration
Piecewise weighted log-rank test (PWLRT)
Allocate different weight to a subset of events
Reduce sample size
Increase power
time-to-event endpoint – motivated by therapeutic vaccine cancer trials with OS endpoint, but the concern could be in any trial that deals with time-to-endpoint with delayed effect, , e.g. cardio-vascular trials.
the regular log-rank test with equal weight across all time points has been commonly used to compare the difference of the endpoint between the experimental group and the control group

6. PWLRT not widely used Challenging and Opportunity! Why not free lunch?
Lack of awareness
Issues
Allocate different weight to a subset of events
Could specify zero weight for some events
- implies that some subjects would not contribute ‘meaningful’ information to the test result
May not be straightforward to interpret the analysis results based on the weighted methods …
Challenging and Opportunity!

7. Concerns Type I probability Intention-to-treat principle
Study results interpretation
Changing point specification
Crossover effect
Others?
-- evaluate and address the above 5 concerns

8. Evaluations/Discussions
Use the PWLRT method in Xu’s paper (Stat in Med 2017; 36(4): ) for evaluation for simplicity
t* denote the hazard ratio changing point
For simplicity, I only use the PWLRT method as proposed by Xu et al. for the evaluation, but the results can also be generalized to other weighted methods that deal with delayed effect.
Inject the product to tumor, melanoma, and break tumor cells so there is an effect. After breaking the tumor cells, they release a lot of antigens that will take some time to stimulate the immune system response, and then you will have a full effect.
Extreme case that hazard ratio=1 or w1 = 0. IF we don’t have issue for w1=0, not effect at all before t*, it should not have the problem for the case that w1>0.

9. Type I probability Inflated? Evaluation
Some subsets may behave differently from others
By allocating zero weight for other subsets, it may inflate Type I probability
Evaluation
No DE under the null of no difference between the two groups – use of different subset should not inflate Type I probability
Simulation

10. Simulation study shows no Type I probability inflation
Sample Size
Regular
log-rank
PWLRT
t*= 3 months t*= 6 months t*= 9 months
300
600
900
0.0502
0.0503
-- 100,000 replications testing at the alpha level of 0.05

11. Intention-to-treat (ITT) principle
Violated?
less weight for some events could mean they contribute less information to the test result
0 weight for some events may imply excluding some subjects so violating ITT principle
Evaluation
What is behind the ITT principle?
Any potential bias?

12. Behind ITT principle ITT principle? Behind ITT principle?
Conduct analysis according to the assigned treatment instead of the actual treatment
Include all randomized subjects
Behind ITT principle?
Preserve the initial randomization in order to prevent bias and provide a secure foundation for statistical tests
Bias with PWLRT?
Should conduct analysis according to the assigned treatment instead of the actual treatment – same as for other methods
Include all randomized subjects with different weights
It’s not difficult to show the initial randomization is preserved even if zero weight is used (in a manuscript to be submitted - not show here in the interest of time)
we have showed that the proposed PWLRT can still preserve the initial randomization with w1=0 as long as subjects are analyzed according to the treatment assignments they received in a manuscript to be submitted.

13. Study results interpretation -- a hypothetical example
Consider a new product:
no survival benefit until 7 months after treatment
Regular log-rank test vs. PWLRT?
Regular log-rank test (p = 0.04)
KM plot for median survival: 25-month (new), 20-month (control)
Show 5-month survival benefit
PWLRT (p = 0.004)
w*1 =0 for subjects died before 7-month, w*2 =1 for others
KM plot for survival - same as above
KM plot for subset of subjects who live > 7-month
Median survival: 35-month (new) vs. 27-month (control)
Show 8-month survival benefit
KM shows the observed median survival 25 vs 20 month….

14. K-M plot with all subjects 
 K-M plot with subjects
who live after 7 months

15. Study results interpretation (2) -- a hypothetical example
Both regular log-rank test and PWLRT can be used
Regular log-rank test: less power and may fail to detect a statistically significant
Interpretations with PWLRT:
5-month survival benefit unconditional on the time point when the new product starts to work – same as for regular log-rank test
No survival benefit for the new product before 7-month
8-month survival benefit instead of month median survival instead of 25 for the treated group if live to the time when the new product kicks in its effect
Which one is more informative or we need both?
Study results based on PWLRT should be interpretable.

16. Changing point specification
Challenging in estimating the changing point (t*) in practice
If miss the changing point
Under powered study?
Worse than using the conventional method?

17. Estimate the changing point in practice
Obtain a close estimate for the changing point
Mechanism of action for the product
Animal/other pre-clinical study data
Literature
Early phase trial data
-- Join adventure from all disciplines

18. Missing the changing point
Xu et al.: Designing therapeutic cancer vaccine trials with delayed treatment effect (Stat in Med 2017; 36(4): )
Assuming fixed changing point
Address the robustness to the mis-specification of t* under various scenarios (Evaluation #4)
Still better than the regular log-rank

19. Xu et al.: Evaluation #4
Table 1. The impact of mis-specifying t* on power calculation
-- assume t*=6-month with 80% power for sample size (see paper for details)

20. Missing the changing point (2)
“Designing cancer immunotherapy trials with random treatment time-log effect”
-- Xu et al. accepted by SIM
Assuming a subject-specific lag duration:
t* ~ Dist(T1, T2)
Incorporating individual heterogeneity in lag time to response to treatment
Further reducing risk of mis-specification

21. Crossover effect
Survival curves for a hypothetical crossover effect trial

22. Crossover effect
Problematic for using PWLRT in trials with crossover effect
Same problem by using other conventional methods such as the regular log-rank test
Conventionally, a clinical trial should be designed with the assumption that there will be no crossover effect
Review issues and sensitivity analyses

23. Summary Can the concerns be addressed? Type I probability
Intention-to-treat principle
Study results interpretation
Changing point specification
Crossover effect

24. Concluding Remark
With careful consideration of all kind of potential biases and appropriate planning, the results based on PWLRT should be valid and interpretable

25. Thank you! My