This document discusses several case studies of dealing with complex study design issues in clinical trials, including non-proportional hazards, cluster randomization, and three-armed trials. The agenda outlines topics on non-proportional hazards modeling and sample size considerations, cluster randomized and stepped-wedge designs, and methods for analyzing data from three-armed trials that include experimental, reference, and placebo groups. Worked examples are provided to illustrate sample size calculations and statistical approaches for each of these complex trial design scenarios.

1. EXTENDING A TRIAL’S DESIGN
Case Studies of Dealing
with Study Design Issues
DEMONSTRATED ON

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3. AGENDA
1. Non-Proportional Hazards
2. Cluster Randomization
3. Three Armed Trials
4. Conclusions and Discussion

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6. Non-proportional HazardsPart 1

7. Survival Analysis is about the expected
duration of time to an event
 Common Methods: Log-rank, Cox Model
Power is related to the number of
events NOT the sample size
 Sample Size = Subjects to get # of events
Flexibility expected in survival analysis
methods and estimation
 Options will depend on model and
endpoint and affect design choices
Sample Size for Survival Analysis
Source: SEER, NCI

8. What is the expected survival curve(s) in the group(s)?
Parametric approximation? Piece-wise curve? Proportional Hazards?
Survival SSD Design Issues
Effect of unequal follow-up due to accrual period?
What accrual pattern to assume? Maximum follow-up same for all?
How to deal with censoring, dropouts or other risks?
Right-censored? Want to model dropout and/or competing risks?
Effect of subjects crossing over or informative censoring?
Appropriate estimand for trial? Enough info to assume before study?

9. Cox PH Model (& log-rank) rely on
proportional hazards assumption
 Ignoring NPH may lead to wrong inference
Non-proportional hazards occurs
where non-constant effect size (HR)
 Increase/decrease over time, crossing,
stratification, delayed effect, “responders”
 Very common issue in immunotherapies
Multiple methods proposed for
analysing NPH data
 Models: Weighted Linear-Rank Test, Max-
Combo, Piecewise Weighted Rank Test
 Model “Free”: Median Survival, RMST, KM
Non-proportional Hazards (NPH)
Source: Satrajit Roychoudhury

10. Piecewise Weighted Log-Rank Test
Piecewise Weighted Log-Rank Test
proposed a model where NPH present
 Piecewise: Different HR per time period
 Weighted: Diff. weight per time period
Simple Delayed Effect Model: HR = 1 until
time t, then constant HR onwards
 Weight = 0 before t, Weight = 1 post-t
 APPLE model as per Xu et al (2017)
Need strong assumption “delayed
duration” (t) and baseline hazard
 Extension: Random time lag - Xu et al (2019) Source: Xu et al (2017)

11. “Using an unstratified log-rank test at the one-sided 2.5%
significance level, a total of 282 events would allow
92.6% power to demonstrate a 33% risk reduction
(hazard ratio for RAD/placebo of about 0.67, as calculated
from an anticipated 50% increase in median PFS, from 6
months in placebo arm to 9 months in the RAD001 arm).
With a uniform accrual of approximately 23 patients per
month over 74 weeks and a minimum follow up of 39
weeks, a total of 352 patients would be required to
obtain 282 PFS events, assuming an exponential
progression-free survival distribution with a median of 6
months in the Placebo arm and of 9 months in RAD001
arm.
With an estimated 10% lost to follow up patients, a total
sample size of 392 patients should be randomized.”
Source:
nejm.org
Parameter Value
Significance Level (One-Sided) 0.025
Placebo Median Survival (months) 6
Everolimus Median Survival (months) 9
Hazard Ratio 0.66667
Accrual Period (Weeks) 74
Minimum Follow-Up (Weeks) 39
Power (%) 92.6
“New” Delayed Duration (Months) 6
Worked Example 1
Source: NEJM (2011)

12. Cluster RandomizationPart 2

13. Cluster Randomized Designs
Trt. randomized by cluster not subject
 Effective unit becomes cluster (Nelder)
 Cluster = Hospital, School, Country
Need to account for affect of cluster
self-similarity in design and analysis
 Measures: ICC, COV, within-cluster variance
Useful for practical and statistical
reasons, though some drawbacks
 Uses: If difficult to randomize by subject,
reduced costs, remove trt. contamination
 Drawbacks: Lower power/lower effective N,
selection bias, high prob. imbalances
Source: Senn (2019)

14. Stepped-Wedge Designs
RCT where subjects move from
treatment to control over time
1-way crossover, all control @ baseline
Often used w/ cluster randomization
“Unit” becomes cluster not subject
Useful for practical and statistical
reasons, though some drawbacks
Uses: Treatment “scarcity”, patient
recruitment, within-cluster analysis
Drawbacks: Allocation bias, expensive,
complex analysis

15. “We estimated that there would be approximately 12
births after 40 weeks gestation per week in each team.
Birth data were collected for each team from 12 weeks
prior to training until 12 weeks following training. Given
this fixed sample size, we determined what difference in
the primary outcome (proportion of women swept)
would be detectable with 80% power. … We were guided
by a review of estimates of ICCs which found that their
values are typically in the range of 0.02–0.1. A small audit
suggested … 32% of nulliparous women and 57% of
multiparous women were currently being swept. … It was
estimated that at 5% significance (two-tailed) and 80%
power, for ICCs in the range of 0.02–0.1 and for baseline
event rates of 20–60%, the study would have power to
detect around a 10% absolute increase in proportion of
women being swept. This was an increase felt to be
clinically worthwhile.”
Source:
nejm.org
Parameter Value
Significance Level (Two-Sided) 0.05
Time Measurements (Weeks) 12
Number of Clusters 10
Measurements Per Cluster per Time 12
ICC 0.02
Baseline Proportion 0.4
Power (%) 80
Worked Example 2
Source: Trials (2017)

17. Stepped-Wedge Design Issues
Model must account for between-cluster & within-cluster
variance, temporal effects and stepped-wedge structure
Different choices available for stepped-wedge design
 Complete, incomplete, “missing” observations, partial effects
For cluster randomization, need to account for “self-
similarity” within clusters using ICC, COV or similar
SSD well-developed for cross-sectional (new subjects per
time) SW-CRT for common endpoints (mean, props, rates)

18. Three Armed TrialsPart 3

19. Non-inferiority Testing
Non-inferiority testing is where hypothesis test that
treatment no worse than standard by a specified margin
Select non-inferiority margin based on expertise & data
 Most often fixed fraction (M2) of active control effect (M1)
Very common for generics or medical devices and can
compare treatment vs control (e.g. RLD) w/o placebo
Simpler design but does it prove both effective (assay
sensitivity) and how to validate using prior placebo data

20. Three Armed Trials
Have Experiment (A), Reference (R) & Placebo (P) groups
• Direct evaluation of assay sensitivity (“gold standard)
• Concurrent placebo only allowable if it is ethical to do so
Need to test H1(a): E/R > P and then H1(b) E > NIM
 Can simplify to a “ratio of differences” test: (E-P)/(R-P) > θ
Framework of Wald-type test for retention of effect
 Can use same approach for means, props, survival, rates
 Can also find optimal allocation for given alternative

21. Three Armed Trials
1 Means (Homoscedatic) Pigeot et al. (2003)
2 Means (Heteroscedatic) Hasler et al. (2008)
3 Proportions Kieser and Friede (2007)
4 Survival/Time-to-Event Mielke et al. (2009)
5 Counts/Rates (Poisson) Mielke and Munk (2009)
6 Counts/Rates (Negative Binomial) Mütze et al. (2016)
7 Non-Parametric Mütze et al. (2016)

22. Worked Example 3
Parameter Value
Significance Level (1-Sided) 0.025
Experimental Arm Mean 1.56
Reference Arm Mean 1.56
Placebo Arm Mean 0
Non-inferiority Ratio 0.5
Common Standard Deviation 2.5
Power 80%
Allocation Proportion (E:R:P) 0.38:0.38:0.24
“It was assumed that the placebo-adjusted effect for
both treatment groups was 1.56% and that the
placebo-adjusted effect for the oral rsCT tablets
must be at least 0.5 times the placebo-adjusted
effect for the ssCT nasal spray for the study to
demonstrate the non-inferiority of the oral rsCT
tablets to the ssCT nasal spray. Thus we wished to
have 95% confidence that the oral tablets were not
less than one-half as effective as nasal spray.
Assuming an SD of 2.5%, power of 80%, and a two-
sided 5% level of significance, it was determined
that approximately 133 patients were required for
each of the active treatment groups and 84 patients
were needed for the placebo treatment group.”

23. Discussion and Conclusions
Trials often require adjustments from standard methods
Need methods which accommodate these complexities
NPH becoming common, especially in immunotherapy
Consider weighted models or model free measures (RMST)
Cluster randomization common constraint in real world
Must adjust for cluster effect but can consider complex CRT
Three armed NI trials give direct comparison to placebo
Flexible framework available for a variety of endpoints

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Designs
Survival/
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Trials
Confidence
Intervals for
Proportions
Three Armed Trials
Non-inferiority

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29. References (Non-Proportional Hazards)
Collett, D., 2015. Modelling survival data in medical research. CRC press.
Schoenfeld, D. A., 1983. Sample-size formula for the proportional-hazards regression model. Biometrics,
pp. 499-503.
Lachin, J. M., & Foulkes, M. A., 1986. Evaluation of sample size and power for analyses of survival with
allowance for nonuniform patient entry, losses to follow-up, noncompliance, and
stratification. Biometrics, pp. 507-519.
Lakatos, E., 1988. Sample sizes based on the log-rank statistic in complex clinical trials. Biometrics, pp.
229-241.
Fleming, T. R. and Harrington, D. P., Counting Process and Survival Analysis. New York, John Wiley and
Sons. 1991.
Yang, S and Prentice, R, 2010. Improved Logrank‐Type Tests for Survival Data Using Adaptive Weights,
Biometrics 66: pp. 30‐38
Public Workshop: Oncology Clinical Trials in the Presence of Non‐Proportional Hazards, The
Duke‐Margolis Center for Health Policy, Feb. 2018

30. References (Non-Proportional Hazards)
Lin, R.S., Lin, J., Roychoudhury, S., Anderson, K.M., Hu, T., Huang, B., Leon, L.F., Liao, J.J., Liu, R., Luo, X.
and Mukhopadhyay, P., 2020. Alternative Analysis Methods for Time to Event Endpoints Under
Nonproportional Hazards: A Comparative Analysis. Statistics in Biopharmaceutical Research, 12(2), pp.
187-198.
Uno, H., Claggett, B., Tian, L., Inoue, E., Gallo, P., Miyata, T., Schrag, D., Takeuchi, M., Uyama, Y., Zhao, L.
and Skali, H., 2014. Moving beyond the hazard ratio in quantifying the between-group difference in
survival analysis. Journal of clinical Oncology, 32(22), pp. 2380.
Xu, Z., Zhen, B., Park, Y., & Zhu, B., 2017. Designing therapeutic cancer vaccine trials with delayed
treatment effect, Statistics in medicine, 36(4), pp. 592-605
Xu, Z., Park, Y., Zhen, B. and Zhu, B., 2018. Designing cancer immunotherapy trials with random
treatment time‐lag effect. Statistics in medicine, 37(30), pp. 4589-4609.
Xu, Z., Park, Y., Liu, K. and Zhu, B., 2020. Treating non-responders: pitfalls and implications for cancer
immunotherapy trial design. Journal of hematology & oncology, 13(1), pp. 1-11.
Yao, J. C., et. al., 2011. Everolimus for advanced pancreatic neuroendocrine tumors. New England
Journal of Medicine, 364(6), pp. 514-523.

31. Donner, A. and Klar, N., 2000. Design and analysis of cluster randomization trials in health research.
Brown, C.A. and Lilford, R.J., 2006. The stepped wedge trial design: a systematic review. BMC medical
research methodology, 6(1), pp. 54.
Hemming, K., Haines, T.P., Chilton, P.J., Girling, A.J. and Lilford, R.J., 2015. The stepped wedge cluster
randomised trial: rationale, design, analysis, and reporting. BMJ, 350, pp. 391.
Hemming, K., Lilford, R., & Girling A.J., 2015. Stepped-wedge cluster randomised controlled trials: a
generic framework including parallel and multiple-level designs, Statistics in Medicine, 34, pp. 181-
196.
Hemming, K., & Girling A., 2014. A menu-driven facility for power and detectable-difference
calculations in stepped-wedge cluster-randomized trials, The Stata Journal, 14, pp. 363-380.
Hussey, M.A., & Hughes, J.P., 2007. Design and analysis of stepped wedge cluster randomized trials,
Contemporary Clinical Trials, 28, pp. 182-191
Kenyon, S., Dann, S., Hope, L., Clarke, P., Hogan, A., Jenkinson, D. and Hemming, K., 2017. Evaluation of
a bespoke training to increase uptake by midwifery teams of NICE Guidance for membrane sweeping
to reduce induction of labour: a stepped wedge cluster randomised design. Trials, 18(1), pp. 357.
References (Cluster Randomization)

32. References (Three Armed Trials)
Food and Drug Administration Non-inferiority clinical trials to establish effectiveness. Guidance for
industry. November 2016. https://www.fda.gov/downloads/Drugs/Guidances/UCM202140.pdf
Blackwelder, W.C., 2002. Showing a Treatment Is Good Because It Is Not Bad: When Does
‘Noninferiority’ Imply Effectiveness?. Control Clinical Trials, 23, pp. 52–54.
Chow, S.C., Shao, J., 2006. On Non-Inferiority Margin and Statistical Tests in Active Control Trial.”
Statistics in Medicine, 25, pp. 1101–1113.
Fleming, T.R., 2008. Current Issues in Non-inferiority Trials. Statistics in Medicine, 27, pp. 317-332.
Althunian, T.A., de Boer, A., Groenwold, R.H. and Klungel, O.H., 2017. Defining the noninferiority margin
and analysing noninferiority: an overview. British journal of clinical pharmacology, 83(8), pp.1636-1642.
I. Pigeot, J. Schäfer, J. Röhmel, D. Hauschke., 2003. Assessing non-inferiority of a new treatment in a
three-arm clinical trial including a placebo. Statistics in Medicine, 22, pp. 883-899.
M. Kieser, T. Friede., 2007. Planning and analysis of three‐arm non‐inferiority trials with binary
endpoints. Statistics in Medicine, 26, pp. 253-273.
M. Hasler, R. Vonk, L.A. Hothorn., 2008. Assessing non-inferiority of a new treatment in a three-arm
trial in the presence of heteroscedasticity. Statistics in Medicine, 27, pp. 490-503.

33. References (Three Armed Trials)
M. Mielke, A. Munk, and A. Schacht., 2008. The assessment of non‐inferiority in a gold standard design
with censored, exponentially distributed endpoints. Statistics in Medicine, 27, pp. 5093-5110.
M. Mielke and A. Munk., 2009. The assessment and planning of non-inferiority trials for retention of
effect hypotheses-towards a general approach. arXiv:0912.4169
Mielke, M., 2010. Maximum Likelihood Theory for Retention of Effect Non-Inferiority Trials (Doctoral
dissertation, Niedersächsische Staats-und Universitätsbibliothek Göttingen).
T. Mütze, A. Munk, T. Friede., 2016. Design and analysis of three‐arm trials with negative binomially
distributed endpoints. Statistics in Medicine, 35, pp. 505-521.
T. Mütze, F. Konietschke, A. Munk, T. Friede., 2017, A studentized permutation test for three-arm trials
in the `gold standard’ design. Statistics in Medicine, 36, pp. 883-898.
Binkley, N., Bolognese, M., Sidorowicz‐Bialynicka, A., Vally, T., Trout, R., Miller, C., Buben, C.E., Gilligan,
J.P., Krause, D.S. and Oral Calcitonin in Postmenopausal Osteoporosis (ORACAL) Investigators, 2012. A
phase 3 trial of the efficacy and safety of oral recombinant calcitonin: the Oral Calcitonin in
Postmenopausal Osteoporosis (ORACAL) trial. Journal of bone and mineral research, 27(8), pp.1821-
1829.