1. On estimating treatment effects under noncompliance in randomized clinical trials
Heejung Bang

2. Background
Randomized clinical trial (RCT)
Groups must be alike in important aspects and differ only in the treatment assigned.
Comparable treatment groups are achieved through randomization
In most trials, “noncompliance” is extremely
difficult to avoid.
---reasons: voluntary refusal, failure to comply, treatment switch, and administrative error

3. Noncompliance & Intent-to-treat (ITT)
With or without protocol deviations, the ITT approach has long served as a standard procedure, preserving the integrity of randomization.
Although the estimate of treatment effect is known to be attenuated under imperfect compliance, the ITT method ignoring post-randomization compliance is strongly advocated because randomization eliminates selection bias.
The approach of comparing policies of “intentions” or the effect of “treatment assignment”.
--- a more pragmatic statement as a measure of effectiveness rather than efficacy.

4. What are other estimators/principles to consider?
As treated (AT)
Per protocol (PP)
Instrumental variables (IV)
Here, we intend to compare these 4 estimators under different compliance scenarios in several RCT settings.

5. AT
Rarely accepted not only for statistical inference but also for decision making.
Well known to give a biased test and inflated type I error even under the null hypothesis.
It can be severely confounded with other nonrandomized prognostic characteristics.
However, it may give some idea about the maximum (or optimal) treatment efficacy if noncompliance is indeed a random phenomenon.

6. PP
It addresses what happened to patients who completed the entire treatment protocol as planned.
This also suffers selection bias and the assessment of bias is difficult.
FDA generally requires that analyses based on PP in addition to ITT be conducted for approval of drug, expecting the results to be logically consistent.

7. IV It aims to establish “causal” interpretation.
Methods making use of IV frequently appear in econometrics literature.
The origin of the IV is traced back to Wald (1940).
The IV estimand was introduced to structural equation modeling by Durbin (1954).
This idea was implicit in the study of RCT with imperfect compliance (Sommer & Zeger, 1991)

8. Notation Zi (treatment indicator) =1 (active treatment) & =0 (control)
where i=1,2,...,N subjects.
“Counterfactual” outcome Yi(Zi) for the i-th subject --- a pair of Yi (1) and Yi (0) are never observed simultaneously.
Treatment effect=μ1- μ0=τ, where
E{Yi(1)}=μ1 & E{Yi(0)}=μ0.
Likewise, it is also assumed that there is counterfactual (binary) dose at each treatment level, Di(Zi).

9. Observed Data
We only get to see the actually received dose di and observed response yi (i=1,2,...,N), where
di=Zi*Di(1)+ (1-Zi)*Di(0) &
yi=di*Yi(1)+ (1-di)*Yi(0).

10. 4 estimators

11. More on IV
IV estimator is the ratio of average ITT effect of Z on y and ITT effect of Z on d.
Underlying idea: any effect of Z on y must be via the effect of Z on d.
It estimates “complier average causal effect (CACE)” as the average causal effect for the subpopulation of compliers (Little & Rubin 2000)
---E{Y(1)-Y(0)|C=1} where C=compliance status.
More assumptions are required.

12. Noncompliance Scenarios
1. Ignorable (i.e., random) noncompliance
2. Nonignorable, symmetric noncompliance:
For example, a study of exercise encouragement for lung disease: healthy subjects (with high lung function score Yi(0)) are more likely to exercise regularly, whereas weak subjects (with low Yi(0)) do not exercise even if encouraged.

13. (conti’) 3. Nonignorable, asymmetric noncompliance:
same as Scenario #2 but only one of the 2 noncompliance patterns is observed.
Remark: implementation of asymmetric nature is important because exact symmetry may not be achieved.

14. Clinical Trial Scenarios
Case 1: Noncompliance can occur in either treatment arm.
e.g., “encouragement” design such as a program for smoking cessation and a physician's encouragement for influenza vaccine or exercise.
Case 2: Participants in the active treatment may not comply, while the ones in the control arm have no choice but to comply
e.g., a new, very expensive or rare intervention

15. (conti’)
Case 3: Partial compliance is permitted in either arm. This is the case when the study treatment is readily available and easily skipped.
e.g., vitamin or multiple pills.

16. Simulation Settings: Case 1
Yi(0) ~ N(μ0,1) and Yi(1)~N(μ1,1)
for i=1,…,1000, where
μ0= μ1=1 (i.e., τ=0) under the null &
μ0=1, μ1=3 (i.e., τ=2) under the alternative.
Z=1 or 0 with equal probability.
Initially, we set Di(0)=0 & Di(1)=1, i.e., complete compliance.

17. (conti’)
Under the “ignorable” noncompliance, a randomly selected 20% of the subjects do not comply.
For the “nonignorable” symmetric noncompliance,
1) if Yi(0)> μ0+0.5, we set Di(0)= Di(1)=1
(named “upper condition”)
2) if Yi(0)< μ - 0.5, we set Di(0)= Di(1)=0
(named “lower condition”).

18. Simulation Settings: Case 2
Same as Case 1 but Di(0)=1 is not allowed.

19. Simulation Settings: Case 3
Among untreated,
- 50% of subjects comply as directed
- 30% take 1/3 amount of the active regimen from some external sources
- 10% take 2/3
- the remaining 10% are able to take an equal amount of a
full dose.
Among treated,
- 50% of subjects take a whole recommended dose
- 30% do not take the regimen at all
- 10% take only 1/3
- the other 10% missed 1/3 of a dose.

20. Dose-Outcome Model
The treatment effect is assumed to be proportional to the dose taken. Specifically,

21. Table 1. Results for Case 1
Simulation results are summarized by the mean of treatment effect estimates and sample standard error (SSE), average confidence interval (CI) and empirical coverage probability (CP, %).
500 simulations.
Warning: ITT estimates the effect of randomization so some results for ITT may not be directly comparable to results for other estimators.

23. Table 2: Results for Case 2

25. Table 3. Results for Case 3

27. Two more analyses
Additive model
Sensitivity analysis

28. Table 4: Results under the additive model (compare with Case 1 results)
Yi(1)-Yi(0)=τ, i.e., a constant additive effect within each individual.
This assumption often appears in Econometrics research.
The condition under which IV always works!

30. Sensitivity analysis for dose-outcome model (compare with Case 3 results)
New model:
IV: 1.99->2.11
AT: 2.00->1.86
PP: 2.00->1.91
Unbiasedness of the IV also relies on linearity between dose and treatment effect.

31. Conclusions
Most of our current knowledge is under the “random noncompliance” setting.
PP and AT estimators are severely biased (esp., overestimation).
ITT and IV estimators carry desirable properties as we assume but not always (esp., under nonignorable noncompliance).
IV is always unbiased under constant treatment effect.

32. Future Extensions
Extensions to binary and survival outcomes are straightforward, we find no reason for different results to be reached.
“Global average treatment effects” or “principal effects” (based on principal stratification) may be of greater public health interest and better suited for more complex study designs (Robins & Greenland 1996; Frangakis & Rubin 2002)
Well-behaved estimators (beyond the standard ITT and IV) along with their analytic properties, especially under our informative asymmetric noncompliance or even more complicated scenarios, would be important contributions.

33. Reference
Bang H. & Davis C.E. On estimating treatment effects under noncompliance in randomized clinical trials: are intent-to-treat or instrumental variables analyses perfect solutions? Statistics in Medicine