# Estimating the treatment effect for adherers using multiple imputation

aa r X i v : . [ s t a t . M E ] F e b Estimating the treatment eﬀect for adherersusing multiple imputation

Junxiang Luo

Biostatistics and Programming, anoﬁ, 55 Corporate Dr, Bridgewater, NJ 08807, USAEmail: [email protected]ﬁ.com

Stephen J. Ruberg

Analytix Thinking, LCC, 11121 Bentgrass Court, Indianapolis, IN 46236, USAEmail: [email protected]

Yongming Qu*

Department of Biometrics, Eli Lilly and Company, Indianapolis, IN 46285, USAEmail: qu [email protected]

February 9, 2021 *Correspondence: Yongming Qu, Department of Biometrics, Eli Lilly and Company, LillyCorporate Center, Indianapolis, IN 46285, U.S.A. Email: qu [email protected] bstract

Randomized controlled trials are considered the gold standard to evaluate the treat-ment eﬀect (estimand) for eﬃcacy and safety. According to the recent InternationalCouncil on Harmonisation (ICH)-E9 addendum (R1), intercurrent events (ICEs) needto be considered when deﬁning an estimand, and principal stratum is one of the ﬁvestrategies used to handle ICEs. Qu et al. (2020, Statistics in Biopharmaceutical Re-search 12:1-18) proposed estimators for the adherer average causal eﬀect (AdACE) forestimating the treatment diﬀerence for those who adhere to one or both treatmentsbased on the causal-inference framework, and demonstrated the consistency of thoseestimators. No variance estimation formula is provided, however, due to the com-plexity of the estimators. In addition, it is diﬃcult to evaluate the performance ofthe bootstrap conﬁdence interval (CI) due to computational intensity in the complexestimation procedure. The current research implements estimators for AdACE us-ing multiple imputation (MI) and constructs CI through bootstrapping. A simulationstudy shows that the MI-based estimators provide consistent estimators with nominalcoverage probability of CIs for the treatment diﬀerence for the adherent populationsof interest. Application to a real dataset is illustrated by comparing two basal insulinsfor patients with type 1 diabetes.

Keywords : Adherer average causal eﬀect, counterfactual eﬀect, principal stratum,tripartite estimands. Introduction

Choosing and deﬁning estimands and constructing corresponding estimators are integralparts of randomized controlled clinical trials. The International Council on Harmonisation(ICH) [1] provides a general framework for choosing and deﬁning estimands using a few keyattributes: treatment of interest, population, handling intercurrent events (ICEs), endpoint,and population-level summary. There are three possible populations in deﬁning estimands:all patients, a subset of patients based on baseline covariates, and a principal stratum basedon post-baseline variables [2, 3]. A principal stratum is a subset of patients deﬁned by thepotential outcome of one or more post-randomization variables [4, 5, 6]. All randomizedpatients are generally used for deﬁning estimands in clinical trials. This approach is popularas it theoretically maintains randomization, but the diﬃculty arises when, as inevitablyhappens, some patients do not provide complete eﬃcacy response data according to theprotocol due to intercurrent events. ICH E9 (R1) deﬁnes intercurrent events as “Eventsoccurring after treatment initiation that aﬀect either the interpretation or the existence ofthe measurements associated with the clinical question of interest. It is necessary to addressintercurrent events when describing the clinical question of interest in order to precisely deﬁnethe treatment eﬀect that is to be estimated.” Deciding which strategy to use for including thedata that will be used in the analysis in response to ICEs can lead to diﬀerent estimands.This is central to the development of ICH E9 (R1) which provides a framework and alanguage for deﬁning the estimand ﬁrst and, subsequently, an appropriate data handlingand analysis approach. As mentioned in ICH E9 (R1), the deﬁnition of the estimand shouldbe guided by the study objective. Deciding the treatment eﬀect for all randomized patientsis an important question, but it may not be the only or the most important question. Theauthors of the recent research [7, 8, 9] provide arguments of importance of the treatmenteﬀect for adherers. The treatment eﬀect for all randomized patients answers the question,“what is the overall average treatment eﬀect before a patient takes the medication?”, whichis an unconditional expectation . The treatment eﬀect for the adherers answers the question,“what is the treatment eﬀect in a patient who can adhere to the treatment?”, which is a3 onditional expectation .The estimation of the treatment eﬀect for a principal stratum has attracted interest fordata analysis in clinical trials previously [1, 10, 11]. There are a variety of approaches forconstructing estimators of the treatment eﬀect in a principal stratum. Most methods requirethe monotonicity assumption and/or using the principal score [12, 13, 14, 15, 16, 17, 18, 19,20, 21, 22, 23]. Monotonicity basically assumes the potential outcome for the stratiﬁcationindicator is a monotone function of the treatment indicator. The monotonicity assumptionimposes a deterministic relationship on the potential outcomes (random variables) of theprincipal stratum variable(s). Qu et al. [24] demonstrate the implausibility of such anassumption in many situations from a theoretic perspective and illustrate this implausibilityusing a real data example in a cross-over study. The principal score is the probabilityof a subject belonging to the principal stratum, via baseline covariates. As an apparentdrawback, methods based on the principal score assume the principal stratum can be fullymodeled through baseline covariates. Recently, Louizos et al. [25] proposed methods directlyestimating the potential outcome of the response variable under the alternative treatmentif the principal stratum can be observed in one treatment group. Qu et al. [8] developedestimators for the adherer average causal eﬀect (AdACE) based on the causal inferenceframework for the treatment eﬀect for those who can adhere to one or both treatmentsby modeling the potential outcome of the response variable and/or the principal score viabaseline covariates and potential post-baseline intermediate measurements. It is important toinclude post-baseline intermediate measurements since patients and their treating physiciansin clinical trials most often make decisions about whether or not to adhere to the randomizedstudy treatment based on their eﬃcacy and safety responses to the assigned treatment.Details on the implementation of estimators for the AdACE are given in Qu et al. [9].Barriers to a wide application of these estimators include the complex estimation process,the time-consuming computation, and the requirement of customized programs for individualclinical trials.Multiple imputation (MI), proposed by Rubin [26], is widely used in handling missing4alues and could be an alternative approach to construct estimators for the AdACE. Theadvantage of using MI is that the estimators can easily be calculated based on the imputedpotential outcomes. In this article, we propose using MI to construct the estimators forthe AdACE. The advantage of this approach is that it can utilize the existing estimationprocedures for “complete” data after MI. In Section 2, we will review the theoretical frame-work and outline the process of the MI-based estimators for adherers. The deﬁnition ofadherence may be study speciﬁc, but generally we consider a patient to be adherent if thepatient predominantly takes their estimand-deﬁned study treatment and conforms to pro-tocol requirements (e.g., no intercurrent events) throughout the intended duration of thetrial. As we are interested to estimate the potential outcome under the intended treatmentregimen for adherers, the (potential) outcomes under a treatment regimen (e.g., treatmentdiscontinuations or treatment switch) are not relevant in our estimation. In Section 3, sim-ulations are conducted to evaluate the performance of the MI-based estimator. In Section4, the application of the MI-based estimator is illustrated with a real clinical study. Finally,Section 5 serves as the summary and discussion.

Let ( X ij , Z ij , Y ij , I ij ) denote the data for assigned treatment i ( i = 0 ,

1) and subject j (1 ≤ j ≤ n i ), where X ij is a vector of baseline covariates, Z ij = ( Z (1) ij , Z (2) ij , . . . , Z ( K − ij ) ′ is avector of intermediate repeated measurements, I ij = ( I (1) ij , I (2) ij , . . . , I ( K − ij ) ′ , and I ( k ) ij (1 ≤ k ≤ K −

1) is the indicator variable for whether a patient is adherent to treatment afterintermediate time point k . Then, the adherence indicator is given by: A ij = K − Y k =1 I ( k ) ij . We use “( t )” following the variable name to denote the potential outcome under the hypo-thetical treatment t ( t = 0 , Y ij ( t ) denotes the potential outcome for subject j randomized to treatment i if taking treatment t . Generally, Y ij ( i ) can be observed but Y ij (1 − i ) cannot be observed in parallel studies.5our principal strata were discussed in Qu et al. [8]: • All randomized patients: S ∗∗ . • Patients who can adhere to the experimental treatment: S ∗ + = { ( i, j ); A ij (1) = 1 }• Patients who can adhere to the control treatment: S + ∗ = { ( i, j ); A ij (0) = 1 }• Patients who can adhere to both treatments: S ++ = { ( i, j ); A ij (0) = 1 , A ij (1) = 1 } Qu et al. [8] provides estimators for S ∗ + , S + ∗ , and S ++ under the following assumptions:A1: Y = Y (1) T + Y (0)(1 − T )A2: Z = Z (1) T + Z (0)(1 − T )A3: A = A (1) T + A (0)(1 − T )A4: T ⊥ { Y (1) , A (1) , Z (1) , Y (0) , A (0) , Z (0) }| X A5: A ( i ) ⊥ { Y (1) , Y (0) , Z (1 − i ) }|{ X, Z ( i ) } , ∀ i = 0 , Y ( i ) ⊥ Z (1 − i ) |{ X, Z ( i ) } , ∀ i = 0 , Z (0) ⊥ Z (1) | X The estimators provided in Qu et al. [8] were rather complex; however, the idea is toestimate the potential response Y ij (1 − i ) or the potential adherence status A ij (1 − i ) underthe alternative treatment.Alternatively, estimators can be achieved naturally with the approach of MI. For a subject j in assigned treatment i , the potential outcome Y ij (1 − i ), Z ij (1 − i ), and I ij (1 − i ) aremissing for the alternative treatment 1 − i and can be imputed using all data from treatment1 − i , i.e., { ( X − i,j , Z − i,j , Y − i,j , I − i,j ) : 1 ≤ j ≤ n − i } , and his/her own baseline value X ij . Let { ( Z ij (1 − i ) ( m ) , Y ij (1 − i ) ( m ) , I ij (1 − i ) ( m ) ) , ≤ m ≤ M } be the M imputed valuesfor the potential outcomes under treatment 1 − i for subjects assigned to treatment i . The6otential adherence indicator under the alternative treatment based on the imputed valuesis calculated as: A ij (1 − i ) ( m ) = K − Y k =1 I ij (1 − i ) ( k,m ) . Table 1: Illustration of MI to impute the potential outcome under treatment T = t forpatients assigned to treatment T = t − X Z (1) Z (2) Z (3) I (1) I (2) I (3) Y t X X X X X t X X X X · t X X X · ·· · ·

101 1 − t X · · · · · · ·

102 1 − t X · · · · · · ·

103 1 − t X · · · · · · · Abbreviations: MI, multiple imputation; “ X ”, non-missing data; “ · ”, missing data.After imputation, for each patient the (potential) outcomes Y and A under both treat-ments are available. Then, the mean response for each treatment can be calculated by simplytaking the average of the (potential) outcome of Y for the (potential) adherers. The esti-mation of the treatment eﬀect for all randomized patients using MI has been extensivelystudied in the literature [27], so we will not discuss it here. Essentially, the estimators for S ∗ + and S + ∗ can be constructed in the exact same way by symmetry. Therefore, we onlyprovide the estimators for the mean response in each treatment on populations S ∗ + and S ++ (Table 2), which are most relevant for placebo-controlled trials and active-comparatortrials, respectively, as argued by Qu et al. [8]. For each treatment, the estimator can beconstructed using patients randomly assigned to treatment T = 0, T = 1, and all patients.Generally, it is preferable to use all patients in constructing the estimators as this mostclosely represents the study population. The treatment diﬀerence can be calculated easily7ased on the estimators for individual treatments.Table 2: Estimators for the mean response on principal strata deﬁned by treatment adherencePS Treatment Patient Estimator E M M P m =1 (cid:26) P n j =1 A j (1) ( m ) ( A j Y j +(1 − A j ) Y j (0) ( m ) ) P n j =1 A j (1) ( m ) (cid:27) T = 0 E M M P m =1 n P n j =1 A j Y j (0) ( m ) P n j =1 A j o S ∗ + E ∪ E M M P m =1 n P j =1 A j (1) ( m ) ( A j Y j +(1 − A j ) Y j (0) ( m ) ) + n P j =1 A j Y j (0) ( m ) n P j =1 A j (1) ( m ) + n P j =1 A j E M M P m =1 n P n j =1 A j (1) ( m ) Y j (1) ( m ) P n j =1 A j (1) ( m ) o T = 1 E P n j =1 A j Y j P n j =1 A j E ∪ E M M P m =1 n P n j =1 A j (1) ( m ) Y j (1) ( m ) + P n j =1 A j Y j P n j =1 A j (1) ( m ) + P n j =1 A j o E M M P m =1 n P n j =1 A j A j (1) ( m ) Y j P n j =1 A j A j (1) ( m ) o T = 0 E M P Mm =1 n P n j =1 A j A j (0) ( m ) Y j (0) ( m ) P n j =1 A j A j (0) ( m ) o E ∪ E M M P m =1 n P n j =1 A j A j (1) ( m ) Y j + P n j =1 A j A j (0) ( m ) Y j (0) ( m ) P n j =1 A j A j (1) ( m ) + P n j =1 A j A j (0) ( m ) o S ++ E M M P m =1 n P n j =1 A j A j (1) ( m ) Y j (1) ( m ) P n j =1 A j A j (1) ( m ) o T = 1 E M M P m =1 n P n j =1 A j A j (0) ( m ) Y j P n j =1 A j A j (0) ( m ) o E ∪ E M M P m =1 n P n j =1 A j A j (1) ( m ) Y j (1) ( m ) + P n j =1 A j A j (0) ( m ) Y j P n j =1 A j A j (1) ( m ) + P n j =1 A j A j (0) ( m ) o Abbreviation: PS, principal stratum; E t ( t = 0 ,

1) is the subset of patients randomized totreatment T = t . In this section, we consider a two-arm, parallel, randomized trial in diabetes with the simula-tion settings as described in Qu et al. [8]. The simulated data are denoted by ( X j , Z j , Y j , I j )8or subject j , where Y j is the primary outcome of HbA1c at Week 24 of treatment, X j isbaseline HbA1c, Z j = ( Z (1) j , Z (2) j , Z (3) j ) ′ is a vector of intermediate repeated measurementsof HbA1c reading at Weeks 6, 12, and 18, and I j = ( I (1) j , I (2) j , I (3) j ) ′ denotes the adherenceto treatment after Weeks 6, 12, and 18, respectively. The data are simulated for treatments T = 0 ,

1, respectively.The baseline value X j , intermediate readings Z j , and primary outcome Y j are generatedby: X j ∼ N ID ( µ x , σ x ) , (1) Z ( k ) j = α k + α k X j + α k T j + η ( k ) j , ≤ k ≤ , (2)and: Y j = β + β X j + β T j + X k =1 β k Z ( k ) j + ǫ j , (3)where NID means normally independently distributed, η ( k ) j ∼ N ID (0 , σ η ) and ǫ j ∼ N ID (0 , σ ǫ ),and η ( k ) j ’s and ǫ j are independent.We assume patients can drop out of the study after the collection of clinical data at timepoint k . The adherence indicator after time point k (1 ≤ k ≤

3) is generated from a logisticmodel: logit { Pr( I ( k ) j = 1 | I ( k − j = 1 , X j , Z ( k ) j ) } = γ + γ X j + γ k Z ( k ) j , (4)where logit( p ) = log( p/ (1 − p )), and by convention we set I (0) j = 1. If the adherence indicatorat any time point is 0, then the data after this time point are set to be missing.To mimic the response and treatment adherence rates in clinical trials for anti-diabetestreatments, we consider two settings in our simulation and the parameters are given: µ x =8 . σ x = 1 . α = α = α = 2 . α = α = α = − . α = − . α = − . α = − . β = 0 . β = − . β = − . β = 0 . β = 0 . β = 0 . σ η = 0 . σ ǫ = 0 . γ = 3, and γ = − . γ = − γ = − γ = − .

5, and j =1 to 150. These simulation parameters are the same for T = 0 , Z ij (1 − i ) , Y ij (1 − i ) , I j (1 − i )) under treatment(1 − i ) for patient j assigned in treatment group i , we need to use the data of X ij and( X − i,j , Z − i,j , Y − i,j , I − i,j ). We ﬁrst create missing records for ( Z ij (1 − i ) , Y ij (1 − i ) , I ij (1 − i )) and then apply an MI procedure to impute these “missing” values. Note the true missingvalues at treatment group (1 − i ) as a result of dropout are also imputed simultaneously.The following three steps are then implemented to impute the data: 1) use regressionmodels to impute Z ij (1 − i ) | X ij based on the relationship between Z − i,j and X − i,j , 2)impute Y ij (1 − i ) | ( X ij , Z ij ( i − Y − i,j ∼ X − i,j + Z − i,j , and3) impute I ij (1 − i ) | ( X ij , Z ij (1 − i )) based on the relationship between I − i,j and X − i,j through multiple logistic regressions. These three steps of multiple imputations can beeasily implemented through SAS PROC MI procedure. SAS code for simulation is providedas a supplementary document.With imputed data, the estimators can easily be calculated through equations providedin Table 2. The true mean responses for each treatment and the treatment diﬀerence for S ∗ + and S ++ can be calculated by numerical integration as described in Qu et al. [8]. To adjustfor baseline covariates, the set of E ∪ E is used in the calculation. The point estimates canbe obtained by averaging the mean estimates from multiple imputed samples. One methodto estimate the variance is to combine the within- and between-imputation variability byRubin [28] and Barnard and Rubin [29]; it has been reported that these methods mayprovide conservative coverage probability [30, 31]. Our simulations also demonstrate thatthe variability achieved through the methods of Barnard and Rubin [29] is too conservativeand the coverage probability of the conﬁdence interval is larger than its nominal value.Therefore, we also use the bootstrap method to estimate the variance of the estimator andreport the simulation in Table 3.Table 3 shows the simulation results based on 3,000 simulated samples. For each simu-lated sample, 200 imputations were implemented based on the multiple imputation proceduredescribed earlier. For the bootstrap approach, 50 bootstrap samples are generated for esti-mating the variability, and then 95% conﬁdence intervals are calculated based on a normal10able 3: Summary of the simulation results for the estimators of treatment eﬀect in twopopulations of adherers (based on 3,000 simulated samples)Bootstrap RubinSetting Parameter True Value Estimate Bias SE CP SE CP µ , ∗ + -0.102 -0.102 -0.001 0.049 0.951 0.052 0.965 µ , ∗ + -1.588 -1.587 0.001 0.046 0.940 0.047 0.950 µ d, ∗ + -1.487 -1.485 0.002 0.057 0.944 0.060 0.9621 µ , ++ -0.192 -0.191 0.001 0.052 0.949 0.058 0.971 µ , ++ -1.638 -1.640 -0.002 0.050 0.944 0.058 0.979 µ d, ++ -1.446 -1.449 -0.003 0.057 0.945 0.065 0.973 µ , ∗ + -0.107 -0.108 -0.001 0.063 0.939 0.069 0.962 µ , ∗ + -1.606 -1.601 0.004 0.052 0.937 0.053 0.948 µ d, ∗ + -1.499 -1.494 0.006 0.069 0.941 0.075 0.9612 µ , ++ -0.272 -0.263 0.009 0.071 0.939 0.088 0.980 µ , ++ -1.679 -1.678 0.000 0.064 0.941 0.088 0.990 µ d, ++ -1.406 -1.415 -0.009 0.069 0.944 0.095 0.990Abbreviations: CP, coverage of probability of the 95% conﬁdence interval; SE, standarderror; µ , ∗ + , population mean for the control group for S ∗ + ; µ , ∗ + , population mean forthe treatment group for S ∗ + ; µ d, ∗ + , population mean for the treatment diﬀerence betweentreatment and control groups for S ∗ + ; µ , ++ , population mean for the control group for S ++ ; µ , ++ , population mean for the treatment group for S ++ ; µ d, ++ , population meanfor the treatment diﬀerence between treatment and control groups for S ++ .approximation, which is appropriate in the simulation since the simulated data are generatedfrom a normal distribution. The reason that the percentile bootstrap was not used for con-structing the conﬁdence interval is that it requires a large number of bootstrap samples andis very time consuming. The estimates from both scenarios have little bias and the empiricalcoverage probability for the 95% conﬁdence intervals is close to the nominal level with thebootstrap approach. Speciﬁcally, by comparing the two scenarios in coverage probability,scenario 2 with lower adherence than scenario 1 has slightly lower but acceptable coverage.The 95% conﬁdence interval based on Rubin’s method has much higher coverage probabilitythan the normal level of 0.95. It is well-known that Rubin’s variance estimation method isconservative. The much wider conﬁdence interval of Rubin’s method is probably due to ahigh proportion of “missing” values to be imputed (at least 50% for Y and A ).11 Application

The application of the proposed method was based on the IMAGINE-3 Study, which hasbeen used by Bergenstal et al. [32] and allows for direct comparison with previous results.IMAGINE-3 was a 52-week treatment trial for patients with type 1 diabetes mellitus todemonstrate basal insulin lispro (BIL) was superior to insulin glargine (GL). In this trial,1,114 adults with type 1 diabetes were randomized to BIL and GL in a 3:2 ratio. The studywas conducted in accordance with the International Conference on Harmonisation Guidelinesfor Good Clinical Practice and the Declaration of Helsinki. This study was registered atclinicaltrials.gov as NCT01454284 and details of the study report have been published.Of the 1,114 randomized patients, 1,112 patients (663 in BIL, 449 in GL) took at least onedose of study drugs. A total of 235 patients permanently discontinued the study treatmentearly due to reasons of lack of eﬃcacy (LoE), adverse events (AE), or adminstration reasons,leaving 877 (78.9%) patients adhering to the treatment.To apply to the proposed methods, we consider the following baseline covariates X thatcould potentially impact treatment adherence: age, gender, HbA1c, low density lipoproteinclolesterol (LDL-C), triglyceride (TG), fasting serum glucose (FSG), and alanine amino-transferase (ALT). The study also collected HbA1c, LDL-C, TG, FSG, and ALT at Week12 and Week 26, and injection site reaction adverse events (a binary variable) that occurredbetween randomization and Week 12 and between Week 12 and Week 26. Those post base-line variables were considered in intermediate covariates Z for Week 12 and Z for Week26, respectively. The primary outcome Y is the HbA1c reading at Week 52.For each stratum of S ∗ + or S ++ , 1,000 imputations were generated and the completedata after imputation were used to estimate the mean response for each treatment groupand the treatment diﬀerence. Due to a large amount of missing values (potential outcomes forthe alternative treatments were not observed), we need large imputations to achieve goodaccuracy for the estimates. Based on our investigation, the 1,000 imputed samples madethe variance due to imputation random error < .

1% of the variance of the ﬁnal estimate(average of the estimates from 1,000 imputations). The variance of the ﬁnal estimate was12stimated using 50 bootstrap samples and the corresponding 95% conﬁdence interval wascalculated using the normal approximation with the bootstrap variance. Due to the relativelylarge sample size, we expect the distributions of the estimators are approximately Gaussian,which was conﬁrmed by normal Q-Q plots of the 50 bootstrap estimates for all parametersfor S ∗ + and S ++ (data not shown here).Table 4: Summary of results of the real data analysis for the estimators of treatment eﬀect inHbA1c for the two populations of adherers using proposed methodsTreatment S ∗ + S ++ Estimate ± SE 95% CI Estimate ± SE 95% CIGL 7.59 ± ± ± ± ± ± Table 4 shows the estimates for HbA1c at 52 weeks for each treatment group and thetreatment diﬀerence for the population S ∗ + and S ++ using the method based on E ∪ E .The estimates were similar to those reported in Qu et al. [9]. It showed BIL was superiorto GL in controlling HbA1c for the two principal strata: S ∗ + and S ++ . In addition to the commonly used estimands for the treatment diﬀerence for all randomizedpatients, the treatment diﬀerence for adherers (a principal stratum) is also important andplays a primary role in assessing the eﬀect of a treatment as described in the so-calledtripartite approach [7, 33]. When making decisions whether to start a new pharmacologictreatment or not, patients and physicians want to know what the eﬀects of that treatment arewhen the patient takes the medication as prescribed. Careful thought and more sophisticatedanalyses are required (i.e., not the na¨ıve completers analysis) so that data from randomized13linical trials can be used to assess this important principal stratum and provide an estimateof the causal eﬀect of the treatment.Furthermore, some aspects of clinical practice remain trial and error; a patient is pre-scribed a medication and follow-up visits are scheduled to assess the status of the patient’sdisease or any resulting side eﬀects. Those observations are used to guide the patient’s sub-sequent treatment with dosage changes or switching to other treatments. In our interactionswith physicians, many start by prescribing a treatment that is highly eﬀective when takenat the recommended dose and frequency and only alter or discontinue that treatment if sideeﬀects are not tolerable. This is considered preferable to be starting with a treatment oflesser eﬃcacy but perhaps greater adherence due to fewer or more acceptable side eﬀects.Such treatments can always be a “fallback” option.These considerations suggest that the treatment eﬀect estimate in the principal stratumof patients who can adhere to treatment is very important, if not more important than theestimate for all randomized patients. Additionally, side eﬀects are most often analyzed anddescribed in the context of what happens when a medication is taken as prescribed, andwe believe this context is most relevant for eﬃcacy as well, especially when assimilatinginformation into beneﬁt-risk assessments.Qu et al. [8] provides the general framework for estimating the AdACE, but the imple-mentation of such estimators is rather complex. In this article, we proposed an MI-basedmethod to construct the estimators, which is much more straightforward than the originalmethod proposed to construct estimators. We evaluated the performance through simula-tions and it showed that the new method provides consistent estimators and has the correctcoverage probability for the bootstrap conﬁdence interval at the nominal alpha level. Wealso applied these MI-based estimators to the same data set as in Qu et al. [9] and yieldedsimilar results as the original estimates.In summary, the MI-based estimation proposed in this article will allow for broader adop-tion and easier estimation of the AdACE, providing estimation for an alternative clinicallymeaningful estimand for adherers. 14 cknowledgements

We would like to thank Dr. Ilya Lipkovich for his scientiﬁc review and useful comments.