Original: Identifying Principal Stratum Causal Effects Conditional on a Post-treatment Intermediate Response 
Revised: Identifying Principal Stratum Causal Effects Conditional on a Post-treatment Intermediate Response XIT 31@ PITT . EDU

Original: Author names withheld
Revised: ABBERBOCK @ GMAIL . COM RASTOGIP @ UPMC .

Original: Editor: Under Review for CLeaR
Revised: EDU GOT 1@ PITT . EDU Editors: Bernhard Sch ¨ olkopf, Caroline Uhler and Kun Zhang

Original: Abstract
Revised: Abstract

Original: In neoadjuvant trials on early-stage breast cancer, patients are usually randomized into a control group and a treatment group with an additional target therapy.
Revised: In neoadjuvant trials on early-stage breast cancer, patients are usually randomized into a control group and a treatment group with an additional target therapy.

Original: Early efficacy of the new regimen is assessed via the binary pathological complete response (pCR) and the eventual efficacy is assessed via long-term clinical outcomes such as survival.
Revised: Early efficacy of the new regimen is assessed via the binary pathological complete response (pCR) and the eventual efficacy is assessed via long-term clinical outcomes such as survival.

Original: Although pCR is strongly associated with survival, it has not been confirmed as a surrogate endpoint.
Revised: Although pCR is strongly associated with survival, it has not been confirmed as a surrogate endpoint.

Original: To fully understand its clinical implication, it is important to establish causal estimands such as the causal effect in survival for patients who would achieve pCR under the new regimen.
Revised: To fully understand its clinical implication, it is important to establish causal estimands such as the causal effect in survival for patients who would achieve pCR under the new regimen.

Original: Under the principal stratification framework, previous works focused on sensitivity analyses by varying model parameters in an imposed model on counterfactual outcomes.
Revised: Under the principal stratification framework, previous works focused on sensitivity analyses by varying model parameters in an imposed model on counterfactual outcomes.

Original: Under mild assumptions, we propose an approach to estimate those model parameters using empirical data and subsequently the causal estimand of interest.
Revised: Under mild assumptions, we propose an approach to estimate those model parameters using empirical data and subsequently the causal estimand of interest.

Original: We also extend our approach to address censored outcome data.
Revised: We also extend our approach to address censored outcome data.

Original: The proposed method is applied to a recent clinical trial and its performance is evaluated via simulation studies.
Revised: The proposed method is applied to a recent clinical trial and its performance is evaluated via simulation studies.

Original: 1. Introduction
Revised: 1. Introduction

Original: We have seen a major shift in the conduct of breast cancer clinical trials in recent years.
Revised: We have seen a major shift in the conduct of breast cancer clinical trials in recent years.

Original: Traditionally, breast cancer patients are randomly assigned to control or treatment after the primary surgery.
Revised: Traditionally, breast cancer patients are randomly assigned to control or treatment after the primary surgery.

Original: Patients from the two groups were then followed over years for comparison of their long-term outcomes such as disease-free survival and overall survival.
Revised: Patients from the two groups were then followed over years for comparison of their long-term outcomes such as disease-free survival and overall survival.

Original: However, in recent years, there have been an increasing number of neoadjuvant trials where many of the systemic therapies are administered prior to the breast surgery (FDA, 2013).
Revised: However, in recent years, there have been an increasing number of neoadjuvant trials where many of the systemic therapies are administered prior to the breast surgery (FDA, 2013).

Original: The primary endpoint in neoadjuvant breast cancer clinical trials is pathological complete response (pCR), a binary indicator of absence of invasive cancer in the breast and axillary nodes (FDA, 2013).
Revised: The primary endpoint in neoadjuvant breast cancer clinical trials is pathological complete response (pCR), a binary indicator of absence of invasive cancer in the breast and axillary nodes (FDA, 2013).

Original: The rationale for using pCR is that efficacy of a treatment can be assessed at the time of surgery instead of the typical 5-10 years of follow-up on survival endpoints in the adjuvant setting.
Revised: The rationale for using pCR is that efficacy of a treatment can be assessed at the time of surgery instead of the typical 5-10 years of follow-up on survival endpoints in the adjuvant setting.

Original: Strong association between pCR and survival has been well documented (Cortazar et al., 2014;
Revised: Strong association between pCR and survival has been well documented (Cortazar et al., 2014;

Original: Von Minckwitz et al., 2012), making pCR an attractive candidate surrogate.
Revised: Von Minckwitz et al., 2012), making pCR an attractive candidate surrogate.

Original: In the latest guidance of the U.S. Food and Drug administration (FDA), pCR was accepted as an endpoint to support accelerated drug approvals, provided certain requirements are met (FDA, 2013).
Revised: In the latest guidance of the U.S. Food and Drug administration (FDA), pCR was accepted as an endpoint to support accelerated drug approvals, provided certain requirements are met (FDA, 2013).

Original: It is important to decipher the causal relationship among treatment, pCR, and survival in order to interpret the efficacy in survival when pCR is involved.
Revised: It is important to decipher the causal relationship among treatment, pCR, and survival in order to interpret the efficacy in survival when pCR is involved.

Original: In the recently published National Surgical Adjuvant Breast and Bowel Project (NSABP) Btrial, patients with operable human epidermal growth factor receptor 2 (HER2)-negative breast cancer were randomly assigned to receive or not to receive bevacizumab along with their neoadjuvant chemotherapy regimens (Bear et al., 2012).
Revised: In the recently published National Surgical Adjuvant Breast and Bowel Project (NSABP) Btrial, patients with operable human epidermal growth factor receptor 2 (HER2)-negative breast cancer were randomly assigned to receive or not to receive bevacizumab along with their neoadjuvant chemotherapy regimens (Bear et al., 2012).

Original: The addition of bevacizumab significantly increased the rate of pCR (28.2% without bevacizumab vs. 34.5% with bevacizumab, p-value = 0 . 02 ).
Revised: The addition of bevacizumab significantly increased the rate of pCR.

Original: In terms of the long-term outcomes, patients on bevacizumab showed improvements in event-free survival (EFS) and overall survival (OS) compared to the control patients(EFS: hazard ratio 0.80, p-value = 0 . OS: hazard ratio 0.65, p-value = 0 . 004 ) (Bear et al., 2015).
Revised: In terms of the long-term outcomes, patients on bevacizumab showed improvements in event-free survival (EFS) and overall survival (OS) compared to the control patients (Bear et al., 2015).

Original: Some investigators are interested in the comparison of survival between pCR patients in the treatment group and pCR patients in the control group.
Revised: Some investigators are interested in the comparison of survival between pCR patients in the treatment group and pCR patients in the control group.

Original: Such comparison, however, is problematic because these two groups of pCR patients are different and any direct comparison between them lacks causal interpretation.
Revised: Such comparison, however, is problematic because these two groups of pCR patients are different and any direct comparison between them lacks causal interpretation.

Original: Under the counterfactual framework (Rubin, 1974), potentially a patient has a pCR status after taking the control regimen and a pCR status after taking the treatment.
Revised: Under the counterfactual framework (Rubin, 1974), potentially a patient has a pCR status after taking the control regimen and a pCR status after taking the treatment.

Original: Similarly, one can define counterfactual outcomes and causal effects in survival status (0/1) after a certain time period such as three years.
Revised: Similarly, one can define counterfactual outcomes and causal effects in survival status (0/1) after a certain time period such as three years.

Original: The principal stratum framework proposed by Frangakis and Rubin (2002) can be used to describe causal effect in long-term outcomes (such as EFS) with an intermediate outcome (such as pCR) involved.
Revised: The principal stratum framework proposed by Frangakis and Rubin (2002) can be used to describe causal effect in long-term outcomes (such as EFS) with an intermediate outcome (such as pCR) involved.

Original: Each principal stratum consists of subjects with the same pair of potential pCR status: the pCR status under the control regimen and the pCR status under the treatment regimen.
Revised: Each principal stratum consists of subjects with the same pair of potential pCR status: the pCR status under the control regimen and the pCR status under the treatment regimen.

Original: One can then define the causal effect of treatment in EFS on each principal stratum.
Revised: One can then define the causal effect of treatment in EFS on each principal stratum.

Original: Here we propose a method to identify and estimate principal stratum causal effects for a binary outcome and later extend our method for censored outcome data.
Revised: Here we propose a method to identify and estimate principal stratum causal effects for a binary outcome and later extend our method for censored outcome data.

Original: The causal estimand of interest is the treatment efficacy in 3-year EFS and OS among patients who would achieve pCR under chemotherapy plus bevacizumab as in our motivating study, the NSABP B-40 trial.
Revised: The causal estimand of interest is the treatment efficacy in 3-year EFS and OS among patients who would achieve pCR under chemotherapy plus bevacizumab as in our motivating study, the NSABP B-40 trial.

Original: A model of counterfactual outcome given the observed data is imposed.
Revised: A model of counterfactual outcome given the observed data is imposed.

Original: Using some probabilistic arguments, we connect the model parameters with quantities that can be empirically estimated from the observed data.
Revised: Using some probabilistic arguments, we connect the model parameters with quantities that can be empirically estimated from the observed data.

Original: The resulting equations allow us to estimate the model parameters and subsequently the causal estimand of interest, and resolve the identifiability issue.
Revised: The resulting equations allow us to estimate the model parameters and subsequently the causal estimand of interest, and resolve the identifiability issue.

Original: Our paper is organized as follows.
Revised: Our paper is organized as follows.

Original: Section 2 presents related works in principal stratum causal effects.
Revised: Section 2 presents related works in principal stratum causal effects.

Original: Section 3 introduces the standard data settings,necessary assumptions, causal estimands of interest, and a regression model in the context of a randomized neoadjuvant trial.
Revised: Section 3 introduces the standard data settings, causal estimands of interest, and a regression model in the context of a randomized neoadjuvant trial.

Original: In Section 4, we propose a method to estimate the model parameters and  the causal estimand.
Revised: In Section 4, we provide key assumptions for identification of the causal estimand and introduce the proposed method.

Original: In Section 5, we conduct a simulation study to assess the performance of our method in terms of bias and coverage of bootstrap confidence intervals.
Revised: In Section 5, we conduct a simulation study to assess the performance of our method in terms of bias and coverage of bootstrap confidence intervals.

Original: In Section 6, we apply the proposed method to the motivating
Revised: In Section 6, we apply the proposed method to the motivating

Original: NSABP B-40 study.
Revised: NSABP B-40 study.

Original: We conclude with a discussion of the proposed method and future works in
Revised: We conclude with a discussion of the proposed method and future works in

Original: Section 7.
Revised: Section 7.

Original: 2. Related Works
Revised: 2. Related Works

Original: Frangakis and Rubin (2002) proposed to split study population into principal strata.
Revised: Frangakis and Rubin (2002) proposed to split study population into principal strata.

Original: Each principal stratum is by definition independent of treatment assignment since it contains information on counterfactual, or potential outcomes rather than the observed outcome for a specific treatment assignment.
Revised: Each principal stratum is by definition independent of treatment assignment since it contains information on counterfactual, or potential outcomes rather than the observed outcome for a specific treatment assignment.

Original: One can then define treatment effects on each principal stratum.
Revised: One can then define treatment effects on each principal stratum.

Original: Additionally, any union of the basic principal strata would also be a valid principal stratum as it leads to comparisons among a common set of individuals.
Revised: Additionally, any union of the basic principal strata would also be a valid principal stratum as it leads to comparisons among a common set of individuals.

Original: Gilbert et al.
Revised: Gilbert et al.

Original: (2015) show the principal stratification framework is useful for evaluating whether and how treatment effects differs across subgroups characterized by the intermediate variable, thus being firmly associated with the utility of the treatment marker.
Revised: (2015) show the principal stratification framework is useful for evaluating whether and how treatment effects differs across subgroups characterized by the intermediate variable, thus being firmly associated with the utility of the treatment marker.

Original: Identification of principal stratum causal effects is in general difficult.
Revised: Identification of principal stratum causal effects is in general difficult.

Original: A major challenge is that we do not observe the individual membership of principal stratum because of its counterfactual nature (Gilbert and Hudgens, 2008; Wolfson and Gilbert, 2010).
Revised: A major challenge is that we do not observe the individual membership of principal stratum because of its counterfactual nature (Gilbert and Hudgens, 2008; Wolfson and Gilbert, 2010).

Original: Under the principal stratification framework, Gilbert et al.
Revised: Under the principal stratification framework, Gilbert et al.

Original: (2003) proposed to perform sensitivity analyses by varying model parameters in an imposed parametric model for counterfactual outcomes.
Revised: (2003) proposed to perform sensitivity analyses by varying model parameters in an imposed parametric model for counterfactual outcomes.

Original: Shepherd et al.
Revised: Shepherd et al.

Original: (2006) and
Revised: (2006) and

Original: Jemiai et al.
Revised: Jemiai et al.

Original: (2007) extended this sensitivity analyses approach by including baseline covariates in the model.
Revised: (2007) extended this sensitivity analyses approach by including baseline covariates in the model.

Revised: These sensitivity analyses can provide researchers with a range of causal estimates under different values of the sensitivity parameters.

Original: In reality, however, it is often unclear what the plausible values are for these sensitivity parameters and the selected combinations may not be exhaustive.
Revised: In reality, however, it is often unclear what the plausible values are for these sensitivity parameters and the selected combinations may not be exhaustive.

Original: Li et al.
Revised: Li et al.

Original: (2010); Zigler and Belin (2012) used Bayesian approaches to model the joint distribution of the counterfactual intermediate outcomes and long-term outcomes and incorporate prior information regarding nonidentifiable associations.
Revised: (2010); Zigler and Belin (2012) used Bayesian approaches to model the joint distribution of the counterfactual intermediate outcomes and long-term outcomes and incorporate prior information regarding nonidentifiable associations.

Original: The lack of identifiability, however, still exists and is reflected by the over-coverage of confidence intervals in their simulation studies.
Revised: The lack of identifiability, however, still exists and is reflected by the over-coverage of confidence intervals in their simulation studies.

Original: Principal stratum causal effects with regards to outcomes truncated by death are not identifiable without further assumptions (Zhang and Rubin, 2003; Kurland et al., 2009; Lee et al., 2010).
Revised: Principal stratum causal effects with regards to outcomes truncated by death are not identifiable without further assumptions (Zhang and Rubin, 2003; Kurland et al., 2009; Lee et al., 2010).

Original: Tchetgen Tchetgen (2014) identified causal effects by borrowing information from post-treatment risk factors of the intermittent outcome and the causal estimand may vary according to the selected risk factors.
Revised: Tchetgen Tchetgen (2014) identified causal effects by borrowing information from post-treatment risk factors of the intermittent outcome and the causal estimand may vary according to the selected risk factors.

Original: Instrumental variables were also introduced to provide information on the unobserved principal strata and the justification of that exclusion restriction assumption is often challenging (Ding et al., 2011; Wang et al., 2017).
Revised: Instrumental variables were also introduced to provide information on the unobserved principal strata and the justification of that exclusion restriction assumption is often challenging (Ding et al., 2011; Wang et al., 2017).

Original: All the above methods either fell into sensitivity analyses or required exclusion restriction assumptions.
Revised: All the above methods either fell into sensitivity analyses or required exclusion restriction assumptions.

Original: In this paper, we propose a method to identify and estimate principal stratum causal effects under data settings as Shepherd et al.
Revised: In this paper, we propose a method to identify and estimate principal stratum causal effects under data settings as Shepherd et al.

Original: (2006) for a binary outcome and later extend our method to address issues of censored outcome data under mild assumptions.
Revised: (2006) for a binary outcome and later extend our method to address issues of censored outcome data under mild assumptions.

Original: Identification of the causal effect is achieved with the bias minimal and the coverage probabilities close to the nominal levels.
Revised: Identification of the causal effect is achieved with the bias minimal and the coverage probabilities close to the nominal levels.

Original: 3. The Principal Stratification Framework of Interest 3.1. Standard Setting for Neoadjuvant Studies
Revised: 3. The Principal Stratification Framework of Interest 3.1. Standard Setting for Neoadjuvant Studies

Original: Consider a neoadjuvant breast cancer clinical trial where patients are randomized to two treatment groups.
Revised: Consider a neoadjuvant breast cancer clinical trial where patients are randomized to two treatment groups.

Original: For subject i = 1 , 2 , . . . , n , let Z i ∈ { 0 , 1 } be the binary treatment assignment;
Revised: For subject i = 1 , 2 , . . . , n , let Z i ∈ { 0 , 1 } be the binary treatment assignment;

Original: X i ∈ Γ = { 0 , 1 , . . .
Revised: X i ∈ Γ = { 0 , 1 , . . .

Original: , K } be a baseline discrete covariate.
Revised: , K } be a baseline discrete covariate.

Original: A continuous baseline variable X
Revised: A continuous baseline variable X

Original: i such as clinical tumor size, would be grouped into K + 1 categories based on scientific knowledge.
Revised: i such as clinical tumor size, would be grouped into K + 1 categories based on scientific knowledge.

Original: We will discuss extensions to the scenarios with a continuous X i in Section 7.
Revised: We will discuss extensions to the scenarios with a continuous X i in Section 7.

Original: Throughout this paper, we assume that the stable unit treatment value assumption (SUTVA) (Rubin, 1980) holds: the potential outcomes of any individual i are unrelated to the treatment assignment of other individuals.
Revised: Throughout this paper, we assume that the stable unit treatment value assumption (SUTVA) (Rubin, 1980) holds: the potential outcomes of any individual i are unrelated to the treatment assignment of other individuals.

Original: Then we can denote S
Revised: Then we can denote S

Original: i ( Z i ) ∈ { 0 , 1 } as a binary post-randomization intermediate response such as the pCR status for subject i under treatment
Revised: i ( Z i ) ∈ { 0 , 1 } as a binary post-randomization intermediate response such as the pCR status for subject i under treatment

Original: Z i (possibly counterfactual).
Revised: Z i (possibly counterfactual).

Original: And denote Y i ( Z i , S i ( Z i ))
Revised: And denote Y i { Z i , S i ( Z i ) }

Original: =
Revised: =

Original: Y i ( Z i ) ∈ { 0 , 1 } as a binary long-term outcome of interest such as the
Revised: Y i ( Z i ) ∈ { 0 , 1 } as a binary long-term outcome of interest such as the

Original: EFS status at 3-year after study entry for subject i under treatment
Revised: EFS status at 3-year after study entry for subject i under treatment

Original: Z i (possibly counterfactual).
Revised: Z i (possibly counterfactual).

Original: For individual i , { Z i , X i , S i ( Z i ) , Y i ( Z i ) } represents the observed data of treatment assignment, baseline covariate, intermediate response and long-term outcome.
Revised: For individual i , { Z i , X i , S i ( Z i ) , Y i ( Z i ) } represents the observed data of treatment assignment, baseline covariate, intermediate response and long-term outcome.

Original: If Z i = 0 , { S i (0) , Y i (0) } are observed and { S i (1) , Y i (1) } are counterfactual. If Z i = 1 , then { S i (1) , Y i (1) } are observed and { S i (0) , Y i (0) } are counterfactual.
Revised: If Z i = 0 , { S i (0) , Y i (0) } are observed and { S i (1) , Y i (1) } are counterfactual. If Z i = 1 , then { S i (1) , Y i (1) } are observed and { S i (0) , Y i (0) } are counterfactual.

Original: Thus for individual i , the complete counterfactual data would be { Z i , X i , S i (0) , S i (1) , Y i (0) , Y i (1) } .
Revised: Thus for individual i , the complete counterfactual data would be { Z i , X i , S i (0) , S i (1) , Y i (0) , Y i (1) } .

Original: Another important assumption is the monotonicity assumption: S i (0) ≤ S i (1) (Angrist et al., 1996), as in the motivating NSABP B-40 study, addition of bevacizumab led to improved pCR (Bear et al., 2012).
Revised: Another important assumption is the monotonicity assumption: S i (0) ≤ S i (1) (Angrist et al., 1996), as in the motivating NSABP B-40 study, addition of bevacizumab led to improved pCR (Bear et al., 2012).

Original: We also assume for subject
Revised: We also assume for subject

Original: i , the treatment assignment Z i is independent of X i and the potential outcomes.
Revised: i , the treatment assignment Z i is independent of X i and the potential outcomes.

Original: Under the principal stratification framework, denote the principal strata to be E jk
Revised: Under the principal stratification framework, denote the principal strata to be E jk

Original: = { i :
Revised: = { i :

Original: S i (0) =
Revised: S i (0) =

Original: j, S i (1) =
Revised: j, S i (1) =

Original: k } , j, k = 0 , 1 .
Revised: k } , j, k = 0 , 1 .

Original: The principal stratum causal effects of interest are
Revised: The principal stratum causal effects of interest are

Original: [Equation]
Revised: [Equation]

Original: Under the monotonicity assumption, the principal stratum E 10 is empty.
Revised: Under the monotonicity assumption, the principal stratum E 10 is empty.

Original: In the NSABP Bstudy, we are interested in the causal effect in E 01 ∪ E 11 , those who would achieve pCR had they been treated with chemotherapy plus bevacizumab:
Revised: In the NSABP Bstudy, we are interested in the causal effect in E 01 ∪ E 11 , those who would achieve pCR had they been treated with chemotherapy plus bevacizumab:

Original: [Equation]
Revised: [Equation]

Original: Other principal stratum causal effects such as θ jk can be estimated using a similar approach as we outline in Section 4.
Revised: Other principal stratum causal effects such as θ jk can be estimated using a similar approach as we outline in Section 4.

Original: 3.2. Modelling a Counterfactual Outcome
Revised: 3.2. Modelling a Counterfactual Outcome

Original: In order to estimate the principal stratum causal effects, Gilbert et al.
Revised: In order to estimate the principal stratum causal effects, Gilbert et al.

Original: (2003) proposed to use a logistic regression model for Pr { S i (1) = 1 | S i (0)
Revised: (2003) proposed to use a logistic regression model for Pr { S i (1) = 1 | S i (0)

Original: = 0 , Y i (0) } as
Revised: = 0 , Y i (0) } as

Original: [Equation]
Revised: [Equation]

Original: Shepherd et al.
Revised: Shepherd et al.

Original: (2006) further extended the logistic regression by incorporating baseline covariates X i as
Revised: (2006) further extended the logistic regression by incorporating baseline covariates X i as

Original: [Equation]
Revised: [Equation]

Original: Jemiai et al.
Revised: Jemiai et al.

Original: (2007) considered a more general model framework:
Revised: (2007) considered a more general model framework:

Original: [Equation]
Revised: [Equation]

Original: where w ( u ) ≡ { 1 + exp ( − u ) } − 1 and g ( · , · ) is a known function.
Revised: where w ( u ) ≡ { 1 + exp ( − u ) } − 1 and g ( · , · ) is a known function.

Original: In the case of Shepherd et al.
Revised: In the case of Shepherd et al.

Original: (2006), g ( u, v )
Revised: (2006), g ( u, v )

Original: =
Revised: =

Original: β 1 u with β 1 known.
Revised: β 1 u with β 1 known.

Original: Jemiai et al.
Revised: Jemiai et al.

Original: (2007) showed that under the monotonicity assumption, inference could be made on θ for any fixed function g and sensitivity analyses could be performed by varying g .
Revised: (2007) showed that under the monotonicity assumption, inference could be made on θ for any fixed function g and sensitivity analyses could be performed by varying g .

Original: 4. The Proposed Method 4.1. Estimation of Causal Estimands
Revised: 4. The Proposed Method 4.1. Key Identification Assumptions

Revised: Identification of causal effects is achieved through two key assumptions.

Revised: First, the monotonicity assumption: S i (0) ≤ S i (1) (Angrist et al., 1996).

Revised: That is, a subject who responds under the control would respond if given the treatment.

Revised: This monotonicity assumption could prove valuable (Bartolucci and Grilli, 2011) and can be justified in many scenarios that the additional therapy would help to improve the response.

Original: The causal estimand of interest is 
Revised: In the motivating NSABP B-40 study, addition of bevacizumab led to improved pCR (Bear et al., 2012). Second, a parametric model is used to describe the counterfactual response under the treatment for a control non-respondent. Both the future long-term outcome and a baseline covariate are predictors in this parametric model. It is required that the level of the covariates is at least of the same dimension of model parameters and the imposed linearity assumption is critical to identify and estimate those regression parameters.

Revised: We will elaborate the second assumption in Section 4.2.

Revised: 4.2. Identification of Model Parameters and causal estimands

Revised: As mentioned in Shepherd et al.

Revised: (2006) and will be described in Section 4.4, when the parameters of model (1) are identified, the causal estimands can be identified.

Revised: Lemma 1 For any x ∈ Γ = { 0 , 1 , . . .

Revised: , K } and y ∈ { 0 , 1 } , let a x = Pr { S (1) =

Revised: 1 | S (0) = 0 , X = x } and b yx =

Revised: Pr { Y (0) =

Revised: y | S (0)

Revised: = 0 ,

Revised: X = x } .

Revised: Let a =

Revised: ( a 0 , a 1 , . .

Revised: . , a K ) T and b y =

Revised: ( b y 0 , b y 1 , . . .

Revised: , b yK ) T .

Revised: Define h x ( β , a , b 0 , b 1 ) =

Original: {+a x − (cid:80) 1 y =
+}
Revised: {+a x − (cid:80) 1 y =
+}

Revised: b yx logit − 1 { β 0 + β 1 y + β 2 x } , and H ( β , a , b 0 , b 1 ) = { h 0 ( β, a , b 0 , b 1 ) , . . .

Revised: , h K ( β, a , b 0 , b 1 ) } T .

Revised: If rank { ∂H ( β , a , b 0 , b 1 ) ∂ β } = 3 , within the neighborhood of β there is a unique solution β = ψ ( a , b 0 , b 1 ) such that H { ψ ( a , b 0 , b 1 ) , a , b 0 , b 1 } = 0 .

Revised: Proof For all x ∈ Γ , we have

Revised: [Equation]

Revised: Hence, H ( β , a , b 0 , b 1 ) = 0 and H ( · ) is a smooth function of β , a , b 0 , and b 1 .

Revised: By invoking the implicit function theorem, when rank( ∂H∂ β ) = 3 , there exists a smooth function ψ such that β = ψ ( a , b 0 , b 1 ) and H { ψ ( a , b 0 , b 1 ) , a , b 0 , b 1 } = 0 .

Revised: The identifiability of model parameter β depends on the availability of a x = Pr { S (1) =

Revised: 1 | S (0) = 0 , X = x } and b yx =

Revised: Pr { Y (0) =

Revised: y | S (0)

Revised: = 0 ,

Revised: X = x } , for x ∈ Γ; y = 0 , 1 .

Revised: The linearity in X = x in model (1) also plays an important role.

Revised: In general, when β 2 ̸ = 0 and K ≥ 2 , there are equal or more equations than the number of unknown parameters in β , Lemma 1 would hold.

Revised: In practice, given ( a , b 0 , b 1 ) , one solves for β such that H ( β , a , b 0 , b 1 ) = 0 .

Revised: Then verify that rank { ∂H ( β , a , b 0 , b 1 ) ∂ β } = 3 at the solution.

Revised: 4.3. Estimation of Causal Estimands

Revised: The causal estimand of interest is

Original: [Equation]
Revised: [Equation]

Original: Because { Y i (1) , S i (1) } are observed for subjects in the treatment arm, Pr { Y i (1) = 1 | S i (1) = 1 } can be estimated by
Revised: Because { Y i (1) , S i (1) } are observed for subjects in the treatment arm, Pr { Y i (1) = 1 | S i (1) = 1 } can be estimated by

Original: [Equation]
Revised: [Equation]

Original: where I ( · ) is the indicator function.
Revised: where I ( · ) is the indicator function.

Original: Meanwhile,
Revised: Meanwhile,

Original: [Equation]
Revised: [Equation]

Original: In equation (5), Pr { X i = x } can be estimated by (cid:99) Pr { X i = x } = (cid:80) i I( X
Revised: In equation (5), Pr { X i = x } can be estimated by (cid:99) Pr { X i = x } = (cid:80) i I

Original: [Equation]

Original:  = x ) nand
Revised: ( X i = x ) n

Original: Pr
Revised: and

Original: [Equation] S [Equation] [Equation]{+i (1) = 1 , Y i (0) = 
+}
Revised: Pr { S {+i (1) = 1 , Y i (0) = 
+}

Original: [Equation] Y [Equation] = 1 | X i = x } = {+(cid:80) 
+}
Revised: | X i = x } = {+(cid:80) 
+}

Original: [Equation] [Equation] [Equation] Y{+j =
+}
Revised: {+j =
+}

Original: [Equation] [Equation] S i [Equation]
Revised: Pr { S i (1) = 1 , Y i (0) = 1 , S i (0) =

Original: = j | X [Equation] = x } = (cid:80) {+
+}
Revised: j | X i = x } = (cid:80) {+
+}

Original: [Equation] Y [Equation]{+j =
+}
Revised: {+j =
+}

Original: = 1 | [Equation] = j
Revised: Pr {

Original: [Equation]
Revised: [Equation]

Original: X [Equation] =x } · Pr [Equation]
Revised: i (1) = 1 , Y i (0) = 1 | S i (0) =

Original: [Equation] = j | X [Equation]
Revised: j, X i = x } · Pr { S i (0) =

Original:  = x } = (cid:80) [Equation] · Pr { Y i [Equation]{+
+}
Revised: j | X i = x } = (cid:80) {+
+}

Original: = 1 | S i

Original: [Equation] ) = j [Equation]
Revised: j =0 (cid:104)

Original: X i [Equation] x }
Revised: Pr { S i (1) = 1 | S i (0) =

Original: [Equation]
Revised: j, Y i (0)

Original: | [Equation] x 
Revised: = 1 , X i = x } · Pr { Y i (0) =

Original: [Equation]
Revised: [Equation]

Original: In equation (6), Pr { Y i (0) = 

Revised: In equation (6), Pr { Y i (0) =

Original: | S i (0) =
Revised: | S i (0) =

Original: j, X i = x } , j = 0 , 1 , can be estimated by(cid:99) Pr { Y i (0) = 1 | S i (0) = j, X i = x } = (cid:80) i I ( Z i =
Revised: j, X i = x } , j = 0 , 1 , can be estimated by

Original: [Equation]
Revised: [Equation]

Original: By the monotonicity assumption, Pr { S i (1) = 1 | S i (0) = 1 , Y i (0)
Revised: By the monotonicity assumption, Pr { S i (1) = 1 | S i (0) = 1 , Y i (0)

Original: = 1 , X i = x } ≡ 1 .
Revised: = 1 , X i = x } ≡ 1 .

Original: The estimation of Pr { S i ( j ) = 1 | X i = x } , j = 0 , 1 , is described in Lemma 1.
Revised: The estimation of Pr { S i ( j ) = 1 | X i = x } , j = 0 , 1 , is described in Lemma 2.

Original: Lemma 1 Under the monotonicity assumption, for any x , we denote
Revised: Lemma 2 Under the monotonicity assumption, for any x , we denote

Original: [Equation]
Revised: [Equation]

Original: the observed proportions of responders in the control group and the treatment group with X = x , respectively.
Revised: the observed proportions of responders in the control group and the treatment group with X = x , respectively.

Original: We use maximum likelihood estimation to estimate Pr { S i ( j ) = 1 | X i = x } , j = 0 , 1 .
Revised: We use maximum likelihood estimation to estimate Pr { S i ( j ) = 1 | X i = x } , j = 0 , 1 .

Original: (a) when (cid:98) q 1 ( x ) ≥ (cid:98) q 0 ( x ) , the maximum likelihood estimate of Pr { S i ( j ) = 

Revised: (a) when (cid:98) q 1 ( x ) ≥ (cid:98) q 0 ( x ) , the maximum likelihood estimate of Pr { S i ( j ) =

Original: | X i = x } is (cid:98) q j ( x ) , j = 0 , 1; (b) when (cid:98) q 1 ( x ) < (cid:98) q 0 ( x ) , the maximum likelihood estimate of Pr { S i ( j ) = 

Revised: | X i = x } is (cid:98) q j ( x ) , j = 0 , 1; (b) when (cid:98) q 1 ( x ) < (cid:98) q 0 ( x ) , the maximum likelihood estimate of Pr { S i ( j ) =

Original: |
Revised: |

Original: X i = x } is (cid:80) i
Revised: X i = x } is (cid:80) i

Original: I ( S i = 1 , X i = x ) (cid:80) i I ( X i = x ) , j = 0 , 1 .
Revised: I ( S i = 1 , X i = x ) (cid:80) i I ( X i = x ) , j = 0 , 1 .

Original: In the second scenario, the estimates are the same as the pooled proportion of responders among patients with X = x .
Revised: In the second scenario, the estimates are the same as the pooled proportion of responders among patients with X = x .

Original: The proof of Lemma 1 is presented in Appendix A.
Revised: The proof of Lemma 2 is presented in Appendix A.

Original: The last item in equation (5) needed for estimating the causal estimand is Pr { S i (1) = 

Revised: The last item in equation (5) needed for estimating the causal estimand is Pr { S i (1) =

Original: | S i (0) = 0 , Y i (0)
Revised: | S i (0) = 0 , Y i (0)

Original: = 1 , X i = x } .
Revised: = 1 , X i = x } .

Original: Gilbert et al.
Revised: Gilbert et al.

Original: (2003) and Shepherd et al.
Revised: (2003) and Shepherd et al.

Original: (2006) conducted sensitivity analyses by varying the values of the β in equation (1).
Revised: (2006) conducted sensitivity analyses by varying the values of the β in model (1).

Original: In Section 4.2, we discuss how to estimate β using a probabilistic equation.
Revised: In Section 4.4, we discuss how to estimate β using a probabilistic equation.

Original: 4.2. Estimation of Model Parameters
Revised: 4.4. Estimation of Model Parameters

Original: For any x ∈ Γ = { 0 , 1 , . . .

Original: , K } , consider the following probabilistic equation

Original: [Equation]

Original: Let
Revised: Let

Original: [Equation]
Revised: [Equation]

Original: This leads to an equation system:
Revised: This leads to an equation system:

Original: [Equation]
Revised: [Equation]

Original: We can estimate G L ( x ) with the following empirical estimates from the observed data by
Revised: We can estimate G L ( x ) with the following empirical estimates from the observed data by

Original: [Equation]
Revised: [Equation]

Original: where the numerator and the denominator are derived from Lemma 1.
Revised: where the numerator and the denominator are derived from Lemma 2.

Original: The details are presented in
Revised: The details are presented in

Original: Appendix A
Revised: Appendix A.

Original: [Equation]

Original: Because { X i , S i (0) , Y i (0) } are observed for subjects in the control arm, G R ( x, y ) can be estimated by
Revised: Because { X i , S i (0) , Y i (0) } are observed for subjects in the control arm, G R ( x, y ) can be estimated by

Original: [Equation]
Revised: [Equation]

Original: With (cid:98) G L ( x ) and (cid:98) G R ( x, y ) estimated from the observed data and G M ( x, y ; β ) specified as the regression model in equation (1), we have
Revised: With (cid:98) G L ( x ) and (cid:98) G R ( x, y ) estimated from the observed data and G M ( x, y ; β ) specified as the regression model in equation (1), we have

Original: [Equation]
Revised: [Equation]

Original: The number of unknown parameters β in system of equations (7) is three and the number of equations is ( K + 1) , for X i ∈ Γ = { 0 , 1 , . .
Revised: The number of unknown parameters β in system of equations (7) is three and the number of equations is ( K + 1) , for X i ∈ Γ = { 0 , 1 , . .

Original: . , K } .
Revised: . , K } .

Original: For (7), when K + 1 < 3 , we cannot uniquely solve for β .
Revised: For (7), when K + 1 < 3 , we cannot uniquely solve for β .

Original: When K + 1 = 3 , the number of equations is the same as the number of unknown parameters and in general we can solve for β .
Revised: When K + 1 = 3 , the number of equations is the same as the number of unknown parameters and in general we can solve for β .

Original: When K + 1 > 3 , there are more equations than the number of unknown parameters, and there are generally no exact solutions to the equation systems (7).
Revised: When K + 1 > 3 , there are more equations than the number of unknown parameters, and there are generally no exact solutions to the equation systems (7).

Original: In that case, we propose to estimate β by
Revised: In that case, we propose to estimate β by

Original: [Equation]
Revised: [Equation]

Original: where (cid:98) G L ( x ) , (cid:98) G R ( x, y ) and G M ( x, y ; β ) are probabilities bounded between 0 and 1.
Revised: where (cid:98) G L ( x ) , (cid:98) G R ( x, y ) and G M ( x, y ; β ) are probabilities bounded between 0 and 1.

Original: With β estimated, we can estimate the causal estimand θ via the procedure outlined in Section 4.1.
Revised: With β estimated, we can estimate the causal estimand θ via the procedure outlined in Section 4.3.

Original: 4.3. Consistency of (cid:98) β and (cid:98) θ
Revised: 4.5. Consistency of (cid:98) β and (cid:98) θ

Original: Here we provide the theoretical guarantee of our estimators β and θ .
Revised: Here we provide the theoretical guarantee of our estimators β and θ .

Original: Let
Revised: Let

Original: [Equation]
Revised: [Equation]

Original: Theorem 2 Under the following conditions:
Revised: Theorem 3 Under the following conditions:

Original: (a) β satisfies Q ( x ) 0 ( β ) = 0 , ∀ x ∈ Γ = { 0 , 1 , . . .
Revised: (a) β satisfies Q ( x ) 0 ( β ) = 0 , ∀ x ∈ Γ = { 0 , 1 , . . .

Original: , K } .
Revised: , K } .

Original: (b) rank (cid:12)(cid:12)(cid:12)(cid:12) ∂
Revised: (b) rank (cid:12)(cid:12)(cid:12)(cid:12) ∂

Original: ˜ Q 0 ( β ) ∂ β (cid:12)(cid:12)(cid:12)(cid:12) ≥ dim( β ) .
Revised: ˜ Q 0 ( β ) ∂ [Equation] (cid:12)(cid:12)(cid:12)(cid:12) ≥ dim( β ) .

Original: (c) (cid:98) G L ( x ) p → G L ( x ) , (cid:98) G R ( x, y ) p → G R ( x, y ) , as n → ∞ , ∀ x ∈ Γ;
Revised: (c) (cid:98) G L ( x ) p → G L ( x ) , (cid:98) G R ( x, y ) p → G R ( x, y ) , as n → ∞ , ∀ x ∈ Γ;

Original: ∀ y = 0 , 1 .
Revised: ∀ y = 0 , 1 .

Original: Then (cid:98) β = arg min β
Revised: Then (cid:98) β = arg min β

Original: Q n ( β ) p → β and the causal estimand (cid:98) θ p→ θ as n → ∞
Revised: Q n ( β ) p → β and the causal estimand (cid:98) [Equation] p

Revised: [Equation]

Original:  The detailed proof of Theorem 2 is presented in Appendix B.
Revised: ∞ . The detailed proof of Theorem 3 is presented in Appendix B.

Original: 4.4. Extension to Censored Data
Revised: 4.6. Extension to Censored Data

Original: As in the motivating NSABP B-40 study, the long-term outcome Y i may be subject to right censoring.
Revised: As in the motivating NSABP B-40 study, the long-term outcome Y i may be subject to right censoring.

Original: For any time T = t 0 of interest, the binary counterfactual outcomes would be { Y i (0; t 0 ) , Y i (1; t 0 ) } and the causal estimand can be formulated as
Revised: For any time T = t 0 of interest, the binary counterfactual outcomes would be { Y i (0; t 0 ) , Y i (1; t 0 ) } and the causal estimand can be formulated as

Original: [Equation]
Revised: [Equation]

Original: With Y i subject to censoring, Pr[ Y i (1; t 0 ) = 1 | i ∈ E +1 ] can be estimated by the Kaplan-Meier (KM) estimates at time T = t 0 .
Revised: With Y i subject to censoring, Pr { Y i (1; t 0 ) = 1 | i ∈ E +1 } can be estimated by the Kaplan-Meier (KM) estimates at time T = t 0 .

Original: The estimation is similar for other relevant quantities such as
Revised: The estimation is similar for other relevant quantities such as

Original: Pr[ Y i (0; t 0 ) = 1 | S i (0) =
Revised: Pr { Y i (0; t 0 ) = 1 | S i (0) =

Original: j, X i = x ] in equation (6) under the scenario where Y i ( Z i ) is always observed.
Revised: j, X i = x } in equation (6) under the scenario where Y i ( Z i ) is always observed.

Original: 5. Simulation Studies
Revised: 5. Simulation Studies

Original: A simulation study is used to assess the performance of the proposed method.
Revised: A simulation study is used to assess the performance of the proposed method.

Original: The setup is chosen to resemble the NSABP B-40 study by simulating treatment assignment, baseline tumor size category, binary pCR response status, and binary survival status, specifically:
Revised: The setup is chosen to resemble the NSABP B-40 study by simulating treatment assignment, baseline tumor size category, binary pCR response status, and binary survival status, specifically:

Original: [Equation]
Revised: [Equation]

Original: We simulate the subject-level data as follows.
Revised: We simulate the subject-level data as follows.

Original: First, we simulate the categorical baseline tumor category
Revised: First, we simulate the categorical baseline tumor category

Original: X i from a multinomial distribution with Pr { X i = x } = 0 .
Revised: X i from a multinomial distribution with Pr { X i = x } = 0 .

Original: 25 , x ∈ { 0 , 1 , 2 , 3 } .
Revised: 25 , x ∈ { 0 , 1 , 2 , 3 } .

Original: Next, we simulate S i (0) given X i from a Bernoulli distribution with Pr { S i (0)
Revised: Next, we simulate S i (0) given X i from a Bernoulli distribution with Pr { S i (0)

Original: = 1 | X i = x } = p ( x ) with p (0) , p (1) , p (2) , p (3) = 0 .
Revised: = 1 | X i = x } = p ( x ) with p (0) , p (1) , p (2) , p (3) = 0 .

Original: 25 , 0 . 2 , respectively.
Revised: 25 , 0 . 2 , respectively.

Original: We then simulate the survival status under control, Y i (0) , with a Bernoulli draw with Pr { Y i (0) =
Revised: We then simulate the survival status under control, Y i (0) , with a Bernoulli draw with Pr { Y i (0) =

Original: 1 | S i (0) = 0 , X i = x } = 0 .
Revised: 1 | S i (0) = 0 , X i = x } = 0 .

Original: 55 for x = 0 , 1 , 2 , 3 , respectively and Pr { Y i (0)
Revised: 55 for x = 0 , 1 , 2 , 3 , respectively and Pr { Y i (0)

Original: = 1 | S i (0)
Revised: = 1 | S i (0)

Original: = 1 , X i = x } = 0 .
Revised: = 1 , X i = x } = 0 .

Original: 66 for x = 0 , 1 , 2 , 3 , respectively.
Revised: 66 for x = 0 , 1 , 2 , 3 , respectively.

Original: The choice of these numbers reflects a 20% improvement in 3-year EFS for respondents over nonrespondents under the control regimen.
Revised: The choice of these numbers reflects a 20% improvement in 3-year EFS for respondents over nonrespondents under the control regimen.

Original: Next, we simulate the conditional distribution [ S i (1) | S i (0) , Y i (0) , X i ] .
Revised: Next, we simulate the conditional distribution { S i (1) | S i (0) , Y i (0) , X i } .

Original: For subjects with S i (0) =1 we set S i (1) to be 1 to enforce the monotonicity assumption.
Revised: For subjects with S i (0) = we set S i (1) to be 1 to enforce the monotonicity assumption.

Original: For subjects with S i (0)
Revised: For subjects with S i (0)

Original: = 

Revised: =

Original: we draw S i (1) from a Bernoulli distribution:
Revised: we draw S i (1) from a Bernoulli distribution:

Original: Pr { S i (1) = 

Revised: Pr { S i (1) =

Original: | S i (0) = 0 , Y i (0) =
Revised: | S i (0) = 0 , Y i (0) =

Original: y, X i = x } = exp( β 0 + β 1 y + β 2 x ) 

Revised: y, X i = x } = exp( β 0 + β 1 y + β 2 x )

Original: + exp( β 0 + β 1 y + β 2 x ) .
Revised: + exp( β 0 + β 1 y + β 2 x ) .

Original: We try different settings for β
Revised: We try different settings for β

Original: =
Revised: =

Original: (-3, -5, 0.2), (-5, -1, -2), and (-7, 3, 0.2).
Revised: (-3, -5, 0.2), (-5, -1, -2), and (-7, 3, 0.2).

Original: We then simulate the survival status under treatment, Y i (1) , according to the following probability distributions:[Equation] [Equation]
Revised: We then simulate the survival status under treatment, Y i (1) , according to the following probability distributions:

Original: [Equation] [Equation] 0 , Y [Equation]
Revised: Pr { Y i (1) = 1 | S i (0) = 0 , S i (1) = 0 , Y i (0) = 0 } = 0 .

Original: [Equation] [Equation] Pr { Y i (1) = 1 | S i (0) = 0 , S i (1) = 0 , Y i (0) = 1 } = 0 .
Revised: Pr { Y i (1) = 1 | S i (0) = 0 , S i (1) = 0 , Y i (0) = 1 } = 0 .

Original: Pr { Y i (1) = 1 | S i (0) = 0 , S i (1) = 1 , Y i (0) = 0 } = 0 . 85 ,
Revised: Pr { Y i (1) = 1 | S i (0) = 0 , S i (1) = 1 , Y i (0) = 0 } = 0 . 85 ,

Original: Pr { Y i (1) = 1 | S i (0) = 0 , S i (1) = 1 , Y i (0) = 1 } = 0 . 9 ,
Revised: Pr { Y i (1) = 1 | S i (0) = 0 , S i (1) = 1 , Y i (0) = 1 } = 0 . 9 ,

Original: Pr { Y i (1) = 1 | S i (0) = 1 , S i (1) = 1 , Y i (0) = 0 } = 0 . 85 ,
Revised: Pr { Y i (1) = 1 | S i (0) = 1 , S i (1) = 1 , Y i (0) = 0 } = 0 . 85 ,

Original: Pr { Y i (1) = 1 | S i (0) = 1 , S i (1) = 1 , Y i (0) = 1 } = 0 . 9 .
Revised: Pr { Y i (1) = 1 | S i (0) = 1 , S i (1) = 1 , Y i (0) = 1 } = 0 . 9 .

Original: These probabilities are chosen to make the 3-year EFS under treatment greater for those who would obtain pCR under treatment than those who would not, and have a greater 3-year EFS for those patients who would be event-free under control than those who would not be event-free under control.
Revised: These probabilities are chosen to make the 3-year EFS under treatment greater for those who would obtain pCR under treatment than those who would not, and have a greater 3-year EFS for those patients who would be event-free under control than those who would not be event-free under control.

Original: We set these probabilities to be independent of the baseline tumor size given the potential outcomes { S i (0) , S i (1) , Y i (0) } .
Revised: We set these probabilities to be independent of the baseline tumor size given the potential outcomes { S i (0) , S i (1) , Y i (0) } .

Original: Lastly we simulate the treatment assignment with equal probability for each arm as a Bernoulli draw with Pr { Z i = 0 } and Pr { Z i = 1 } both equal to 0.5 to ensure that independence between potential outcomes and treatment assignment.
Revised: Lastly we simulate the treatment assignment with equal probability for each arm as a Bernoulli draw with Pr { Z i = 0 } and Pr { Z i = 1 } both equal to 0.5 to ensure that independence between potential outcomes and treatment assignment.

Original: For the simulated data the true average causal effect for principal stratum S i (1) = 1 , E [ Y i (1) − Y
Revised: For the simulated data the true average causal effect for principal stratum S i (1) = 1 , E { Y i (1) − Y

Original: i (0) | S i (1) = 1] , can be calculated using the above parameters for simulations.
Revised: i (0) | S i (1) = 1 } , can be calculated using the above parameters for simulations.

Original: The detailed calculations is given in Appendix C. Under the three parameter settings the true values of the causal estimands are θ =0.179, 0.130, and 0.120, respectively.
Revised: The detailed calculations is given in Appendix C. Under the three parameter settings the true values of the causal estimands are θ =0.179, 0.130, and 0.120, respectively.

Original: This means that under the three different settings, if the treatment was administered to all subjects who would achieve pCR under treatment there would be a 17.9%, 13.0%, 12.0% increment in survival respectively, within the time frame under consideration, than had all of them taken the control instead.
Revised: This means that under the three different settings, if the treatment was administered to all subjects who would achieve pCR under treatment there would be a 17.9%, 13.0%, 12.0% increment in survival respectively, within the time frame under consideration, than had all of them taken the control instead.

Original: Under each parameter setting and a chosen sample size n =1000, 2000, or 4000, we simulate
Revised: Under each parameter setting and a chosen sample size n =1000, 2000, or 4000, we simulate

Original: R =1000 replicates.
Revised: R =1000 replicates.

Original: A quasi-Newton method, the Broyden-Fletcher-Goldfarb-Shanno(BFGS) algorithm, is used for the optimization.
Revised: A quasi-Newton method, the Broyden-Fletcher-Goldfarb-Shanno algorithm, is used for the optimization.

Original: We create B =500 bootstrap samples to obtain the 95% confidence interval for the causal estimates.
Revised: We create B =500 bootstrap samples to obtain the 95% confidence interval for the causal estimates.

Original: Let (cid:98) θ ( r ) be the mean estimate among bootstrap samples from the r replicate, r = 1 , . . .
Revised: Let (cid:98) θ ( r ) be the mean estimate among bootstrap samples from the r replicate, r = 1 , . . .

Original: , R .
Revised: , R .

Original: We construct bootstrap confidence intervals to account for the variability introduced by estimating model parameters.
Revised: We construct bootstrap confidence intervals to account for the variability introduced by estimating model parameters.

Original: We use the basic bootstrap CI, or the pivotal CI (Davison and Hinkley, 1997) for constructing CIs from bootstrap estimates.
Revised: We use the basic bootstrap CI, or the pivotal CI (Davison and Hinkley, 1997) for constructing CIs from bootstrap estimates.

Original: Let { (cid:98) θ (1) , (cid:98) θ (2) , . . .
Revised: Let { (cid:98) θ (1) , (cid:98) θ (2) , . . .

Original: , (cid:98) θ ( B ) } are the causal effect estimates from B bootstrap samples.
Revised: , (cid:98) θ ( B ) } are the causal effect estimates from B bootstrap samples.

Original: Denote θ ∗ (1 − α/ 2) and θ ∗ ( α/ 2) as the 100(1 − α/ 2)% and 100( α/ 2)% of the bootstrap causal effect estimates.
Revised: Denote θ ∗ (1 − α/ 2) and θ ∗ ( α/ 2) as the 100(1 − α/ 2)% and 100( α/ 2)% of the bootstrap causal effect estimates.

Original: The 100(1 − α )% bootstrap confidence interval (Davison and Hinkley, 1997) is (2 (cid:98) θ − θ ∗ (1 − α/ 2) , 2 (cid:98) θ − θ ∗ ( α/ 2) ) where (cid:98) θ is the estimate from the data.
Revised: The 100(1 − α )% bootstrap confidence interval (Davison and Hinkley, 1997) is (2 (cid:98) θ − θ ∗ (1 − α/ 2) , 2 (cid:98) θ − θ ∗ ( α/ 2) ) where (cid:98) θ is the estimate from the data.

Original: We report the empirical bias, mean squared error (MSE), average length of 95% CIs, and the coverage of those CIs.
Revised: We report the empirical bias, mean squared error (MSE), average length of 95% CIs, and the coverage of those CIs.

Original: Bias ( (cid:98) θ ) = 1 R (cid:80)
Revised: Bias ( (cid:98) θ ) = 1 R (cid:80)

Original: Rr =1  (cid:98) θ ( r ) − θ  , MSE ( (cid:98) θ ) = 1 R (cid:80)
Revised: Rr =1 { (cid:98) θ ( r ) − θ } , MSE ( (cid:98) θ ) = 1 R (cid:80)

Original: Rr =1 ( (cid:98) θ ( r ) − θ ) 2 , 95% CI width = 1 R (cid:80)
Revised: Rr =1 { (cid:98) θ ( r ) − θ } 2 , 95% CI width = 1 R (cid:80)

Original: Rr =1 | (cid:98) θ ( r ) U, 0 .
Revised: Rr =1 | (cid:98) θ ( r ) U, 0 .

Original: 05 − (cid:98) θ ( r ) L, 0 .
Revised: 05 − (cid:98) θ ( r ) L, 0 .

Original: | , and 95% CI coverage
Revised: | , and 95% CI coverage

Original: = 1 R (cid:80)
Revised: = 1 R (cid:80)

Original: Rr =

Revised: Rr =

Original: I { θ ∈ ( (cid:98) θ
Revised: I { θ ∈ ( (cid:98) θ

Original: ( r ) L, 0 .
Revised: ( r ) L, 0 .

Original: 05 , (cid:98) θ
Revised: 05 , (cid:98) θ

Original: ( r ) U, 0 . 05 ) } with (cid:98) θ ( r ) L, 0 .and (cid:98) θ ( r ) U, 0 . 

Revised: ( r ) U, 0 . 05 ) } with (cid:98) θ ( r ) L, 0 .and (cid:98) θ ( r ) U, 0 .

Original: the lower bound and upper bound of the 95% bootstrap CIs of (cid:98) θ from the r th simulated dataset.
Revised: the lower bound and upper bound of the 95% bootstrap CIs of (cid:98) θ from the r th simulated dataset.

Original:  1 shows the simulation results of the proposed method under three different parameter settings and various sample sizes.
Revised: Table 1 shows the simulation results of the proposed method under three different parameter settings and various sample sizes.

Original: Our simulation results show the identification of causal effects is achieved with the bias negligible and the coverage probabilities close to the nominal levels.
Revised: Our simulation results show the identification of causal effects is achieved with the bias negligible and the coverage probabilities close to the nominal levels.

Original: Code implementing simulations presented in this section is provided in the supplementary materials.

Original: [Table]
Revised: [Table]

Original: 6. Application to NSABP B-40 Trial 6.1. B-40 Data Analysis
Revised: 6. Application to NSABP B-40 Trial 6.1. B-40 Data Analysis

Original: Here we apply the proposed method to the NSABP B-40 study (Bear et al., 2012, 2015).
Revised: Here we apply the proposed method to the NSABP B-40 study (Bear et al., 2012, 2015).

Original: Among the 1206 enrolled participants, 13 withdrew consent, 7 had missing data and 2 had had inoperable disease after chemotherapy.
Revised: Among the 1206 enrolled participants, 13 withdrew consent, 7 had missing data and 2 had had inoperable disease after chemotherapy.

Original: Another 15 patients did not have nodal assessment so their pCR status were not ascertained.
Revised: Another 15 patients did not have nodal assessment so their pCR status were not ascertained.

Original: We conducted our analysis among the rest 1169 patients.
Revised: We conducted our analysis among the rest 1169 patients.

Original: Our purpose is to estimate the causal treatment effect in 3-year EFS and OS among patients who would obtain a pCR had bevacizumab been added to their treatment regimen.
Revised: Our purpose is to estimate the causal treatment effect in 3-year EFS and OS among patients who would obtain a pCR had bevacizumab been added to their treatment regimen.

Original: KM estimates were used since there were 61 patients censored at 3 years.
Revised: KM estimates were used since there were 61 patients censored at 3 years.

Original: To apply our method, the clinical tumor size was used as the baseline auxiliary covariate X .
Revised: To apply our method, the clinical tumor size was used as the baseline auxiliary covariate X .

Original: Patients were grouped into four nearly equal-sized groups: 2-3 cm, 3.1-4 cm, 4.1-6 cm and > 6 cm, based on breast cancer expert knowledge.
Revised: Patients were grouped into four nearly equal-sized groups: 2-3 cm, 3.1-4 cm, 4.1-6 cm and > 6 cm, based on breast cancer expert knowledge.

Original: We coded these four tumor size groups into { 0 , 1 , 2 , 3 } , respectively.
Revised: We coded these four tumor size groups into { 0 , 1 , 2 , 3 } , respectively.

Original: Among the 589 patients in the control arm, the proportions of those who achieved pCR in each patient group are 28%, 23%, 22% and 17%, respectively; among the 580 patients in the treatment arm, the proportions of those who achieved pCR are 31%, 26%, 25% and 27%, respectively.
Revised: Among the 589 patients in the control arm, the proportions of those who achieved pCR in each patient group are 28%, 23%, 22% and 17%, respectively; among the 580 patients in the treatment arm, the proportions of those who achieved pCR are 31%, 26%, 25% and 27%, respectively.

Original: This does not violate the monotonicity assumption S i (0) ≤ S i (1) .
Revised: This does not violate the monotonicity assumption S i (0) ≤ S i (1) .

Original: The 3-year longterm outcome status Y i = 1 if the patient i survived within the first 3 years and 0 otherwise.
Revised: The 3-year longterm outcome status Y i = 1 if the patient i survived within the first 3 years and 0 otherwise.

Original: We calculated the 95% bootstrap confidence intervals from 500 bootstrap samples.
Revised: We calculated the 95% bootstrap confidence intervals from 500 bootstrap samples.

Original: The estimated causal treatment effect in 3-year EFS among those who would obtained pCR under treatment is (cid:98) θ EFS
Revised: The estimated causal treatment effect in 3-year EFS among those who would obtained pCR under treatment is (cid:98) θ EFS

Original: = 0 .
Revised: = 0 .

Original: 180 (95% CI=(0.056, 0.377)) with (cid:98) β
Revised: 180 (95% CI=(0.056, 0.377)) with (cid:98) β

Original: = ( − 1 .
Revised: = ( − 1 .

Original: 797 , − 5 .
Revised: 797 , − 5 .

Original: The estimated causal treatment effect in 3-year OS among those who would obtained pCR under treatment is (cid:98) θ OS = 0 .(

Revised: The estimated causal treatment effect in 3-year OS among those who would obtained pCR under treatment is (cid:98) θ OS = 0 .(

Original: % CI=(0.062, 0.354)) with (cid:98) β
Revised: % CI=(0.062, 0.354)) with (cid:98) β

Original: = ( − 1 . 85 , − 4 .
Revised: = ( − 1 . 85 , − 4 .

Original: For both scenarios, because 0 is outside of the 95% CIs, we would claim that the addition of bevacizumab improves 3-year EFS and
Revised: For both scenarios, because 0 is outside of the 95% CIs, we would claim that the addition of bevacizumab improves 3-year EFS and

Original: OS among patients who would respond to neoadjuvant chemotherapy plus bevacizumab at a 95% confidence level.
Revised: OS among patients who would respond to neoadjuvant chemotherapy plus bevacizumab at a 95% confidence level.

Original: 6.2. Sensitivity of Inital Parameters in Optimization
Revised: 6.2. Sensitivity of Inital Parameters in Optimization

Original: For the real data application, the initial estimate β init =
Revised: For the real data application, the initial estimate β init =

Original: ( β 0 , β 1 , β 2 ) was set at (0 , 0 , 0) .
Revised: ( β 0 , β 1 , β 2 ) was set at (0 , 0 , 0) .

Original: To see the sensitivity of initial parameters, we tried 9261 = 21 × 21 × 21 different initial values of β init , with β 0 , β 1 , and β 2 on the integer grids of [ − 10 , 10] × [ − 10 , 10] ×
Revised: To see the sensitivity of initial parameters, we tried 9261 = 21 × 21 × 21 different initial values of β init , with β 0 , β 1 , and β 2 on the integer grids of [ − 10 , 10] × [ − 10 , 10] ×

Original: [ − 10 , 10] .
Revised: [ − 10 , 10] .

Original: The corresponding histograms of causal estimates in 3-year EFS and 3-year OS at convergence are presented in Figure 1.
Revised: The corresponding histograms of causal estimates in 3-year EFS and 3-year OS at convergence are presented in Figure 1.

Original: Our estimated model parameters (cid:98) β in Section 6.1 achieves the minimum loss of equation (8).
Revised: Our estimated model parameters (cid:98) β in Section 6.1 achieves the minimum loss of equation (8).

Original: Except for some extreme initialization such as (10,10,10), most of the (cid:98) θ are the same or very close to the causal estimates calculated by using β init = (0,0,0) as initial parameters.
Revised: Except for some extreme initialization such as (10,10,10), most of the (cid:98) θ are the same or very close to the causal estimates calculated by using β init = (0,0,0) as initial parameters.

Original: Therefore, we conclude that the causal estimand is not sensitive to the initial parameter settings in optimization.
Revised: Therefore, we conclude that the causal estimand is not sensitive to the initial parameter settings in optimization.

Original: In practice, we suggest running optimization with various initial values and identify the right estimate.
Revised: In practice, we suggest running optimization with various initial values and identify the right estimate.

Original: 6.3. Comparisons to Sensitivity Analysis Method
Revised: 6.3. Comparisons to Sensitivity Analysis Method

Original: We compared the performance of our method with that of the sensitivity analysis similar to Gilbert et al.
Revised: We compared the performance of our method with that of the sensitivity analysis similar to Gilbert et al.

Original: (2003) and Shepherd et al.
Revised: (2003) and Shepherd et al.

Original: Recall that for X = x ∈ Γ = { 0 , 1 , . .
Revised: Recall that for X = x ∈ Γ = { 0 , 1 , . .

Original: , K } , we have an equation system:
Revised: , K } , we have an equation system:

Original: [Equation]
Revised: [Equation]

Original: where G M ( x, y ; β ) =
Revised: where G M ( x, y ; β ) =

Original: exp( β 0 + β 1 y + β 2 x ) 1 + exp( β 0 + β 1 y + β 2 x ) .
Revised: exp( β 0 + β 1 y + β 2 x ) 1 + exp( β 0 + β 1 y + β 2 x ) .

Original: In the sensitivity analysis we varied the value of β 1 from -7 to -3.
Revised: In the sensitivity analysis we varied the value of β 1 from -7 to -3.

Original: Then for each category of x we define β x = β 0 + β 2 x .
Revised: Then for each category of x we define β x = β 0 + β 2 x .

Original: Under this reparameterization we have only one unknown parameter, β x , for each equation.
Revised: Under this reparameterization we have only one unknown parameter, β x , for each equation.

Original: We then solved for β x for each equation independently and obtained the causal estimand subsequently.B), respectively. The β β 1 and β the integer [ −
Revised: We then solved for β x for each equation independently and obtained the causal estimand subsequently.

Original: 10] × [ −

Original: 10] × [ − extreme initialization such as β init = (

Original: 10), most of the are the same very close the estimate

Original: By varying values of β 1 around the estimated (cid:98) β 1 from Section 6.1, the corresponding causal estimands in 3-year EFS and 3-year OS are presented in Table 2.
Revised: By varying values of β 1 around the estimated (cid:98) β 1 from Section 6.1, the corresponding causal estimands in 3-year EFS and 3-year OS are presented in Table 2.

Original: The estimated causal effects  in 3-year EFS varied from 0.159 to 0.181 with none of the 95% CIs including 0; the estimated causal effects in 3-year OS varied from 0.132 to 0.176 with none of the 95% CIs including 0.
Revised: The estimated causal effects B in 3-year EFS varied from 0.159 to 0.181 with none of the 95% CIs including 0; the estimated causal effects in 3-year OS varied from 0.132 to 0.176 with none of the 95% CIs including 0.

Original: These intervals overlapped a lot with the confidence intervals of real data.
Revised: These intervals overlapped a lot with the confidence intervals of real data.

Original: These results suggest the addition of bevacizumab improved 3-year EFS and 3-year OS among patients who would respond to neoadjuvant chemotherapy plus bevacizumab. 
Revised: These results suggest the addition of bevacizumab improved 3-year EFS and 3-year OS among patients who would respond to neoadjuvant chemotherapy plus bevacizumab. [Table] [Table] [Table]

Original: [Table]
Revised: [Table]

Original: 7. Discussion and Future Works
Revised: [Table] [Table] [Table] [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Original:  We have proposed a method under the principal stratification framework to estimate causal effects of a treatment on a binary long-term endpoint conditional on a post-treatment binary marker in randomized controlled clinical trials.
Revised: OS0. [Table] 7. Discussion and Future Works We have proposed a method under the principal stratification framework to estimate causal effects of a treatment on a binary long-term endpoint conditional on a post-treatment binary marker in randomized controlled clinical trials.

Original: We also extend our method to address censored outcome data.
Revised: We also extend our method to address censored outcome data.

Original: In our motivating study, we demonstrate the causal effect of the new regimen in the longterm survival for patients who would achieve pCR.
Revised: In our motivating study, we demonstrate the causal effect of the new regimen in the longterm survival for patients who would achieve pCR.

Original: Other principal stratum causal effects can be estimated in a similar fashion.
Revised: Other principal stratum causal effects can be estimated in a similar fashion.

Original: Our approach can play an important role in a sensitivity analysis.
Revised: Our approach can play an important role in a sensitivity analysis.

Original: Identification of causal effects is achieved through two assumptions.
Revised: Identification of causal effects is achieved through two assumptions.

Original: First, a subject who responds under the control would respond if given the treatment.
Revised: First, a subject who responds under the control would respond if given the treatment.

Original: This monotonicity assumption could prove valuable (Bartolucci and Grilli, 2011) and can be justified in many scenarios that the additional therapy would help to improve the response.
Revised: This monotonicity assumption could prove valuable (Bartolucci and Grilli, 2011) and can be justified in many scenarios that the additional therapy would help to improve the response.

Revised: When the auxiliary variable X is discrete, we can identify and estimate Pr { S (1) = 1 | S (0)

Revised: = 0 , X } under the monotonicity assumption.

Original: Second, a parametric model is used to describe the counterfactual response under the treatment for a control non-respondent.
Revised: Second, a parametric model is used to describe the counterfactual response under the treatment for a control non-respondent (Shepherd et al., 2006).

Original: Both the future longterm outcome and a baseline covariate are predictors in this parametric model.
Revised: Both the future long-term outcome and a baseline covariate are predictors in this parametric model.

Original: It is required that the level of the covariates is at least of the same dimension of model  parameters and the imposed linearity assumption is critical to identify and estimate those regression parameters.
Revised: Shepherd et al. (2006) did not consider when the auxiliary X is discrete, the parameters of model (1) can be identified when the level of the discrete covariate is at least of the same dimension of model parameters. Instead they performed sensitivity analyses by varying the values of those model parameters in order to estimate the causal estimands. It is recognized that no diagnostic tool is available to verify the validity of this counterfactual model.

Original: In the motivating dataset, we discretized a continuous baseline variable into several levels.
Revised: In the motivating dataset, we discretized a continuous baseline variable into several levels.

Original: In practice, the linearity assumption may not hold.
Revised: In practice, the linearity assumption may not hold.

Original: We would consider a two-pronged approach: (1) to estimate G L ( x ) and G R ( x, y ) by nonparametric estimates such as spline or kernel density estimates for a univariate continuous X ; (2) to use a more flexible model for the counterfactual response such as a logistic regression with natural cubic spline with fixed and even-spaced knots along the domain of X .
Revised: We would consider a two-pronged approach: 1) to estimate G L ( x ) and G R ( x, y ) by nonparametric estimates such as spline or kernel density estimates for a univariate continuous X ; 2) to use a more flexible model for the counterfactual response such as a logistic regression with natural cubic spline with fixed and even-spaced knots along the domain of X .

Original: For each given x , we can still use the same probabilistic argument to link those estimates and the model parameters.
Revised: For each given x , we can still use the same probabilistic argument to link those estimates and the model parameters.

Original: The objective function would be a weighted sum of the squared difference of those probabilistic estimates. 
Revised: The objective function would be a weighted sum of the squared difference of those probabilistic estimates. Acknowledgments

Revised: This work is supported by the National Cancer Institute at the National Institutes of Health, U.S.

Revised: Department of Health and Human Services, Public Health Service grants U10-CA180868 (NCTN),

Revised: U10-CA180822 (NRG SDMC).

Original: Appendix A. Estimation of Pr { S i (0) = 1 | X i = x } and Pr { S i (1) = 1 | X i = x }
Revised: Appendix A. Estimation of Pr { S i (0) = 1 | X i = x } and Pr { S i (1) = 1 | X i = x }

Original: We use the maximum likelihood approach to estimate Pr { S i (0) = 

Revised: We use the maximum likelihood approach to estimate Pr { S i (0) =

Original: | X i = x } , Pr { S i (0) =
Revised: | X i = x } , Pr { S i (0) =

Original: | X i = x } and Pr { S i (1) = 

Revised: | X i = x } and Pr { S i (1) =

Original: | X i = x } .
Revised: | X i = x } .

Original: Let
Revised: Let

Original: E jkx
Revised: E jkx

Original: =
Revised: =

Original: { i : S i (0) =
Revised: { i : S i (0) =

Original: j, S i (1) =
Revised: j, S i (1) =

Original: k | X i = x } , j, k = 0 , 1 , x ∈ Γ be the principal stratum under each category X = x .
Revised: k | X i = x } , j, k = 0 , 1 , x ∈ Γ be the principal stratum under each category X = x .

Original: Because of the monotonicity assumption,
Revised: Because of the monotonicity assumption,

Original: E 10 x is empty.
Revised: E 10 x is empty.

Original: Let
Revised: Let

Original: [Equation]
Revised: [Equation]

Original: Therefore, p 00 x + p 01 x + p 11 x = 1 for all x ∈ Γ .
Revised: Therefore, p 00 x + p 01 x + p 11 x = 1 for all x ∈ Γ .

Original: For each x , Pr { E jkx } can be estimated from the observed data { Z i , X i , S i
Revised: For each x , Pr { E jkx } can be estimated from the observed data { Z i , X i , S i

Original: ( Z i ) , i = 1 , 2 , . . . , n } via maximum likelihood.
Revised: ( Z i ) , i = 1 , 2 , . . . , n } via maximum likelihood.

Original: Let N zsx be the total number of subjects with Z
Revised: Let N zsx be the total number of subjects with Z

Original: = z, S ( Z )
Revised: = z, S ( Z )

Original: =
Revised: =

Original: s and baseline category x with (cid:80) Z ; S =0 , 1; X N zsx = n .
Revised: s and baseline category x with (cid:80) Z ; S =0 , 1; X N zsx = n .

Original: Then the likelihood function for ( p 00 x , p 01 x , p 11 x ) is given by
Revised: Then the likelihood function for ( p 00 x , p 01 x , p 11 x ) is given by

Original: L ( p 00 x , p 01 x , p 11 x | N 00 x , N 01 x , N 10 x , N 11 x ) ∝ f ( N zsx )
Revised: L [Equation] p 00 x , p 01 x , p 11 x | N 00 x , N 01 x , N 10 x , N 11 x ) ∝ f ( N zsx )

Original: [Equation]
Revised: [Equation]

Original: N 10 x 00 x · ( p 01 x + p 11 x ) N 11 x (by monotonicity assumption) = 
Revised: ( p 01 x + p 11 x ) N 11 x (by monotonicity assumption) = [Equation] p [Equation]

Original: (1 − p 11 x ) N 00 x · p N01 x 11 x · p N 10 x 00 x · (
Revised: [Equation] p 11 x ) [Equation] p N

Original: [Equation]
Revised: [Equation]

Original: p 00 x ) N 11 x = (1 − p 11 x ) N 00 x · p N 01 x 11 x · (1 − p +1 x ) N 10 x · p N 11 x +1 x (1) When N 00 x · N 11 x ≥ N 01 x · N 10 x , the resulting MLEs for ( p 00 x , p 01 x , p 11 x ) are given by (cid:98) p 00 x = (cid:99)
Revised: (1) When N 00 x · N 11 x ≥ N 01 x · N 10 x , the resulting MLEs for ( p 00 x , p 01 x , p 11 x ) are given by (cid:98) p 00 x = (cid:99)

Original: Pr
Revised: Pr

Original: [Equation]
Revised: [Equation]

Original: S i
Revised: S i

Original: [Equation][Equation] [Equation] [Equation]
Revised: [Equation]

Original: p 01 x = (cid:99) P r { S i (0)

Original: = 0 ,

Original: [-S i (1) = 
-]
Revised: [-S i (1) = 
-]

Original: | X i = x } = 1 −

Original: [Equation]
Revised: [Equation]

Original: p

Original: [Equation]
Revised: [Equation]

Original: p 11 x Obviously for each x ∈ Γ , (cid:98) p 

Revised: Obviously for each x ∈ Γ , (cid:98) p

Original: x is the proportion of non-respondents in the treatment arm with
Revised: x is the proportion of non-respondents in the treatment arm with

Original: X = x ; (cid:98) p 11 x is the proportion of respondents in the control arm with X = x .
Revised: X = x ; (cid:98) p 11 x is the proportion of respondents in the control arm with X = x .

Original: (2) When N 00 x · N 11 x < N 01 x · N 10 x , (cid:98) p 11 x = (cid:98) p +1 x .
Revised: (2) When N 00 x · N 11 x < N 01 x · N 10 x , (cid:98) p 11 x = (cid:98) p +1 x .

Original: The likelihood function is given by
Revised: The likelihood function is given by

Original: [Equation]
Revised: [Equation]

Original: The resulting MLEs for ( p 00 x , p 01 x , p 11 x ) are given by
Revised: The resulting MLEs for ( p 00 x , p 01 x , p 11 x ) are given by

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Then (cid:98) p 

Revised: Then (cid:98) p

Original: x is the proportion of non-respondents among all subjects with X
Revised: x is the proportion of non-respondents among all subjects with X

Original: = x ; (cid:98) p 11 x is the proportion of respondents among all subjects with X = x .
Revised: = x ; (cid:98) p 11 x is the proportion of respondents among all subjects with X = x .

Original: Appendix B. Proof of Consistency of (cid:98) β and (cid:98) θ
Revised: Appendix B. Proof of Consistency of (cid:98) β and (cid:98) θ

Original: Here we show our estimator (cid:98) β is a consistent estimator for β .
Revised: Here we show our estimator (cid:98) β is a consistent estimator for β .

Original: We first show that (cid:98) β can be considered as an extremum estimator as defined by Hayashi (2000).
Revised: We first show that (cid:98) β can be considered as an extremum estimator as defined by Hayashi (2000).

Original: Then we prove that the conditions set forth by Hayashi (2000) for consistency of an extremum estimator are satisfied by our estimator.
Revised: Then we prove that the conditions set forth by Hayashi (2000) for consistency of an extremum estimator are satisfied by our estimator.

Original: Then by
Revised: Then by

Original: Slutsky’s theorem, the causal estimand (cid:98) θ is a consistent estimator for θ .
Revised: Slutsky’s theorem, the causal estimand (cid:98) θ is a consistent estimator for θ .

Original: Definition 3 (Extremum Estimator) An estimator (cid:98) η is an extremum estimator if there is a function
Revised: Definition 4 (Extremum Estimator) An estimator (cid:98) η is an extremum estimator if there is a function

Original: Q n ( η ) such that
Revised: Q n ( η ) such that

Original: (Hayashi, 2000)
Revised: (Hayashi, 2000)

Original: [Equation]
Revised: [Equation]

Original: One example of an extremum estimator is the maximum likelihood estimator where
Revised: One example of an extremum estimator is the maximum likelihood estimator where

Original: [Equation]
Revised: [Equation]

Original: Here we minimize the objective function,
Revised: Here we minimize the objective function,

Original: [Equation]
Revised: [Equation]

Original: [ (cid:98) G L ( x ) − 1 (cid:88) y = G M ( x, y ; β ) · (cid:98) G R x, y )] 2 ; x ∈ Γ which is equivalent to maximizing − Q n ( β ) .
Revised: which is equivalent to maximizing − Q n ( β ) .

Original: Therefore (cid:98) β is an extremum estimator.
Revised: Therefore (cid:98) β is an extremum estimator.

Original: Let
Revised: Let

Original: Q 0 ( β ) =

Original: K

Original: [Equation]
Revised: [Equation]

Original: =

Original: Q ( [Equation] [Equation] x ∈ Γ where Q ( x ) 0 ( β ) =
Revised: where Q ( x ) 0 ( β ) =

Original: [ G L ( x ) −1 (cid:88) y
Revised: { G L ( x ) −

Original: =
Revised: [Equation]

Original: G M ( x, y ; β ) · G R ( x, y )] 2 .
Revised: G M ( x, y ; β ) · G R ( x, y ) } 2 .

Original: We present sufficient conditions for the existence of a unique local minimizer of Q 0 ( β ) in Lemma 4.
Revised: We present sufficient conditions for the existence of a unique local minimizer of Q 0 ( β ) in Lemma 5.

Original: Lemma 4 There exists a unique local minimizer β 0 for Q 0 ( β ) if:
Revised: Lemma 5 There exists a unique local minimizer β 0 for Q 0 ( β ) if:

Original: (a) Q ( x ) 0 ( β 0 ) =
Revised: (a) Q ( x ) 0 ( β 0 ) =

Original: 0 , ∀ x ∈ Γ
Revised: 0 , ∀ x ∈ Γ

Original: = { 0 , 1 , 2 , . .
Revised: = { 0 , 1 , 2 , . .

Original: . , K } .
Revised: . , K } .

Original: [Figure]
Revised: [Figure]

Original: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]

Original: (cid:12)(cid:12)(cid:12)(cid:12)
Revised: (cid:12)(cid:12)(cid:12)(cid:12)

Original: β

Original: [Equation]
Revised: [Table]

Revised: ∂ β (cid:12)(cid:12)(cid:12)(cid:12) β

Original: ≥ dim( β ) where ˜ Q 0 ( β ) =
Revised: = β≥ dim( β ) where ˜ Q 0 ( β ) =

Original: [ Q (0)0 ( β ) , Q (1)0 ( β ) , . . .
Revised: { Q (0)0 ( β ) , Q (1)0 ( β ) , . . .

Original: , Q ( K )( β )] T .
Revised: , Q ( K )( β ) } T .

Original: Proof From (a) we have that β 0 minimizes Q 0 ( β ) since Q 0 ( β ) ≥ 0 , ∀ β and Q 0 ( β 0 )
Revised: Proof From (a) we have that β 0 minimizes Q 0 ( β ) since Q 0 ( β ) ≥ 0 , ∀ β and Q 0 ( β 0 )

Original: = 0 .
Revised: = 0 .

Original: Then from (b) and the Implicit Function Theorem, there exists a unique function g ( G L ( x ) ,
Revised: Then from (b) and the Implicit Function Theorem, there exists a unique function g { G L ( x ) ,

Original: G R ( x, y ) ) such that g ( G L ( x ) , G R ( x, y ) ) = β 0 , in the neighborhood of ( G L ( x ) , G R ( x, y ) ) where ( G L ( x ) , G R ( x, y ) )
Revised: G R ( x, y ) } such that g { G L ( x ) , G R ( x, y ) } = β 0 , in the neighborhood of { G L ( x ) , G R ( x, y ) } where { G L ( x ) , G R ( x, y ) } =

Original: = { G L ( x ) , G R ( x, y ); x ∈ { 0 , 1 , . . .
Revised: [ G L ( x ) , G R ( x, y ); x ∈ { 0 , 1 , . . .

Original: , K } , y = 0 , 1 } .
Revised: , K } , y = 0 , 1] .

Original: Thus, β 0 is a unique local minimizer for Q 0 ( β ) .
Revised: Thus, β 0 is a unique local minimizer for Q 0 ( β ) .

Original: This completes the proof of Lemma 4.

Original: The proof of Theorem 2 is given as below.
Revised: The proof of Theorem 3 is given as below.

Original: Proof From Proposition 7.1 in (Hayashi, 2000): an extremum estimator (cid:98) η is a consistent estimator for η if there is a function Q 0 ( η ) satisfying the following two conditions:
Revised: Proof From Proposition 7.1 in (Hayashi, 2000): an extremum estimator (cid:98) η is a consistent estimator for η if there is a function Q 0 ( η ) satisfying the following two conditions:

Original: (I) Identification: Q 0 ( η ) is uniquely maximized on H at η 0 ∈ H .
Revised: (I) Identification: Q 0 ( η ) is uniquely maximized on H at η 0 ∈ H .

Original: (II) Uniform convergence: Q n ( · ) converges uniformly in probability to Q 0 ( · ) .
Revised: (II) Uniform convergence: Q n ( · ) converges uniformly in probability to Q 0 ( · ) .

Original: The condition (I) is satisfied according to Lemma 4.
Revised: The condition (I) is satisfied according to Lemma 5.

Original: To show that the condition (II) is satisfied here, let
Revised: To show that the condition (II) is satisfied here, let

Original: [Equation]
Revised: [Equation]

Original: G L ( x ) − 1 (cid:88) y =

Original: G M ( x, y ; β ) · (cid:98) G R

Original: x, y )]

Original: Q 0 ( β ) =

Original: K (cid:88) x =

Original: Q ( x ) 0 ( β )=

Original: K (cid:88) x

Original: = [ G L ( x ) − 1 (cid:88)

Original: y =

Original: G M ( x, y ; β ) · G R ( x, y )]

Original: From | Q n
Revised: From | Q n

Original: [Equation]
Revised: [Equation]

Original: ( x

Original: [Equation]
Revised: [Equation]

Original: · | [Equation] [Equation] |
Revised: x

Original: [Equation]
Revised: [Equation]

Original: x ∈ Γ since 0 ≤ | Q ( x ) n ( β ) | ≤ 1 and 0 ≤ | Q ( x ) 

Revised: Γ because 0 ≤ | Q ( x ) n ( β ) | ≤ 1 and 0 ≤ | Q ( x )

Original: ( β )
Revised: ( β )

Original: | ≤ 1 , each of which is a difference of two probability estimates.
Revised: | ≤ 1 , each of which is a difference of two probability estimates.

Original: Therefore,
Revised: Therefore,

Original: [Equation]
Revised: [Equation]

Original: since G M ( x, y ; β ) is a probability bounded between 0 and 1.
Revised: because G M ( x, y ; β ) is a probability bounded between 0 and 1.

Original: Because (cid:98) G L ( x ) and (cid:98) G R ( x, y ) are either sample proportions or their ratios, 
Revised: Since (cid:98) G L ( x ) and (cid:98) G R ( x, y ) are either sample proportions or their ratios, [Equation] ∞ (cid:98) G [Equation]

Original: [Equation]
Revised: [Equation]

Revised: G

Original: Because (cid:98) G L ( x ) and (cid:98) G R ( x, y ) do not involve β , from (9) we have
Revised: [Equation] [Equation] , as n → ∞ As (cid:98) G L ( x ) and (cid:98) G R ( x, y ) do not involve β , from (9) we have

Original: [Equation]
Revised: [Equation]

Original: where p = ⇒ denotes uniform convergence in probability.
Revised: where p = ⇒ denotes uniform convergence in probability.

Original: This confirms condition (II) and completes the proof of (cid:98)
Revised: This confirms condition (II) and completes the proof of (cid:98)

Original: β p → β as n → ∞ .
Revised: β p → β as n → ∞ .

Original: Because the causal estimate (cid:98) θ is a continuously differentiable function of (cid:98) β and relevant sample proportions, by Slutsky’s theorem, (cid:98) θ p → θ as n → ∞ .
Revised: Because the causal estimate (cid:98) θ is a continuously differentiable function of (cid:98) β and relevant sample proportions, by Slutsky’s theorem, (cid:98) θ p → θ as n → ∞ .

Original: This completes the proof of Theorem 2.

Original: Appendix C. Calculation for the True Principal Stratum Causal Effect
Revised: Appendix C. Calculation for the True Principal Stratum Causal Effect

Original: For the simulated data, the true average causal effect for principal stratum S i (1) = 1 can be calculated by
Revised: For the simulated data, the true average causal effect for principal stratum S i (1) = 1 can be calculated by

Original: [Equation]
Revised: [Equation]

Revised: Y i (0) | S i (1) = 1 } =

Original: {+E { Y i (1) = 
+}
Revised: {+E { Y i (1) = 
+}

Original: {+| S i (1) = 1 } − E { Y i (0) = 
+}
Revised: {+| S i (1) = 1 } − E { Y i (0) = 
+}

Original:  where
Revised: | S i (1) = 1 } = Pr { Y i (1) = 1 , S i (1) = 1 } − Pr { Y i (0) = 1 , S i (1) = 1 } Pr { S i (1) = 1 } where

Original: [Equation]
Revised: [Equation]

Revised: Pr { Y i (0) = 1 , S i (1) = 1 } = (cid:88) x

Revised: (cid:104) Pr { X i

Original: , X i = x } · Pr { S i (1) = 
 
Revised: = x } · Pr { S i (0) = 
 | X i = x } · Pr { Y i (0) = 1 | S i (0)

Original: {+= 1 , X i = x } + Pr { X i = x } · Pr { S i (0) = 
+}
Revised: {+= 1 , X i = x } + Pr { X i = x } · Pr { S i (0) = 
+}

Revised: | X i = x } · Pr { Y i (0)

Revised: =

Original: | S i (0) = 0 , Y{+X i = x } · Pr { S i (1) = 
 | S+} i (0) = 
Revised: | S i (0) = 0 , {+X i = x } · Pr { S i (1) = 
 | S+} i (0) = 0 , Y i (0)

Revised: = 1 , X i = x } (cid:105)

Revised: Pr { Y i (1) = 1 , S i (1) = 1 } = (cid:88) x (cid:88) y

Revised: (cid:104) Pr { X i

Revised: =

Original: {+x } · Pr { S i (0) = 
+}
Revised: {+x } · Pr { S i (0) = 
+}

Revised: | X i = x } · Pr { Y i (0) =

Revised: y | S i (0)

Original: y, X i = x } · Pr { Y i (1) =  
 
Revised: = 1 , X i = x } · Pr { Y i (1) = 1 | Y i (0) = y, S i (0) = 1 , X i = x } + Pr { X i = x } · Pr { S i (0) = 
 | X i = x } · Pr { Y i (0) =

Original:  | S i (0) = 0 ,  S i (1) = 1{+
 | S i (0) = 0+} , Y i (0) =
Revised: y | S i (0) = 0 , X i = x } · Pr { S i (1) = {+
 | S i (0) = 0+} , Y i (0) =

Original: {+y, X i = x } · Pr { Y i (1) = 
+}
Revised: {+y, X i = x } · Pr { Y i (1) = 
+}

Revised: | S i (0) = 0 , S i (1) = 1 , Y i (0) =

Original: y, X i = x } (cid:111)
Revised: y, X i = x } (cid:105)