Survivor Average Causal Effect Research Articles

A mixed model approach to estimate the survivor average causal effect in cluster-randomized trials.

In many medical studies, the outcome measure (such as quality of life, QOL) for some study participants becomes informatively truncated (censored, missing, or unobserved) due to death or other forms of dropout, creating a nonignorable missing data problem. In such cases, the use of a composite outcome or imputation methods that fill in unmeasurable QOL values for those who died rely on strong and untestable assumptions and may be conceptually unappealing to certain stakeholders when estimating a treatment effect. The survivor average causal effect (SACE) is an alternative causal estimand that surmounts some of these issues. While principal stratification has been applied to estimate the SACE in individually randomized trials, methods for estimating the SACE in cluster-randomized trials are currently limited. To address this gap, we develop a mixed model approach along with an expectation-maximization algorithm to estimate the SACE in cluster-randomized trials. We model the continuous outcome measure with a random intercept to account for intracluster correlations due to cluster-level randomization, and model the principal strata membership both with and without a random intercept. In simulations, we compare the performance of our approaches with an existing fixed-effects approach to illustrate the importance of accounting for clustering in cluster-randomized trials. The methodology is then illustrated using a cluster-randomized trial of telecare and assistive technology on health-related QOL in the elderly.

Matching methods for truncation by death problems

Abstract Even in a carefully designed randomised trial, outcomes for some study participants can be missing, or more precisely, ill defined, because participants had died prior to outcome collection. This problem, known as truncation by death, means that the treated and untreated are no longer balanced with respect to covariates determining survival. Therefore, researchers often utilise principal stratification and focus on the Survivor Average Causal Effect (SACE). We present matching-based methods for SACE identification and estimation. We provide identification results motivating the use of matching and discuss practical issues, including the choice of distance measures, matching with replacement, and post-matching estimators. Because the assumptions needed for SACE identification can be too strong, we also present sensitivity analysis techniques and illustrate their use in real data analysis.

A sensitivity analysis approach for the causal hazard ratio in randomized and observational studies.

The hazard ratio (HR) is often reported as the main causal effect when studying survival data. Despite its popularity, the HR suffers from an unclear causal interpretation. As already pointed out in the literature, there is a built-in selection bias in the HR, because similarly to the truncation by death problem, the HR conditions on post-treatment survival. A recently proposed alternative, inspired by the Survivor Average Causal Effect, is the causal HR, defined as the ratio between hazards across treatment groups among the study participants that would have survived regardless of their treatment assignment. We discuss the challenge in identifying the causal HR and present a sensitivity analysis identification approach in randomized controlled trials utilizing a working frailty model. We further extend our framework to adjust for potential confounders using inverse probability of treatment weighting. We present a Cox-based and a flexible non-parametric kernel-based estimation under right censoring. We study the finite-sample properties of the proposed estimation methods through simulations. We illustrate the utility of our framework using two real-dataexamples.

Detailed statistical analysis plan for a randomised controlled trial of the effects of a modified muscle sparing posterior technique (SPAIRE) in hip hemiarthroplasty for displaced intracapsular fractures on post-operative function compared to a standard lateral approach: HemiSPAIRE

BackgroundThe HemiSPAIRE trial is being conducted to determine whether a modified muscle sparing technique (SPAIRE-“Save Piriformis and Internus, Repairing Externus”) in hip hemiarthroplasty brings clinical benefits compared to the standard lateral technique in adults aged 60 years or older, with a displaced intracapsular hip fracture. This article describes the detailed statistical analysis plan for the trial. Methods and designHemiSPAIRE is a definitive, pragmatic, superiority, multicentre, randomised controlled trial (with internal pilot) with two parallel groups. Participants, ward staff and all research staff involved in post-operative assessments are blinded to allocation. This article describes in detail (1) the primary and secondary outcomes; (2) the statistical analysis principles, including a survivor average causal effect (SACE) method chosen specifically to address the issue of potential bias from differential survival between trial arms, which was seen from data review by the Trial Steering Committee, the participants that will be included in each analysis, the covariates that will be included in each analysis, and how the results will be presented; (3) planned main analysis of the primary outcome; (4) planned analyses of the secondary outcomes; and (5) planned additional analyses of the primary and secondary outcomes.Trial registrationClinicalTrials.gov NCT04095611. Registered on 19 September 2019.

Estimating the Effect of Tafamidis on Cardiovascular-Related Hospitalization in NYHA Class III Patients with Transthyretin Amyloid Cardiomyopathy in the Presence of Death

Background: The Tafamidis in Transthyretin Cardiomyopathy Clinical Trial (ATTR-ACT) demonstrated the effectiveness of tafamidis for the treatment of patients with transthyretin amyloid cardiomyopathy (ATTR-CM). Tafamidis reduced mortality in all subgroups of patients studied. Tafamidis also reduced observed frequency of cardiovascular (CV)-related hospitalizations in all subgroups except those who were New York Heart Association (NYHA) class III at baseline who, paradoxically, had a higher frequency of CV-related hospitalizations than placebo. Given the greater mortality rate with placebo, this analysis assessed the impact of the confounding effect of death on the frequency of CV-related hospitalization in ATTR-ACT. Methods: In ATTR-ACT, patients with ATTR-CM were randomized to tafamidis (n = 264) or placebo (n = 177) for 30 months. Post hoc analyses first defined and compared the effect of tafamidis treatment in the subset of NYHA class III patients from each treatment arm alive at month 30. The impact of a potential survivor bias was then adjusted for using principal stratification, estimating the frequency of CV-related hospitalization in NYHA class III patients who would have survived regardless of assigned treatment group (defined as the survivor average causal effect [SACE]). Results: In the subset of NYHA class III patients alive at month 30, tafamidis reduced the relative risk of CV-related hospitalization versus placebo (relative risk: 0.95 [95% CI: 0.55–1.65]). In the principal stratification analyses of those patients who would survive to 30 months regardless of treatment, tafamidis treatment was associated with a 24% lower risk of CV-related hospitalization (relative risk: 0.76 [95% CI: 0.45–1.24]). Similarly, there was a larger reduction in CV-related hospitalization frequency with tafamidis in NYHA class I or II patients in the SACE than was initially observed in ATTR-ACT. Conclusions: Initial data from ATTR-ACT likely underestimated the effect of tafamidis on CV-related hospitalizations due to the confounding effect of death. When SACE was used to adjust for survivor bias, there was a 24% reduction in the frequency of CV-related hospitalization in NYHA class III patients treated with tafamidis.

Conditional Separable Effects

Researchers are often interested in treatment effects on outcomes that are only defined conditional on posttreatment events. For example, in a study of the effect of different cancer treatments on quality of life at end of follow-up, the quality of life of individuals who die during the study is undefined. In these settings, naive contrasts of outcomes conditional on posttreatment events are not average causal effects, even in randomized experiments. Therefore, the effect in the principal stratum of those who would have the same value of the posttreatment variable regardless of treatment (such as the survivor average causal effect) is often advocated for causal inference. While principal stratum effects are average causal effects, they refer to a subset of the population that cannot be observed and may not exist. Therefore, it is not clear how these effects inform decisions or policies. Here we propose the conditional separable effects, quantifying causal effects of modified versions of the study treatment in an observable subset of the population. These effects, which may quantify direct effects of the study treatment, require transparent reasoning about candidate modified treatments and their mechanisms. We provide identifying conditions and various estimators of these effects along with an applied example. Supplementary materials for this article are available online.

Without Commitment to an Ontology, There Could Be No Causal Inference.

Without Commitment to an Ontology, There Could Be No Causal Inference.

Estimating treatment effect of tafamidis on hospitalisation in NYHA class III ATTR-CM patients in the presence of death using principal stratification

Abstract Background Hospitalisation has been widely used as a measure of morbidity in clinical trials for chronic heart failure. As previously reported in the Transthyretin Amyloidosis Cardiomyopathy Clinical Trial (ATTR-ACT), the observed frequency of cardiovascular (CV)-related hospitalisations favoured tafamidis over placebo in all analyses except in the NYHA Class III baseline subgroup with a relative risk of 1.41 [95% CI: 1.05, 1.90]1. However, the evaluation of the causal effect of treatment on hospitalisation is complicated in the presence of death as death precludes further hospitalisation and thus limits the evaluation of hospitalisation rates between groups. This confounding effect of death is particularly relevant in NYHA Class III participants who had higher mortality rates compared to Class I or II participants. Purpose This aim of this study was to assess the impact of the confounding effect of death on CV-related hospitalisation. Methods The effect of tafamidis treatment on CV-related hospitalisations was defined and compared in the subset of participants from either treatment arm who were alive at the end of the study (ie, Month 30); however the subset of participants who survive on the tafamidis arm would be expected to have different characteristics from the subset of participants who survive on the control arm and thus these analyses do not estimate the causal effect. To address this possible survivor bias, we additionally conducted a post hoc principal stratification analysis. Principal stratification is a statistical method to adjust for post treatment events, in this case mortality, by classifying participants according to their potential to survive under each treatment arm. Specifically, we sought to estimate the principal strata effect on CV-related hospitalisations in NYHA Class III participants who would survive irrespective of treatment assignment. Sensitivity analyses were performed to assess the model assumptions. Results The analysis within the subset of participants who survived at Month 30 demonstrated a relative risk of 1.1 (95% CI: 0.56, 2.16) between tafamidis and placebo. The principal stratification analyses of subjects who would survive to 30 months irrespective of treatment estimated a 0.75 (95% CI: 0.44, 1.23) relative risk between tafamidis and placebo, suggesting a 25% reduction in frequency of CV-related hospitalisations in the tafamidis arm. Conclusion The initial data reported in ATTR-ACT likely underestimated the true effect of tafamidis on CV-related hospitalisations due to the confounding effect of death. The analysis of treatment group among observed survivors estimated a smaller relative risk for NYHA Class III; however, this analysis does not fully address the possible survivor bias. The survivor average causal effect, which adjusts for the survival bias demonstrated a reduction in the frequency of CV-related hospitalisation in the tafamidis treated arm for NYHA Class III participants. Funding Acknowledgement Type of funding sources: Private company. Main funding source(s): Pfizer

Causal Inference in Studying the Long-Term Health Effects of Disasters: Challenges and Potential Solutions.

Two frequently encountered but underrecognized challenges for causal inference in studying the long-term health effects of disasters among survivors include 1) time-varying effects of disasters on a time-to-event outcome and 2) selection bias due to selective attrition. In this paper, we review approaches for overcoming these challenges and demonstrate application of the approaches to a real-world longitudinal data set of older adults who were directly affected by the 2011 Great East Japan Earthquake and Tsunami (n=4,857). To illustrate the problem of time-varying effects of disasters, we examined the association between degree of damage due to the tsunami and all-cause mortality. We compared results from Cox regression analysis assuming proportional hazards with those derived using adjusted parametric survival curves allowing for time-varying hazard ratios. To illustrate the problem of selection bias, we examined the association between proximity to the coast (a proxy for housing damage from the tsunami) and depressive symptoms. We corrected for selection bias due to attrition in the 2 postdisaster follow-up surveys (conducted in 2013 and 2016) using multivariable adjustment, inverse probability of censoring weighting, and survivor average causal effect estimation. Our results demonstrate that analytical approaches which ignore time-varying effects on mortality and selection bias due to selective attrition may underestimate the long-term health effects of disasters.

Quantile regression to estimate the survivor average causal effect of periodontal treatment effects on birthweight and gestational age.

Survival average causal effect (SACE) can give valid estimates of the periodontal treatment effect on birth outcomes in randomized controlled trials when fetal losses are unequal across the treatment arms. A regression-based method to estimate SACE using ordinary least squares (OLS) regression can be biased if the treatment effect varies across the outcome distribution. In this case quantile regression may be a suitable alternative. We compared OLS and quantile regression models estimating SACE to calculate the effect of periodontal treatment on birthweight and gestational age in secondary analyses of publicly available Obstetrics and Periodontal Therapy (OPT) trial data. Periodontal treatment tended to increase birthweight and gestational age at the lowest quantiles, remained flat in the middle quantiles, and trended to decrease both birthweight and gestational age in the highest quantiles. In quantile regression models estimating SACE the β-coefficients: 95% confidence intervals (CI) for the 5th, 50th, and 95th percentiles were 277.5: -141.0 to 696.0g, 1.4: -107 to 110.3g, and -84: -344 to 175.3g for birthweight, and 0.6: -1.0 to 2.2weeks, -0.1: -0.5 to 0.2weeks, and -0.6: -1.0 to -0.1weeks for gestational age. Estimates from OLS models estimating SACE were close to the null, β: 95% CI -4.7: 132.3 to 123.0g for birthweight, and 0.03: -0.72 to 0.78weeks for gestational age. OLS models to evaluate SACE for periodontal treatment effects on birthweight and gestational age may be biased towards the null. Quantile regression may be a preferable alternative.

Post-randomization Biomarker Effect Modification Analysis in an HIV Vaccine Clinical Trial.

While the HVTN 505 trial showed no overall efficacy of the tested vaccine to prevent HIV infection over placebo, markers measuring immune response to vaccination were strongly correlated with infection. This finding generated the hypothesis that some marker-defined vaccinated subgroups were partially protected whereas others had their risk increased. This hypothesis can be assessed using the principal stratification framework (Frangakis and Rubin, 2002) for studying treatment effect modification by an intermediate response variable, using methods in the sub-field of principal surrogate (PS) analysis that studies multiple principal strata. Unfortunately, available methods for PS analysis require an augmented study design not available in HVTN 505, and make untestable structural risk assumptions, motivating a need for more robust PS methods. Fortunately, another sub-field of principal stratification, survivor average causal effect (SACE) analysis (Rubin, 2006) - which studies effects in a single principal stratum - provides many methods not requiring an augmented design and making fewer assumptions. We show how, for a binary intermediate response variable, methods developed for SACE analysis can be adapted to PS analysis, providing new and more robust PS methods. Application to HVTN 505 supports that the vaccine partially protected individuals with vaccine-induced T-cells expressing certain combinations of functions.

A comparison of methods to estimate the survivor average causal effect in the presence of missing data: a simulation study

BackgroundAttrition due to death and non-attendance are common sources of bias in studies of age-related diseases. A simulation study is presented to compare two methods for estimating the survivor average causal effect (SACE) of a binary exposure (sex-specific dietary iron intake) on a binary outcome (age-related macular degeneration, AMD) in this setting.MethodsA dataset of 10,000 participants was simulated 1200 times under each scenario with outcome data missing dependent on measured and unmeasured covariates and survival. Scenarios differed by the magnitude and direction of effect of an unmeasured confounder on both survival and the outcome, and whether participants who died following a protective exposure would also die if they had not received the exposure (validity of the monotonicity assumption). The performance of a marginal structural model (MSM, weighting for exposure, survival and missing data) was compared to a sensitivity approach for estimating the SACE. As an illustrative example, the SACE of iron intake on AMD was estimated using data from 39,918 participants of the Melbourne Collaborative Cohort Study.ResultsThe MSM approach tended to underestimate the true magnitude of effect when the unmeasured confounder had opposing directions of effect on survival and the outcome. Overestimation was observed when the unmeasured confounder had the same direction of effect on survival and the outcome. Violation of the monotonicity assumption did not increase bias. The estimates were similar between the MSM approach and the sensitivity approach assessed at the sensitivity parameter of 1 (assuming no survival bias). In the illustrative example, high iron intake was found to be protective of AMD (adjusted OR 0.57, 95% CI 0.40–0.82) using complete case analysis via traditional logistic regression. The adjusted SACE odds ratio did not differ substantially from the complete case estimate, ranging from 0.54 to 0.58 for each of the SACE methods.ConclusionsOn average, MSMs with weighting for exposure, missing data and survival produced biased estimates of the SACE in the presence of an unmeasured survival-outcome confounder. The direction and magnitude of effect of unmeasured survival-outcome confounders should be considered when assessing exposure-outcome associations in the presence of attrition due to death.

Assessing the ratio of means as a causal estimand in clinical endpoint bioequivalence studies in the presence of intercurrent events.

In clinical endpoint bioequivalence studies, the observed per-protocol (PP) population (compliers and completers in general) is usually used in the primary analysis for equivalence assessment. However, intercurrent events, ie, missingness and noncompliance, are not properly handled. The resulting estimand is not causal. Previously, we proposed the first causal framework to assess equivalence in the presence of missing data and noncompliance. We proposed a causal survivor average causal effect (SACE) estimand for the difference of means (DOM). In equivalence assessment, DOM is not as widely used as the ratio of means (ROM). However, no existing formula links the observed PP estimand to the SACE estimand for ROM as exists for DOM. Herein, we propose a similar causal framework for ROM using the principal stratification approach, one of the strategies recommended by the International Conference on Harmonisation (ICH) E9 R1 addendum. We quantify the bias of the observed ROM PP estimand for the SACE estimand, which provides a basis to identify three conditions under which the two estimands are equal. We propose a sensitivity analysis method to evaluate the robustness of the current PP estimator to estimate the SACE estimand. We extend Fieller's confidence interval for the SACE estimand using ROM, which can be applied to many settings. Simulation demonstrates that the PP estimator is biased in either directions and may inflate type 1 error and/or change power when the three identified conditions are violated. Our work can be applied to comparative clinical biosimilar studies.

An Investigation of Selection Bias in Estimating Racial Disparity in Stroke Risk Factors

Selection due to survival or attrition might bias estimates of racial disparities in health, but few studies quantify the likely magnitude of such bias. In a large national cohort with moderate loss to follow-up, we contrasted racial differences in 2 stroke risk factors, incident hypertension and incident left ventricular hypertrophy, estimated by complete-case analyses, inverse probability of attrition weighting, and the survivor average causal effect. We used data on 12,497 black and 17,660 white participants enrolled in the United States (2003-2007) and collected incident risk factor data approximately 10 years after baseline. At follow-up, 21.0% of white participants and 23.0% of black participants had died; additionally 22.0% of white participants and 28.4% of black participants had withdrawn. Individual probabilities of completing the follow-up visit were estimated using baseline demographic and health characteristics. Adjusted risk ratio estimates of racial disparities from complete-case analyses in both incident hypertension (1.11, 95% confidence interval: 1.02, 1.21) and incident left ventricular hypertrophy (1.02, 95% confidence interval: 0.84, 1.24) were virtually identical to estimates from inverse probability of attrition weighting and survivor average causal effect. Despite racial differences in mortality and attrition, we found little evidence of selection bias in the estimation of racial differences for these incident risk factors.

Estimation of causal effects in clinical endpoint bioequivalence studies in the presence of intercurrent events: noncompliance and missing data

ABSTRACTIn clinical endpoint bioequivalence (BE) studies, the primary analysis for assessing equivalence between a generic and an innovator product is based on the observed per-protocol (PP) population (usually completers and compliers). However, missing data and noncompliance are post-randomization intercurrent events and may introduce selection bias. Therefore, PP analysis is generally not causal. The FDA Missing Data Working Group recommended using “causal estimands of primary interest.” In this paper, we propose a principal stratification causal framework and co-primary causal estimands to test equivalence, which was also recommended by the recently published ICH E9 (R1) addendum to address intercurrent events. We identify three conditions under which the current PP estimator is unbiased for one of the proposed co-primary causal estimands – the “Survivor Average Causal Effect” (SACE) estimand. Simulation shows that when these three conditions are not met, the PP estimator is biased and may inflate Type 1 error and/or change power. We also propose a tipping point sensitivity analysis to evaluate the robustness of the current PP estimator in testing equivalence when the sensitivity parameters deviate from the three identified conditions, but stay within a clinically meaningful range. Our work is the first causal equivalence assessment in equivalence studies with intercurrent events.

Periodontal treatment among mothers with mild to moderate periodontal disease and preterm birth: reanalysis of OPT trial data accounting for selective survival.

The Obstetrics and Periodontal Therapy (OPT) study, a randomized controlled trial, reported no effect of periodontal treatment on preterm birth. Even though there were more spontaneous abortions or stillbirths in the control group, sensitivity analyses using conventional approaches did not change the results. The development of newer epidemiological methods to assess bias caused by the truncated outcome, and the availability of OPT study data in the public domain, allowed us to reanalyse these data. We used the survivor average causal effect (SACE), also known as the principal strata effect, to correct potential bias resulting from unequal survival of fetuses in the treatment and control arms of the OPT study. The risks of preterm and spontaneous abortions or stillbirths were respectively 49/413 (11.86%) and 5/413 (1.21%) in the periodontal treatment group, and 52/410 (12.68%) and 14/410 (3.33%) respectively in the control group. The risk differences (%) comparing periodontal treatment and control groups were -0.82%: 95% confidence interval (CI) -5.30% to 3.67% for preterm birth, and -2.12%: 95% CI -4.14% to -0.11% for spontaneous abortions or stillbirths before correction. Risk differences for preterm birth comparing periodontal treatment and control groups increased in magnitude, reached statistical significance and showed a beneficial effect of periodontal treatment after correction for bias using SACE. Periodontal treatment provided to mothers with mild to moderate periodontal disease before 21 weeks of gestation may prevent preterm births.

Identification and estimation of causal effects with outcomes truncated by death.

SummaryIt is common in medical studies that the outcome of interest is truncated by death, meaning that a subject has died before the outcome could be measured. In this case, restricted analysis among survivors may be subject to selection bias. Hence, it is of interest to estimate the survivor average causal effect, defined as the average causal effect among the subgroup consisting of subjects who would survive under either exposure. In this paper, we consider the identification and estimation problems of the survivor average causal effect. We propose to use a substitution variable in place of the latent membership in the always-survivor group. The identification conditions required for a substitution variable are conceptually similar to conditions for a conditional instrumental variable, and may apply to both randomized and observational studies. We show that the survivor average causal effect is identifiable with use of such a substitution variable, and propose novel model parameterizations for estimation of the survivor average causal effect under our identification assumptions. Our approaches are illustrated via simulation studies and a data analysis.

Survival Bias When Assessing Risk Factors for Age-Related Macular Degeneration: A Tutorial with Application to the Exposure of Smoking

ABSTRACTPurpose: We illustrate the effect of survival bias when investigating risk factors for eye disease in elderly populations for whom death is a competing risk. Our investigation focuses on the relationship between smoking and late age-related macular degeneration (AMD) in an observational study impacted by censoring due to death.Methods: Statistical methodology to calculate the survivor average causal effect (SACE) as a sensitivity analysis is described, including example statistical computing code for Stata and R. To demonstrate this method, we examine the causal effect of smoking history at baseline (1990–1994) on the presence of late AMD at the third study wave (2003–2007) using data from the Melbourne Collaborative Cohort Study.Results: Of the 40,506 participants eligible for inclusion, 38,092 (94%) survived until the start of the third study wave, 20,752 (51%) were graded for AMD (60% female, aged 47–85 years, mean 65 ± 8.7 years). Late AMD was detected in 122 participants. Logistic regression showed strong evidence of an increased risk of late AMD for current smokers compared to non-smokers (adjusted naïve odds ratio 2.99, 95% confidence interval, CI, 1.74–5.13). Among participants expected to be alive at the start of follow-up regardless of their smoking status, the estimated SACE odds ratio comparing current smokers to non-smokers was at least 3.42 (95% CI 1.57–5.15).Conclusions: Survival bias can attenuate associations between harmful exposures and diseases of aging. Estimation of the SACE using a sensitivity analysis approach should be considered when conducting epidemiological research within elderly populations.

Estimating the palliative effect of percutaneous endoscopic gastrostomy in an observational registry using principal stratification and generalized propensity scores.

Clinical disease registries offer a rich collection of valuable patient information but also pose challenges that require special care and attention in statistical analyses. The goal of this paper is to propose a statistical framework that allows for estimating the effect of surgical insertion of a percutaneous endogastrostomy (PEG) tube for patients living with amyotrophic lateral sclerosis (ALS) using data from a clinical registry. Although all ALS patients are informed about PEG, only some patients agree to the procedure which, leads to the potential for selection bias. Assessing the effect of PEG is further complicated by the aggressively fatal disease, such that time to death competes directly with both the opportunity to receive PEG and clinical outcome measurements. Our proposed methodology handles the “censoring by death” phenomenon through principal stratification and selection bias for PEG treatment through generalized propensity scores. We develop a fully Bayesian modeling approach to estimate the survivor average causal effect (SACE) of PEG on BMI, a surrogate outcome measure of nutrition and quality of life. The use of propensity score methods within the principal stratification framework demonstrates a significant and positive effect of PEG treatment, particularly when time of treatment is included in the treatment definition.

Sharp Bounds on Survivor Average Causal Effects When the Outcome Is Binary and Truncated by Death

In randomized trials with follow-up, outcomes may be undefined for individuals who die before the follow-up is complete. In such settings, Frangakis and Rubin [2002] proposed the “principal stratum effect” or “Survivor Average Causal Effect” (SACE), which is a fair treatment comparison in the subpopulation that would have survived under either treatment arm. Many of the existing results for estimating the SACE are difficult to carry out in practice. In this article, when the outcome is binary, we apply the symbolic Balke-Pearl linear programming method to derive simple formulas for the sharp bounds on the SACE under the monotonicity assumption commonly used by many researchers.