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1.  Accommodating Missingness When Assessing Surrogacy Via Principal Stratification 
Clinical trials (London, England)  2013;10(3):363-377.
When an outcome of interest in a clinical trial is late-occurring or difficult to obtain, surrogate markers can extract information about the effect of the treatment on the outcome of interest. Understanding associations between the causal effect of treatment on the outcome and the causal effect of treatment on the surrogate is critical to understanding the value of a surrogate from a clinical perspective.
Traditional regression approaches to determine the proportion of the treatment effect explained by surrogate markers suffer from several shortcomings: they can be unstable, and can lie outside of the 0–1 range. Further, they do not account for the fact that surrogate measures are obtained post-randomization, and thus the surrogate-outcome relationship may be subject to unmeasured confounding. Methods to avoid these problem are of key importance.
Frangakis C, Rubin DM. Principal stratification in causal inference. Biometrics 2002; 58:21–9 suggested assessing the causal effect of treatment within pre-randomization “principal strata” defined by the counterfactual joint distribution of the surrogate marker under the different treatment arms, with the proportion of the overall outcome causal effect attributable to subjects for whom the treatment affects the proposed surrogate as the key measure of interest. Li Y, Taylor JMG, Elliott MR. Bayesian approach to surrogacy assessment using principal stratification in clinical trials. Biometrics 2010; 66:523–31 developed this “principal surrogacy” approach for dichotomous markers and outcomes, utilizing Bayesian methods that accommodated non-identifiability in the model parameters. Because the surrogate marker is typically observed early, outcome data is often missing. Here we extend Li, Taylor, and Elliott to accommodate missing data in the observable final outcome under ignorable and non-ignorable settings. We also allow for the possibility that missingness has a counterfactual component, a feature that previous literature has not addressed.
We apply the proposed methods to a trial of glaucoma control comparing surgery versus medication, where intraocular pressure (IOP) control at 12 months is a surrogate for IOP control at 96 months. We also conduct a series of simulations to consider the impacts of non-ignorability, as well as sensitivity to priors and the ability of the Decision Information Criterion to choose the correct model when parameters are not fully identified.
Because model parameters cannot be fully identified from data, informative priors can introduce non-trivial bias in moderate sample size settings, while more non-informative priors can yield wide credible intervals.
Assessing the linkage between causal effects of treatment on a surrogate marker and causal effects of a treatment on an outcome is important to understanding the value of a marker. These causal effects are not fully identifiable: hence we explore the sensitivity and identifiability aspects of these models and show that relatively weak assumptions can still yield meaningful results.
PMCID: PMC4096330  PMID: 23553326
Causal Inference; Surrogate Marker; Bayesian Analysis; dentifiability; Non-response; Counterfactual
2.  Intermediate outcomes in randomized clinical trials: an introduction 
Trials  2013;14:78.
Intermediate outcomes are common and typically on the causal pathway to the final outcome. Some examples include noncompliance, missing data, and truncation by death like pregnancy (e.g. when the trial intervention is given to non-pregnant women and the final outcome is preeclampsia, defined only on pregnant women). The intention-to-treat approach does not account properly for them, and more appropriate alternative approaches like principal stratification are not yet widely known. The purposes of this study are to inform researchers that the intention-to-treat approach unfortunately does not fit all problems we face in experimental research, to introduce the principal stratification approach for dealing with intermediate outcomes, and to illustrate its application to a trial of long term calcium supplementation in women at high risk of preeclampsia.
Principal stratification and related concepts are introduced. Two ways for estimating causal effects are discussed and their application is illustrated using the calcium trial, where noncompliance and pregnancy are considered as intermediate outcomes, and preeclampsia is the main final outcome.
The limitations of traditional approaches and methods for dealing with intermediate outcomes are demonstrated. The steps, assumptions and required calculations involved in the application of the principal stratification approach are discussed in detail in the case of our calcium trial.
The intention-to-treat approach is a very sound one but unfortunately it does not fit all problems we find in randomized clinical trials; this is particularly the case for intermediate outcomes, where alternative approaches like principal stratification should be considered.
PMCID: PMC3610291  PMID: 23510143
Intermediate outcomes; Intention-to-treat approach; Principal stratification; Causal effects
3.  Surrogacy Assessment Using Principal Stratification and a Gaussian Copula Model 
In clinical trials, a surrogate outcome (S) can be measured before the outcome of interest (T) and may provide early information regarding the treatment (Z) effect on T. Many methods of surrogacy validation rely on models for the conditional distribution of T given Z and S. However, S is a post-randomization variable, and unobserved, simultaneous predictors of S and T may exist, resulting in a non-causal interpretation. Frangakis and Rubin1 developed the concept of principal surrogacy, stratifying on the joint distribution of the surrogate marker under treatment and control to assess the association between the causal effects of treatment on the marker and the causal effects of treatment on the clinical outcome. Working within the principal surrogacy framework, we address the scenario of an ordinal categorical variable as a surrogate for a censored failure time true endpoint. A Gaussian copula model is used to model the joint distribution of the potential outcomes of T, given the potential outcomes of S. Because the proposed model cannot be fully identified from the data, we use a Bayesian estimation approach with prior distributions consistent with reasonable assumptions in the surrogacy assessment setting. The method is applied to data from a colorectal cancer clinical trial, previously analyzed by Burzykowski et al..2
PMCID: PMC4272338  PMID: 24947559
Causal inference; Gaussian copula; Potential outcomes; Surrogate endpoint
4.  Estimating Causal Effects in Trials Involving Multi-Treatment Arms Subject to Non-compliance: A Bayesian framework 
Data analysis for randomized trials including multi-treatment arms is often complicated by subjects who do not comply with their treatment assignment. We discuss here methods of estimating treatment efficacy for randomized trials involving multi-treatment arms subject to non-compliance. One treatment effect of interest in the presence of non-compliance is the complier average causal effect (CACE) (Angrist et al. 1996), which is defined as the treatment effect for subjects who would comply regardless of the assigned treatment. Following the idea of principal stratification (Frangakis & Rubin 2002), we define principal compliance (Little et al. 2009) in trials with three treatment arms, extend CACE and define causal estimands of interest in this setting. In addition, we discuss structural assumptions needed for estimation of causal effects and the identifiability problem inherent in this setting from both a Bayesian and a classical statistical perspective. We propose a likelihood-based framework that models potential outcomes in this setting and a Bayes procedure for statistical inference. We compare our method with a method of moments approach proposed by Cheng & Small (2006) using a hypothetical data set, and further illustrate our approach with an application to a behavioral intervention study (Janevic et al. 2003).
PMCID: PMC3104736  PMID: 21637737
Causal Inference; Complier Average Causal Effect; Multi-arm Trials; Non-compliance; Principal Compliance; Principal Stratification
5.  Marginalized models for longitudinal ordinal data with application to quality of life studies 
Statistics in medicine  2008;27(21):4359-4380.
Random effects are often used in generalized linear models to explain the serial dependence for longitudinal categorical data. Marginalized random effects models (MREMs) for the analysis of longitudinal binary data have been proposed to permit likelihood-based estimation of marginal regression parameters. In this paper, we introduce an extension of the MREM to accommodate longitudinal ordinal data. Maximum marginal likelihood estimation is implemented utilizing quasi-Newton algorithms with Monte Carlo integration of the random effects. Our approach is applied to analyze the quality of life data from a recent colorectal cancer clinical trial. Dropout occurs at a high rate and is often due to tumor progression or death. To deal with progression/death, we use a mixture model for the joint distribution of longitudinal measures and progression/death times and principal stratification to draw causal inferences about survivors.
PMCID: PMC2858760  PMID: 18613246
marginalized likelihood-based models; ordinal data models; dropout
6.  Using latent outcome trajectory classes in causal inference* 
Statistics and its interface  2009;2(4):403-412.
In longitudinal studies, outcome trajectories can provide important information about substantively and clinically meaningful underlying subpopulations who may also respond differently to treatments or interventions. Growth mixture analysis is an efficient way of identifying heterogeneous trajectory classes. However, given its exploratory nature, it is unclear how involvement of latent classes should be handled in the analysis when estimating causal treatment effects. In this paper, we propose a 2-step approach, where formulation of trajectory strata and identification of causal effects are separated. In Step 1, we stratify individuals in one of the assignment conditions (reference condition) into trajectory strata on the basis of growth mixture analysis. In Step 2, we estimate treatment effects for different trajectory strata, treating the stratum membership as partly known (known for individuals assigned to the reference condition and missing for the rest). The results can be interpreted as how subpopulations that differ in terms of outcome prognosis under one treatment condition would change their prognosis differently when exposed to another treatment condition. Causal effect estimation in Step 2 is consistent with that in the principal stratification approach (Frangakis and Rubin, 2002) in the sense that clarified identifying assumptions can be employed and therefore systematic sensitivity analyses are possible. Longitudinal development of attention deficit among children from the Johns Hopkins School Intervention Trial (Ialongo et al., 1999) will be presented as an example.
PMCID: PMC2863041  PMID: 20445809
Causal inference; Latent trajectory class; Longitudinal outcome prognosis; Growth mixture modeling; Principal stratification; Reference stratification
7.  A Bayesian Approach to Improved Estimation of Causal Effect Predictiveness for a Principal Surrogate Endpoint 
Biometrics  2012;68(3):10.1111/j.1541-0420.2011.01736.x.
The literature on potential outcomes has shown that traditional methods for characterizing surrogate endpoints in clinical trials based only on observed quantities can fail to capture causal relationships between treatments, surrogates, and outcomes. Building on the potential-outcomes formulation of a principal surrogate, we introduce a Bayesian method to estimate the Causal Effect Predictiveness (CEP) surface and quantify a candidate surrogate’s utility for reliably predicting clinical outcomes. In considering the full joint distribution of all potentially-observable quantities, our Bayesian approach has the following features. First, our approach illuminates implicit assumptions embedded in previously-used estimation strategies that have been shown to result in poor performance. Second, our approach provides tools for making explicit and scientifically-interpretable assumptions regarding associations about which observed data are not informative. Through simulations based on an HIV vaccine trial, we found that the Bayesian approach can produce estimates of the CEP surface with improved performance compared to previous methods. Third, our approach can extend principal-surrogate estimation beyond the previously-considered setting of a vaccine trial where the candidate surrogate is constant in one arm of the study. We illustrate this extension through an application to an AIDS therapy trial where the candidate surrogate varies in both treatment arms.
PMCID: PMC3860118  PMID: 22348277
Biomarker; Causal effect predictiveness; principal stratification; surrogate endpoint
8.  Statistical identifiability and the surrogate endpoint problem, with application to vaccine trials 
Biometrics  2010;66(4):1153-1161.
Given a randomized treatment Z, a clinical outcome Y, and a biomarker S measured some fixed time after Z is administered, we may be interested in addressing the surrogate endpoint problem by evaluating whether S can be used to reliably predict the effect of Z on Y. Several recent proposals for the statistical evaluation of surrogate value have been based on the framework of principal stratification. In this paper, we consider two principal stratification estimands: joint risks and marginal risks. Joint risks measure causal associations of treatment effects on S and Y, providing insight into the surrogate value of the biomarker, but are not statistically identifiable from vaccine trial data. While marginal risks do not measure causal associations of treatment effects, they nevertheless provide guidance for future research, and we describe a data collection scheme and assumptions under which the marginal risks are statistically identifiable. We show how different sets of assumptions affect the identifiability of these estimands; in particular, we depart from previous work by considering the consequences of relaxing the assumption of no individual treatment effects on Y before S is measured. Based on algebraic relationships between joint and marginal risks, we propose a sensitivity analysis approach for assessment of surrogate value, and show that in many cases the surrogate value of a biomarker may be hard to establish, even when the sample size is large.
PMCID: PMC3597127  PMID: 20105158
Estimated likelihood; Identifiability; Principal stratification; Sensitivity analysis; Surrogate endpoint; Vaccine trials
9.  Causal Vaccine Effects on Binary Postinfection Outcomes 
The effects of vaccine on postinfection outcomes, such as disease, death, and secondary transmission to others, are important scientific and public health aspects of prophylactic vaccination. As a result, evaluation of many vaccine effects condition on being infected. Conditioning on an event that occurs posttreatment (in our case, infection subsequent to assignment to vaccine or control) can result in selection bias. Moreover, because the set of individuals who would become infected if vaccinated is likely not identical to the set of those who would become infected if given control, comparisons that condition on infection do not have a causal interpretation. In this article we consider identifiability and estimation of causal vaccine effects on binary postinfection outcomes. Using the principal stratification framework, we define a postinfection causal vaccine efficacy estimand in individuals who would be infected regardless of treatment assignment. The estimand is shown to be not identifiable under the standard assumptions of the stable unit treatment value, monotonicity, and independence of treatment assignment. Thus selection models are proposed that identify the causal estimand. Closed-form maximum likelihood estimators (MLEs) are then derived under these models, including those assuming maximum possible levels of positive and negative selection bias. These results show the relations between the MLE of the causal estimand and two commonly used estimators for vaccine effects on postinfection outcomes. For example, the usual intent-to-treat estimator is shown to be an upper bound on the postinfection causal vaccine effect provided that the magnitude of protection against infection is not too large. The methods are used to evaluate postinfection vaccine effects in a clinical trial of a rotavirus vaccine candidate and in a field study of a pertussis vaccine. Our results show that pertussis vaccination has a significant causal effect in reducing disease severity.
PMCID: PMC2603579  PMID: 19096723
Causal inference; Infectious disease; Maximum likelihood; Principal stratification; Sensitivity analysis
The annals of applied statistics  2008;2(3):1034-1055.
Participants in longitudinal studies on the effects of drug treatment and criminal justice system interventions are at high risk for institutionalization (e.g., spending time in an environment where their freedom to use drugs, commit crimes, or engage in risky behavior may be circumscribed). Methods used for estimating treatment effects in the presence of institutionalization during follow-up can be highly sensitive to assumptions that are unlikely to be met in applications and thus likely to yield misleading inferences. In this paper, we consider the use of principal stratification to control for institutionalization at follow-up. Principal stratification has been suggested for similar problems where outcomes are unobservable for samples of study participants because of dropout, death, or other forms of censoring. The method identifies principal strata within which causal effects are well defined and potentially estimable. We extend the method of principal stratification to model institutionalization at follow-up and estimate the effect of residential substance abuse treatment versus outpatient services in a large scale study of adolescent substance abuse treatment programs. Additionally, we discuss practical issues in applying the principal stratification model to data. We show via simulation studies that the model can only recover true effects provided the data meet strenuous demands and that there must be caution taken when implementing principal stratification as a technique to control for post-treatment confounders such as institutionalization.
PMCID: PMC2749670  PMID: 19779599
Principal Stratification; Post-Treatment Confounder; Institutionalization; Causal Inference
11.  Sensitivity Analysis and Bounding of Causal Effects With Alternative Identifying Assumptions 
When identification of causal effects relies on untestable assumptions regarding nonidentified parameters, sensitivity of causal effect estimates is often questioned. For proper interpretation of causal effect estimates in this situation, deriving bounds on causal parameters or exploring the sensitivity of estimates to scientifically plausible alternative assumptions can be critical. In this paper, we propose a practical way of bounding and sensitivity analysis, where multiple identifying assumptions are combined to construct tighter common bounds. In particular, we focus on the use of competing identifying assumptions that impose different restrictions on the same non-identified parameter. Since these assumptions are connected through the same parameter, direct translation across them is possible. Based on this cross-translatability, various information in the data, carried by alternative assumptions, can be effectively combined to construct tighter bounds on causal effects. Flexibility of the suggested approach is demonstrated focusing on the estimation of the complier average causal effect (CACE) in a randomized job search intervention trial that suffers from noncompliance and subsequent missing outcomes.
PMCID: PMC3150587  PMID: 21822369
alternative assumptions; bounds; causal inference; missing data; noncompliance; principal stratification; sensitivity analysis
12.  Causal Inference for Bivariate Longitudinal Quality of Life Data in Presence of Death Using Global Odds Ratios 
Statistics in medicine  2013;32(24):4275-4284.
In longitudinal clinical trials, if a subject drops out due to death, certain responses, such as those measuring quality of life (QOL), will not be defined after the time of death. Thus, standard missing data analyses, e.g., under ignorable dropout, are problematic because these approaches implicitly ‘impute’ values of the response after death. In this paper we define a new survivors average causal effect for a bivariate response in a longitudinal quality of life study that had a high dropout rate with the dropout often due to death (or tumor progression). We show how principal stratification, with a few sensitivity parameters, can be used to draw causal inferences about the joint distribution of these two ordinal quality of life measures.
PMCID: PMC3935993  PMID: 23720372
13.  Surrogacy assessment using principal stratification when surrogate and outcome measures are multivariate normal 
Biostatistics (Oxford, England)  2013;15(2):266-283.
In clinical trials, a surrogate outcome variable (S) can be measured before the outcome of interest (T) and may provide early information regarding the treatment (Z) effect on T. Using the principal surrogacy framework introduced by Frangakis and Rubin (2002. Principal stratification in causal inference. Biometrics 58, 21–29), we consider an approach that has a causal interpretation and develop a Bayesian estimation strategy for surrogate validation when the joint distribution of potential surrogate and outcome measures is multivariate normal. From the joint conditional distribution of the potential outcomes of T, given the potential outcomes of S, we propose surrogacy validation measures from this model. As the model is not fully identifiable from the data, we propose some reasonable prior distributions and assumptions that can be placed on weakly identified parameters to aid in estimation. We explore the relationship between our surrogacy measures and the surrogacy measures proposed by Prentice (1989. Surrogate endpoints in clinical trials: definition and operational criteria. Statistics in Medicine 8, 431–440). The method is applied to data from a macular degeneration study and an ovarian cancer study.
PMCID: PMC4023321  PMID: 24285772
Bayesian estimation; Principal stratification; Surrogate endpoints
14.  Effectiveness of Electronic Reminders to Improve Medication Adherence in Tuberculosis Patients: A Cluster-Randomised Trial 
PLoS Medicine  2015;12(9):e1001876.
Mobile text messaging and medication monitors (medication monitor boxes) have the potential to improve adherence to tuberculosis (TB) treatment and reduce the need for directly observed treatment (DOT), but to our knowledge they have not been properly evaluated in TB patients. We assessed the effectiveness of text messaging and medication monitors to improve medication adherence in TB patients.
Methods and Findings
In a pragmatic cluster-randomised trial, 36 districts/counties (each with at least 300 active pulmonary TB patients registered in 2009) within the provinces of Heilongjiang, Jiangsu, Hunan, and Chongqing, China, were randomised using stratification and restriction to one of four case-management approaches in which patients received reminders via text messages, a medication monitor, combined, or neither (control). Patients in the intervention arms received reminders to take their drugs and reminders for monthly follow-up visits, and the managing doctor was recommended to switch patients with adherence problems to more intensive management or DOT. In all arms, patients took medications out of a medication monitor box, which recorded when the box was opened, but the box gave reminders only in the medication monitor and combined arms. Patients were followed up for 6 mo. The primary endpoint was the percentage of patient-months on TB treatment where at least 20% of doses were missed as measured by pill count and failure to open the medication monitor box. Secondary endpoints included additional adherence and standard treatment outcome measures. Interventions were not masked to study staff and patients. From 1 June 2011 to 7 March 2012, 4,292 new pulmonary TB patients were enrolled across the 36 clusters. A total of 119 patients (by arm: 33 control, 33 text messaging, 23 medication monitor, 30 combined) withdrew from the study in the first month because they were reassessed as not having TB by their managing doctor (61 patients) or were switched to a different treatment model because of hospitalisation or travel (58 patients), leaving 4,173 TB patients (by arm: 1,104 control, 1,008 text messaging, 997 medication monitor, 1,064 combined). The cluster geometric mean of the percentage of patient-months on TB treatment where at least 20% of doses were missed was 29.9% in the control arm; in comparison, this percentage was 27.3% in the text messaging arm (adjusted mean ratio [aMR] 0.94, 95% CI 0.71, 1.24), 17.0% in the medication monitor arm (aMR 0.58, 95% CI 0.42, 0.79), and 13.9% in the combined arm (aMR 0.49, 95% CI 0.27, 0.88). Patient loss to follow-up was lower in the text messaging arm than the control arm (aMR 0.42, 95% CI 0.18–0.98). Equipment malfunction or operation error was reported in all study arms. Analyses separating patients with and without medication monitor problems did not change the results. Initiation of intensive management was underutilised.
This study is the first to our knowledge to utilise a randomised trial design to demonstrate the effectiveness of a medication monitor to improve medication adherence in TB patients. Reminders from medication monitors improved medication adherence in TB patients, but text messaging reminders did not. In a setting such as China where universal use of DOT is not feasible, innovative approaches to support patients in adhering to TB treatment, such as this, are needed.
Trial Registration
Current Controlled Trials, ISRCTN46846388
In a cluster-randomized controlled trial, Katherine Fielding and colleagues examine the effectiveness of electronic reminders for improving tuberculosis medication adherence.
Editors' Summary
Tuberculosis—a contagious bacterial disease that usually infects the lungs—is a major global public health problem. Every year, about 9 million people develop tuberculosis and at least 1.3 million people die as a result. Mycobacterium tuberculosis, the organism that causes tuberculosis, is spread in airborne droplets when people with tuberculosis cough or sneeze. The symptoms of tuberculosis include cough, weight loss, and fever. Diagnostic tests for tuberculosis include sputum smear microscopy (microscopic analysis of mucus coughed up from the lungs), the growth of M. tuberculosis from sputum samples, and chest X-rays. Tuberculosis can be cured by taking antibiotics daily for several months (usually isoniazid, rifampicin, ethambutol, and pyrazinamide for two months followed by isoniazid and rifampicin for a further four months), but the emergence of multidrug-resistant M. tuberculosis is making tuberculosis increasingly hard to treat.
Why Was This Study Done?
Because tuberculosis treatment is long and unpleasant, patients often fail to take all their drugs. To improve medication adherence, the World Health Organization recommends that health care workers supervise patients while they take their medication (directly observed treatment, DOT). However, DOT can be hard to implement. In China, for example, where 11% of tuberculosis cases occur, DOT cannot be implemented in many parts of the country, and the national tuberculosis control policy permits self-administered treatment and treatment monitored by family members. It is estimated that over half of individuals with tuberculosis in China self-administer their treatment, but, in 2010, 20% of patients treated using nationally recommended case-management approaches were lost to follow-up or failed to take their medications regularly. In this pragmatic cluster-randomized trial, the researchers investigate whether reminders delivered by mobile phone or by medication monitor boxes (which provide audio reminders to patients and record when the box is opened) might improve tuberculosis medication adherence in China. A pragmatic trial asks whether an intervention works under real-life conditions; a cluster-randomized trial randomly assigns groups of people (here, people living in different counties/districts) to receive alternative interventions and compares outcomes in the differently treated “clusters.”
What Did the Researchers Do and Find?
The researchers assigned people newly diagnosed with tuberculosis in counties/districts to receive reminders about taking their antibiotics and about monthly follow-up visits via text messaging, a medication monitor, or both text messaging and a medication monitor (the intervention arms), or to receive standard nationally recommended care without electronic reminders (the control arm). All the trial participants (about 1,000 per arm) took their drugs out of a medication monitor box, but the box’s audio reminder function was switched off in the text messaging only and control arms. In the intervention arms, doctors were advised to switch participants with poor medication adherence (evaluated at follow-up visits) to either more intensive management or DOT, depending on the level of missed treatment doses. Compared to the control arm, the percentage of patient-months with at least 20% of the drug doses missed (called “poor adherence” and measured by pill counts and data from the medication monitor) was not significantly reduced in the text messaging arm, whereas poor adherence was significantly reduced by 42% and 51% in the medication monitor and the combined arms, respectively (a significant reduction is unlikely to have happened by chance). Notably, fewer patients were switched to intensive management or DOT than expected based on medication adherence evaluations.
What Do These Findings Mean?
These findings show that, in China, the use of an electronic medication monitor box to remind patients to take their anti-tuberculosis drugs improved medication adherence. Interestingly, text messaging alone, which has been shown to improve adherence to antiretroviral therapy among HIV-positive individuals, did not improve medication adherence among patients with tuberculosis, possibly because the messages were too frequent or too impersonal, although this intervention (but none of the others) did reduce patient loss to follow-up. Battery problems with the medication monitor may have resulted in over-estimation of poor adherence to treatment. Moreover, the researchers’ assumption that opening the medication monitor box is synonymous with taking the medication may have introduced some inaccuracies into these findings. Despite these limitations and the underuse of more intensive case management in patients with poor adherence, these findings suggest that using medication monitors to deliver electronic drug reminders to patients might improve medication adherence among patients with tuberculosis in China and in other settings.
Additional Information
This list of resources contains links that can be accessed when viewing the PDF on a device or via the online version of the article at This study is further discussed in a PLOS Medicine Perspective by John Metcalfe, Max R. O’Donnell, and David R. BangsbergThe World Health Organization provides information (in several languages) on tuberculosis and on its Directly Observed Treatment Short Course (DOTS) strategy; the Global Tuberculosis Report 2014 provides information about tuberculosis around the worldThe Stop TB Partnership is working towards tuberculosis elimination and provides personal stories about tuberculosis (in English and Spanish); the Tuberculosis Vaccine Initiative (a not-for-profit organization) also provides personal stories about tuberculosisThe US Centers for Disease Control and Prevention provides information about tuberculosis and about treatment for tuberculosis (in English and Spanish)The US National Institute of Allergy and Infectious Diseases also has detailed information on all aspects of tuberculosisMedlinePlus has links to further information about tuberculosis (in English and Spanish)More information about this trial is available
PMCID: PMC4570796  PMID: 26372470
15.  Evaluating Candidate Principal Surrogate Endpoints 
Biometrics  2008;64(4):1146-1154.
Frangakis and Rubin (2002, Biometrics 58, 21–29) proposed a new definition of a surrogate endpoint (a “principal” surrogate) based on causal effects. We introduce an estimand for evaluating a principal surrogate, the causal effect predictiveness (CEP) surface, which quantifies how well causal treatment effects on the biomarker predict causal treatment effects on the clinical endpoint. Although the CEP surface is not identifiable due to missing potential outcomes, it can be identified by incorporating a baseline covariate(s) that predicts the biomarker. Given case–cohort sampling of such a baseline predictor and the biomarker in a large blinded randomized clinical trial, we develop an estimated likelihood method for estimating the CEP surface. This estimation assesses the “surrogate value” of the biomarker for reliably predicting clinical treatment effects for the same or similar setting as the trial. A CEP surface plot provides a way to compare the surrogate value of multiple biomarkers. The approach is illustrated by the problem of assessing an immune response to a vaccine as a surrogate endpoint for infection.
PMCID: PMC2726718  PMID: 18363776
Case cohort; Causal inference; Clinical trial; HIV vaccine; Postrandomization selection bias; Structural model; Prentice criteria; Principal stratification
16.  Partially hidden Markov model for time-varying principal stratification in HIV prevention trials 
It is frequently of interest to estimate the intervention effect that adjusts for post-randomization variables in clinical trials. In the recently completed HPTN 035 trial, there is differential condom use between the three microbicide gel arms and the No Gel control arm, so that intention to treat (ITT) analyses only assess the net treatment effect that includes the indirect treatment effect mediated through differential condom use. Various statistical methods in causal inference have been developed to adjust for post-randomization variables. We extend the principal stratification framework to time-varying behavioral variables in HIV prevention trials with a time-to-event endpoint, using a partially hidden Markov model (pHMM). We formulate the causal estimand of interest, establish assumptions that enable identifiability of the causal parameters, and develop maximum likelihood methods for estimation. Application of our model on the HPTN 035 trial reveals an interesting pattern of prevention effectiveness among different condom-use principal strata.
PMCID: PMC3649016  PMID: 23667279
microbicide; causal inference; posttreatment variables; direct effect
17.  Semiparametric estimation of the average causal effect of treatment on an outcome measured after a postrandomization event, with missing outcome data 
In the past decade, several principal stratification–based statistical methods have been developed for testing and estimation of a treatment effect on an outcome measured after a postrandomization event. Two examples are the evaluation of the effect of a cancer treatment on quality of life in subjects who remain alive and the evaluation of the effect of an HIV vaccine on viral load in subjects who acquire HIV infection. However, in general the developed methods have not addressed the issue of missing outcome data, and hence their validity relies on a missing completely at random (MCAR) assumption. Because in many applications the MCAR assumption is untenable, while a missing at random (MAR) assumption is defensible, we extend the semiparametric likelihood sensitivity analysis approach of Gilbert and others (2003) and Jemiai and Rotnitzky (2005) to allow the outcome to be MAR. We combine these methods with the robust likelihood–based method of Little and An (2004) for handling MAR data to provide semiparametric estimation of the average causal effect of treatment on the outcome. The new method, which does not require a monotonicity assumption, is evaluated in a simulation study and is applied to data from the first HIV vaccine efficacy trial.
PMCID: PMC2800161  PMID: 19815692
Causal inference; HIV vaccine trial; Missing at random; Posttreatment selection bias; Principal stratification; Sensitivity analysis
18.  Causal inference in longitudinal studies with history-restricted marginal structural models 
A new class of Marginal Structural Models (MSMs), History-Restricted MSMs (HRMSMs), was recently introduced for longitudinal data for the purpose of defining causal parameters which may often be better suited for public health research or at least more practicable than MSMs (6, 2). HRMSMs allow investigators to analyze the causal effect of a treatment on an outcome based on a fixed, shorter and user-specified history of exposure compared to MSMs. By default, the latter represent the treatment causal effect of interest based on a treatment history defined by the treatments assigned between the study’s start and outcome collection. We lay out in this article the formal statistical framework behind HRMSMs. Beyond allowing a more flexible causal analysis, HRMSMs improve computational tractability and mitigate statistical power concerns when designing longitudinal studies. We also develop three consistent estimators of HRMSM parameters under sufficient model assumptions: the Inverse Probability of Treatment Weighted (IPTW), G-computation and Double Robust (DR) estimators. In addition, we show that the assumptions commonly adopted for identification and consistent estimation of MSM parameters (existence of counterfactuals, consistency, time-ordering and sequential randomization assumptions) also lead to identification and consistent estimation of HRMSM parameters.
PMCID: PMC3475192  PMID: 23087778
causal inference; counterfactual; marginal structural model; longitudinal study; IPTW; G-computation; Double Robust
19.  The Project Data Sphere Initiative: Accelerating Cancer Research by Sharing Data 
The Oncologist  2015;20(5):464-e20.
By providing access to large, late-phase, cancer-trial data sets, the Project Data Sphere initiative has the potential to transform cancer research by optimizing research efficiency and accelerating progress toward meaningful improvements in cancer care. This type of platform provides opportunities for unique research projects that can examine relatively neglected areas and that can construct models necessitating large amounts of detailed data.
In this paper, we provide background and context regarding the potential for a new data-sharing platform, the Project Data Sphere (PDS) initiative, funded by financial and in-kind contributions from the CEO Roundtable on Cancer, to transform cancer research and improve patient outcomes. Given the relatively modest decline in cancer death rates over the past several years, a new research paradigm is needed to accelerate therapeutic approaches for oncologic diseases. Phase III clinical trials generate large volumes of potentially usable information, often on hundreds of patients, including patients treated with standard of care therapies (i.e., controls). Both nationally and internationally, a variety of stakeholders have pursued data-sharing efforts to make individual patient-level clinical trial data available to the scientific research community.
Potential Benefits and Risks of Data Sharing.
For researchers, shared data have the potential to foster a more collaborative environment, to answer research questions in a shorter time frame than traditional randomized control trials, to reduce duplication of effort, and to improve efficiency. For industry participants, use of trial data to answer additional clinical questions could increase research and development efficiency and guide future projects through validation of surrogate end points, development of prognostic or predictive models, selection of patients for phase II trials, stratification in phase III studies, and identification of patient subgroups for development of novel therapies. Data transparency also helps promote a public image of collaboration and altruism among industry participants. For patient participants, data sharing maximizes their contribution to public health and increases access to information that may be used to develop better treatments. Concerns about data-sharing efforts include protection of patient privacy and confidentiality. To alleviate these concerns, data sets are deidentified to maintain anonymity. To address industry concerns about protection of intellectual property and competitiveness, we illustrate several models for data sharing with varying levels of access to the data and varying relationships between trial sponsors and data access sponsors.
The Project Data Sphere Initiative.
PDS is an independent initiative of the CEO Roundtable on Cancer Life Sciences Consortium, built to voluntarily share, integrate, and analyze comparator arms of historical cancer clinical trial data sets to advance future cancer research. The aim is to provide a neutral, broad-access platform for industry and academia to share raw, deidentified data from late-phase oncology clinical trials using comparator-arm data sets. These data are likely to be hypothesis generating or hypothesis confirming but, notably, do not take the place of performing a well-designed trial to address a specific hypothesis. Prospective providers of data to PDS complete and sign a data sharing agreement that includes a description of the data they propose to upload, and then they follow easy instructions on the website for uploading their deidentified data. The SAS Institute has also collaborated with the initiative to provide intrinsic analytic tools accessible within the website itself.
As of October 2014, the PDS website has available data from 14 cancer clinical trials covering 9,000 subjects, with hopes to further expand the database to include more than 25,000 subject accruals within the next year. PDS differentiates itself from other data-sharing initiatives by its degree of openness, requiring submission of only a brief application with background information of the individual requesting access and agreement to terms of use. Data from several different sponsors may be pooled to develop a comprehensive cohort for analysis. In order to protect patient privacy, data providers in the U.S. are responsible for deidentifying data according to standards set forth by the Privacy Rule of the U.S. Health Insurance Portability and Accountability Act of 1996.
Using Data Sharing to Improve Outcomes in Cancer: The “Prostate Cancer Challenge.”
Control-arm data of several studies among patients with metastatic castration-resistant prostate cancer (mCRPC) are currently available through PDS. These data sets have multiple potential uses. The “Prostate Cancer Challenge” will ask the cancer research community to use clinical trial data deposited in the PDS website to address key research questions regarding mCRPC.
General themes that could be explored by the cancer community are described in this article: prognostic models evaluating the influence of pretreatment factors on survival and patient-reported outcomes; comparative effectiveness research evaluating the efficacy of standard of care therapies, as illustrated in our companion article comparing mitoxantrone plus prednisone with prednisone alone; effects of practice variation in dose, frequency, and duration of therapy; level of patient adherence to elements of trial protocols to inform the design of future clinical trials; and age of subjects, regional differences in health care, and other confounding factors that might affect outcomes.
Potential Limitations and Methodological Challenges.
The number of data sets available and the lack of experimental-arm data limit the potential scope of research using the current PDS. The number of trials is expected to grow exponentially over the next year and may include multiple cancer settings, such as breast, colorectal, lung, hematologic malignancy, and bone marrow transplantation. Other potential limitations include the retrospective nature of the data analyses performed using PDS and its generalizability, given that clinical trials are often conducted among younger, healthier, and less racially diverse patient populations. Methodological challenges exist when combining individual patient data from multiple clinical trials; however, advancements in statistical methods for secondary database analysis offer many tools for reanalyzing data arising from disparate trials, such as propensity score matching. Despite these concerns, few if any comparable data sets include this level of detail across multiple clinical trials and populations.
Access to large, late-phase, cancer-trial data sets has the potential to transform cancer research by optimizing research efficiency and accelerating progress toward meaningful improvements in cancer care. This type of platform provides opportunities for unique research projects that can examine relatively neglected areas and that can construct models necessitating large amounts of detailed data. The full potential of PDS will be realized only when multiple tumor types and larger numbers of data sets are available through the website.
PMCID: PMC4425388  PMID: 25876994
Project Data Sphere; Data sharing; Prostate cancer; Comparative effectiveness research
20.  The Role of Adiposity in Cardiometabolic Traits: A Mendelian Randomization Analysis 
Fall, Tove | Hägg, Sara | Mägi, Reedik | Ploner, Alexander | Fischer, Krista | Horikoshi, Momoko | Sarin, Antti-Pekka | Thorleifsson, Gudmar | Ladenvall, Claes | Kals, Mart | Kuningas, Maris | Draisma, Harmen H. M. | Ried, Janina S. | van Zuydam, Natalie R. | Huikari, Ville | Mangino, Massimo | Sonestedt, Emily | Benyamin, Beben | Nelson, Christopher P. | Rivera, Natalia V. | Kristiansson, Kati | Shen, Huei-yi | Havulinna, Aki S. | Dehghan, Abbas | Donnelly, Louise A. | Kaakinen, Marika | Nuotio, Marja-Liisa | Robertson, Neil | de Bruijn, Renée F. A. G. | Ikram, M. Arfan | Amin, Najaf | Balmforth, Anthony J. | Braund, Peter S. | Doney, Alexander S. F. | Döring, Angela | Elliott, Paul | Esko, Tõnu | Franco, Oscar H. | Gretarsdottir, Solveig | Hartikainen, Anna-Liisa | Heikkilä, Kauko | Herzig, Karl-Heinz | Holm, Hilma | Hottenga, Jouke Jan | Hyppönen, Elina | Illig, Thomas | Isaacs, Aaron | Isomaa, Bo | Karssen, Lennart C. | Kettunen, Johannes | Koenig, Wolfgang | Kuulasmaa, Kari | Laatikainen, Tiina | Laitinen, Jaana | Lindgren, Cecilia | Lyssenko, Valeriya | Läärä, Esa | Rayner, Nigel W. | Männistö, Satu | Pouta, Anneli | Rathmann, Wolfgang | Rivadeneira, Fernando | Ruokonen, Aimo | Savolainen, Markku J. | Sijbrands, Eric J. G. | Small, Kerrin S. | Smit, Jan H. | Steinthorsdottir, Valgerdur | Syvänen, Ann-Christine | Taanila, Anja | Tobin, Martin D. | Uitterlinden, Andre G. | Willems, Sara M. | Willemsen, Gonneke | Witteman, Jacqueline | Perola, Markus | Evans, Alun | Ferrières, Jean | Virtamo, Jarmo | Kee, Frank | Tregouet, David-Alexandre | Arveiler, Dominique | Amouyel, Philippe | Ferrario, Marco M. | Brambilla, Paolo | Hall, Alistair S. | Heath, Andrew C. | Madden, Pamela A. F. | Martin, Nicholas G. | Montgomery, Grant W. | Whitfield, John B. | Jula, Antti | Knekt, Paul | Oostra, Ben | van Duijn, Cornelia M. | Penninx, Brenda W. J. H. | Davey Smith, George | Kaprio, Jaakko | Samani, Nilesh J. | Gieger, Christian | Peters, Annette | Wichmann, H.-Erich | Boomsma, Dorret I. | de Geus, Eco J. C. | Tuomi, TiinaMaija | Power, Chris | Hammond, Christopher J. | Spector, Tim D. | Lind, Lars | Orho-Melander, Marju | Palmer, Colin Neil Alexander | Morris, Andrew D. | Groop, Leif | Järvelin, Marjo-Riitta | Salomaa, Veikko | Vartiainen, Erkki | Hofman, Albert | Ripatti, Samuli | Metspalu, Andres | Thorsteinsdottir, Unnur | Stefansson, Kari | Pedersen, Nancy L. | McCarthy, Mark I. | Ingelsson, Erik | Prokopenko, Inga
PLoS Medicine  2013;10(6):e1001474.
In this study, Prokopenko and colleagues provide novel evidence for causal relationship between adiposity and heart failure and increased liver enzymes using a Mendelian randomization study design.
Please see later in the article for the Editors' Summary
The association between adiposity and cardiometabolic traits is well known from epidemiological studies. Whilst the causal relationship is clear for some of these traits, for others it is not. We aimed to determine whether adiposity is causally related to various cardiometabolic traits using the Mendelian randomization approach.
Methods and Findings
We used the adiposity-associated variant rs9939609 at the FTO locus as an instrumental variable (IV) for body mass index (BMI) in a Mendelian randomization design. Thirty-six population-based studies of individuals of European descent contributed to the analyses.
Age- and sex-adjusted regression models were fitted to test for association between (i) rs9939609 and BMI (n = 198,502), (ii) rs9939609 and 24 traits, and (iii) BMI and 24 traits. The causal effect of BMI on the outcome measures was quantified by IV estimators. The estimators were compared to the BMI–trait associations derived from the same individuals. In the IV analysis, we demonstrated novel evidence for a causal relationship between adiposity and incident heart failure (hazard ratio, 1.19 per BMI-unit increase; 95% CI, 1.03–1.39) and replicated earlier reports of a causal association with type 2 diabetes, metabolic syndrome, dyslipidemia, and hypertension (odds ratio for IV estimator, 1.1–1.4; all p<0.05). For quantitative traits, our results provide novel evidence for a causal effect of adiposity on the liver enzymes alanine aminotransferase and gamma-glutamyl transferase and confirm previous reports of a causal effect of adiposity on systolic and diastolic blood pressure, fasting insulin, 2-h post-load glucose from the oral glucose tolerance test, C-reactive protein, triglycerides, and high-density lipoprotein cholesterol levels (all p<0.05). The estimated causal effects were in agreement with traditional observational measures in all instances except for type 2 diabetes, where the causal estimate was larger than the observational estimate (p = 0.001).
We provide novel evidence for a causal relationship between adiposity and heart failure as well as between adiposity and increased liver enzymes.
Please see later in the article for the Editors' Summary
Editors' Summary
Cardiovascular disease (CVD)—disease that affects the heart and/or the blood vessels—is a major cause of illness and death worldwide. In the US, for example, coronary heart disease—a CVD in which narrowing of the heart's blood vessels by fatty deposits slows the blood supply to the heart and may eventually cause a heart attack—is the leading cause of death, and stroke—a CVD in which the brain's blood supply is interrupted—is the fourth leading cause of death. Globally, both the incidence of CVD (the number of new cases in a population every year) and its prevalence (the proportion of the population with CVD) are increasing, particularly in low- and middle-income countries. This increasing burden of CVD is occurring in parallel with a global increase in the incidence and prevalence of obesity—having an unhealthy amount of body fat (adiposity)—and of metabolic diseases—conditions such as diabetes in which metabolism (the processes that the body uses to make energy from food) is disrupted, with resulting high blood sugar and damage to the blood vessels.
Why Was This Study Done?
Epidemiological studies—investigations that record the patterns and causes of disease in populations—have reported an association between adiposity (indicated by an increased body mass index [BMI], which is calculated by dividing body weight in kilograms by height in meters squared) and cardiometabolic traits such as coronary heart disease, stroke, heart failure (a condition in which the heart is incapable of pumping sufficient amounts of blood around the body), diabetes, high blood pressure (hypertension), and high blood cholesterol (dyslipidemia). However, observational studies cannot prove that adiposity causes any particular cardiometabolic trait because overweight individuals may share other characteristics (confounding factors) that are the real causes of both obesity and the cardiometabolic disease. Moreover, it is possible that having CVD or a metabolic disease causes obesity (reverse causation). For example, individuals with heart failure cannot do much exercise, so heart failure may cause obesity rather than vice versa. Here, the researchers use “Mendelian randomization” to examine whether adiposity is causally related to various cardiometabolic traits. Because gene variants are inherited randomly, they are not prone to confounding and are free from reverse causation. It is known that a genetic variant (rs9939609) within the genome region that encodes the fat-mass- and obesity-associated gene (FTO) is associated with increased BMI. Thus, an investigation of the associations between rs9939609 and cardiometabolic traits can indicate whether obesity is causally related to these traits.
What Did the Researchers Do and Find?
The researchers analyzed the association between rs9939609 (the “instrumental variable,” or IV) and BMI, between rs9939609 and 24 cardiometabolic traits, and between BMI and the same traits using genetic and health data collected in 36 population-based studies of nearly 200,000 individuals of European descent. They then quantified the strength of the causal association between BMI and the cardiometabolic traits by calculating “IV estimators.” Higher BMI showed a causal relationship with heart failure, metabolic syndrome (a combination of medical disorders that increases the risk of developing CVD), type 2 diabetes, dyslipidemia, hypertension, increased blood levels of liver enzymes (an indicator of liver damage; some metabolic disorders involve liver damage), and several other cardiometabolic traits. All the IV estimators were similar to the BMI–cardiovascular trait associations (observational estimates) derived from the same individuals, with the exception of diabetes, where the causal estimate was higher than the observational estimate, probably because the observational estimate is based on a single BMI measurement, whereas the causal estimate considers lifetime changes in BMI.
What Do These Findings Mean?
Like all Mendelian randomization studies, the reliability of the causal associations reported here depends on several assumptions made by the researchers. Nevertheless, these findings provide support for many previously suspected and biologically plausible causal relationships, such as that between adiposity and hypertension. They also provide new insights into the causal effect of obesity on liver enzyme levels and on heart failure. In the latter case, these findings suggest that a one-unit increase in BMI might increase the incidence of heart failure by 17%. In the US, this corresponds to 113,000 additional cases of heart failure for every unit increase in BMI at the population level. Although additional studies are needed to confirm and extend these findings, these results suggest that global efforts to reduce the burden of obesity will likely also reduce the occurrence of CVD and metabolic disorders.
Additional Information
Please access these websites via the online version of this summary at
The American Heart Association provides information on all aspects of cardiovascular disease and tips on keeping the heart healthy, including weight management (in several languages); its website includes personal stories about stroke and heart attacks
The US Centers for Disease Control and Prevention has information on heart disease, stroke, and all aspects of overweight and obesity (in English and Spanish)
The UK National Health Service Choices website provides information about cardiovascular disease and obesity, including a personal story about losing weight
The World Health Organization provides information on obesity (in several languages)
The International Obesity Taskforce provides information about the global obesity epidemic
Wikipedia has a page on Mendelian randomization (note: Wikipedia is a free online encyclopedia that anyone can edit; available in several languages)
MedlinePlus provides links to other sources of information on heart disease, on vascular disease, on obesity, and on metabolic disorders (in English and Spanish)
The International Association for the Study of Obesity provides maps and information about obesity worldwide
The International Diabetes Federation has a web page that describes types, complications, and risk factors of diabetes
PMCID: PMC3692470  PMID: 23824655
21.  Causal Effect Models for Realistic Individualized Treatment and Intention to Treat Rules* 
Marginal structural models (MSM) are an important class of models in causal inference. Given a longitudinal data structure observed on a sample of n independent and identically distributed experimental units, MSM model the counterfactual outcome distribution corresponding with a static treatment intervention, conditional on user-supplied baseline covariates. Identification of a static treatment regimen-specific outcome distribution based on observational data requires, beyond the standard sequential randomization assumption, the assumption that each experimental unit has positive probability of following the static treatment regimen. The latter assumption is called the experimental treatment assignment (ETA) assumption, and is parameter-specific. In many studies the ETA is violated because some of the static treatment interventions to be compared cannot be followed by all experimental units, due either to baseline characteristics or to the occurrence of certain events over time. For example, the development of adverse effects or contraindications can force a subject to stop an assigned treatment regimen.
In this article we propose causal effect models for a user-supplied set of realistic individualized treatment rules. Realistic individualized treatment rules are defined as treatment rules which always map into the set of possible treatment options. Thus, causal effect models for realistic treatment rules do not rely on the ETA assumption and are fully identifiable from the data. Further, these models can be chosen to generalize marginal structural models for static treatment interventions. The estimating function methodology of Robins and Rotnitzky (1992) (analogue to its application in Murphy, et. al. (2001) for a single treatment rule) provides us with the corresponding locally efficient double robust inverse probability of treatment weighted estimator.
In addition, we define causal effect models for “intention-to-treat” regimens. The proposed intention-to-treat interventions enforce a static intervention until the time point at which the next treatment does not belong to the set of possible treatment options, at which point the intervention is stopped. We provide locally efficient estimators of such intention-to-treat causal effects.
PMCID: PMC2613338  PMID: 19122793
counterfactual; causal effect; causal inference; double robust estimating function; dynamic treatment regimen; estimating function; individualized stopped treatment regimen; individualized treatment rule; inverse probability of treatment weighted estimating functions; locally efficient estimation; static treatment intervention
22.  Mediation Analysis With Intermediate Confounding: Structural Equation Modeling Viewed Through the Causal Inference Lens 
American Journal of Epidemiology  2014;181(1):64-80.
The study of mediation has a long tradition in the social sciences and a relatively more recent one in epidemiology. The first school is linked to path analysis and structural equation models (SEMs), while the second is related mostly to methods developed within the potential outcomes approach to causal inference. By giving model-free definitions of direct and indirect effects and clear assumptions for their identification, the latter school has formalized notions intuitively developed in the former and has greatly increased the flexibility of the models involved. However, through its predominant focus on nonparametric identification, the causal inference approach to effect decomposition via natural effects is limited to settings that exclude intermediate confounders. Such confounders are naturally dealt with (albeit with the caveats of informality and modeling inflexibility) in the SEM framework. Therefore, it seems pertinent to revisit SEMs with intermediate confounders, armed with the formal definitions and (parametric) identification assumptions from causal inference. Here we investigate: 1) how identification assumptions affect the specification of SEMs, 2) whether the more restrictive SEM assumptions can be relaxed, and 3) whether existing sensitivity analyses can be extended to this setting. Data from the Avon Longitudinal Study of Parents and Children (1990–2005) are used for illustration.
PMCID: PMC4383385  PMID: 25504026
eating disorders; estimation by combination; G-computation; parametric identification; path analysis; sensitivity analysis
23.  Joint modeling of longitudinal ordinal data and competing risks survival times and analysis of the NINDS rt-PA stroke trial 
Statistics in medicine  2010;29(5):546-557.
Existing joint models for longitudinal and survival data are not applicable for longitudinal ordinal outcomes with possible non-ignorable missing values caused by multiple reasons. We propose a joint model for longitudinal ordinal measurements and competing risks failure time data, in which a partial proportional odds model for the longitudinal ordinal outcome is linked to the event times by latent random variables. At the survival endpoint, our model adopts the competing risks framework to model multiple failure types at the same time. The partial proportional odds model, as an extension of the popular proportional odds model for ordinal outcomes, is more flexible and at the same time provides a tool to test the proportional odds assumption. We use a likelihood approach and derive an EM algorithm to obtain the maximum likelihood estimates of the parameters. We further show that all the parameters at the survival endpoint are identifiable from the data. Our joint model enables one to make inference for both the longitudinal ordinal outcome and the failure times simultaneously. In addition, the inference at the longitudinal endpoint is adjusted for possible non-ignorable missing data caused by the failure times. We apply the method to the NINDS rt-PA stroke trial. Our study considers the modified Rankin Scale only. Other ordinal outcomes in the trial, such as the Barthel and Glasgow scales can be treated in the same way.
PMCID: PMC2822130  PMID: 19943331
24.  A Bayesian Approach to Surrogacy Assessment Using Principal Stratification in Clinical Trials 
Biometrics  2009;66(2):523-531.
A surrogate marker (S) is a variable that can be measured earlier and often easier than the true endpoint (T) in a clinical trial. Most previous research has been devoted to developing surrogacy measures to quantify how well S can replace T or examining the use of S in predicting the effect of a treatment (Z). However, the research often requires one to fit models for the distribution of T given S and Z. It is well known that such models do not have causal interpretations because the models condition on a post-randomization variable S. In this paper, we directly model the relationship among T, S and Z using a potential outcomes framework introduced by Frangakis and Rubin (2002). We propose a Bayesian estimation method to evaluate the causal probabilities associated with the cross-classification of the potential outcomes of S and T when S and T are both binary. We use a log-linear model to directly model the association between the potential outcomes of S and T through the odds ratios. The quantities derived from this approach always have causal interpretations. However, this causal model is not identifiable from the data without additional assumptions. To reduce the non-identifiability problem and increase the precision of statistical inferences, we assume monotonicity and incorporate prior belief that is plausible in the surrogate context by using prior distributions. We also explore the relationship among the surrogacy measures based on traditional models and this counterfactual model. The method is applied to the data from a glaucoma treatment study.
PMCID: PMC3365598  PMID: 19673864
Bayesian Estimation; Counterfactual Model; Randomized Trial; Surrogate Marker
25.  Principal Stratification in Causal Inference 
Biometrics  2002;58(1):21-29.
Many scientific problems require that treatment comparisons be adjusted for posttreatment variables, but the estimands underlying standard methods are not causal effects. To address this deficiency, we propose a general framework for comparing treatments adjusting for posttreatment variables that yields principal effects based on principal stratification. Principal stratification with respect to a posttreatment variable is a cross-classification of subjects defined by the joint potential values of that posttreatment variable under each of the treatments being compared. Principal effects are causal effects within a principal stratum. The key property of principal strata is that they are not affected by treatment assignment and therefore can be used just as any pretreatment covariate, such as age category. As a result, the central property of our principal effects is that they are always causal effects and do not suffer from the complications of standard posttreatment-adjusted estimands. We discuss briefly that such principal causal effects are the link between three recent applications with adjustment for posttreatment variables: (i) treatment noncompliance, (ii) missing outcomes (dropout) following treatment noncompliance, and (iii) censoring by death. We then attack the problem of surrogate or biomarker endpoints, where we show, using principal causal effects, that all current definitions of surrogacy, even when perfectly true, do not generally have the desired interpretation as causal effects of treatment on outcome. We go on to formulate estimands based on principal stratification and principal causal effects and show their superiority.
PMCID: PMC4137767  PMID: 11890317
Biomarker; Causal inference; Censoring by death; Missing data; Posttreatment variable; Principal stratification; Quality of life; Rubin causal model; Surrogate

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