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1.  Accommodating Missingness When Assessing Surrogacy Via Principal Stratification 
Clinical trials (London, England)  2013;10(3):363-377.
Background
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.
Purpose
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.
Methods
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.
Results
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.
Limitations
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.
Conclusions
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.
doi:10.1177/1740774513479522
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.
Background
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.
Methods
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.
Results
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.
Conclusions
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.
doi:10.1186/1745-6215-14-78
PMCID: PMC3610291  PMID: 23510143
Intermediate outcomes; Intention-to-treat approach; Principal stratification; Causal effects
3.  Estimating Causal Effects in Trials Involving Multi-Treatment Arms Subject to Non-compliance: A Bayesian framework 
Summary
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).
doi:10.1111/j.1467-9876.2009.00709.x
PMCID: PMC3104736  PMID: 21637737
Causal Inference; Complier Average Causal Effect; Multi-arm Trials; Non-compliance; Principal Compliance; Principal Stratification
4.  Marginalized models for longitudinal ordinal data with application to quality of life studies 
Statistics in medicine  2008;27(21):4359-4380.
SUMMARY
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.
doi:10.1002/sim.3352
PMCID: PMC2858760  PMID: 18613246
marginalized likelihood-based models; ordinal data models; dropout
5.  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
6.  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.
Summary
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.
doi:10.1111/j.1541-0420.2011.01736.x
PMCID: PMC3860118  PMID: 22348277
Biomarker; Causal effect predictiveness; principal stratification; surrogate endpoint
7.  Statistical identifiability and the surrogate endpoint problem, with application to vaccine trials 
Biometrics  2010;66(4):1153-1161.
Summary
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.
doi:10.1111/j.1541-0420.2009.01380.x
PMCID: PMC3597127  PMID: 20105158
Estimated likelihood; Identifiability; Principal stratification; Sensitivity analysis; Surrogate endpoint; Vaccine trials
8.  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.
doi:10.1198/016214505000000970
PMCID: PMC2603579  PMID: 19096723
Causal inference; Infectious disease; Maximum likelihood; Principal stratification; Sensitivity analysis
9.  AN APPLICATION OF PRINCIPAL STRATIFICATION TO CONTROL FOR INSTITUTIONALIZATION AT FOLLOW-UP IN STUDIES OF SUBSTANCE ABUSE TREATMENT PROGRAMS* 
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.
doi:10.1214/08-AOAS179
PMCID: PMC2749670  PMID: 19779599
Principal Stratification; Post-Treatment Confounder; Institutionalization; Causal Inference
10.  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.
SUMMARY
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.
doi:10.1002/sim.5857
PMCID: PMC3935993  PMID: 23720372
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.
doi:10.3102/1076998610383985
PMCID: PMC3150587  PMID: 21822369
alternative assumptions; bounds; causal inference; missing data; noncompliance; principal stratification; sensitivity analysis
12.  Evaluating Candidate Principal Surrogate Endpoints 
Biometrics  2008;64(4):1146-1154.
Summary
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.
doi:10.1111/j.1541-0420.2008.01014.x
PMCID: PMC2726718  PMID: 18363776
Case cohort; Causal inference; Clinical trial; HIV vaccine; Postrandomization selection bias; Structural model; Prentice criteria; Principal stratification
13.  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.
doi:10.1080/01621459.2011.643743
PMCID: PMC3649016  PMID: 23667279
microbicide; causal inference; posttreatment variables; direct effect
14.  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.
doi:10.1093/biostatistics/kxp034
PMCID: PMC2800161  PMID: 19815692
Causal inference; HIV vaccine trial; Missing at random; Posttreatment selection bias; Principal stratification; Sensitivity analysis
15.  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.
doi:10.1214/07-EJS050
PMCID: PMC3475192  PMID: 23087778
causal inference; counterfactual; marginal structural model; longitudinal study; IPTW; G-computation; Double Robust
16.  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
17.  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
Background
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).
Conclusions
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 http://dx.doi.org/10.1371/journal.pmed.1001474.
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
doi:10.1371/journal.pmed.1001474
PMCID: PMC3692470  PMID: 23824655
18.  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.
SUMMARY
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.
doi:10.1002/sim.3798
PMCID: PMC2822130  PMID: 19943331
19.  A Bayesian Approach to Surrogacy Assessment Using Principal Stratification in Clinical Trials 
Biometrics  2009;66(2):523-531.
Summary
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.
doi:10.1111/j.1541-0420.2009.01303.x
PMCID: PMC3365598  PMID: 19673864
Bayesian Estimation; Counterfactual Model; Randomized Trial; Surrogate Marker
20.  Causal Inference in Longitudinal Comparative Effectiveness Studies With Repeated Measures of A Continuous Intermediate Variable 
Statistics in medicine  2014;33(20):3509-3527.
We propose a principal stratification approach to assess causal effects in non-randomized longitudinal comparative effectiveness studies with a binary endpoint outcome and repeated measures of a continuous intermediate variable. Our method is an extension of the principal stratification approach by Lin et al. [10,11], originally proposed for a longitudinal randomized study to assess the treatment effect of a continuous outcome adjusting for the heterogeneity of a repeatedly measured binary intermediate variable. Our motivation for this work comes from a comparison of the effect of two glucose-lowering medications on a clinical cohort of patients with type 2 diabetes. Here we consider a causal inference problem assessing how well the two medications work relative to one another on two binary endpoint outcomes: cardiovascular disease related hospitalization and all-cause mortality. Clinically, these glucose-lowering medications can have differential effects on the intermediate outcome, glucose level over time. Ultimately we want to compare medication effects on the endpoint outcomes among individuals in the same glucose trajectory stratum while accounting for the heterogeneity in baseline covariates (i.e., to obtain “principal effects” on the endpoint outcomes). The proposed method involves a 3-step model estimation procedure. Step 1 identifies principal strata associated with the intermediate variable using hybrid growth mixture modeling analyses [13]. Step 2 obtains the stratum membership using the pseudoclass technique [17,18], and derives propensity scores for treatment assignment. Step 3 obtains the stratum-specific treatment effect on the endpoint outcome weighted by inverse propensity probabilities derived from Step 2.
doi:10.1002/sim.6120
PMCID: PMC4122661  PMID: 24577715
Causal inference; Comparative effectiveness studies; Growth mixture model; Principal stratification; Propensity score
21.  Principal Stratification in Causal Inference 
Biometrics  2002;58(1):21-29.
Summary
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
22.  Metabolic Signatures of Adiposity in Young Adults: Mendelian Randomization Analysis and Effects of Weight Change 
PLoS Medicine  2014;11(12):e1001765.
In this study, Wurtz and colleagues investigated to what extent elevated body mass index (BMI) within the normal weight range has causal influences on the detailed systemic metabolite profile in early adulthood using Mendelian randomization analysis.
Please see later in the article for the Editors' Summary
Background
Increased adiposity is linked with higher risk for cardiometabolic diseases. We aimed to determine to what extent elevated body mass index (BMI) within the normal weight range has causal effects on the detailed systemic metabolite profile in early adulthood.
Methods and Findings
We used Mendelian randomization to estimate causal effects of BMI on 82 metabolic measures in 12,664 adolescents and young adults from four population-based cohorts in Finland (mean age 26 y, range 16–39 y; 51% women; mean ± standard deviation BMI 24±4 kg/m2). Circulating metabolites were quantified by high-throughput nuclear magnetic resonance metabolomics and biochemical assays. In cross-sectional analyses, elevated BMI was adversely associated with cardiometabolic risk markers throughout the systemic metabolite profile, including lipoprotein subclasses, fatty acid composition, amino acids, inflammatory markers, and various hormones (p<0.0005 for 68 measures). Metabolite associations with BMI were generally stronger for men than for women (median 136%, interquartile range 125%–183%). A gene score for predisposition to elevated BMI, composed of 32 established genetic correlates, was used as the instrument to assess causality. Causal effects of elevated BMI closely matched observational estimates (correspondence 87%±3%; R2 = 0.89), suggesting causative influences of adiposity on the levels of numerous metabolites (p<0.0005 for 24 measures), including lipoprotein lipid subclasses and particle size, branched-chain and aromatic amino acids, and inflammation-related glycoprotein acetyls. Causal analyses of certain metabolites and potential sex differences warrant stronger statistical power. Metabolite changes associated with change in BMI during 6 y of follow-up were examined for 1,488 individuals. Change in BMI was accompanied by widespread metabolite changes, which had an association pattern similar to that of the cross-sectional observations, yet with greater metabolic effects (correspondence 160%±2%; R2 = 0.92).
Conclusions
Mendelian randomization indicates causal adverse effects of increased adiposity with multiple cardiometabolic risk markers across the metabolite profile in adolescents and young adults within the non-obese weight range. Consistent with the causal influences of adiposity, weight changes were paralleled by extensive metabolic changes, suggesting a broadly modifiable systemic metabolite profile in early adulthood.
Please see later in the article for the Editors' Summary
Editors' Summary
Background
Adiposity—having excessive body fat—is a growing global threat to public health. Body mass index (BMI, calculated by dividing a person's weight in kilograms by their height in meters squared) is a coarse indicator of excess body weight, but the measure is useful in large population studies. Compared to people with a lean body weight (a BMI of 18.5–24.9 kg/m2), individuals with higher BMI have an elevated risk of developing life-shortening cardiometabolic diseases—cardiovascular diseases that affect the heart and/or the blood vessels (for example, heart failure and stroke) and metabolic diseases that affect the cellular chemical reactions that sustain life (for example, diabetes). People become unhealthily fat by consuming food and drink that contains more energy (calories) than they need for their daily activities. So adiposity can be prevented and reversed by eating less and exercising more.
Why Was This Study Done?
Epidemiological studies, which record the patterns of risk factors and disease in populations, suggest that the illness and death associated with excess body weight is partly attributable to abnormalities in how individuals with high adiposity metabolize carbohydrates and fats, leading to higher blood sugar and cholesterol levels. Further, adiposity is also associated with many other deviations in the metabolic profile than these commonly measured risk factors. However, epidemiological studies cannot prove that adiposity causes specific changes in a person's systemic (overall) metabolic profile because individuals with high BMI may share other characteristics (confounding factors) that are the actual causes of both adiposity and metabolic abnormalities. Moreover, having a change in some aspect of metabolism could also lead to adiposity, rather than vice versa (reverse causation). Importantly, if there is a causal effect of adiposity on cardiometabolic risk factor levels, it might be possible to prevent the progression towards cardiometabolic diseases by weight loss. Here, the researchers use “Mendelian randomization” to examine whether increased BMI within the normal and overweight range is causally influencing the metabolic risk factors from many biological pathways during early adulthood. Because gene variants are inherited randomly, they are not prone to confounding and are free from reverse causation. Several gene variants are known to lead to modestly increased BMI. Thus, an investigation of the associations between these gene variants and risk factors across the systemic metabolite profile in a population of healthy individuals can indicate whether higher BMI is causally related to known and novel metabolic risk factors and higher cardiometabolic disease risk.
What Did the Researchers Do and Find?
The researchers measured the BMI of 12,664 adolescents and young adults (average BMI 24.7 kg/m2) living in Finland and the blood levels of 82 metabolites in these young individuals at a single time point. Statistical analysis of these data indicated that elevated BMI was adversely associated with numerous cardiometabolic risk factors. For example, elevated BMI was associated with raised levels of low-density lipoprotein, “bad” cholesterol that increases cardiovascular disease risk. Next, the researchers used a gene score for predisposition to increased BMI, composed of 32 gene variants correlated with increased BMI, as an “instrumental variable” to assess whether adiposity causes metabolite abnormalities. The effects on the systemic metabolite profile of a 1-kg/m2 increment in BMI due to genetic predisposition closely matched the effects of an observed 1-kg/m2 increment in adulthood BMI on the metabolic profile. That is, higher levels of adiposity had causal effects on the levels of numerous blood-based metabolic risk factors, including higher levels of low-density lipoprotein cholesterol and triglyceride-carrying lipoproteins, protein markers of chronic inflammation and adverse liver function, impaired insulin sensitivity, and elevated concentrations of several amino acids that have recently been linked with the risk for developing diabetes. Elevated BMI also causally led to lower levels of certain high-density lipoprotein lipids in the blood, a marker for the risk of future cardiovascular disease. Finally, an examination of the metabolic changes associated with changes in BMI in 1,488 young adults after a period of six years showed that those metabolic measures that were most strongly associated with BMI at a single time point likewise displayed the highest responsiveness to weight change over time.
What Do These Findings Mean?
These findings suggest that increased adiposity has causal adverse effects on multiple cardiometabolic risk markers in non-obese young adults beyond the effects on cholesterol and blood sugar. Like all Mendelian randomization studies, the reliability of the causal association reported here depends on several assumptions made by the researchers. Nevertheless, these findings suggest that increased adiposity has causal adverse effects on multiple cardiometabolic risk markers in non-obese young adults. Importantly, the results of both the causal effect analyses and the longitudinal study suggest that there is no threshold below which a BMI increase does not adversely affect the metabolic profile, and that a systemic metabolic profile linked with high cardiometabolic disease risk that becomes established during early adulthood can be reversed. Overall, these findings therefore highlight the importance of weight reduction as a key target for metabolic risk factor control among young adults.
Additional Information
Please access these websites via the online version of this summary at http://dx.doi.org/10.1371/journal.pmed.1001765.
The Computational Medicine Research Team of the University of Oulu has a webpage that provides further information on metabolite profiling by high-throughput NMR metabolomics
The World Health Organization provides information on obesity (in several languages)
The Global Burden of Disease Study website provides the latest details about global obesity trends
The UK National Health Service Choices website provides information about obesity, cardiovascular disease, and type 2 diabetes (including some personal stories)
The American Heart Association provides information on all aspects of cardiovascular disease and diabetes and on keeping healthy; its website includes personal stories about heart attacks, stroke, and diabetes
The US Centers for Disease Control and Prevention has information on all aspects of overweight and obesity and information about heart disease, stroke, and diabetes
MedlinePlus provides links to other sources of information on heart disease, vascular disease, and obesity (in English and Spanish)
Wikipedia has a page on Mendelian randomization (note: Wikipedia is a free online encyclopedia that anyone can edit; available in several languages)
doi:10.1371/journal.pmed.1001765
PMCID: PMC4260795  PMID: 25490400
23.  Using Cure Models and Multiple Imputation to Utilize Recurrence as an Auxiliary Variable for Overall Survival 
Background
Intermediate outcome variables can often be used as auxiliary variables for the true outcome of interest in randomized clinical trials. For many cancers, time to recurrence is an informative marker in predicting a patient’s overall survival outcome, and could provide auxiliary information for the analysis of survival times.
Purpose
To investigate whether models linking recurrence and death combined with a multiple imputation procedure for censored observations can result in efficiency gains in the estimation of treatment effects, and be used to shorten trial lengths.
Methods
Recurrence and death times are modeled using data from 12 trials in colorectal cancer. Multiple imputation is used as a strategy for handling missing values arising from censoring. The imputation procedure uses a cure model for time to recurrence and a time-dependent Weibull proportional hazards model for time to death. Recurrence times are imputed, and then death times are imputed conditionally on recurrence times. To illustrate these methods, trials are artificially censored 2-years after the last accrual, the imputation procedure is implemented, and a log-rank test and Cox model are used to analyze and compare these new data with the original data.
Results
The results show modest, but consistent gains in efficiency in the analysis by using the auxiliary information in recurrence times. Comparison of analyses show the treatment effect estimates and log rank test results from the 2-year censored imputed data to be in between the estimates from the original data and the artificially censored data, indicating that the procedure was able to recover some of the lost information due to censoring.
Limitations
The models used are all fully parametric, requiring distributional assumptions of the data.
Conclusions
The proposed models may be useful to improve the efficiency in estimation of treatment effects in cancer trials and shortening trial length.
doi:10.1177/1740774511414741
PMCID: PMC3197975  PMID: 21921063
Auxiliary Variables; Colon Cancer; Cure Models; Multiple Imputation; Surrogate Endpoints
24.  Identification and estimation of survivor average causal effects 
Statistics in Medicine  2014;33(21):3601-3628.
In longitudinal studies, outcomes ascertained at follow-up are typically undefined for individuals who die prior to the follow-up visit. In such settings, outcomes are said to be truncated by death and inference about the effects of a point treatment or exposure, restricted to individuals alive at the follow-up visit, could be biased even if as in experimental studies, treatment assignment were randomized. To account for truncation by death, the survivor average causal effect (SACE) defines the effect of treatment on the outcome for the subset of individuals who would have survived regardless of exposure status. In this paper, the author nonparametrically identifies SACE by leveraging post-exposure longitudinal correlates of survival and outcome that may also mediate the exposure effects on survival and outcome. Nonparametric identification is achieved by supposing that the longitudinal data arise from a certain nonparametric structural equations model and by making the monotonicity assumption that the effect of exposure on survival agrees in its direction across individuals. A novel weighted analysis involving a consistent estimate of the survival process is shown to produce consistent estimates of SACE. A data illustration is given, and the methods are extended to the context of time-varying exposures. We discuss a sensitivity analysis framework that relaxes assumptions about independent errors in the nonparametric structural equations model and may be used to assess the extent to which inference may be altered by a violation of key identifying assumptions. © 2014 The Authors. Statistics in Medicine published by John Wiley & Sons, Ltd.
doi:10.1002/sim.6181
PMCID: PMC4131726  PMID: 24889022
truncation by death; principal stratification; double robust; sensitivity analysis
25.  Longitudinal Data with Follow-up Truncated by Death: Match the Analysis Method to Research Aims 
Diverse analysis approaches have been proposed to distinguish data missing due to death from nonresponse, and to summarize trajectories of longitudinal data truncated by death. We demonstrate how these analysis approaches arise from factorizations of the distribution of longitudinal data and survival information. Models are illustrated using cognitive functioning data for older adults. For unconditional models, deaths do not occur, deaths are independent of the longitudinal response, or the unconditional longitudinal response is averaged over the survival distribution. Unconditional models, such as random effects models fit to unbalanced data, may implicitly impute data beyond the time of death. Fully conditional models stratify the longitudinal response trajectory by time of death. Fully conditional models are effective for describing individual trajectories, in terms of either aging (age, or years from baseline) or dying (years from death). Causal models (principal stratification) as currently applied are fully conditional models, since group differences at one timepoint are described for a cohort that will survive past a later timepoint. Partly conditional models summarize the longitudinal response in the dynamic cohort of survivors. Partly conditional models are serial cross-sectional snapshots of the response, reflecting the average response in survivors at a given timepoint rather than individual trajectories. Joint models of survival and longitudinal response describe the evolving health status of the entire cohort. Researchers using longitudinal data should consider which method of accommodating deaths is consistent with research aims, and use analysis methods accordingly.
doi:10.1214/09-STS293
PMCID: PMC2812934  PMID: 20119502
Censoring; Generalized estimating equations; Longitudinal data; Missing data; Quality of life; Random effects models; Truncation by death

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