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1.  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
2.  Commentary on “Principal Stratification — a Goal or a Tool?” by Judea Pearl 
This commentary takes up Pearl's welcome challenge to clearly articulate the scientific value of principal stratification estimands that we and colleagues have investigated, in the area of randomized placebo-controlled preventive vaccine efficacy trials, especially trials of HIV vaccines. After briefly arguing that certain principal stratification estimands for studying vaccine effects on post-infection outcomes are of genuine scientific interest, the bulk of our commentary argues that the “causal effect predictiveness” (CEP) principal stratification estimand for evaluating immune biomarkers as surrogate endpoints is not of ultimate scientific interest, because it evaluates surrogacy restricted to the setting of a particular vaccine efficacy trial, but is nevertheless useful for guiding the selection of primary immune biomarker endpoints in Phase I/II vaccine trials and for facilitating assessment of transportability/bridging surrogacy.
doi:10.2202/1557-4679.1341
PMCID: PMC3204668  PMID: 22049267
principal stratification; causal inference; vaccine trial
3.  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
4.  Causal assessment of surrogacy in a meta-analysis of colorectal cancer trials 
Biostatistics (Oxford, England)  2011;12(3):478-492.
When the true end points (T) are difficult or costly to measure, surrogate markers (S) are often collected in clinical trials to help predict the effect of the treatment (Z). There is great interest in understanding the relationship among S, T, and Z. A principal stratification (PS) framework has been proposed by Frangakis and Rubin (2002) to study their causal associations. In this paper, we extend the framework to a multiple trial setting and propose a Bayesian hierarchical PS model to assess surrogacy. We apply the method to data from a large collection of colon cancer trials in which S and T are binary. We obtain the trial-specific causal measures among S, T, and Z, as well as their overall population-level counterparts that are invariant across trials. The method allows for information sharing across trials and reduces the nonidentifiability problem. We examine the frequentist properties of our model estimates and the impact of the monotonicity assumption using simulations. We also illustrate the challenges in evaluating surrogacy in the counterfactual framework that result from nonidentifiability.
doi:10.1093/biostatistics/kxq082
PMCID: PMC3114655  PMID: 21252079
Bayesian estimation; Counterfactual model; Identifiability; Multiple trials; Principal stratification; Surrogate marker
5.  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
6.  Clarifying the Role of Principal Stratification in the Paired Availability Design 
The paired availability design for historical controls postulated four classes corresponding to the treatment (old or new) a participant would receive if arrival occurred during either of two time periods associated with different availabilities of treatment. These classes were later extended to other settings and called principal strata. Judea Pearl asks if principal stratification is a goal or a tool and lists four interpretations of principal stratification. In the case of the paired availability design, principal stratification is a tool that falls squarely into Pearl's interpretation of principal stratification as “an approximation to research questions concerning population averages.” We describe the paired availability design and the important role played by principal stratification in estimating the effect of receipt of treatment in a population using data on changes in availability of treatment. We discuss the assumptions and their plausibility. We also introduce the extrapolated estimate to make the generalizability assumption more plausible. By showing why the assumptions are plausible we show why the paired availability design, which includes principal stratification as a key component, is useful for estimating the effect of receipt of treatment in a population. Thus, for our application, we answer Pearl's challenge to clearly demonstrate the value of principal stratification.
doi:10.2202/1557-4679.1338
PMCID: PMC3114955  PMID: 21686085
principal stratification; causal inference; paired availability design
7.  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
8.  A tutorial on principal stratification-based sensitivity analysis: Application to smoking cessation studies 
Background
One problem with assessing effects of smoking cessation interventions on withdrawal symptoms is that symptoms are affected by whether participants abstain from smoking during trials. Those who enter a randomized trial but do not change smoking behavior might not experience withdrawal related symptoms.
Purpose
We present a tutorial of how one can use a principal stratification sensitivity analysis to account for abstinence in the estimation of smoking cessation intervention effects. The paper is intended to introduce researchers to principal stratification and describe how they might implement the methods.
Methods
We provide a hypothetical example that demonstrates why estimating effects within observed abstention groups is problematic. We demonstrate how estimation of effects within groups defined by potential abstention that an individual would have in either arm of a study can provide meaningful inferences. We describe a sensitivity analysis method to estimate such effects, and use it to investigate effects of a combined behavioral and nicotine replacement therapy intervention on withdrawal symptoms in a female prisoner population.
Results
Overall, the intervention was found to reduce withdrawal symptoms but the effect was not statistically significant in the group that was observed to abstain. More importantly, the intervention was found to be highly effective in the group that would abstain regardless of intervention assignment. The effectiveness of the intervention in other potential abstinence strata depends on the sensitivity analysis assumptions.
Limitations
We make assumptions to narrow the range of our sensitivity parameter estimates. While appropriate in this situation, such assumptions might not be plausible in all situations.
Conclusions
A principal stratification sensitivity analysis provides a meaningful method of accounting for abstinence effects in the evaluation of smoking cessation interventions on withdrawal symptoms. Smoking researchers have previously recommended analyses in subgroups defined by observed abstention status in the evaluation of smoking cessation interventions. We believe that principal stratification analyses should replace such analyses as the preferred means of accounting for post-randomization abstinence effects in the evaluation of smoking cessation programs.
doi:10.1177/1740774510367811
PMCID: PMC2874094  PMID: 20423924
9.  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
10.  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
11.  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
12.  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
13.  Principal Stratification — a Goal or a Tool? 
Principal stratification has recently become a popular tool to address certain causal inference questions, particularly in dealing with post-randomization factors in randomized trials. Here, we analyze the conceptual basis for this framework and invite response to clarify the value of principal stratification in estimating causal effects of interest.
doi:10.2202/1557-4679.1322
PMCID: PMC3083139  PMID: 21556288
causal inference; principal stratification; surrogate endpoints; direct effect; mediation
14.  An Introduction to Causal Inference* 
This paper summarizes recent advances in causal inference and underscores the paradigmatic shifts that must be undertaken in moving from traditional statistical analysis to causal analysis of multivariate data. Special emphasis is placed on the assumptions that underlie all causal inferences, the languages used in formulating those assumptions, the conditional nature of all causal and counterfactual claims, and the methods that have been developed for the assessment of such claims. These advances are illustrated using a general theory of causation based on the Structural Causal Model (SCM) described in Pearl (2000a), which subsumes and unifies other approaches to causation, and provides a coherent mathematical foundation for the analysis of causes and counterfactuals. In particular, the paper surveys the development of mathematical tools for inferring (from a combination of data and assumptions) answers to three types of causal queries: those about (1) the effects of potential interventions, (2) probabilities of counterfactuals, and (3) direct and indirect effects (also known as "mediation"). Finally, the paper defines the formal and conceptual relationships between the structural and potential-outcome frameworks and presents tools for a symbiotic analysis that uses the strong features of both. The tools are demonstrated in the analyses of mediation, causes of effects, and probabilities of causation.
doi:10.2202/1557-4679.1203
PMCID: PMC2836213  PMID: 20305706
structural equation models; confounding; graphical methods; counterfactuals; causal effects; potential-outcome; mediation; policy evaluation; causes of effects
15.  Network modeling of the transcriptional effects of copy number aberrations in glioblastoma 
DNA copy number aberrations (CNAs) are a characteristic feature of cancer genomes. In this work, Rebecka Jörnsten, Sven Nelander and colleagues combine network modeling and experimental methods to analyze the systems-level effects of CNAs in glioblastoma.
We introduce a modeling approach termed EPoC (Endogenous Perturbation analysis of Cancer), enabling the construction of global, gene-level models that causally connect gene copy number with expression in glioblastoma.On the basis of the resulting model, we predict genes that are likely to be disease-driving and validate selected predictions experimentally. We also demonstrate that further analysis of the network model by sparse singular value decomposition allows stratification of patients with glioblastoma into short-term and long-term survivors, introducing decomposed network models as a useful principle for biomarker discovery.Finally, in systematic comparisons, we demonstrate that EPoC is computationally efficient and yields more consistent results than mRNA-only methods, standard eQTL methods, and two recent multivariate methods for genotype–mRNA coupling.
Gains and losses of chromosomal material (DNA copy number aberrations; CNAs) are a characteristic feature of cancer genomes. At the level of a single locus, it is well known that increased copy number (gene amplification) typically leads to increased gene expression, whereas decreased copy number (gene deletion) leads to decreased gene expression (Pollack et al, 2002; Lee et al, 2008; Nilsson et al, 2008). However, CNAs also affect the expression of genes located outside the amplified/deleted region itself via indirect mechanisms. To fully understand the action of CNAs, it is therefore necessary to analyze their action in a network context. Toward this goal, improved computational approaches will be important, if not essential.
To determine the global effects on transcription of CNAs in the brain tumor glioblastoma, we develop EPoC (Endogenous Perturbation analysis of Cancer), a computational technique capable of inferring sparse, causal network models by combining genome-wide, paired CNA- and mRNA-level data. EPoC aims to detect disease-driving copy number aberrations and their effect on target mRNA expression, and stratify patients into long-term and short-term survivors. Technically, EPoC relates CNA perturbations to mRNA responses by matrix equations, derived from a steady-state approximation of the transcriptional network. Patient prognostic scores are obtained from singular value decompositions of the network matrix. The models are constructed by solving a large-scale, regularized regression problem.
We apply EPoC to glioblastoma data from The Cancer Genome Atlas (TCGA) consortium (186 patients). The identified CNA-driven network comprises 10 672 genes, and contains a number of copy number-altered genes that control multiple downstream genes. Highly connected hub genes include well-known oncogenes and tumor supressor genes that are frequently deleted or amplified in glioblastoma, including EGFR, PDGFRA, CDKN2A and CDKN2B, confirming a clear association between these aberrations and transcriptional variability of these brain tumors. In addition, we identify a number of hub genes that have previously not been associated with glioblastoma, including interferon alpha 1 (IFNA1), myeloid/lymphoid or mixed-lineage leukemia translocated to 10 (MLLT10, a well-known leukemia gene), glutamate decarboxylase 2 GAD2, a postulated glutamate receptor GPR158 and Necdin (NDN). Furthermore, we demonstrate that the network model contains useful information on downstream target genes (including stem cell regulators), and possible drug targets.
We proceed to explore the validity of a small network region experimentally. Introducing experimental perturbations of NDN and other targets in four glioblastoma cell lines (T98G, U-87MG, U-343MG and U-373MG), we confirm several predicted mechanisms. We also demonstrate that the TCGA glioblastoma patients can be stratified into long-term and short-term survivors, using our proposed prognostic scores derived from a singular vector decomposition of the network model. Finally, we compare EPoC to existing methods for mRNA networks analysis and expression quantitative locus methods, and demonstrate that EPoC produces more consistent models between technically independent glioblastoma data sets, and that the EPoC models exhibit better overlap with known protein–protein interaction networks and pathway maps.
In summary, we conclude that large-scale integrative modeling reveals mechanistically and prognostically informative networks in human glioblastoma. Our approach operates at the gene level and our data support that individual hub genes can be identified in practice. Very large aberrations, however, cannot be fully resolved by the current modeling strategy.
DNA copy number aberrations (CNAs) are a hallmark of cancer genomes. However, little is known about how such changes affect global gene expression. We develop a modeling framework, EPoC (Endogenous Perturbation analysis of Cancer), to (1) detect disease-driving CNAs and their effect on target mRNA expression, and to (2) stratify cancer patients into long- and short-term survivors. Our method constructs causal network models of gene expression by combining genome-wide DNA- and RNA-level data. Prognostic scores are obtained from a singular value decomposition of the networks. By applying EPoC to glioblastoma data from The Cancer Genome Atlas consortium, we demonstrate that the resulting network models contain known disease-relevant hub genes, reveal interesting candidate hubs, and uncover predictors of patient survival. Targeted validations in four glioblastoma cell lines support selected predictions, and implicate the p53-interacting protein Necdin in suppressing glioblastoma cell growth. We conclude that large-scale network modeling of the effects of CNAs on gene expression may provide insights into the biology of human cancer. Free software in MATLAB and R is provided.
doi:10.1038/msb.2011.17
PMCID: PMC3101951  PMID: 21525872
cancer biology; cancer genomics; glioblastoma
16.  Hard, harder, hardest: principal stratification, statistical identifiability, and the inherent difficulty of finding surrogate endpoints 
In many areas of clinical investigation there is great interest in identifying and validating surrogate endpoints, biomarkers that can be measured a relatively short time after a treatment has been administered and that can reliably predict the effect of treatment on the clinical outcome of interest. However, despite dramatic advances in the ability to measure biomarkers, the recent history of clinical research is littered with failed surrogates. In this paper, we present a statistical perspective on why identifying surrogate endpoints is so difficult. We view the problem from the framework of causal inference, with a particular focus on the technique of principal stratification (PS), an approach which is appealing because the resulting estimands are not biased by unmeasured confounding. In many settings, PS estimands are not statistically identifiable and their degree of non-identifiability can be thought of as representing the statistical difficulty of assessing the surrogate value of a biomarker. In this work, we examine the identifiability issue and present key simplifying assumptions and enhanced study designs that enable the partial or full identification of PS estimands. We also present example situations where these assumptions and designs may or may not be feasible, providing insight into the problem characteristics which make the statistical evaluation of surrogate endpoints so challenging.
doi:10.1186/1742-7622-11-14
PMCID: PMC4171402  PMID: 25342953
Surrogate endpoint; Principal stratification; Causal inference; Statistical identifiability
17.  Assessing the sensitivity of methods for estimating principal causal effects 
Statistical methods in medical research  2011;10.1177/0962280211421840.
The framework of principal stratification provides a way to think about treatment effects conditional on post-randomization variables, such as level of compliance. In particular, the complier average causal effect (CACE)–the effect of the treatment for those individuals who would comply with their treatment assignment under either treatment condition–is often of substantive interest. However, estimation of the CACE is not always straightforward, with a variety of estimation procedures and underlying assumptions, but little advice to help researchers select between methods. In this paper we discuss and examine two methods that rely on very different assumptions to estimate the CACE: a maximum likelihood (“joint”) method that assumes the “exclusion restriction,” and a propensity score based method that relies on “principal ignorability.” We detail the assumptions underlying each approach, and assess each method’s sensitivity to both its own assumptions and those of the other method using both simulated data and a motivating example. We find that the exclusion restriction based joint approach appears somewhat less sensitive to its assumptions, and that the performance of both methods is significantly improved when there are strong predictors of compliance. Interestingly, we also find that each method performs particularly well when the assumptions of the other approach are violated. These results highlight the importance of carefully selecting an estimation procedure whose assumptions are likely to be satisfied in practice and of having strong predictors of principal stratum membership.
doi:10.1177/0962280211421840
PMCID: PMC3253203  PMID: 21971481
Complier average causal effect; Intermediate outcomes; Noncompliance; Principal stratification; Propensity scores
18.  Can we apply the Mendelian randomization methodology without considering epigenetic effects? 
Introduction
Instrumental variable (IV) methods have been used in econometrics for several decades now, but have only recently been introduced into the epidemiologic research frameworks. Similarly, Mendelian randomization studies, which use the IV methodology for analysis and inference in epidemiology, were introduced into the epidemiologist's toolbox only in the last decade.
Analysis
Mendelian randomization studies using instrumental variables (IVs) have the potential to avoid some of the limitations of observational epidemiology (confounding, reverse causality, regression dilution bias) for making causal inferences. Certain limitations of randomized controlled trials, such as problems with generalizability, feasibility and ethics for some exposures, and high costs, also make the use of Mendelian randomization in observational studies attractive. Unlike conventional randomized controlled trials (RCTs), Mendelian randomization studies can be conducted in a representative sample without imposing any exclusion criteria or requiring volunteers to be amenable to random treatment allocation.
Within the last decade, epigenetics has gained recognition as an independent field of study, and appears to be the new direction for future research into the genetics of complex diseases. Although previous articles have addressed some of the limitations of Mendelian randomization (such as the lack of suitable genetic variants, unreliable associations, population stratification, linkage disequilibrium (LD), pleiotropy, developmental canalization, the need for large sample sizes and some potential problems with binary outcomes), none has directly characterized the impact of epigenetics on Mendelian randomization. The possibility of epigenetic effects (non-Mendelian, heritable changes in gene expression not accompanied by alterations in DNA sequence) could alter the core instrumental variable assumptions of Mendelian randomization.
This paper applies conceptual considerations, algebraic derivations and data simulations to question the appropriateness of Mendelian randomization methods when epigenetic modifications are present.
Conclusion
Given an inheritance of gene expression from parents, Mendelian randomization studies not only need to assume a random distribution of alleles in the offspring, but also a random distribution of epigenetic changes (e.g. gene expression) at conception, in order for the core assumptions of the Mendelian randomization methodology to remain valid. As an increasing number of epidemiologists employ Mendelian randomization methods in their research, caution is therefore needed in drawing conclusions from these studies if these assumptions are not met.
doi:10.1186/1742-7622-6-3
PMCID: PMC2698894  PMID: 19432981
19.  Mediation Analysis with Principal Stratification 
Statistics in medicine  2009;28(7):1108-1130.
In assessing the mechanism of treatment efficacy in randomized clinical trials, investigators often perform mediation analyses by analyzing if the significant intent-to-treat treatment effect on outcome occurs through or around a third intermediate or mediating variable: indirect and direct effects, respectively. Standard mediation analyses assume sequential ignorability, i.e., conditional on covariates the intermediate or mediating factor is randomly assigned, as is the treatment in a randomized clinical trial. This research focuses on the application of the principal stratification approach for estimating the direct effect of a randomized treatment but without the standard sequential ignorability assumption. This approach is used to estimate the direct effect of treatment as a difference between expectations of potential outcomes within latent sub-groups of participants for whom the intermediate variable behavior would be constant, regardless of the randomized treatment assignment. Using a Bayesian estimation procedure, we also assess the sensitivity of results based on the principal stratification approach to heterogeneity of the variances among these principal strata. We assess this approach with simulations and apply it to two psychiatric examples. Both examples and the simulations indicated robustness of our findings to the homogeneous variance assumption. However, simulations showed that the magnitude of treatment effects derived under the principal stratification approach were sensitive to model mis-specification.
doi:10.1002/sim.3533
PMCID: PMC2669107  PMID: 19184975
Principal stratification; mediating variables; direct effects; principal strata probabilities; heterogeneous variances
20.  Principal Stratification and Attribution Prohibition: Good Ideas Taken Too Far 
Pearl’s article provides a useful springboard for discussing further the benefits and drawbacks of principal stratification and the associated discomfort with attributing effects to post-treatment variables. The basic insights of the approach are important: pay close attention to modification of treatment effects by variables not observable before treatment decisions are made, and be careful in attributing effects to variables when counterfactuals are ill-defined. These insights have often been taken too far in many areas of application of the approach, including instrumental variables, censoring by death, and surrogate outcomes. A novel finding is that the usual principal stratification estimand in the setting of censoring by death is by itself of little practical value in estimating intervention effects.
doi:10.2202/1557-4679.1367
PMCID: PMC3204670  PMID: 22049269
principal stratification; causal inference
21.  Comparison of Population-Based Association Study Methods Correcting for Population Stratification 
PLoS ONE  2008;3(10):e3392.
Population stratification can cause spurious associations in population–based association studies. Several statistical methods have been proposed to reduce the impact of population stratification on population–based association studies. We simulated a set of stratified populations based on the real haplotype data from the HapMap ENCODE project, and compared the relative power, type I error rates, accuracy and positive prediction value of four prevailing population–based association study methods: traditional case-control tests, structured association (SA), genomic control (GC) and principal components analysis (PCA) under various population stratification levels. Additionally, we evaluated the effects of sample sizes and frequencies of disease susceptible allele on the performance of the four analytical methods in the presence of population stratification. We found that the performance of PCA was very stable under various scenarios. Our comparison results suggest that SA and PCA have comparable performance, if sufficient ancestral informative markers are used in SA analysis. GC appeared to be strongly conservative in significantly stratified populations. It may be better to apply GC in the stratified populations with low stratification level. Our study intends to provide a practical guideline for researchers to select proper study methods and make appropriate inference of the results in population-based association studies.
doi:10.1371/journal.pone.0003392
PMCID: PMC2562035  PMID: 18852890
22.  Causal Inference in Randomized Experiments With Mediational Processes 
Psychological methods  2008;13(4):314-336.
This article links the structural equation modeling (SEM) approach with the principal stratification (PS) approach, both of which have been widely used to study the role of intermediate posttreatment outcomes in randomized experiments. Despite the potential benefit of such integration, the 2 approaches have been developed in parallel with little interaction. This article proposes the cross-model translation (CMT) approach, in which parameter estimates are translated back and forth between the PS and SEM models. First, without involving any particular identifying assumptions, translation between PS and SEM parameters is carried out on the basis of their close conceptual connection. Monte Carlo simulations are used to further clarify the relation between the 2 approaches under particular identifying assumptions. The study concludes that, under the common goal of causal inference, what makes a practical difference is the choice of identifying assumptions, not the modeling framework itself. The CMT approach provides a common ground in which the PS and SEM approaches can be jointly considered, focusing on their common inferential problems.
doi:10.1037/a0014207
PMCID: PMC2927874  PMID: 19071997
cross-model translation; mediational process; principal stratification; randomized experiment; structural equation modeling
23.  Response to Pearl's Comments on Principal Stratification 
Dr. Pearl invites researchers to justify their use of principal stratification. This comment explains how the use of principal stratification simplified a complex mediational problem encountered when evaluating a smoking cessation intervention's effect on reducing smoking withdrawal symptoms.
doi:10.2202/1557-4679.1330
PMCID: PMC3114954  PMID: 21686084
causal inference; principal stratification; mediation; smoking cessation interventions
24.  Links between analysis of surrogate endpoints and endogeneity 
Statistics in medicine  2010;29(28):2869-2879.
Summary
There has been substantive interest in the assessment of surrogate endpoints in medical research. These are measures which could potentially replace “true” endpoints in clinical trials and lead to studies that require less follow-up. Recent research in the area has focused on assessments using causal inference frameworks. Beginning with a simple model for associating the surrogate and true endpoints in the population, we approach the problem as one of endogenous covariates. An instrumental variables estimator and general two-stage algorithm is proposed. Existing surrogacy frameworks are then evaluated in the context of the model. In addition, we define an extended relative effect estimator as well as a sensitivity analysis for assessing what we term the treatment instrumentality assumption. A numerical example is used to illustrate the methodology.
doi:10.1002/sim.4027
PMCID: PMC2997195  PMID: 20803482
Clinical Trial; Counterfactual; Nonlinear response; Prentice Criterion; Structural equations model
25.  Bounding the Infectiousness Effect in Vaccine Trials 
Epidemiology (Cambridge, Mass.)  2011;22(5):686-693.
In vaccine trials, the vaccination of one person might prevent the infection of another; a distinction can be drawn between the ways such a protective effect might arise. Consider a setting with 2 persons per household in which one of the 2 is vaccinated. Vaccinating the first person may protect the second person by preventing the first from being infected and passing the infection on to the second. Alternatively, vaccinating the first person may protect the second by rendering the infection less contagious even if the first is infected. This latter mechanism is sometimes referred to as an “infectiousness effect” of the vaccine. Crude estimators for the infectiousness effect will be subject to selection bias due to stratification on a postvaccination event, namely the infection status of the first person. We use theory concerning causal inference under interference along with a principal-stratification framework to show that, although the crude estimator is biased, it is, under plausible assumptions, conservative for what one might define as a causal infectiousness effect. This applies to bias from selection due to the persons in the comparison, and also to selection due to pathogen virulence. We illustrate our results with an example from the literature.
doi:10.1097/EDE.0b013e31822708d5
PMCID: PMC3792580  PMID: 21753730

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