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1.  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
2.  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
3.  Principal Stratification — Uses and Limitations 
Pearl (2011) asked for the causal inference community to clarify the role of the principal stratification framework in the analysis of causal effects. Here, I argue that the notion of principal stratification has shed light on problems of non-compliance, censoring-by-death, and the analysis of post-infection outcomes; that it may be of use in considering problems of surrogacy but further development is needed; that it is of some use in assessing “direct effects”; but that it is not the appropriate tool for assessing “mediation.” There is nothing within the principal stratification framework that corresponds to a measure of an “indirect” or “mediated” effect.
doi:10.2202/1557-4679.1329
PMCID: PMC3154088  PMID: 21841939
causal inference; mediation; non-compliance; potential outcomes; principal stratification; surrogates
4.  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
5.  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
6.  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
7.  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
8.  THE POTENTIAL FOR BIAS IN PRINCIPAL CAUSAL EFFECT ESTIMATION WHEN TREATMENT RECEIVED DEPENDS ON A KEY COVARIATE1 
The annals of applied statistics  2011;5(3):1876-1892.
Motivated by a potential-outcomes perspective, the idea of principal stratification has been widely recognized for its relevance in settings susceptible to posttreatment selection bias such as randomized clinical trials where treatment received can differ from treatment assigned. In one such setting, we address subtleties involved in inference for causal effects when using a key covariate to predict membership in latent principal strata. We show that when treatment received can differ from treatment assigned in both study arms, incorporating a stratum-predictive covariate can make estimates of the “complier average causal effect” (CACE) derive from observations in the two treatment arms with different covariate distributions. Adopting a Bayesian perspective and using Markov chain Monte Carlo for computation, we develop posterior checks that characterize the extent to which incorporating the pretreatment covariate endangers estimation of the CACE. We apply the method to analyze a clinical trial comparing two treatments for jaw fractures in which the study protocol allowed surgeons to overrule both possible randomized treatment assignments based on their clinical judgment and the data contained a key covariate (injury severity) predictive of treatment received.
doi:10.1214/11-AOAS477
PMCID: PMC3269822  PMID: 22308190
Complier average causal effect; noncompliance; principal effect; principal stratification
9.  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
10.  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
11.  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
12.  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
13.  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
14.  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
15.  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
16.  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
17.  Empirical Power and Sample Size Calculations for Cluster-Randomized and Cluster-Randomized Crossover Studies 
PLoS ONE  2012;7(4):e35564.
In recent years, the number of studies using a cluster-randomized design has grown dramatically. In addition, the cluster-randomized crossover design has been touted as a methodological advance that can increase efficiency of cluster-randomized studies in certain situations. While the cluster-randomized crossover trial has become a popular tool, standards of design, analysis, reporting and implementation have not been established for this emergent design. We address one particular aspect of cluster-randomized and cluster-randomized crossover trial design: estimating statistical power. We present a general framework for estimating power via simulation in cluster-randomized studies with or without one or more crossover periods. We have implemented this framework in the clusterPower software package for R, freely available online from the Comprehensive R Archive Network. Our simulation framework is easy to implement and users may customize the methods used for data analysis. We give four examples of using the software in practice. The clusterPower package could play an important role in the design of future cluster-randomized and cluster-randomized crossover studies. This work is the first to establish a universal method for calculating power for both cluster-randomized and cluster-randomized clinical trials. More research is needed to develop standardized and recommended methodology for cluster-randomized crossover studies.
doi:10.1371/journal.pone.0035564
PMCID: PMC3338707  PMID: 22558168
18.  Causal Mediation of a Human Immunodeficiency Virus Preventive Intervention 
Nursing Research  2012;61(3):224-230.
Background
Assessing mediation is important because most interventions are specifically designed to affect an intermediate variable or mediator; this mediator, in turn, is hypothesized to affect outcome behaviors. Although there may be randomization to the intervention, randomization to levels of the mediator is not generally possible. Therefore, drawing causal inferences about the effect of the mediator on the outcome is not straightforward.
Objectives
We introduce an approach to causal mediation analysis that uses the potential outcomes framework for causal inference, and then discuss this approach in terms of the scientific questions addressed and the assumptions needed for identifying and estimating the effects.
Method
We illustrate the approach using data from the Criminal Justice Drug Abuse Treatment studies: Reducing Risky Relationships HIV intervention (RRR-HIV) implemented with 243 incarcerated women reentering the community. The intervention was designed to affect various mediators at 30 days post-intervention including risky relationship thoughts, beliefs, and attitudes, which were then hypothesized to affect engagement in risky sexual behaviors such as unprotected sex at 90 days post-intervention.
Results
Using propensity score weighting, we found the intervention resulted in a significant decrease in risky relationship thoughts (−0.529, p = .03); risky relationship thoughts resulted in an increase in the odds of unprotected sex (.447, p < .001). However, the direct effect of the intervention on unprotected sex was not significant (0.388, p = .479).
Discussion
By reducing bias, propensity score models improve the accuracy of statistical analysis of interventions with mediators and allow researchers to determine not only if their intervention works, but also how it works.
doi:10.1097/NNR.0b013e318254165c
PMCID: PMC3377683  PMID: 22551997
causal inference; potential outcomes framework; incarcerated women
19.  Causal Mediation Analyses for Randomized Trials 
SUMMARY
In the context of randomized intervention trials, we describe causal methods for analyzing how post-randomization factors constitute the process through which randomized baseline interventions act on outcomes. Traditionally, such mediation analyses have been undertaken with great caution, because they assume that the mediating factor is also randomly assigned to individuals in addition to the randomized baseline intervention (i.e., sequential ignorability). Because the mediating factors are typically not randomized, such analyses are unprotected from unmeasured confounders that may lead to biased inference. We review several causal approaches that attempt to reduce such bias without assuming that the mediating factor is randomized. However, these causal approaches require certain interaction assumptions that may be assessed if there is enough treatment heterogeneity with respect to the mediator. We describe available estimation procedures in the context of several examples from the literature and provide resources for software code.
doi:10.1007/s10742-008-0028-9
PMCID: PMC2688317  PMID: 19484136
Structural mean models; principal stratification; direct effects; unmeasured confounding; baseline randomization; sequential ignorability
20.  Adjusting for Population Stratification in a Fine Scale with Principal Components and Sequencing Data 
Genetic epidemiology  2013;37(8):10.1002/gepi.21764.
Population stratification is of primary interest in genetic studies to infer human evolution history and to avoid spurious findings in association testing. Although it is well studied with high-density single nucleotide polymorphisms (SNPs) in genome-wide association studies (GWASs), next-generation sequencing brings both new opportunities and challenges to uncovering population structures in finer scales. Several recent studies have noticed different confounding effects from variants of different minor allele frequencies (MAFs). In this paper, using a low-coverage sequencing dataset from the 1000 Genomes Project, we compared a popular method, principal component analysis (PCA), with a recently proposed spectral clustering technique, called spectral dimensional reduction (SDR), in detecting and adjusting for population stratification at the level of ethnic subgroups. We investigated the varying performance of adjusting for population stratification with different types and sets of variants when testing on different types of variants. One main conclusion is that principal components based on all variants or common variants were generally most effective in controlling inflations caused by population stratification; in particular, contrary to many speculations on the effectiveness of rare variants, we did not find much added value with the use of only rare variants. In addition, SDR was confirmed to be more robust than PCA, especially when applied to rare variants.
doi:10.1002/gepi.21764
PMCID: PMC3864649  PMID: 24123217
1000 Genomes Project; Association testing; Common variants; Principal component analysis; Rare variants; Spectral analysis
21.  Estimating causal effects of air quality regulations using principal stratification for spatially correlated multivariate intermediate outcomes 
Biostatistics (Oxford, England)  2012;13(2):289-302.
Methods for causal inference regarding health effects of air quality regulations are met with unique challenges because (1) changes in air quality are intermediates on the causal pathway between regulation and health, (2) regulations typically affect multiple pollutants on the causal pathway towards health, and (3) regulating a given location can affect pollution at other locations, that is, there is interference between observations. We propose a principal stratification method designed to examine causal effects of a regulation on health that are and are not associated with causal effects of the regulation on air quality. A novel feature of our approach is the accommodation of a continuously scaled multivariate intermediate response vector representing multiple pollutants. Furthermore, we use a spatial hierarchical model for potential pollution concentrations and ultimately use estimates from this model to assess validity of assumptions regarding interference. We apply our method to estimate causal effects of the 1990 Clean Air Act Amendments among approximately 7 million Medicare enrollees living within 6 miles of a pollution monitor.
doi:10.1093/biostatistics/kxr052
PMCID: PMC3297824  PMID: 22267524
Air pollution; Bayesian statistics; Causal inference; Principal stratification; Spatial data
22.  Accounting for Population Stratification in Practice: A Comparison of the Main Strategies Dedicated to Genome-Wide Association Studies 
PLoS ONE  2011;6(12):e28845.
Genome-Wide Association Studies are powerful tools to detect genetic variants associated with diseases. Their results have, however, been questioned, in part because of the bias induced by population stratification. This is a consequence of systematic differences in allele frequencies due to the difference in sample ancestries that can lead to both false positive or false negative findings. Many strategies are available to account for stratification but their performances differ, for instance according to the type of population structure, the disease susceptibility locus minor allele frequency, the degree of sampling imbalanced, or the sample size. We focus on the type of population structure and propose a comparison of the most commonly used methods to deal with stratification that are the Genomic Control, Principal Component based methods such as implemented in Eigenstrat, adjusted Regressions and Meta-Analyses strategies. Our assessment of the methods is based on a large simulation study, involving several scenarios corresponding to many types of population structures. We focused on both false positive rate and power to determine which methods perform the best. Our analysis showed that if there is no population structure, none of the tests led to a bias nor decreased the power except for the Meta-Analyses. When the population is stratified, adjusted Logistic Regressions and Eigenstrat are the best solutions to account for stratification even though only the Logistic Regressions are able to constantly maintain correct false positive rates. This study provides more details about these methods. Their advantages and limitations in different stratification scenarios are highlighted in order to propose practical guidelines to account for population stratification in Genome-Wide Association Studies.
doi:10.1371/journal.pone.0028845
PMCID: PMC3244428  PMID: 22216125
23.  Risk Stratification for Sudden Cardiac Death: Current Approaches and Predictive Value 
Current Cardiology Reviews  2009;5(1):56-64.
Sudden cardiac death (SCD) is a serious public health problem; the annual incidence of out-of-hospital cardiac arrest in North America is approximately 166,200. Identifying patients at risk is a difficult proposition. At the present time, left ventricular ejection fraction (LVEF) remains the single most important marker for risk stratification. According to current guidelines, most patients with LVEF <35% could benefit from prophylactic ICD implantation, particularly in the setting of symptomatic heart failure. Current risk stratification strategies fail to identify patients at risk of SCD in larger population groups encompassing a greater number of potential SCD victims. However, the best approach to identifying patients and the value of various risk stratification tools is not entirely clear. The goal of this review is to discuss the problem of SCD and the value of the different risk stratification markers and their potential clinical use either alone or in combination with other risk stratification markers.
doi:10.2174/157340309787048130
PMCID: PMC2803290  PMID: 20066150
Sudden cardiac death; risk stratification markers.
24.  ASSESSING SURROGATE ENDPOINTS IN VACCINE TRIALS WITH CASE-COHORT SAMPLING AND THE COX MODEL1 
The annals of applied statistics  2008;2(1):386-407.
Assessing immune responses to study vaccines as surrogates of protection plays a central role in vaccine clinical trials. Motivated by three ongoing or pending HIV vaccine efficacy trials, we consider such surrogate endpoint assessment in a randomized placebo-controlled trial with case-cohort sampling of immune responses and a time to event endpoint. Based on the principal surrogate definition under the principal stratification framework proposed by Frangakis and Rubin [Biometrics 58 (2002) 21–29] and adapted by Gilbert and Hudgens (2006), we introduce estimands that measure the value of an immune response as a surrogate of protection in the context of the Cox proportional hazards model. The estimands are not identified because the immune response to vaccine is not measured in placebo recipients. We formulate the problem as a Cox model with missing covariates, and employ novel trial designs for predicting the missing immune responses and thereby identifying the estimands. The first design utilizes information from baseline predictors of the immune response, and bridges their relationship in the vaccine recipients to the placebo recipients. The second design provides a validation set for the unmeasured immune responses of uninfected placebo recipients by immunizing them with the study vaccine after trial closeout. A maximum estimated likelihood approach is proposed for estimation of the parameters. Simulated data examples are given to evaluate the proposed designs and study their properties.
doi:10.1214/07-AOAS132
PMCID: PMC2601643  PMID: 19079758
Clinical trial; discrete failure time model; missing data; potential outcomes; principal stratification; surrogate marker
25.  Control of population stratification by correlation-selected principal components 
Biometrics  2010;67(3):967-974.
Summary
In genome-wide association studies, population stratification is recognized as producing inflated type I error due to the inflation of test statistics. Principal component-based methods applied to genotypes provide information about population structure, and have been widely used to control for stratification. Here we explore the precise relationship between genotype principal components and inflation of association test statistics, thereby drawing a connection between principal component-based stratification control and the alternative approach of genomic control. Our results provide an inherent justification for the use of principal components, but call into question the popular practice of selecting principal components based on significance of eigenvalues alone. We propose a new approach, called EigenCorr, which selects principal components based on both their eigenvalues and their correlation with the (disease) phenotype. Our approach tends to select fewer principal components for stratification control than does testing of eigenvalues alone, providing substantial computational savings and improvements in power. Analyses of simulated and real data demonstrate the usefulness of the proposed approach.
doi:10.1111/j.1541-0420.2010.01520.x
PMCID: PMC3117098  PMID: 21133882
Genomic Control; GWAS; PCA; Population Stratification

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