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.
marginalized likelihood-based models; ordinal data models; dropout
Intermediate outcomes are common and typically on the causal pathway to the final outcome. Some examples include noncompliance, missing data, and truncation by death like pregnancy (e.g. when the trial intervention is given to non-pregnant women and the final outcome is preeclampsia, defined only on pregnant women). The intention-to-treat approach does not account properly for them, and more appropriate alternative approaches like principal stratification are not yet widely known. The purposes of this study are to inform researchers that the intention-to-treat approach unfortunately does not fit all problems we face in experimental research, to introduce the principal stratification approach for dealing with intermediate outcomes, and to illustrate its application to a trial of long term calcium supplementation in women at high risk of preeclampsia.
Principal stratification and related concepts are introduced. Two ways for estimating causal effects are discussed and their application is illustrated using the calcium trial, where noncompliance and pregnancy are considered as intermediate outcomes, and preeclampsia is the main final outcome.
The limitations of traditional approaches and methods for dealing with intermediate outcomes are demonstrated. The steps, assumptions and required calculations involved in the application of the principal stratification approach are discussed in detail in the case of our calcium trial.
The intention-to-treat approach is a very sound one but unfortunately it does not fit all problems we find in randomized clinical trials; this is particularly the case for intermediate outcomes, where alternative approaches like principal stratification should be considered.
Intermediate outcomes; Intention-to-treat approach; Principal stratification; Causal effects
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.
Estimated likelihood; Identifiability; Principal stratification; Sensitivity analysis; Surrogate endpoint; Vaccine 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.
microbicide; causal inference; posttreatment variables; direct effect
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).
Causal Inference; Complier Average Causal Effect; Multi-arm Trials; Non-compliance; Principal Compliance; Principal Stratification
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.
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.
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.
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).
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.
causal inference; potential outcomes framework; incarcerated women
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.
Biomarker; Causal effect predictiveness; principal stratification; surrogate endpoint
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.
Principal Stratification; Post-Treatment Confounder; Institutionalization; Causal Inference
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.
Case cohort; Causal inference; Clinical trial; HIV vaccine; Postrandomization selection bias; Structural model; Prentice criteria; Principal stratification
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.
Causal inference; Infectious disease; Maximum likelihood; Principal stratification; Sensitivity analysis
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.
alternative assumptions; bounds; causal inference; missing data; noncompliance; principal stratification; sensitivity analysis
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.
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.
Bayesian estimation; Counterfactual model; Identifiability; Multiple trials; Principal stratification; Surrogate marker
In some randomized studies, researchers are interested in determining the effect of treatment assignment on outcomes that may exist only in a subset chosen after randomization. For example, in preventative human immunodeficiency virus (HIV) vaccine efficacy trials, it is of interest to determine whether randomization to vaccine affects postinfection outcomes that may be right-censored. Such outcomes in these trials include time from infection diagnosis to initiation of antiretroviral therapy and time from infection diagnosis to acquired immune deficiency syndrome. Here we present sensitivity analysis methods for making causal comparisons on these postinfection outcomes. We focus on estimating the survival causal effect, defined as the difference between probabilities of not yet experiencing the event in the vaccine and placebo arms, conditional on being infected regardless of treatment assignment. This group is referred to as the always-infected principal stratum. Our key assumption is monotonicity—that subjects randomized to the vaccine arm who become infected would have been infected had they been randomized to placebo. We propose nonparametric, semiparametric, and parametric methods for estimating the survival causal effect. We apply these methods to the first Phase III preventative HIV vaccine trial, VaxGen’s trial of AIDSVAX B/B.
Acquired immune deficiency syndrome; Causal inference; Kaplan–Meier; Principal stratification
In randomized trials with follow-up, outcomes such as quality of life may be undefined for individuals who die before the follow-up is complete. In such settings, restricting analysis to those who survive can give rise to biased outcome comparisons. An alternative approach is to consider the “principal strata effect” or “survivor average causal effect” (SACE), defined as the effect of treatment on the outcome among the subpopulation that would have survived under either treatment arm. The authors describe a very simple technique that can be used to assess the SACE. They give both a sensitivity analysis technique and conditions under which a crude comparison provides a conservative estimate of the SACE. The method is illustrated using data from the ARDSnet (Acute Respiratory Distress Syndrome Network) clinical trial comparing low-volume ventilation and traditional ventilation methods for individuals with acute respiratory distress syndrome.
causal inference; randomized trials; stratification; truncation
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.
Complier average causal effect; noncompliance; principal effect; principal stratification
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.
Censoring; Generalized estimating equations; Longitudinal data; Missing data; Quality of life; Random effects models; Truncation by death
In randomized trials to prevent breast milk transmission of human immunodeficiency virus (HIV) from mother to infant, investigators are often interested in assessing the effect of a treatment or intervention on the cumulative risk of HIV infection by time (age) t in infants who are alive and uninfected at a certain time point τ0 < t. Such comparisons are challenging for two reasons. First, infants are typically randomized at birth (time 0 < τ0) such that comparisons between trial arms among the subset of infants alive and uninfected at τ0 are subject to selection bias. Second, in most mother-to-child transmission (MTCT) trials competing risks are often present, such as death or cessation of breastfeeding prior to HIV infection. In this paper we present methods for assessing the causal effect of a treatment on competing risk outcomes within principal strata. In MTCT trials, the causal effect of interest is that of treatment on the risk of HIV infection by time t > τ0 within the principal stratum of infants who would be alive and uninfected by τ0 regardless of randomization assignment. Large sample non-parametric bounds and a semi-parametric sensitivity analysis model are developed for drawing inference about this causal effect. A simulation study is presented demonstrating that the proposed methods perform well in finite samples. The proposed methods are applied to a large, recent MTCT trial.
Causal inference; Infectious diseases; Principal stratification; Sensitivity analysis
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.
causal inference; counterfactual; marginal structural model; longitudinal study; IPTW; G-computation; Double Robust
Treatment noncompliance and missing outcomes at posttreatment assessments are common problems in field experiments in naturalistic settings. Although the two complications often occur simultaneously, statistical methods that address both complications have not been routinely considered in data analysis practice in the prevention research field. This paper shows that identification and estimation of causal treatment effects considering both noncompliance and missing outcomes can be relatively easily conducted under various missing data assumptions. We review a few assumptions on missing data in the presence of noncompliance, including the latent ignorability proposed by Frangakis and Rubin (Biometrika 86:365–379, 1999), and show how these assumptions can be used in the parametric complier average causal effect (CACE) estimation framework. As an easy way of sensitivity analysis, we propose the use of alternative missing data assumptions, which will provide a range of causal effect estimates. In this way, we are less likely to settle with a possibly biased causal effect estimate based on a single assumption. We demonstrate how alternative missing data assumptions affect identification of causal effects, focusing on the CACE. The data from the Johns Hopkins School Intervention Study (Ialongo et al., Am J Community Psychol 27:599–642, 1999) will be used as an example.
Causal inference; Complier average causal effect; Latent ignorability; Missing at random; Missing data; Noncompliance
This article considers the problem of assessing causal effect moderation in longitudinal settings in which treatment (or exposure) is time-varying and so are the covariates said to moderate its effect. Intermediate Causal Effects that describe time-varying causal effects of treatment conditional on past covariate history are introduced and considered as part of Robins’ Structural Nested Mean Model. Two estimators of the intermediate causal effects, and their standard errors, are presented and discussed: The first is a proposed 2-Stage Regression Estimator. The second is Robins’ G-Estimator. The results of a small simulation study that begins to shed light on the small versus large sample performance of the estimators, and on the bias-variance trade-off between the two estimators are presented. The methodology is illustrated using longitudinal data from a depression study.
Causal inference; Effect modification; Estimating equations; G-Estimation; 2-stage estimation; Time-varying treatment; Time-varying covariates; Bias-variance trade-off
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.
structural equation models; confounding; graphical methods; counterfactuals; causal effects; potential-outcome; mediation; policy evaluation; causes of effects
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.
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
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.
Causal inference; Latent trajectory class; Longitudinal outcome prognosis; Growth mixture modeling; Principal stratification; Reference stratification
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.
Causal inference; HIV vaccine trial; Missing at random; Posttreatment selection bias; Principal stratification; Sensitivity analysis