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Mediation Analysis With Principal Stratification
Abstract
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 (PS) 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 subgroups 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 PS 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 PS approach were sensitive to model mis-specification.
... Principal strata act as pre-treatment covariates, so the causal effects within them, often referred to as principal causal effects (PCEs), are conceptually the same as the standard subgroup causal effects. PCEs are widely used in applied statistics to deal with noncompliance (Angrist et al., 1996;Frumento et al., 2012;Mealli and Pacini, 2013), truncation by death (Rubin, 2006;Ding et al., 2011;Wang et al., 2017), missing data (Frangakis and Rubin, 1999;Mattei et al., 2014), mediation (Rubin, 2004;Gallop et al., 2009;Elliott et al., 2010;Mattei and Mealli, 2011), and surrogate evaluation (Frangakis and Rubin, 2002;Gilbert and Hudgens, 2008;Huang and Gilbert, 2011;Li et al., 2010;Jiang et al., 2016). ...
... Example 3 (Mediation) In mediation analysis, S is the mediator that lies on the causal pathway from the treatment Z to the outcome Y . The subgroup effects τ 11 and τ 00 can assess the direct effect of the treatment on the outcome because the treatment does not change the mediator in these two strata (Rubin, 2004;Gallop et al., 2009;Mattei and Mealli, 2011). In contrast, the subgroup effects τ 10 and τ 01 are less interpretable because they consist of both direct and indirect effects (VanderWeele, 2011). ...
Causal inference concerns not only the average effect of the treatment on the outcome but also the underlying mechanism through an intermediate variable of interest. Principal stratification characterizes such a mechanism by targeting subgroup causal effects within principal strata, which are defined by the joint potential values of an intermediate variable. Due to the fundamental problem of causal inference, principal strata are inherently latent, rendering it challenging to identify and estimate subgroup effects within them. A line of research leverages the principal ignorability assumption that the latent principal strata are mean independent of the potential outcomes conditioning on the observed covariates. Under principal ignorability, we derive various nonparametric identification formulas for causal effects within principal strata in observational studies, which motivate estimators relying on the correct specifications of different parts of the observed‐data distribution. Appropriately combining these estimators yields triply robust estimators for the causal effects within principal strata. These triply robust estimators are consistent if two of the treatment, intermediate variable and outcome models are correctly specified, and moreover, they are locally efficient if all three models are correctly specified. We show that these estimators arise naturally from either the efficient influence functions in the semiparametric theory or the model‐assisted estimators in the survey sampling theory. We evaluate different estimators based on their finite‐sample performance through simulation and apply them to two observational studies.
... Principal strata act as pre-treatment covariates, so the causal effects within them, often referred to as principal causal effects (PCEs), are conceptually the same as the standard subgroup causal effects. PCEs are widely used in applied statistics to deal with noncompliance (Angrist et al., 1996;Frumento et al., 2012;Mealli and Pacini, 2013), truncation by death (Rubin, 2006;Ding et al., 2011;Wang et al., 2017), missing data (Frangakis and Rubin, 1999;Mattei et al., 2014), mediation (Rubin, 2004;Gallop et al., 2009;Elliott et al., 2010;Mattei and Mealli, 2011), and surrogate evaluation (Frangakis and Rubin, 2002;Gilbert and Hudgens, 2008;Huang and Gilbert, 2011;Li et al., 2010;Jiang et al., 2016). ...
... In truncation-by-death problems where Z is the treatment and S is the survival status, Rubin (2006) points out that the only well-defined causal effect is τ 11 , which characterizes the treatment effect for patients who would survive regardless of the treatment. In mediation analysis where S is the mediator, the subgroup effects τ 11 and τ 00 can assess the direct effect of the treatment on the outcome because the treatment does not change the mediator in these two strata (Rubin, 2004;Gallop et al., 2009;Mattei and Mealli, 2011). In contrast, the subgroup effects τ 10 and τ 01 are less interpretable because they consist of both direct and indirect effects (VanderWeele, 2011). ...
Causal inference concerns not only the average effect of the treatment on the outcome but also the underlying mechanism through an intermediate variable of interest. Principal stratification characterizes such mechanism by targeting subgroup causal effects within principal strata, which are defined by the joint potential values of an intermediate variable. Due to the fundamental problem of causal inference, principal strata are inherently latent, rendering it challenging to identify and estimate subgroup effects within them. A line of research leverages the principal ignorability assumption that the latent principal strata are mean independent of the potential outcomes conditioning on the observed covariates. Under principal ignorability, we derive various nonparametric identification formulas for causal effects within principal strata in observational studies, which motivate estimators relying on the correct specifications of different parts of the observed-data distribution. Appropriately combining these estimators further yields new triply robust estimators for the causal effects within principal strata. These new estimators are consistent if two of the treatment, intermediate variable, and outcome models are correctly specified, and they are locally efficient if all three models are correctly specified. We show that these estimators arise naturally from either the efficient influence functions in the semiparametric theory or the model-assisted estimators in the survey sampling theory. We evaluate different estimators based on their finite-sample performance through simulation, apply them to two observational studies, and implement them in an open-source software package.
... Observational data have been historically used in association studies; however, the statistical literature for causal methods has so far mostly focused on methods for randomized experiments. Particular areas of research pertain to methods using propensity scores (Rosenbaum and Rubin 1983;Rubin 2001Rubin , 2004Ho et al. 2007;Pearl 2009;Austin et al. 2005Austin et al. , 2007Austin 2009Austin , 2011Austin and Stuart 2015), principal stratification (Frangakis and Rubin 2002;Gallop et al. 2009;Elliott et al. 2010;Pearl 2011), marginal structural models, and g-estimation (Robins 1999;Hernán et al. 2001Hernán et al. , 2002Robins et al. 2015). ...
... Observational data have been historically used in association studies; however, the statistical literature for causal methods has so far mostly focused on methods for randomized experiments. Particular areas of research pertain to methods using propensity scores (Rosenbaum and Rubin 1983;Rubin 2001Rubin , 2004Ho et al. 2007;Pearl 2009;Austin et al. 2005Austin et al. , 2007Austin 2009Austin , 2011Austin and Stuart 2015), principal stratification (Frangakis and Rubin 2002;Gallop et al. 2009;Elliott et al. 2010;Pearl 2011), marginal structural models, and g-estimation (Robins 1999;Hernán et al. 2001Hernán et al. , 2002Robins et al. 2015). ...
We study causal inference using the framework of potential outcomes in clustered data settings where observational units are clustered in naturally occurring groups (e.g. patients within hospitals). To incorporate the correlated nature of the data, we employ mixed-effects models and a sandwich estimator to make inferences on the average causal effect (ACE). Our methods apply the concept of potential outcomes from the Rubin Causal Model (Holland in J Am Stat Assoc 81(396):945–960, 1986), and extend Schafer and Kang’s methods of estimating the variance of the ACE (Schafer and Kang in Psychol Methods 13(4):279–313, 2008). Particularly, we develop two model-based approaches to estimate the ACE and its variance under a dual-modeling strategy which adjusts for the confounding effect through inverse probability weighting. These two approaches use linear mixed-effects models for the estimation of potential outcomes, but differ in how clustering is handled in the treatment assignment model. We present a summary of our comprehensive simulation study assessing the repetitive sampling properties of the two approaches in a pseudo-random simulation environment. Finally, we report our findings from an application to study the ACE of inadequate prenatal care on birth weight among low-income women in New York State.
... This means that the component-specific variances, σ 2 c0 and σ 2 n0 , are nuisance parameters. We initially assume that the component-specific variances are constant across components: σ 2 " σ 2 c1 " σ 2 c0 " σ 2 n1 " σ 2 n0 (see, for example, Gallop et al., 2009). Since we can directly estimate σ 2 c1 and σ 2 n1 , this means that we no longer need to estimate the component-specific variances in the finite mixture model and can treat these parameters as known. ...
... We now return to the assumption that the component variances are equal, which might be unrealistic in practice (e.g., Gallop et al., 2009). There are two ways to relax this assumption. ...
Principal stratification is a widely used framework for addressing post-randomization complications in a principled way. After using principal stratification to define causal effects of interest, researchers are increasingly turning to finite mixture models to estimate these quantities. Unfortunately, standard estimators of the mixture parameters, like the MLE, are known to exhibit pathological behavior. We study this behavior in a simple but fundamental example: a two-component Gaussian mixture model in which only the component means are unknown. Even though the MLE is asymptotically efficient, we show through extensive simulations that the MLE has undesirable properties in practice. In particular, when mixture components are only weakly separated, we observe "pile up", in which the MLE estimates the component means to be equal, even though they are not. We first show that parametric convergence can break down in certain situations. We then derive a simple moment estimator that displays key features of the MLE and use this estimator to approximate the finite sample behavior of the MLE. Finally, we propose a method to generate valid confidence sets via inverting a sequence of tests, and explore the case in which the component variances are unknown. Throughout, we illustrate the main ideas through an application of principal stratification to the evaluation of JOBS II, a job training program.
... The MRC Pre-school Autism Communication Trial (PACT) was a two-arm RCT of about 150 children with core autism aged 2 years to 4 years 11 months. 22 Its aim was to evaluate a parent-mediated communication-focused treatment in in these children. After an initial orientation meeting, families attended twice-weekly clinic sessions for 6 months followed by a 6-month period of monthly booster sessions. ...
... Previous analysis of PACT data has shown that children and parents assigned to the PACT intervention showed some reduction of a modified Autism Diagnostic Observation Schedule-Generic algorithm score compared with those assigned to TAU, although the effect size was not statistically significant. 22 However, the between-group effect size for the secondary outcomes of parental synchronous acts (as a proportion of total parent communication acts) and child initiations (as a proportion of total child acts) were substantial and statistically significant. In the evaluation of treatment effect mechanisms the goal is to understand the two-step mechanism by which the intervention influences the child behavioural outcome with the parent and then, in turn, generalises to behaviour with the external Autism Diagnostic Observation Schedule-Generic assessor. ...
The development of the capability and capacity to evaluate the outcomes of trials of complex interventions is a key priority of the National Institute for Health Research (NIHR) and the Medical Research Council (MRC). The evaluation of complex treatment programmes for mental illness (e.g. cognitive–behavioural therapy for depression or psychosis) not only is a vital component of this research in its own right but also provides a well-established model for the evaluation of complex interventions in other clinical areas. In the context of efficacy and mechanism evaluation (EME) there is a particular need for robust methods for making valid causal inference in explanatory analyses of the mechanisms of treatment-induced change in clinical outcomes in randomised clinical trials.
Objectives
The key objective was to produce statistical methods to enable trial investigators to make valid causal inferences about the mechanisms of treatment-induced change in these clinical outcomes. The primary objective of this report is to disseminate this methodology, aiming specifically at trial practitioners.
Methods
The three components of the research were (1) the extension of instrumental variable (IV) methods to latent growth curve models and growth mixture models for repeated-measures data; (2) the development of designs and regression methods for parallel trials; and (3) the evaluation of the sensitivity/robustness of findings to the assumptions necessary for model identifiability. We illustrate our methods with applications from psychological and psychosocial intervention trials, keeping the technical details to a minimum, leaving the reporting of the more theoretical and mathematically demanding results for publication in appropriate specialist journals.
Results
We show how to estimate treatment effects and introduce methods for EME. We explain the use of IV methods and principal stratification to evaluate the role of putative treatment effect mediators and therapeutic process measures. These results are extended to the analysis of longitudinal data structures. We consider the design of EME trials. We focus on designs to create convincing IVs, bearing in mind assumptions needed to attain model identifiability. A key area of application that has become apparent during this work is the potential role of treatment moderators (predictive markers) in the evaluation of treatment effect mechanisms for personalised therapies (stratified medicine). We consider the role of targeted therapies and multiarm trials and the use of parallel trials to help elucidate the evaluation of mediators working in parallel.
Conclusions
In order to demonstrate both efficacy and mechanism, it is necessary to (1) demonstrate a treatment effect on the primary (clinical) outcome, (2) demonstrate a treatment effect on the putative mediator (mechanism) and (3) demonstrate a causal effect from the mediator to the outcome. Appropriate regression models should be applied for (3) or alternative IV procedures, which account for unmeasured confounding, provided that a valid instrument can be identified. Stratified medicine may provide a setting where such instruments can be designed into the trial. This work could be extended by considering improved trial designs, sample size considerations and measurement properties.
Funding
The project presents independent research funded under the MRC–NIHR Methodology Research Programme (grant reference G0900678).
... Angrist et al. (1996) and Imbens and Rubin (1997) developed a framework to provide a more principled approach for the analyses of experiments with nonadherence, and Frangakis and Rubin (2002) extended those frameworks to develop the more general principal stratification framework. This framework has been applied to a wide variety of real-life problems involving complications such as censoring or truncation due to death (Zhang and Rubin, 2003) and the occurrence of intermediate variables that are thought to mediate the effects of treatments on the outcome (Gallop et al., 2009). VanderWeele (2011) provides a detailed review of principal stratification. ...
... A dominant type of LCA approaches is to adopt latent class mixture modeling, which views the observed data as a manifestation of multiple latent classes or subgroups. Such LCA methods have been well studied for various kinds of data, including standard uncensored data (Wedel et al., 1993;Gallop et al., 2009;Lim et al., 2014, for example), censored data (Farewell, 1982;Jedidi et al., 1993;Mair and Hudec, 2009;Qu et al., 2015;Egleston et al., 2017, for example), longitudinal data (Muthén and Shedden, 1999;Nagin, 1999;Muthén, 2004;Reinecke and Seddig, 2011;Lai et al., 2016;Jo et al., 2017;Bacci et al., 2019, among others), and longitudinal data in combination with survival data (Lin et al., 2002(Lin et al., , 2004Altstein et al., 2011;Proust-Lima et al., 2016;Hilton et al., 2018;Han et al., 2007;Han, 2009). However, LCA methods tailored to delineate the heterogeneity in recurrent event trajectory are limited. ...
Recurrent event data frequently arise in chronic disease studies, providing rich information on disease progression. The concept of latent class offers a sensible perspective to characterize complex population heterogeneity in recurrent event trajectories that may not be adequately captured by a single regression model. However, the development of latent class methods for recurrent event data has been sparse, typically requiring strong parametric assumptions and involving algorithmic issues. In this work, we investigate latent class analysis of recurrent event data based on flexible semiparametric multiplicative modelling. We derive a robust estimation procedure through novelly adapting the conditional score technique and utilizing the special characteristics of multiplicative intensity modelling. The proposed estimation procedure can be stably and efficiently implemented based on existing computational routines. We provide solid theoretical underpinnings for the proposed method, and demonstrate its satisfactory finite sample performance via extensive simulation studies. An application to a dataset from research participants at Goizueta Alzheimer's Disease Research Center illustrates the practical utility of our proposals.
... Based on the framework of counterfactuals, modern mediation approaches interpret the mediation effect as natural effects, controlled effects, and principal stratification effects, all of which can be interpreted causally because they are based 10 on the contrast among the potential outcomes within the same subject. Such approaches include Angrist et al. (1996), who apply two-stage least squares to estimate principal stratification effects among compliers; Ten Have et al. (2007), who propose rank preserving models (RPM) for controlled effects, Gallop et al. (2009), who focus on Bayesian approaches for principal stratification effects, and Imai et al. (2010a,b), who propose nonparametric identification of natural direct and indirect 15 effects. A comprehensive review of these causal approaches can be found in Coffman et al. (2016). ...
Causal mediation analysis has become popular in recent years. The goal of mediation analyses is to learn the direct effects of exposure on outcome as well as mediated effects on the pathway from exposure to outcome. A set of generalized structural equations to estimate the direct and indirect effects for mediation analysis is proposed when the number of mediators is of high-dimensionality. Specifically, a two-step procedure is considered where the penalization framework can be adopted to perform variable selection. A partial linear model is used to account for a nonlinear relationship among pre-treatment confounders and the response variable in each model. Procedures for estimating the coefficients for the treatment and the mediators in the structural models are developed. The obtained estimators can be interpreted as causal effects without imposing a linear assumption on the model structure. The performance of Sobel's method in obtaining the standard error and confidence interval for the estimated joint indirect effect is also evaluated in simulation studies. Simulation results show a superior performance of the proposed method. It is applied to an epidemiologic study in which the goal is to understand how DNA methylation mediates the effect of childhood trauma on regulation of human stress reactivity.
... In surrogate evaluation problems, principal stratification helps to clarify criteria for good surrogate endpoints (Frangakis andRubin, 2002, Gilbert andHudgens, 2008). In mediation analysis, principal stratification with respect to the mediator represents different causal mechanisms from the treatment to the outcome (Rubin, 2004, Gallop et al., 2009, Elliott, Raghunathan and Li, 2010, Mattei and Mealli, 2011. VanderWeele (2008) and Forastiere, Mattei and Ding (2018) linked the principal stratification approach with the direct and indirect effect approach and Jo (2008) linked the principal stratification approach with structural equation model for mediation analysis. ...
... Mixture models have been studied vastly for classifying subjects into different categories and inferring their association patterns or category-specific properties (Zeng et al., 2018). In the context of mediation analysis, previous mixture model approaches have primarily been proposed in the form of a principal stratification model (Gallop et al., 2009). Here, we introduce a Gaussian mixture model for the joint modeling of β mj and α aj and the subsequent inference of the composite association patterns. ...
Causal mediation analysis aims to characterize an exposure's effect on an outcome and quantify the indirect effect that acts through a given mediator or a group of mediators of interest. With the increasing availability of measurements on a large number of potential mediators, like the epigenome or the microbiome, new statistical methods are needed to simultaneously accommodate high‐dimensional mediators while directly target penalization of the natural indirect effect (NIE) for active mediator identification. Here, we develop two novel prior models for identification of active mediators in high‐dimensional mediation analysis through penalizing NIEs in a Bayesian paradigm. Both methods specify a joint prior distribution on the exposure‐mediator effect and mediator‐outcome effect with either (a) a four‐component Gaussian mixture prior or (b) a product threshold Gaussian prior. By jointly modelling the two parameters that contribute to the NIE, the proposed methods enable penalization on their product in a targeted way. Resultant inference can take into account the four‐component composite structure underlying the NIE. We show through simulations that the proposed methods improve both selection and estimation accuracy compared to other competing methods. We applied our methods for an in‐depth analysis of two ongoing epidemiologic studies: the Multi‐Ethnic Study of Atherosclerosis (MESA) and the LIFECODES birth cohort. The identified active mediators in both studies reveal important biological pathways for understanding disease mechanisms.
... Identification of causal mediation mechanisms in the absence of sequential ignorability can sometimes be achieved through the principal stratification approach (Gallop et al., 2009;Mattei and Mealli, 2011) or by leveraging ancillary variables that satisfy certain exclusion restrictions (Imai et al., 2013;Burgess et al., 2015;Frölich and Huber, 2017). A major strand of work in the health sciences uses baseline covariates interacted with random exposure assignment as instrumental variables for the effect of the mediator on the outcome (Ten Have et al., 2007;Dunn and Bentall, 2007;Albert, 2008;Small, 2012;Zheng and Zhou, 2015); see also the commentary by Ogburn (2012). ...
Although the exposure can be randomly assigned in studies of mediation effects, any form of direct intervention on the mediator is often infeasible. As a result, unmeasured mediator-outcome confounding can seldom be ruled out. We propose semiparametric identification of natural direct and indirect effects in the presence of unmeasured mediator-outcome confounding by leveraging heteroskedasticity restrictions on the observed data law. For inference, we develop semiparametric estimators that remain consistent under partial misspecification of the observed data model. We illustrate the robustness of the proposed estimators through both simulations and an application to evaluate the effect of self-efficacy on fatigue among health care workers during the COVID-19 outbreak.
... Mixture models have been studied vastly for classifying subjects into different categories and inferring their association patterns or category-specific properties (Zeng et al., 2018). In the context of mediation analysis, previous mixture model approaches have primarily been proposed in the form of a principal stratification model (Gallop et al., 2009). Here, we introduce a Gaussian mixture model for the joint modeling of β mj and α aj and the subsequent inference of the composite association patterns. ...
Causal mediation analysis aims to characterize an exposure's effect on an outcome and quantify the indirect effect that acts through a given mediator or a group of mediators of interest. With the increasing availability of measurements on a large number of potential mediators, like the epigenome or the microbiome, new statistical methods are needed to simultaneously accommodate high-dimensional mediators while directly target penalization of the natural indirect effect (NIE) for active mediator identification. Here, we develop two novel prior models for identification of active mediators in high-dimensional mediation analysis through penalizing NIEs in a Bayesian paradigm. Both methods specify a joint prior distribution on the exposure-mediator effect and mediator-outcome effect with either (a) a four-component Gaussian mixture prior or (b) a product threshold Gaussian prior. By jointly modeling the two parameters that contribute to the NIE, the proposed methods enable penalization on their product in a targeted way. Resultant inference can take into account the four-component composite structure underlying the NIE. We show through simulations that the proposed methods improve both selection and estimation accuracy compared to other competing methods. We applied our methods for an in-depth analysis of two ongoing epidemiologic studies: the Multi-Ethnic Study of Atherosclerosis (MESA) and the LIFECODES birth cohort. The identified active mediators in both studies reveal important biological pathways for understanding disease mechanisms.
... Since we do not observe both potential outcomes for each customer, these strata are latent, but the size of these groups and their average response can be estimated using a mixture model for the customers in the treatment group who are observed to purchase. This is a new adaptation of principal stratification (Frangakis and Rubin 2002), which has been used previously to analyze experiments with treatment non-compliance (Roy et al. 2007, Gallop et al. 2009, Barajas et al. 2016) and truncation by death (Rubin et al. 2006, Zhang et al. 2009, Ding et al. 2011. The details of this model are laid out in Section 2. ...
Advertising experiments often suffer from noisy responses making precise estimation of the average treatment effect (ATE) and evaluating ROI difficult. We develop a principal stratification model that improves the precision of the ATE by dividing the customers into three strata - those who buy regardless of ad exposure, those who buy only if exposed to ads and those who do not buy regardless. The method decreases the variance of the ATE by separating out the typically large share of customers who never buy and therefore have individual treatment effects that are exactly zero. Applying the procedure to 5 catalog mailing experiments with sample sizes around 140,000 shows a reduction of 36-57% in the variance of the estimate. When we include pre-randomization covariates that predict stratum membership, we find that estimates of customers' past response to similar advertising are a good predictor of stratum membership, even if such estimates are biased because past advertising was targeted. Customers who have not purchased recently are also more likely to be in the "never purchase" stratum. We provide simple summary statistics that firms can compute from their own experiment data to determine if the procedure is expected to be beneficial before applying it.
... Apart from interventions on the mechanism, we could also mitigate problems in causal interpretation of the mechanism to outcome path by other methods. Suggested statistical methods include analyzing the sensitivity of results on indirect effects to confounders of the mechanism to outcome relationship (Imai, Keele, & Tingley, 2010;Imai, Keele, & Yamamoto, 2010;VanderWeele, 2010), use of principal stratification ( Gallop et al., 2009;Joffe, Small, & Hsu, 2007) or inverse probability weighting (Coffman, 2011;Coffman & Zhong, 2012). In our review in Study I, we identified very few examples of any of these other techniques employed in studies of mechanisms of change in treatment of PTSS. ...
Background: For many who experience them, repeated traumatic events lead to chronic posttraumatic stress symptoms (PTSS). Forms of trauma-focused cognitive-behavioral therapy are able to treat PTSS among adults as well as children and adolescents. However, not all those suffering from PTSS benefit from such treatment. Availability and utilization of treatment are also significant problems both globally and in Finland. For further developing, better targeting, and effective implementation of treatments, it would be important to understand the underlying mechanisms of change by which they are able to reduce PTSS.
Theories of posttraumatic stress disorder suggest two psychological mechanisms of change in particular to be key to treating PTSS: improvements in overly negative posttraumatic cognitions and the integration, contextualization, or normalization of traumatic memories. This dissertation examined the role of changes in negative posttraumatic cognitions and traumatic memories in two interventions aimed at children and adolescents traumatized by war or violence and evaluated the overall level of evidence available for different mechanisms of change in the treatment of PTSS. Additionally, with a randomized controlled trial, we studied the feasibility, acceptability, and effectiveness of narrative exposure therapy in the treatment of multiply traumatized children and adolescents within the Finnish healthcare system.
Method: The empirical research in this dissertation is based on two sets of data. The first data set, collected from Gaza, Palestine, concerned 482 school-aged children randomized to take part in either a four-week group intervention, Teaching Recovery Techniques, or a waitlist. Using self-report measures, we collected information on the children’s experiences of war trauma, mental health, and posttraumatic cognitions before the intervention, during it, after it, and as follow-up six months later. The second data set concerned 50 children and adolescents 9–17 years of age living in Finland, who entered treatment at different healthcare units due to significant PTSS because of exposure to war or violence in the family. Half were randomized to receive narrative exposure therapy, while the rest received treatment as usual. Using mainly self-report measures, we collected information on their mental health, posttraumatic cognitions, and traumatic memories before and after treatment. In addition, the dissertation contains a systematic review collecting all available empirical evidence from randomized, controlled trials on the role of different mechanisms of change in psychological treatment of PTSS.
Results: The Teaching Recovery Techniques group intervention was not able to change the posttraumatic cognitions of Gazan children significantly. Such changes did not act as its mechanism of change. This may also explain its limited effects on PTSS. High levels of PTSS and depressive symptoms predicted particularly severe, stable posttraumatic cognitions.
Clinicians were able to implement narrative exposure therapy successfully in the treatment of multiply traumatized children and adolescents at different units within the Finnish healthcare system. Narrative exposure therapy was at least as effective in treating PTSS as treatment as usual. Some results suggested it was slightly more effective. Positive changes in posttraumatic cognitions and traumatic memories were both associated with amelioration of PTSS. However, we found no evidence of such changes acting as mechanisms of change specific to narrative exposure therapy. We found no evidence of overall significant changes in posttraumatic cognitions during treatment. Overall changes in traumatic memories were relatively small.
The systematic review found evidence for the importance of changes in negative posttraumatic cognitions in several forms of treatment. Negative cognitions related to the self may be especially relevant. As of yet, there is next to no empirical evidence on changes in traumatic memories as a mechanism of change. Some preliminary findings suggest mindfulness-based interventions can reduce PTSS by increasing levels of dispositional mindfulness. The evidence for other specific mechanisms of change is still very limited and mixed.
Conclusions: Exposure-based treatment is suitable for treatment of multiply traumatized children and adolescents within the Finnish healthcare system and does reduce PTSS. Changing the negative posttraumatic cognitions of children and adolescents traumatized by war and violence may be especially challenging. Still, it is a probable pathway to treating PTSS among them, as well. The effectiveness of group psychosocial interventions in reducing PTSS may be limited by their inability to affect posttraumatic cognitions. Changes in traumatic memories may be linked to recovery from PTSS, but evidence for such a link is still very limited.
... While both of these assumptions are plausible in the noncompliance application, they are questionable in more general mediation applications. Recent work, however, has extended the principal stratification framework to more general mediation analysis applications (Gallop et al., 2009;Jo, 2008;Page, 2012). ...
In this paper we examine ways to conceptualize and address potential bias that can arise when the mechanism for missing outcome data is at least partially associated with treatment assignment, an issue we refer to as treatment confounded missingness (TCM). In discussing TCM, we bring together concepts from the methodological literature on missing data, mediation, and principal stratification. We use a pair of simulation studies to demonstrate the main biasing properties of TCM and test different analytic approaches for estimating treatment effects given this missing data problem. We also demonstrate TCM and the different analytic approaches with empirical data from a study of a traditional high school that was converted to a charter school. The empirical illustration highlights the need to investigate possible TCM bias in high school intervention evaluations, where there is often an interest in studying the effects of an intervention or reform on both school persistence and academic achievement.
... We formulate the post-treatment effect modifiers analysis as a single principal stratification analysis. Principal stratification analysis is an analysis of causal effects among subgroups defined by the potential values of post-treatment variables (Frangakis and Rubin, 2002;Mealli and Rubin, 2002;Joffe et al., 2007;Gallop et al., 2009;Elliott et al., 2010;Gilbert et al., 2011;Page et al., 2015). Single principal stratification is a special case in which subgroups are defined by the potential value of the post-treatment variable under only one treatment condition (Angrist et al., 1996;Mattei and Mealli, 2007;Jin and Rubin, 2009;Jin et al., 2010;Schwartz et al., 2011;Dai et al., 2012;Frumento et al., 2012;Feller et al., 2016). ...
Class size reduction represents an appealing but costly education policy lever. A crucial question, therefore, is whether certain types of students benefit particularly from small classes, i.e. are there effect modifiers for the benefit of small classes? We use data from the Tennessee ‘Student/teacher achievement ratio’ study, which is a large randomized experiment of class size to address this question. In the Tennessee study, a challenge is that relatively few potential effect modifiers were measured at baseline. Yet many proxies for effect modifiers were measured after baseline. In an experiment, treatment effect modification based on pretreatment variables can be assessed by using standard regression techniques. For post‐treatment variables, however, such regression approaches are valid only under a strong sequential ignorability assumption. We develop two instrumental‐variable‐based approaches for studying effect modification based on post‐treatment variables. The key advantage of the method proposed is that it allows relaxation of the strong sequential ignorability assumption. However, this advantage comes at the cost of reliance on alternative assumptions that can be challenging to verify. We provide evidence that students who are not academically engaged outside the classroom benefit more from small classes than students who are academically engaged.
... VanderWeele, 2008; Gallop et al., 2009;Elliott et al., 2010), and fuzzy regression discontinuity designs (e.g. Li et al., 2015;Chib and Jacobi, 2016). ...
Inferring causal effects of treatments is a central goal in many disciplines. The potential outcomes framework is a main statistical approach to causal inference, in which a causal effect is defined as a comparison of the potential outcomes of the same units under different treatment conditions. Because for each unit at most one of the potential outcomes is observed and the rest are missing, causal inference is inherently a missing data problem. Indeed, there is a close analogy in the terminology and the inferential framework between causal inference and missing data. Despite the intrinsic connection between the two subjects, statistical analyses of causal inference and missing data also have marked differences in aims, settings and methods. This article provides a systematic review of causal inference from the missing data perspective. Focusing on ignorable treatment assignment mechanisms, we discuss a wide range of causal inference methods that have analogues in missing data analysis, such as imputation, inverse probability weighting and doubly-robust methods. Under each of the three modes of inference--Frequentist, Bayesian, and Fisherian randomization--we present the general structure of inference for both finite-sample and super-population estimands, and illustrate via specific examples. We identify open questions to motivate more research to bridge the two fields.
... The CE and CA frameworks have tradeoffs in terms of assumptions, bias in parameter estimation and variability [39]. Estimation methods within the CA framework have been shown to have less bias, but more variability than standard methods within the CE framework [40]. As the parameters of the proposed structural model have a direct mapping to the parameters of the CA model, these models offer the potential for assumptions that are reasonable to make in one framework to aid in informing the parameter values within the alternative framework. ...
Two paradigms for the evaluation of surrogate markers in randomized clinical trials have been proposed: the causal effects paradigm and the causal association paradigm. Each of these paradigms rely on assumptions that must be made to proceed with estimation and to validate a candidate surrogate marker (S) for the true outcome of interest (T). We consider the setting in which S and T are Gaussian and are generated from structural models that include an unobserved confounder. Under the assumed structural models, we relate the quantities used to evaluate surrogacy within both the causal effects and causal association frameworks. We review some of the common assumptions made to aid in estimating these quantities and show that assumptions made within one framework can imply strong assumptions within the alternative framework. We demonstrate that there is a similarity, but not exact correspondence between the quantities used to evaluate surrogacy within each framework, and show that the conditions for identifiability of the surrogacy parameters are different from the conditions, which lead to a correspondence of these quantities.
... The mixture models are used to estimate outcomes and probabilities of latent survival class membership within intervention surgery arms; the mixture models are not used to estimate the probability of principal stratum membership directly. Examples of the mixture formulation of principal stratification include the likelihood based methods of Frumento et al. (2012) and equations 10-14 in Gallop et al. (2009). Here, we do not estimate the probability of principal stratum membership directly through a model based on mixtures of principal strata. ...
1Abstract
Rates of kidney cancer have been increasing, with small incidental tumors experiencing the fastest growth rates. Much of the increase could be due to increased use of CT scans, MRIs, and ultrasounds for unrelated conditions. Many tumors might never have been detected or become symptomatic in the past. This suggests that many patients might benefit from less aggressive therapy, such as active surveillance by which tumors are surgically removed only if they become sufficiently large. However, it has been difficult for clinicians to identify subgroups of patients for whom treatment might be especially beneficial or harmful. In this work, we use a principal stratification framework to estimate the proportion and characteristics of individuals who have large or small hazard rates of death in two treatment arms. This allows us to assess who might be helped or harmed by aggressive treatment. We also use Weibull mixture models. This work differs from much previous work in that the survival classes upon which principal stratification is based are latent variables. That is, survival class is not an observed variable. We apply this work using Surveillance Epidemiology and End Results-Medicare claims data. Clinicians can use our methods for investigating treatments with heterogeneous effects.
... In other words, differential impacts across strata could also be due to differences in observed or unobserved characteristics other than care type. See Gallop et al. (2009) for a discussion of using principal stratification for mediation analysis, which generally requires much stronger assumptions than those presented here. ...
Early childhood education research often compares a group of children who receive the intervention of interest to a group of children who receive care in a range of different care settings. In this paper, we estimate differential impacts of an early childhood intervention by alternative care type, using data from the Head Start Impact Study, a large-scale randomized evaluation. To do so, we utilize a Bayesian principal stratification framework to estimate separate impacts for two types of Compliers: those children who would otherwise be in other center-based care when assigned to control and those who would otherwise be in home-based care. We find strong, positive short-term effects of Head Start on receptive vocabulary for those Compliers who would otherwise be in home-based care. By contrast, we find no meaningful impact of Head Start on vocabulary for those Compliers who would otherwise be in other center-based care. Our findings suggest that alternative care type is a potentially important source of variation in early childhood education interventions.
... To establish the causal interpretation of mediation analysis, several frameworks have been developed within Rubin's causal model (Rubin, 1974), including principle stratification models (Gallop et al., 2009) and counterfac-tual models, such as structural mean model (Robins and Greenland, 1992). Principal strata method is often used in the context of compliance of the intervention and often needs the level of mediator and exposure to be small, for example, binary. ...
Causal mediation analysis is an important statistical method in social and medical studies, as it can provide insights about why an intervention works and inform the development of future interventions. Currently, most causal mediation methods focus on mediation effects defined on a mean scale. However, in health-risk studies, such as alcohol or risky sex, outcomes are typically count data and heavily skewed. Thus, mediation effects in these setting would be natural on a rate ratio scale, such as in Poisson and negative binomial regression methods. Existing methods also mainly rely on the assumption of no unmeasured confounding between mediator and outcome. To allow for potential confounders between the mediator and outcome, we define the direct and mediator effects on a new scale and propose a multiplicative structural mean model for mediation analysis with count outcomes. The estimator is compared with both Poisson and negative binomial regression methods assuming sequential ignorability using a simulation study and a real world example about an alcohol-related intervention study. Mediation analyses using the new methods confirm the study hypothesis that the intervention decreases drinking by decreasing individual's normative perceptions of alcohol use.
... The compliance behavior defined by the potential treatment acceptances is used to address the noncompliance problem (Angrist et al., 1996), the potential survival status is used to evaluate the effect on the quality of life with truncation by death (Rubin, 2006;Ding et al., 2011;Yang and Small, 2015), and the potential employment status is used to deal with the truncation of wages due to unemployment in the evaluation of job-training programs (Zhang and Rubin, 2003;Zhang et al., 2009;Frumento et al., 2012). The potential response indicators are used to address non-ignorable nonresponse problems (Frangakis and Rubin, 1999;Mealli and Pacini, 2008;Mattei et al., 2014), and the potential intermediate variables are used to define direct and indirect effects (Rubin, 2004;Gallop et al., 2009;Mattei and Mealli, 2011). ...
... The compliance behavior defined by the potential treatment acceptances is used to address the noncompliance problem (Angrist et al., 1996), the potential survival status is used to evaluate the effect on the quality of life with truncation by death (Rubin, 2006;Ding et al., 2011;Yang and Small, 2015), and the potential employment status is used to deal with the truncation of wages due to unemployment in the evaluation of job-training programs (Zhang and Rubin, 2003;Zhang et al., 2009;Frumento et al., 2012). The potential response indicators are used to address non-ignorable nonresponse problems (Frangakis and Rubin, 1999;Mealli and Pacini, 2008;Mattei et al., 2014), and the potential intermediate variables are used to define direct and indirect effects (Rubin, 2004;Gallop et al., 2009;Mattei and Mealli, 2011). ...
Principal stratification is a causal framework to analyze randomized
experiments with a post-treatment variable between the treatment and endpoint
variables. Because the principal strata defined by the potential outcomes of
the post-treatment variable are not observable, we generally cannot identify
the causal effects within principal strata. Motivated by a real data set of
phase III adjuvant colon clinical trials, we propose approaches to identifying
and estimating the principal causal effects via multiple trials. For the
identifiability, we remove the commonly-used exclusion restriction assumption
by stipulating that the principal causal effects are homogeneous across these
trials. To remove another commonly-used monotonicity assumption, we give a
necessary condition for the local identifiability, which requires at least
three trials. Applying our approaches to the data from adjuvant colon clinical
trials, we find that the commonly-used monotonicity assumption is untenable,
and disease-free survival with three-year follow-up is a valid surrogate
endpoint for overall survival with five-year follow-up, which satisfies both
the causal necessity and the causal sufficiency. We also propose a sensitivity
analysis approach based on Bayesian hierarchical models to investigate the
impact of the deviation from the homogeneity assumption.
... Based on the framework of counterfactuals, more recent approaches interpret the mediation effect as natural effects, controlled effects and principal stratification effects, all of which can be interpreted causally because they are based on the difference among the potential outcomes within the same subject. Such approaches include Imai et al. (2010b), who propose nonparametric identification of natural direct and indirect effects; Angrist et al. (1996), who apply two-stage least squares to estimate principal stratification effects among compliers; Ten Have et al. (2007), who propose rank preserving models (RPM) for controlled effects, and Gallop et al. (2009), who focus on Bayesian approaches for principal stratification effects. ...
We examine the causal effect of parental restrictive feeding practices on children's weight status. An important mediator is children's self-regulation status. Recent approaches interpret mediation effects on the basis of the potential outcomes framework. Inverse probability weighting based on propensity scores are used to adjust for confounding and to reduce the dimensionality of confounders simultaneously. We show that combining machine learning algorithms and logistic regression to estimate the propensity scores can be more accurate and efficient in estimating the controlled direct effects than using logistic regression alone. A data application shows that the causal effect of mother's restrictive feeding differs according to whether the daughter eats in the absence of hunger.
... This is similar to the instrumental variable approach to noncompliance in randomized experiments (Angrist et al., 1996;Imbens and Rubin, 1997;Hirano et al., 2000), a special case of principal stratification. Recently a rapid growing literature has extended principal stratification to a wide range of settings in both experimental and observational studies, including "censoring by death" (Rubin, 2006;Mattei and Mealli, 2007;Zhang et al., 2009), missing data (Mattei et al., 2014), surrogate endpoints (Gilbert et al., 2003;Li et al., 2009Li et al., , 2011, mediation analysis (Gallop et al., 2009;Elliott et al., 2010), and designs (Mattei and Mealli, 2011). More complex settings such as ordinal or continuous intermediate variables have also been explored (Frangakis et al., 2004;Jin and Rubin, 2008;Griffin et al., 2008;Schwartz et al., 2011). ...
It has been argued that innovation in transaction technology may modify the
cash holding behaviour of agents, as debit card holders may either withdraw
cash from ATMs or purchase items using POS devices at retailers. In this paper,
within the Rubin Causal Model, we investigate the causal effects of the use of
debit cards on the cash inventories held by households using data from the
Italy Survey of Household Income and Wealth (SHIW). We adopt the principal
stratification approach to incorporate the share of debit card holders who do
not use this payment instrument. We use a regression model with the propensity
score as the single predictor to adjust for the imbalance in observed
covariates. We further develop a sensitivity analysis approach to assess the
sensitivity of the proposed model to violation to the key unconfoundedness
assumption. Our empirical results suggest statistically significant negative
effects of debit cards on the household cash level in Italy.
The study of treatment effects is often complicated by noncompliance and missing data. In the one-sided noncompliance setting where of interest are the complier and noncomplier average causal effects, we address outcome missingness of the latent missing at random type (LMAR, also known as latent ignorability). That is, conditional on covariates and treatment assigned, the missingness may depend on compliance type. Within the instrumental variable (IV) approach to noncompliance, methods have been proposed for handling LMAR outcome that additionally invoke an exclusion restriction-type assumption on missingness, but no solution has been proposed for when a non-IV approach is used. This article focuses on effect identification in the presence of LMAR outcomes, with a view to flexibly accommodate different principal identification approaches. We show that under treatment assignment ignorability and LMAR only, effect nonidentifiability boils down to a set of two connected mixture equations involving unidentified stratum-specific response probabilities and outcome means. This clarifies that (except for a special case) effect identification generally requires two additional assumptions: a specific missingness mechanism assumption and a principal identification assumption. This provides a template for identifying effects based on separate choices of these assumptions. We consider a range of specific missingness assumptions, including those that have appeared in the literature and some new ones. Incidentally, we find an issue in the existing assumptions, and propose a modification of the assumptions to avoid the issue. Results under different assumptions are illustrated using data from the Baltimore Experience Corps Trial.
The draft ICH E9 (R1) addendum by the International Conference on Harmonisation working group opens for the use of a principal stratum in the analysis of study data for regulatory purpose, if a relevant estimand can be justified. Inspired by the so-called complier average causal effect and work within this framework, we propose a new estimator – Extrapolation based on propensity to comply – that estimates the treatment effect of an active treatment A relative to a comparator B (active or placebo), in the principal stratum of patients who would comply, if they were treated with treatment A. Sensitivity of the approach to the number of covariates and their ability to predict principal stratum membership is investigated based on data from a placebo-controlled study of brexpiprazole in schizophrenia. The performance of the estimator is compared with another estimator that is also based on principal stratification. A simulation study supports that the proposed estimator has a negligible bias even with a small sample size, except when the covariate predicting compliance is very weak. Not surprisingly, precision of the estimate increases substantially with stronger predictors of compliance. It is shown analytically that the estimator used for comparison is biased in general, and the bias is explicitly derived in a simple case with a binary predictor of compliance. While the proposed methodology is technically easy to implement, choosing predictors for modelling compliance is a key issue in practice.
Mediator models have been developed primarily under the assumption of no-unmeasured-confounding. In many situations, this assumption is violated and may lead to the identification of mediator variables that actually are statistical artifacts. The rank preserving model (RPM) is an alternative approach to estimate controlled direct and mediator effects. It is based on the structural mean models framework and a no-effect-modifier assumption. The RPM assumes that unobserved confounders do not interact with treatment or mediators. This assumption is often more plausible to hold than the no-unmeasured-confounder assumption. So far, models using the no-effect-modifier assumption have been rarely used, which might be due to its low power and inefficiency in many scenarios. Here, a semi-parametric nonlinear extension, the nRPM, is proposed that overcomes this inefficiency using thin plate regression splines that both increase the predictive power of the model and decrease the misspecification present in many situations. In a simulation study, it is shown that the nRPM provides estimates that are robust against the violation of the no-effect-modifier assumption and that are substantively more efficient than those of the RPM. The model is illustrated using a data set on CD4 cell counts in a context of the human immunodeficiency virus (HIV).
This paper revisits results from a field experiment, conducted in Florence, Italy, to study the effects of incentives provided to high school teens to motivate them to visit art museums. In the experiment, different classes of students were randomized to three types of encouragement and were offered a free visit to a main museum in the city. Using the principal stratification framework, the paper explores causal pathways that may lead students to increase future visits, as induced by the encouragement received, or by the individual experience of the proposed free museum visit, or by the spillover of classmates’ experience. We do so by estimating and interpreting the causal effects of the three forms of encouragement within the principal strata defined by compliance behaviors. Bayesian inferential methods are used to derive the posterior distributions of weakly identified causal parameters.
In randomized clinical trials, it is often of interest to estimate the effect of treatment on quality of life (QOL), in addition to those on the event itself. When an event occurs in some patients prior to QOL score assessment, investigators may compare QOL scores between patient subgroups defined by the event after randomization. However, owing to postrandomization selection bias, this analysis can mislead investigators about treatment efficacy and result in paradoxical findings. The recent Japanese Osteoporosis Intervention Trial (JOINT‐02), which compared the benefits of a combination therapy for fracture prevention with those of a monotherapy, exemplifies the case in point; the average QOL score was higher in the combination therapy arm for the unfractured subgroup but was lower for the fractured subgroup. To address this issue, principal strata effects (PSEs), which are treatment effects estimated within subgroups of individuals stratified by potential intermediate variable, have been discussed in the literature. In this paper, we describe a simple procedure for estimating the PSEs using marginal structural models. This procedure utilizes SAS code for the estimation. In addition, we present a simple sensitivity analysis method for examining the resulting estimates. The analyses of JOINT‐02 data using these methods revealed that QOL scores were higher in the combination therapy arm than in the monotherapy arm for both subgroups.
Objective:
To introduce readers to instrumental variable analyses for causal inferences using as an example a test of the hypothesis that the quality of the therapeutic alliance has a causal role in relation to the outcome of psychotherapy.
Method:
We used data from a recent non-inferiority trial of cognitive and dynamic therapies for major depressive disorder in a community mental health setting. The data (N = 161) were analyzed using standard approaches as well as a multilevel 2-stage instrumental variables approach that allows for causal interpretations by removing the influence of unmeasured confounds.
Results:
Instrumental variables were created at the patient and therapist level using baseline patient and therapist variables. These baseline variables predicted the alliance but were otherwise unrelated to treatment outcome other than through their effects on the alliance. Standard multilevel mixed effects analyses revealed statistically significant associations of the alliance with outcome at the therapist level of analysis. The therapist level effect remained statistically significant when using the instrumental variables approach.
Conclusion:
Our results support the hypothesis that, at least at the therapist level, the alliance plays a causal role in producing better outcomes. Instrumental variable analyses can be a useful tool to supplement standard analyses.
Mediation analysis seeks to understand the mechanism by which a treatment affects an outcome. Many conventional and causal mediation approaches have been developed for continuous outcomes. When the outcome of interest is non-continuous (e.g., binary, count, or zero-inflated count), mediation approaches relying on linear models may not be appropriate. This chapter introduces the framework and definition of direct and indirect (mediation) treatment effects on non-continuous outcomes around and through intermediate variables (mediators). Different mediation approaches for non-continuous outcomes are discussed under a variety of settings where post-baseline confounders and unmeasured confounders may be a concern, and under a series of assumptions with and without sequential ignorability. Methods are illustrated with application to a randomized dental caries prevention trial.
Objective:
In recent years, a shift has occurred from identifying variables that mediate the effects of psychosocial interventions for problematic alcohol use to trying to identify effective mechanisms of behavior change (MOBCs) that lead to better drinking-related and other outcomes. Although implementing interventions targeting specific MOBCs has considerable conceptual and intuitive appeal, it is important that strong empirical evidence be available that supports such implementation. One aim of this article is to consider some prior mediational analyses of the effects of alcohol-focused psychosocial treatments to illustrate findings and the types of evidence that typically have been presented. A second aim is to consider methodological approaches, specifically certain statistical analyses of observational data and experimental designs, that could provide more rigorous evidence regarding the causal effects of mechanisms of change (mediators) in producing desired drinking-related outcomes.
Results:
Prior analyses that have examined the association of a mediator with an outcome, or the relationship between a mediator and an outcome with assigned treatment controlled, do not provide strong evidence regarding the causal effects of mediators on outcomes. Unfortunately, experimental designs that directly manipulate mediators seem difficult to apply in the alcohol treatment field as an approach to examine mediator/MOBC effects.
Conclusion:
To produce more compelling evidence, we need to investigate interventions that target a specific MOBC and then capitalize on random assignment to intervention/control conditions by using principal stratification or instrumental variable analyses to identify the effects of intervention-induced mediator change on outcomes.
In assessing causal mediation effects in randomized studies, a challenge is that the direct and indirect effects can vary across participants due to different measured and unmeasured characteristics. In that case, the population effect estimated from standard approaches implicitly averages over and does not estimate the heterogeneous direct and indirect effects. We propose a Bayesian semiparametric method to estimate heterogeneous direct and indirect effects via clusters, where the clusters are formed by both individual covariate profiles and individual effects due to unmeasured characteristics. These cluster-specific direct and indirect effects can be estimated through a set of regression models where specific coefficients are clustered by a stick-breaking prior. To let clustering be appropriately informed by individual direct and indirect effects, we specify a data-dependent prior. We conduct simulation studies to assess performance of the proposed method compared to other methods. We use this approach to estimate heterogeneous causal direct and indirect effects of an expressive writing intervention for patients with renal cell carcinoma.
Treatment effects may be moderated by individual or contextual characteristics or by other concurrent or consecutive treatments. This chapter reveals the conceptual confusion in past literature and aims to clarify the definitions of “moderated treatment effects” in terms of potential outcomes. The chapter then reviews experimental designs including randomized block designs, factorial designs, and multisite randomized trials and the corresponding analytic methods that are suitable for evaluating moderated treatment effects. It also introduces principal stratification, a relatively new approach for disclosing the heterogeneity of treatment effects across latent subpopulations.
This chapter discusses extensions of ratio-of-mediator-probability weighting (RMPW) to studies of moderated mediation, that is, mediation mechanisms moderated by individual or contextual pretreatment characteristics. Additionally, RMPW is extended to investigations of complex mediation mechanisms that involve concurrent or consecutive mediators. Instead of defining each causal effect as a function of path coefficients, this chapter emphasizes decomposing the total treatment effect into causal effects defined in terms of potential outcomes. RMPW is utilized to estimate each of the population average counterfactual outcomes. This strategy minimizes the reliance on arbitrary assumptions about the structural form of the outcome model.
This chapter reviews and compares the existing analytic methods for investigating causal mediation with experimental or quasi-experimental data. Experimental data typically involve the randomization of treatment only; quasi-experimental data do not involve experimental manipulation of either the treatment or the mediator. To ease the comparison across these various analytic methods, this review is restricted to the case of a single mediator. The causal effects of key interest are the natural direct effect and the natural indirect effect as defined in Chapter 9 that decompose the total effect. Some of the analytic methods are well-known whereas others have been proposed only recently. A comparison of identification and model-based assumptions across all these methods are summarized in tables.
It has been argued that the use of debit cards may modify cash holding behaviour, as debit card holders may either withdraw cash from automated teller machines or purchase items by using point-of-sale devices at retailers. Within the Rubin causal model, we investigate the causal effects of the use of debit cards on the cash inventories held by households by using data from the Italy Survey of Household Income and Wealth. We adopt the principal stratification approach to incorporate the share of debit card holders who do not use this payment instrument. We use a regression model with the propensity score as the single predictor to adjust for the imbalance in observed covariates. We further develop a sensitivity analysis approach to assess the sensitivity of the proposed model to violation of the key unconfoundedness assumption. Our empirical results suggest statistically significant negative effects of debit cards on the household cash level in Italy.
Mediation occurs as part of a hypothesized causal chain of events: An intervention or treatment, T, has an effect on the mediator, M, which then affects an outcome variable, Y. Within the potential outcomes framework for causal inference, three different definitions of the mediation effects have been proposed: principal strata effects (e.g., Rubin, Scand. J. Stat. 31:161–170, 2004; Jo, Psychol. Methods 13:314–336, 2008), natural effects (e.g., Pearl, Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, 2001; Imai et al., Psychol. Methods 15:309–334, 2010), and controlled effects (e.g., Robins and Greenland, Epidemiology 3:143–155, 1992; VanderWeele, Epidemiology 20:18–26, 2009). We illustrate that each of these definitions answers a different scientific question. We examine five different estimators of the various definitions and discuss identifying assumptions about unmeasured confounding, the existence of direct effects (i.e., the effect of T on Y that is not due to M), iatrogenic effects of T on M, the existence of post-treatment confounders, and the existence of interactions. We assess the robustness of each of the estimators to violations of the assumptions using a simulation study that systematically challenges different aspects of these assumptions. We found that when no assumptions were violated, as may be expected, each approach was unbiased for its respective population value and 95 % confidence interval (CI) coverage was maintained. However, when assumptions are violated, the effects may be severely biased and 95 % CI coverage is not maintained. We suggest that researchers choose the appropriate definition based on the scientific question to be addressed and the identifying assumptions that are plausible given their data.
IntroductionPotential mediators in psychological treatment trialsMethods for mediation in psychological treatment trialsCausal mediation analysis using instrumental variables estimationCausal mediation analysis using principal stratificationOur motivating example: The SoCRATES trialConclusions
AcknowledgementsReferences
Mediation: Direct and indirect effectsThe mediation formula: A simple solution to a thorny problemRelation to other methodsConclusions
AcknowledgmentsReferences
Exploration of causal mechanisms is often important for researchers and policymakers to understand how an intervention works and how it can be improved. This task can be crucial in clustered encouragement designs (CED). Encouragement design studies arise frequently when the treatment cannot be enforced because of ethical or practical constrains and an encouragement intervention (information campaigns, incentives, etc) is conceived with the purpose of increasing the uptake of the treatment of interest. By design, encouragements always entail the complication of non-compliance. Encouragements can also give rise to a variety of mechanisms, particularly when encouragement is assigned at cluster level. Social interactions among units within the same cluster can result in spillover effects. Disentangling the effect of encouragement through spillover effects from that through the enhancement of the treatment would give better insight into the intervention and it could be compelling for planning the scaling-up phase of the program. Building on previous works on CEDs and non-compliance, we use the principal stratification framework to define stratum-specific causal effects, that is, effects for specific latent subpopulations, defined by the joint potential compliance statuses under both encouragement conditions. We show how the latter stratum-specific causal effects are related to the decomposition commonly used in the literature and provide flexible homogeneity assumptions under which an extrapolation across principal strata allows one to disentangle the effects. Estimation of causal estimands can be performed with Bayesian inferential methods using hierarchical models to account for clustering. We illustrate the proposed methodology by analyzing a cluster randomized experiment implemented in Zambia and designed to evaluate the impact on malaria prevalence of an agricultural loan program intended to increase the bed net coverage. Farmer households assigned to the program could take advantage of a deferred payment and a discount in the purchase of new bed nets. Our analysis shows a lack of evidence of an effect of the offering of the program to a cluster of households through spillover effects, that is through a greater bed net coverage in the neighborhood.
Mediation analyses are critical to understanding why behavioral interventions work. To yield a causal interpretation, common mediation approaches must make an assumption of “sequential ignorability.” The current article describes an alternative approach to causal mediation called structural mean models (SMMs). A specific SMM called a rank-preserving model (RPM) is introduced in the context of an applied example. Particular attention is given to the assumptions of both approaches to mediation. Applying both mediation approaches to the college student drinking data yield notable differences in the magnitude of effects. Simulated examples reveal instances in which the traditional approach can yield strongly biased results, whereas the RPM approach remains unbiased in these cases. At the same time, the RPM approach has its own assumptions that must be met for correct inference, such as the existence of a covariate that strongly moderates the effect of the intervention on the mediator and no unmeasured confounders that also serve as a moderator of the effect of the intervention or the mediator on the outcome. The RPM approach to mediation offers an alternative way to perform mediation analysis when there may be unmeasured confounders. 2015
We develop a model to estimate the causal effect of living arrangement (living alone versus living with someone) on cognitive decline based on a 15-year prospective cohort study, where episodic memory function is measured every 5 years. One key feature of the model is the combination of propensity score matching to balance confounding variables between the two living arrangement groups—to reduce bias due to unbalanced covariates at baseline, with a pattern–mixture model for longitudinal data—to deal with non-ignorable dropout. A fully Bayesian approach allows us to convey the uncertainty in the estimation of the propensity score and subsequent matching in the inference of the causal effect of interest. The analysis conducted adds to previous studies in the literature concerning the protective effect of living with someone, by proposing a modelling approach treating living arrangement as an exposure.
A Monte Carlo study compared 14 methods to test the statistical significance of the intervening variable effect. An intervening variable (mediator) transmits the effect of an independent variable to a dependent variable. The commonly used R. M. Baron and D. A. Kenny (1986) approach has low statistical power. Two methods based on the distribution of the product and 2 difference-in-coefficients methods have the most accurate Type I error rates and greatest statistical power except in 1 important case in which Type I error rates are too high. The best balance of Type I error and statistical power across all cases is the test of the joint significance of the two effects comprising the intervening variable effect.
In this article, we attempt to distinguish between the properties of moderator and mediator variables at a number of levels. First, we seek to make theorists and researchers aware of the importance of not using the terms moderator and mediator interchangeably by carefully elaborating, both conceptually and strategically, the many ways in which moderators and mediators differ. We then go beyond this largely pedagogical function and delineate the conceptual and strategic implications of making use of such distinctions with regard to a wide range of phenomena, including control and stress, attitudes, and personality traits. We also provide a specific compendium of analytic procedures appropriate for making the most effective use of the moderator and mediator distinction, both separately and in terms of a broader causal system that includes both moderators and mediators. (46 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The literature contains much discussion about which criteria should be applied to the selection of candidates for expatriate assignments. We designed an executive teaching exercise around this problem and obtained rankings of criteria from 12 of the leading international scholars in intercultural management, comparing these with data gathered from almost 700 business participants. Results indicate that their ratings converge with the expert rankings in many respects, especially in the high importance attached to personological factors; but diverge from experts in some particulars, such as undervaluing spouse support and overvaluing achievement drive. The article goes on to examine factors influencing their judgements. Of these, nationality has the most marked effects, with some particularly strong contrasts between Eastern and Western cultures. From our analysis we conclude that even so-called neutral criteria carry associated cultural value content, and that even the most sophisticated global executives are subject to biased assessments of these criteria.
We outline a framework for causal inference in setting where assignment to a binary treatment is ignorable, but compliance with the assignment is not perfect so that the receipt of treatment is nonignorable. To address the problems associated with comparing subjects by the ignorable assignment--an "intention-to-treat analysis"--we make use of instrumental variables, which have long been used by economists in the context of regression models with constant treatment effects. We show that the instrumental variables (IV) estimand can be embedded within the Rubin Causal Model (RCM) and that under some simple and easily interpretable assumptions, the IV estimand is the average causal effect for a subgroup of units, the compliers. Without these assumptions, the IV estimand is simply the ratio of intention-to-treat causal estimands with no interpretation as an average causal effect. The advantages of embedding the IV approach in the RCM are that it clarifies the nature of critical assumptions needed for a causal interpretation, and moreover allows us to consider sensitivity of the results to deviations from key assumptions in a straightforward manner. We apply our analysis to estimate the effect of veteran status in the Vietnam era on mortality, using the lottery number assigned priority for the draft as an instrument, and we use our results to investigate the sensitivity of the conclusions to critical assumptions. Statistics Version of Record
In the portion of the paper translated here, Neyman introduces a model for the analysis of field experiments conducted for the purpose of comparing a number of crop varieties, which makes use of a double-indexed array of unknown potential yields, one index corresponding to varieties and the other to plots. The yield corresponding to only one variety will be observed on any given plot, but through an urn model embodying sampling without replacement from this doubly indexed array, Neyman obtains a formula for the variance of the difference between the averages of the observed yields of two varieties. This variance involves the variance over all plots of the potential yields and the correlation coefficient $r$ between the potential yields of the two varieties on the same plot. Since it is impossible to estimate $r$ directly, Neyman advises taking $r = 1$, observing that in practice this may lead to using too large an estimated standard deviation, when comparing two variety means.
For most of this century, randomization has been a cornerstone of scientific experimentation, especially when dealing with humans as experimental units. In practice, however, noncompliance is relatively common with human subjects, complicating traditional theories of inference that require adherence to the random treatment assignment. In this paper we present Bayesian inferential methods for causal estimands in the presence of noncompliance, when the binary treatment assignment is random and hence ignorable, but the binary treatment received is not ignorable. We assume that both the treatment assigned and the treatment received are observed. We describe posterior estimation using EM and data augmentation algorithms. Also, we investigate the role of two assumptions often made in econometric instrumental variables analyses, the exclusion restriction and the monotonicity assumption, without which the likelihood functions generally have substantial regions of maxima. We apply our procedures to real and artificial data, thereby demonstrating the technology and showing that our new methods can yield valid inferences that differ in practically important ways from those based on previous methods for analysis in the presence of noncompliance, including intention-to-treat analyses and analyses based on econometric instrumental variables techniques. Finally, we perform a simulation to investigate the operating characteristics of the competing procedures in a simple setting, which indicates relatively dramatic improvements in frequency operating characteristics attainable using our Bayesian procedures.
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.
In this article, we attempt to distinguish between the properties of moderator and mediator variables at a number of levels. First, we seek to make theorists and researchers aware of the importance of not using the terms moderator and mediator interchangeably by carefully elaborating, both conceptually and strategically, the many ways in which moderators and mediators differ. We then go beyond this largely pedagogical function and delineate the conceptual and strategic implications of making use of such distinctions with regard to a wide range of phenomena, including control and stress, attitudes, and personality traits. We also provide a specific compendium of analytic procedures appropriate for making the most effective use of the moderator and mediator distinction, both separately and in terms of a broader causal system that includes both moderators and mediators.
A Monte Carlo study compared 14 methods to test the statistical significance of the intervening variable effect. An intervening variable (mediator) transmits the effect of an independent variable to a dependent variable. The commonly used R. M. Baron and D. A. Kenny (1986) approach has low statistical power. Two methods based on the distribution of the product and 2 difference-in-coefficients methods have the most accurate Type I error rates and greatest statistical power except in 1 important case in which Type I error rates are too high. The best balance of Type I error and statistical power across all cases is the test of the joint significance of the two effects comprising the intervening variable effect.
Many randomized experiments suffer from noncompliance. Some of these experiments, so-called encouragement designs, can be expected to have especially large amounts of noncompliance, because encouragement to take the treatment rather than the treatment itself is randomly assigned to individuals. We present an extended framework for the analysis of data from such experiments with a binary treatment, binary encouragement, and background covariates. There are two key features of this framework: we use an instrumental variables approach to link intention-to-treat effects to treatment effects and we adopt a Bayesian approach for inference and sensitivity analysis. This framework is illustrated in a medical example concerning the effects of inoculation for influenza. In this example, the analyses suggest that positive estimates of the intention-to-treat effect need not be due to the treatment itself, but rather to the encouragement to take the treatment: the intention-to-treat effect for the subpopulation who would be inoculated whether or not encouraged is estimated to be approximately as large as the intention-to-treat effect for the subpopulation whose inoculation status would agree with their (randomized) encouragement status whether or not encouraged. Thus, our methods suggest that global intention-to-treat estimates, although often regarded as conservative, can be too coarse and even misleading when taken as summarizing the evidence in the data for the effects of treatments.
Recently, instrumental variables methods have been used to address non-compliance in randomized experiments. Complicating such analyses is often the presence of missing data. The standard model for missing data, missing at random (MAR), has some unattractive features in this context. In this paper we compare MAR-based estimates of the complier average causal effect (CACE) with an estimator based on an alternative, nonignorable model for the missing data process, developed by Frangakis and Rubin (1999, Biometrika, 86, 365-379). We also introduce a new missing data model that, like the Frangakis-Rubin model, is specially suited for models with instrumental variables, but makes different substantive assumptions. We analyze these issues in the context of a randomized trial of breast self-examination (BSE). In the study two methods of teaching BSE, consisting of either mailed information about BSE (the standard treatment) or the attendance of a course involving theoretical and practical sessions (the new treatment), were compared with the aim of assessing whether teaching programs could increase BSE practice and improve examination skills. The study was affected by the two sources of bias mentioned above: only 55% of women assigned to receive the new treatment complied with their assignment and 35% of the women did not respond to the post-test questionnaire. Comparing the causal estimand of the new treatment using the MAR, Frangakis-Rubin, and our new approach, the results suggest that for these data the MAR assumption appears least plausible, and that the new model appears most plausible among the three choices.
Suicide attempts constitute a major risk factor for completed suicide, yet few interventions specifically designed to prevent suicide attempts have been evaluated.
To determine the effectiveness of a 10-session cognitive therapy intervention designed to prevent repeat suicide attempts in adults who recently attempted suicide.
Randomized controlled trial of adults (N = 120) who attempted suicide and were evaluated at a hospital emergency department within 48 hours of the attempt. Potential participants (N = 350) were consecutively recruited from October 1999 to September 2002; 66 refused to participate and 164 were ineligible. Participants were followed up for 18 months.
Cognitive therapy or enhanced usual care with tracking and referral services.
Incidence of repeat suicide attempts and number of days until a repeat suicide attempt. Suicide ideation (dichotomized), hopelessness, and depression severity at 1, 3, 6, 12, and 18 months.
From baseline to the 18-month assessment, 13 participants (24.1%) in the cognitive therapy group and 23 participants (41.6%) in the usual care group made at least 1 subsequent suicide attempt (asymptotic z score, 1.97; P = .049). Using the Kaplan-Meier method, the estimated 18-month reattempt-free probability in the cognitive therapy group was 0.76 (95% confidence interval [CI], 0.62-0.85) and in the usual care group was 0.58 (95% CI, 0.44-0.70). Participants in the cognitive therapy group had a significantly lower reattempt rate (Wald chi2(1) = 3.9; P = .049) and were 50% less likely to reattempt suicide than participants in the usual care group (hazard ratio, 0.51; 95% CI, 0.26-0.997). The severity of self-reported depression was significantly lower for the cognitive therapy group than for the usual care group at 6 months (P= .02), 12 months (P = .009), and 18 months (P = .046). The cognitive therapy group reported significantly less hopelessness than the usual care group at 6 months (P = .045). There were no significant differences between groups based on rates of suicide ideation at any assessment point.
Cognitive therapy was effective in preventing suicide attempts for adults who recently attempted suicide.
We present a random effects logistic approach for estimating the efficacy of treatment for compliers in a randomized trial with treatment non-adherence and longitudinal binary outcomes. We use our approach to analyse a primary care depression intervention trial. The use of a random effects model to estimate efficacy supplements intent-to-treat longitudinal analyses based on random effects logistic models that are commonly used in primary care depression research. Our estimation approach is an extension of Nagelkerke et al.'s instrumental variables approximation for cross-sectional binary outcomes. Our approach is easily implementable with standard random effects logistic regression software. We show through a simulation study that our approach provides reasonably accurate inferences for the setting of the depression trial under model assumptions. We also evaluate the sensitivity of our approach to model assumptions for the depression trial.
Suicide rates are highest in late life; the majority of older adults who die by suicide have seen a primary care physician in preceding months. Depression is the strongest risk factor for late-life suicide and for suicide's precursor, suicidal ideation.
To determine the effect of a primary care intervention on suicidal ideation and depression in older patients.
Randomized controlled trial known as PROSPECT (Prevention of Suicide in Primary Care Elderly: Collaborative Trial) with patient recruitment from 20 primary care practices in New York City, Philadelphia, and Pittsburgh regions, May 1999 through August 2001.
Two-stage, age-stratified (60-74, > or =75 years) depression screening of randomly sampled patients; enrollment included patients who screened positive and a random sample of screened negative patients. This analysis included patients with a depression diagnosis (N = 598).
Treatment guidelines tailored for the elderly with care management compared with usual care.
Assessment of suicidal ideation and depression severity at baseline, 4 months, 8 months, and 12 months.
Rates of suicidal ideation declined faster (P =.01) in intervention patients compared with usual care patients; at 4 months, in the intervention group, raw rates of suicidal ideation declined 12.9% points (29.4% to 16.5%) compared with 3.0% points (20.1% to 17.1% in usual care [P =.01]). Among patients reporting suicidal ideation, resolution of ideation was faster among intervention patients (P =.03); differences peaked at 8 months (70.7% vs 43.9% resolution; P =.005). Intervention patients had a more favorable course of depression in both degree and speed of symptom reduction; group difference peaked at 4 months. The effects on depression were not significant among patients with minor depression unless suicidal ideation was present.
Evidence of the intervention's effectiveness in community-based primary care with a heterogeneous sample of depressed patients introduces new challenges related to its sustainability and dissemination. The intervention's effectiveness in reducing suicidal ideation, regardless of depression severity, reinforces its role as a prevention strategy to reduce risk factors for suicide in late life.
Mediating variables are prominent in psychological theory and research. A mediating variable transmits the effect of an independent variable on a dependent variable. Differences between mediating variables and confounders, moderators, and covariates are outlined. Statistical methods to assess mediation and modern comprehensive approaches are described. Future directions for mediation analysis are discussed.
We consider estimation of the received treatment effect on a dichotomous outcome in randomised trials with non-compliance. We explore inference about the parameters of the structural mean models of Robins (1994, 1997) and Robins et al. (1999). We show that, in contrast to the additive and multiplicative structural mean models for continuous and count outcomes, unbiased estimating functions for a nonzero (structural) treatment effect parameter do not exist in the presence of many continuous and discrete baseline covariates, even when the randomisation probabilities are known. The best that can be hoped for are estimators, such as those proposed in this paper, that are guaranteed both to estimate consistently the (null) treatment effect when the null hypothesis of no treatment effect is true and to have small bias when the true treatment effect is close to but not equal to zero. Copyright 2004, Oxford University Press.
We consider studies for evaluating the short-term effect of a treatment of interest on a time-to-event outcome. The studies we consider are only partially controlled in the following sense: (1) Subjects' exposure to the treatment of interest can vary over time, but this exposure is not directly controlled by the study; (2) subjects' follow-up time is not directly controlled by the study; and (3) the study directly controls another factor that can affect subjects' exposure to the treatment of interest as well as subjects' follow-up time. When factors 1 and 2 are both present in the study, evaluating the treatment of interest using standard methods, including instrumental variables, does not generally estimate treatment effects. We develop the methodology for estimating the effect of treatment 1 in this setting of partially controlled studies under explicit assumptions using the framework for principal stratification for causal inference. We illustrate our methods by a study to evaluate the efficacy of the Baltimore Needle Exchange Program to reduce the risk of human immunodeficiency virus (HIV) transmission, using data on distance of the program's sites from the subjects.
Causal inference is best understood using potential outcomes. This use is particularly important in more complex settings, that is, observational studies or randomized experiments with complications such as noncompliance. The topic of this lecture, the issue of estimating the causal effect of a treatment on a primary outcome that is ``censored'' by death, is another such complication. For example, suppose that we wish to estimate the effect of a new drug on Quality of Life (QOL) in a randomized experiment, where some of the patients die before the time designated for their QOL to be assessed. Another example with the same structure occurs with the evaluation of an educational program designed to increase final test scores, which are not defined for those who drop out of school before taking the test. A further application is to studies of the effect of job-training programs on wages, where wages are only defined for those who are employed. The analysis of examples like these is greatly clarified using potential outcomes to define causal effects, followed by principal stratification on the intermediated outcomes (e.g., survival).
For most of this century, randomization has been a cornerstone of scientific experimentation, especially when dealing with humans as experimental units. In practice, however, noncompliance is relatively common with human subjects, complicating traditional theories of inference that require adherence to the random treatment assignment. In this paper we present Bayesian inferential methods for causal estimands in the presence of noncompliance, when the binary treatment assignment is random and hence ignorable, but the binary treatment received is not ignorable. We assume that both the treatment assigned and the treatment received are observed. We describe posterior estimation using EM and data augmentation algorithms. Also, we investigate the role of two assumptions often made in econometric instrumental variables analyses, the exclusion restriction and the monotonicity assumption, without which the likelihood functions generally have substantial regions of maxima. We apply our procedures to real and artificial data, thereby demonstrating the technology and sheering that our new methods can yield valid inferences that differ in practically important ways from those based on previous methods for analysis in the presence of noncompliance, including intention-to-treat analyses and analyses based on econometric instrumental variables techniques. Finally, we perform a simulation to investigate the operating characteristics of the competing procedures in a simple setting, which indicates relatively dramatic improvements in frequency operating characteristics attainable using our Bayesian procedures.
Rejoinder on Causal Inference Through Potential Outcomes and Principal Stratification: Application to Studies with ``Censoring'' Due to Death by D. B. Rubin [math.ST/0612783]
We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.
We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.
The Gibbs sampler, the algorithm of Metropolis and similar iterative simulation methods are potentially very helpful for summarizing multivariate distributions. Used naively, however, iterative simulation can give misleading answers. Our methods are simple and generally applicable to the output of any iterative simulation; they are designed for researchers primarily interested in the science underlying the data and models they are analyzing, rather than for researchers interested in the probability theory underlying the iterative simulations themselves. Our recommended strategy is to use several independent sequences, with starting points sampled from an overdispersed distribution. At each step of the iterative simulation, we obtain, for each univariate estimand of interest, a distributional estimate and an estimate of how much sharper the distributional estimate might become if the simulations were continued indefinitely. Because our focus is on applied inference for Bayesian posterior distributions in real problems, which often tend toward normality after transformations and marginalization, we derive our results as normal-theory approximations to exact Bayesian inference, conditional on the observed simulations. The methods are illustrated on a random-effects mixture model applied to experimental measurements of reaction times of normal and schizophrenic patients.
Causal effects are comparisons among values that would have been observed under all possible assignments of treatments to experimental units. In an experiment, one assignment of treatments is chosen and only the values under that assignment can be observed. Bayesian inference for causal effects follows from finding the predictive distribution of the values under the other assignments of treatments. This perspective makes clear the role of mechanisms that sample experimental units, assign treatments and record data. Unless these mechanisms are ignorable (known probabilistic functions of recorded values), the Bayesian must model them in the data analysis and, consequently, confront inferences for causal effects that are sensitive to the specification of the prior distribution of the data. Moreover, not all ignorable mechanisms can yield data from which inferences for causal effects are insensitive to prior specifications. Classical randomized designs stand out as especially appealing assignment mechanisms designed to make inference for causal effects straightforward by limiting the sensitivity of a valid Bayesian analysis.
SUMMARY A generalization of the sampling method introduced by Metropolis et al. (1953) is presented along with an exposition of the relevant theory, techniques of application and methods and difficulties
of assessing the error in Monte Carlo estimates. Examples of the methods, including the generation of random orthogonal matrices
and potential applications of the methods to numerical problems arising in statistics, are discussed.
The Hamilton Depression Rating Scale (HDRS) is the most widely used scale for patient selection and follow-up in research studies of treatments of depression. Despite extensive study of the reliability and validity of the total scale score, the psychometric characteristics of the individual items have not been well studied. In the only reliability study to report agreement on individual items using a test-retest interview method, most of the items had only fair or poor agreement. Because this is due in part to variability in the way the information is obtained to make the various rating distinctions, the Structured Interview Guide for the HDRS (SIGH-D) was developed to standardize the manner of administration of the scale. A test-retest reliability study conducted on a series of psychiatric inpatients demonstrated that the use of the SIGH-D results in a substantially improved level of agreement for most of the HDRS items.
In many areas of medical research, such as psychiatry and gerontology, latent class variables are used to classify individuals into disease categories, often with the intention of hierarchical modeling. Problems arise when it is not clear how many disease classes are appropriate, creating a need for model selection and diagnostic techniques. Previous work has shown that the Pearson chi 2 statistic and the log-likelihood ratio G2 statistic are not valid test statistics for evaluating latent class models. Other methods, such as information criteria, provide decision rules without providing explicit information about where discrepancies occur between a model and the data. Identifiability issues further complicate these problems. This paper develops procedures for assessing Markov chain Monte Carlo convergence and model diagnosis and for selecting the number of categories for the latent variable based on evidence in the data using Markov chain Monte Carlo techniques. Simulations and a psychiatric example are presented to demonstrate the effective use of these methods.
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 tinder 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 forrmulate estimands based on principal stratification and principal causal effects and show their superiority.
We review and compare two causal modeling approaches that correspond to two major and distinct classes of inference - efficacy and intervention-based inference - in the context of randomized trials with subject noncompliance.
We review the definitions of efficacy and intervention-based effects in the clinical trials literature and relate these to two separate and distinct causal modeling approaches: the structural mean modeling (SMM) approach and the principal stratification, instrumental variable approach.
The SMM-based efficacy approach focuses on the effect of actually receiving treatment. In contrast, the principal stratification method addresses the effect of treatment assignment within partially unobserved latent subgroups defined by compliance behavior. While these approaches differ in terms of philosophy, model definitions, and estimation, they estimate the same causal effect under certain assumptions, but estimate very different causal effects when those assumptions are relaxed. We illustrate these results using a randomized psychiatry trial where the focus is physician compliance to the designated protocol and the other examines patient compliance to the designated protocol, both from the same trial.
The validity of the models under the instrumental variable, SMM and principal stratification approaches depends on modeling assumptions, some of which may not be verifiable from the observed data and potentially less realistic than the no-confounding assumption made by non-causal approaches.
This comparison in terms of efficacy versus intervention-based effects in causal modeling parallels the explanatory versus pragmatic approaches in clinical trials research; therefore researchers should weigh carefully when choosing causal modeling methodology based on whether efficacy or intervention-based effects are of interest.
We describe instrumental variable (IV) methods for the estimation of the 'dose'-response effects of psychological interventions in randomized controlled trials in which there is variability in the patients' adherence to the allocated therapy (that is, variability in the actual number of sessions of therapy attended) and also variability in the strength of the therapeutic alliance between patients and their therapists. The effect of the therapy on outcome is assumed to be a function of both the number of sessions attended and the strength of the therapeutic alliance, with no intervention effects in the absence of any sessions attended (an exclusion restriction) and the effect of the strength of the alliance being represented by a multiplicative term (interaction) in the treatment-effect model. The IV methods that are described allow for: (a) hidden confounding between sessions, alliance and outcome; (b) measurement errors in the alliance; and (c) that alliance is only measured in those receiving treatment. Three two-stage estimation procedures are illustrated, and their equivalence demonstrated, through Monte Carlo simulation and analysis of the results of an actual trial.
We present a linear rank preserving model (RPM) approach for analyzing mediation of a randomized baseline intervention's effect on a univariate follow-up outcome. Unlike standard mediation analyses, our approach does not assume that the mediating factor is also randomly assigned to individuals in addition to the randomized baseline intervention (i.e., sequential ignorability), but does make several structural interaction assumptions that currently are untestable. The G-estimation procedure for the proposed RPM represents an extension of the work on direct effects of randomized intervention effects for survival outcomes by Robins and Greenland (1994, Journal of the American Statistical Association 89, 737-749) and on intervention non-adherence by Ten Have et al. (2004, Journal of the American Statistical Association 99, 8-16). Simulations show good estimation and confidence interval performance by the proposed RPM approach under unmeasured confounding relative to the standard mediation approach, but poor performance under departures from the structural interaction assumptions. The trade-off between these assumptions is evaluated in the context of two suicide/depression intervention studies.
This article discusses a nested latent class model for analyzing longitudinal randomized trials when subjects do not always adhere to the treatment to which they are randomized. In the Prevention of Suicide in Primary Care Elderly: Collaborative Trial, subjects were randomized to either the control treatment, where they received standard care, or to the intervention, where they received standard care in addition to meeting with depression health specialists. The health specialists educate patients, their families, and physicians about depression and monitor their treatment. Those randomized to the control treatment have no access to the health specialists: however, those randomized to the intervention could choose not to meet with the health specialists, hence receiving only the standard care. Subjects participated in the study for two years where depression severity and adherence to meeting with health specialists were measured at each follow-up. The outcome of interest is the effect of meeting with the health specialists on depression severity. Traditional intention-to-treat and as-treated analyses may produce biased causal effect estimates in the presence of subject noncompliance. Utilizing a nested latent class model that uses subject-specific and time-invariant "superclasses" allows us to summarize longitudinal trends of compliance patterns and to estimate the effect of the intervention using intention-to-treat contrasts within principal strata that correspond to longitudinal compliance behavior patterns. Analyses show that subjects with more severe depression are more likely to adhere to treatment randomization, and those that are compliant and meet with health specialists benefit from the meetings and show improvement in depression. Simulation results show that our estimation procedure produces reasonable parameter estimates under correct model assumptions.
The causal relationships between wealth and health status were discussed. The causal inference and the direct and indirect causal effects were also presented. The coefficients of current health and wealth status were significant in predicting current mortality. But the coefficients of prior and current general health in distribution were not significant for predicting current wealth.
The use of the concept of 'direct' versus 'indirect' causal effects is common, not only in statistics but also in many areas of social and economic sciences. The related terms of 'biomarkers' and 'surrogates' are common in pharmacological and biomedical sciences. Sometimes this concept is represented by graphical displays of various kinds. The view here is that there is a great deal of imprecise discussion surrounding this topic and, moreover, that the most straightforward way to clarify the situation is by using potential outcomes to define causal effects. In particular, I suggest that the use of principal stratification is key to understanding the meaning of direct and indirect causal effects. A current study of anthrax vaccine will be used to illustrate ideas. Copyright 2004 Board of the Foundation of the Scandinavian Journal of Statistics..
This article addresses unique causal issues in the context of a randomized study on improving adherence to best practice guidelines by primary care physicians (PCP's) in treating their depressed patients. The study assessed an encouragement strategy to improve PCP guideline adherence. In this context, we compare two causal approaches: the conditional-compliance (CC) Bayesian latent class and the conditional-observable (CO) structural mean model methods. The CC methods estimate contrasts between randomized encouragement and no-encouragement arms [intent-to-treat (ITT) estimand] given latent PCP guideline complier classes. The CO methods estimate contrasts between PCP guideline adherence and nonadherence conditions (as-treated estimand) given observed PCP adherence status. The CC ITT estimand for patients with PCP compliers equals the CO as-treated estimand depending on assumptions. One such assumption pertains to the absence of physician defiers, who do the opposite of what they are encouraged to do in treating patients for depression. We relate these two estimands to each other in our clinical context when the no-defier assumption is not plausible. In other contexts, previous statistical literature has appropriately assumed the absence of defiers. However, indications in the behavioral literature, anecdotal evidence in the study, and results from the data analysis and simulations suggest that defiers do exist in the context of physician-based interventions in primary care. Both simulation and empirical results show that even with a small estimated proportion of defiers under Bayesian model assumptions, inference is sensitive to different assumptions about this class of PCP noncompliers.
Manual for the BDI-II. The Psychological Corporation
- Beck At Ra Steer
- Brown
Beck AT, Steer RA, Brown GK. Manual for the BDI-II. The Psychological Corporation: San Antonio, TX, 1996.