Estimating causal contrasts involving intermediate variables in the presence of selection bias

An important goal across the biomedical and social sciences is the quantification of the role of intermediate factors in explaining how an exposure exerts an effect on an outcome. Selection bias has the potential to severely undermine the validity of inferences on direct and indirect causal effects in observational as well as in randomized studies. The phenomenon of selection may arise through several mechanisms, and we here focus on instances of missing data. We study the sign and magnitude of selection bias in the estimates of direct and indirect effects when data on any of the factors involved in the analysis is either missing at random or not missing at random. Under some simplifying assumptions, the bias formulae can lead to nonparametric sensitivity analyses. These sensitivity analyses can be applied to causal effects on the risk difference and risk‐ratio scales irrespectively of the estimation approach employed. To incorporate parametric assumptions, we also develop a sensitivity analysis for selection bias in mediation analysis in the spirit of the expectation–maximization algorithm. The approaches are applied to data from a health disparities study investigating the role of stage at diagnosis on racial disparities in colorectal cancer survival. Copyright © 2016 John Wiley & Sons, Ltd.     

Boosting the precision of mediation analyses of randomised experiments through covariate adjustment

Analyses of randomised experiments frequently include attempts to decompose the intention‐to‐treat effect into a direct and indirect effect, mediated by given intermediaries, with the aim to shed light onto the treatment mechanism. Methods from causal mediation analysis have facilitated this by allowing for arbitrary models for the outcome and the mediator. They thereby generalise the traditional approach to direct and indirect effects, which is essentially limited to linear models. The default maximum likelihood methods make use of a model for the conditional distribution of the mediator, given treatment and baseline covariates, but are prone to bias when that model is misspecified. In randomised experiments, specification of such model can be easily avoided, but at the expense of a sometimes major efficiency loss when those baseline covariates are predictive of the mediator. In this article, we develop a compromise approach: it makes use of a model for the mediator to optimally extract information from the baseline covariate data but is insulated from the impact of misspecification of that model; it achieves this by exploiting the known randomisation probabilities. Simulation studies and the analysis of a randomised study show major efficiency gains and confirm our theoretical findings that the default methods from causal mediation analysis are sometimes, although not always, reasonably robust to model misspecification. Copyright © 2017 John Wiley & Sons, Ltd.     

Mediation and mechanism

The concepts of mediation and mechanism are contrasted and logical implications holding between theses two concepts are described. The concept of mediation can be formalized using counterfactual definitions of indirect effects; the concept of mechanism can be formalized within the sufficient cause framework. It is shown that both concepts can be illustrated using a single causal diagram. It is also shown that mediation implies mechanism but mechanism need not imply mediation. Discussion is given regarding how the distinction between “statistical causality” and “mechanistic causality” is blurred by recent work in causal inference concerning methods for testing for mediation and mechanism.     

Continuous-time causal mediation analysis

While causal mediation analysis has seen considerable recent development for a single measured mediator (M) and final outcome (Y), less attention has been given to repeatedly measured M and Y. Previous methods have typically involved discrete-time models that limit inference to the particular measurement times used and do not recognize the continuous nature of the mediation process over time. To overcome such limitations, we present a new continuous-time approach to causal mediation analysis that uses a differential equations model in a potential outcomes framework to describe the causal relationships among model variables over time. A connection between the differential equation models and standard repeated measures models is made to provide convenient model formulation and fitting. A continuous-time extension of the sequential ignorability assumption allows for identifiable natural direct and indirect effects as functions of time, with estimation based on a two-step approach to model fitting in conjunction with a continuous-time mediation formula. Novel features include a measure of an overall mediation effect based on the “area between the curves,” and an approach for predicting the effects of new interventions. Simulation studies show good properties of estimators and the new methodology is applied to data from a cohort study to investigate sugary drink consumption as a mediator of the effect of socioeconomic status on dental caries in children.     

Mediation analysis when a continuous mediator is measured with error and the outcome follows a generalized linear model

Mediation analysis is a popular approach to examine the extent to which the effect of an exposure on an outcome is through an intermediate variable (mediator) and the extent to which the effect is direct. When the mediator is mis‐measured, the validity of mediation analysis can be severely undermined. In this paper, we first study the bias of classical, non‐differential measurement error on a continuous mediator in the estimation of direct and indirect causal effects in generalized linear models when the outcome is either continuous or discrete and exposure–mediator interaction may be present. Our theoretical results as well as a numerical study demonstrate that in the presence of non‐linearities, the bias of naive estimators for direct and indirect effects that ignore measurement error can take unintuitive directions. We then develop methods to correct for measurement error. Three correction approaches using method of moments, regression calibration, and SIMEX are compared. We apply the proposed method to the Massachusetts General Hospital lung cancer study to evaluate the effect of genetic variants mediated through smoking on lung cancer risk. Copyright © 2014 John Wiley & Sons, Ltd.     

Direct and indirect effects in a survival context

A cornerstone of epidemiologic research is to understand the causal pathways from an exposure to an outcome. Mediation analysis based on counterfactuals is an important tool when addressing such questions. However, none of the existing techniques for formal mediation analysis can be applied to survival data. This is a severe shortcoming, as many epidemiologic questions can be addressed only with censored survival data. A solution has been to use a number of Cox models (with and without the potential mediator), but this approach does not allow a causal interpretation and is not mathematically consistent. In this paper, we propose a simple measure of mediation in a survival setting. The measure is based on counterfactuals, and measures the natural direct and indirect effects. The method allows a causal interpretation of the mediated effect (in terms of additional cases per unit of time) and is mathematically consistent. The technique is illustrated by analyzing socioeconomic status, work environment, and long-term sickness absence. A detailed implementation guide is included in an online eAppendix (http://links.lww.com/EDE/A476).     

Mediation analysis in a case-control study when the mediator is a censored variable

Mediation analysis is an approach for assessing the direct and indirect effects of an initial variable on an outcome through a mediator. In practice, mediation models can involve a censored mediator (eg, a woman's age at menopause). The current research for mediation analysis with a censored mediator focuses on scenarios where outcomes are continuous. However, the outcomes can be binary (eg, type 2 diabetes). Another challenge when analyzing such a mediation model is to use data from a case-control study, which results in biased estimations for the initial variable-mediator association if a standard approach is directly applied. In this study, we propose an approach (denoted as MAC-CC) to analyze the mediation model with a censored mediator given data from a case-control study, based on the semiparametric accelerated failure time model along with a pseudo-likelihood function. We adapted the measures for assessing the indirect and direct effects using counterfactual definitions. We conducted simulation studies to investigate the performance of MAC-CC and compared it to those of the naïve approach and the complete-case approach. MAC-CC accurately estimates the coefficients of different paths, the indirect effects, and the proportions of the total effects mediated. We applied the proposed and existing approaches to the mediation study of genetic variants, a woman's age at menopause, and type 2 diabetes based on a case-control study of type 2 diabetes. Our results indicate that there is no mediating effect from the age at menopause on the association between the genetic variants and type 2 diabetes.     

Estimation of causal mediation effects for a dichotomous outcome in multiple‐mediator models using the mediation formula

Mediators are intermediate variables in the causal pathway between an exposure and an outcome. Mediation analysis investigates the extent to which exposure effects occur through these variables, thus revealing causal mechanisms. In this paper, we consider the estimation of the mediation effect when the outcome is binary and multiple mediators of different types exist. We give a precise definition of the total mediation effect as well as decomposed mediation effects through individual or sets of mediators using the potential outcomes framework. We formulate a model of joint distribution (probit‐normal) using continuous latent variables for any binary mediators to account for correlations among multiple mediators. A mediation formula approach is proposed to estimate the total mediation effect and decomposed mediation effects based on this parametric model. Estimation of mediation effects through individual or subsets of mediators requires an assumption involving the joint distribution of multiple counterfactuals. We conduct a simulation study that demonstrates low bias of mediation effect estimators for two‐mediator models with various combinations of mediator types. The results also show that the power to detect a nonzero total mediation effect increases as the correlation coefficient between two mediators increases, whereas power for individual mediation effects reaches a maximum when the mediators are uncorrelated. We illustrate our approach by applying it to a retrospective cohort study of dental caries in adolescents with low and high socioeconomic status. Sensitivity analysis is performed to assess the robustness of conclusions regarding mediation effects when the assumption of no unmeasured mediator‐outcome confounders is violated. Copyright © 2013 John Wiley & Sons, Ltd.     

Mediation analysis via potential outcomes models

This paper develops a causal or manipulation model framework for mediation analysis based on the concept of potential outcome. Using this framework, we provide new definitions and measures of mediation. Effects of manipulations are modeled via the linear structural model. Corresponding structural equation models (SEMs), in conjunction with two-stage least-squares estimation and the delta method, are used to perform inference. The methods are applied to data from a study of nursing interventions for postoperative pain. We address the cases of more than two treatment groups, and an interaction among mediators. For the latter, a sensitivity analysis approach to handle unidentified parameters is described. Interpretative advantages of the potential outcomes framework for mediation are emphasized.     

Six-way decomposition of causal effects: Unifying mediation and mechanistic interaction

The sufficient component cause (SCC) model and counterfactual model are two common methods for causal inference, each with their own advantages: the SCC model allows the mechanistic interaction to be detailed, whereas the counterfactual model features a systemic framework for quantifying causal effects. Hence, integrating the SCC and counterfactual models may facilitate the conceptualization of causation. Based on the marginal SCC (mSCC) model, we propose a novel counterfactual mSCC framework that includes the steps of definition, identification, and estimation. We further propose a six-way effect decomposition for assessing mediation and the mechanistic interaction. The results demonstrate that when all variables are binary, the six-way decomposition is an extension of four-way decomposition and that without agonism, the six-way decomposition is reduced to four-way decomposition. To illustrate the utility of the proposed decomposition, we apply it to a Taiwanese cohort to examine the mechanism of hepatitis C virus (HCV)-induced hepatocellular carcinoma (HCC) with liver inflammation measured by alanine aminotransferase (ALT) as a mediator. Among the HCV-induced HCC cases, 62.27% are not explained by either mediation or interaction in relation to ALT; 9.32% are purely mediated by ALT; 16.53% are caused by the synergistic effect of HCV and ALT; and 9.31% are due to the mediated synergistic effect of HCV and ALT. In summary, we introduce an SCC model framework based on counterfactual theory and detail the required identification assumptions and estimation procedures; we also propose a six-way effect decomposition to unify mediation and mechanistic interaction analyses.     

Mediation analysis for common binary outcomes

Mediation analysis provides an attractive causal inference framework to decompose the total effect of an exposure on an outcome into natural direct effects and natural indirect effects acting through a mediator. For binary outcomes, mediation analysis methods have been developed using logistic regression when the binary outcome is rare. These methods will not hold in practice when a disease is common. In this paper, we develop mediation analysis methods that relax the rare disease assumption when using logistic regression. We calculate the natural direct and indirect effects for common diseases by exploiting the relationship between logit and probit models. Specifically, we derive closed-form expressions for the natural direct and indirect effects on the odds ratio scale. Mediation models for both continuous and binary mediators are considered. We demonstrate through simulation that the proposed method performs well for common binary outcomes. We apply the proposed methods to analyze the Normative Aging Study to identify DNA methylation sites that are mediators of smoking behavior on the outcome of obstructed airway function.     

A Bayesian semiparametric latent variable approach to causal mediation

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.     

Using mediation analysis to identify causal mechanisms in disease management interventions

For over two decades, disease management (DM) has been touted as an intervention capable of producing large scale cost savings for health care purchasers. However, the preponderance of scientific evidence suggests that these programs do not save money. This finding is not surprising given that the theorized causal mechanism by which the intervention supposedly influences the outcome has not been systematically assessed. Mediation analysis is a statistical approach to identifying causal pathways by testing the relationships between the treatment, the outcome, and an intermediate variable that is posited to mediate the relationship between the treatment and outcome. This analysis can therefore help identify how to make DM interventions effective by determining the causal mechanisms between intervention components and the desired outcome. DM interventions can then be optimized by eliminating those activities that are ineffective or even counter-productive. In this article we seek to promote the application of mediation analysis to DM program evaluation by describing the two principal frameworks generally followed in causal mediation analysis; structural equation modeling and potential outcomes. After comparing several approaches within these frameworks using real and simulated data, we find that some methods perform better than others under the conditions imposed upon the models. We conclude that mediation analysis can assist DM programs in developing and testing the causal pathways that enable interventions to be effective in achieving desired outcomes.     

Confounding of indirect effects: a sensitivity analysis exploring the range of bias due to a cause common to both the mediator and the outcome

Several investigators have demonstrated that the assessment of indirect and direct effects is biased in the presence of a cause that is common to both the mediator and the outcome if one has not controlled for this variable in the analysis. However, little work has been done to quantify the bias caused by this type of unmeasured confounding and determine whether this bias will materially affect conclusions regarding mediation. The author developed a sensitivity analysis program to address this crucial issue. Data from 2 well-known studies in the methodological literature on mediation were reanalyzed using this program. The results of mediation analyses were found not to be as vulnerable to the impact of confounding as previously described; however, these findings varied sharply between the 2 studies. Although the indirect effect observed in one study could potentially be due to a cause common to both the mediator and the outcome, such confounding could not feasibly explain the results of the other study. These disparate results demonstrate the utility of the current sensitivity analysis when assessing mediation.     

New proposal to address mediation analysis interrogations by using genetic variants as instrumental variables

The application of causal mediation analysis (CMA) considering the mediation effect of a third variable is increasing in epidemiological studies; however, this requires fitting strong assumptions on confounding bias. To address this limitation, we propose an extension of CMA combining it with Mendelian randomization (MRinCMA). We applied the new approach to analyse the causal effect of obesity and diabetes on pancreatic cancer, considering each factor as potential mediator. To check the performance of MRinCMA under several conditions/scenarios, we used it in different simulated data sets and compared it with structural equation models. For continuous variables, MRinCMA and structural equation models performed similarly, suggesting that both approaches are valid to obtain unbiased estimates. When noncontinuous variables were considered, MRinCMA presented, overall, lower bias than structural equation models. By applying MRinCMA, we did not find any evidence of causality of obesity or diabetes on pancreatic cancer. With this new methodology, researchers would be able to address CMA hypotheses by appropriately accounting for the confounding bias assumption regardless of the conditions used in their studies in different settings.     

A Bayesian regularized mediation analysis with multiple exposures

Mediation analysis assesses the effect of study exposures on an outcome both through and around specific mediators. While mediation analysis involving multiple mediators has been addressed in recent literature, the case of multiple exposures has received little attention. With the presence of multiple exposures, we consider regularizations that allow simultaneous effect selection and estimation while stabilizing model fit and accounting for model selection uncertainty. In the framework of linear structural-equation models, we analytically show that a two-stage approach regularizing regression coefficients does not guarantee a unimodal posterior distribution and that a product-of-coefficient approach regularizing direct and indirect effects tends to penalize excessively. We propose a regularized difference-of-coefficient approach that bypasses these limitations. Using the connection between regularizations and Bayesian hierarchical models with Laplace prior, we develop an efficient Markov chain Monte Carlo algorithm for posterior estimation and inference. Through simulations, we show that the proposed approach has better empirical performances compared to some alternatives. The methodology is illustrated using data from two epidemiological studies in human reproduction.     

Identifiability and estimation of causal mediation effects with missing data

Mediation analysis is a standard approach to understanding how and why an intervention works in social and medical sciences. However, the presence of missing data, especially missing not at random data, poses a great challenge for the applicability of this approach in practice. Current methods for handling such missingness are still lacking in causal mediation analysis. In this article, we first show the identifiability of causal mediation effects with different types of missing outcomes under different missingness mechanisms. We then provide corresponding approaches for estimation and inference. Especially for missing not at random data, we develop an estimating equation–based approach to estimate causal mediation effects, which can easily handle different types of mediators and outcomes, and we also establish the asymptotic results of the estimators. Simulation results show good performance for the proposed estimators in finite samples. Finally, we use a real data set from the Clinical Antipsychotic Trials of Intervention Effectiveness Research for Alzheimer disease to illustrate our approach.     

Inverse odds ratio‐weighted estimation for causal mediation analysis

An important scientific goal of studies in the health and social sciences is increasingly to determine to what extent the total effect of a point exposure is mediated by an intermediate variable on the causal pathway between the exposure and the outcome. A causal framework has recently been proposed for mediation analysis, which gives rise to new definitions, formal identification results and novel estimators of direct and indirect effects. In the present paper, the author describes a new inverse odds ratio‐weighted approach to estimate so‐called natural direct and indirect effects. The approach, which uses as a weight the inverse of an estimate of the odds ratio function relating the exposure and the mediator, is universal in that it can be used to decompose total effects in a number of regression models commonly used in practice. Specifically, the approach may be used for effect decomposition in generalized linear models with a nonlinear link function, and in a number of other commonly used models such as the Cox proportional hazards regression for a survival outcome. The approach is simple and can be implemented in standard software provided a weight can be specified for each observation. An additional advantage of the method is that it easily incorporates multiple mediators of a categorical, discrete or continuous nature. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimation of direct causal effects

Many common problems in epidemiologic and clinical research involve estimating the effect of an exposure on an outcome while blocking the exposure's effect on an intermediate variable. Effects of this kind are termed direct effects. Estimation of direct effects is typically the goal of research aimed at understanding mechanistic pathways by which an exposure acts to cause or prevent disease, as well as in many other settings. Although multivariable regression is commonly used to estimate direct effects, this approach requires assumptions beyond those required for the estimation of total causal effects. In addition, when the exposure and intermediate variables interact to cause disease, multivariable regression estimates a particular type of direct effect-the effect of an exposure on an outcome when the intermediate is fixed at a specified level. Using the counterfactual framework, we distinguish this definition of a direct effect (controlled direct effect) from an alternative definition, in which the effect of the exposure on the intermediate is blocked, but the intermediate is otherwise allowed to vary as it would in the absence of exposure (natural direct effect). We illustrate the difference between controlled and natural direct effects using several examples. We present an estimation approach for natural direct effects that can be implemented using standard statistical software, and we review the assumptions underlying our approach (which are less restrictive than those proposed by previous authors).     

Two‐step estimation in ratio‐of‐mediator‐probability weighted causal mediation analysis

This study investigates appropriate estimation of estimator variability in the context of causal mediation analysis that employs propensity score‐based weighting. Such an analysis decomposes the total effect of a treatment on the outcome into an indirect effect transmitted through a focal mediator and a direct effect bypassing the mediator. Ratio‐of‐mediator‐probability weighting estimates these causal effects by adjusting for the confounding impact of a large number of pretreatment covariates through propensity score‐based weighting. In step 1, a propensity score model is estimated. In step 2, the causal effects of interest are estimated using weights derived from the prior step's regression coefficient estimates. Statistical inferences obtained from this 2‐step estimation procedure are potentially problematic if the estimated standard errors of the causal effect estimates do not reflect the sampling uncertainty in the estimation of the weights. This study extends to ratio‐of‐mediator‐probability weighting analysis a solution to the 2‐step estimation problem by stacking the score functions from both steps. We derive the asymptotic variance‐covariance matrix for the indirect effect and direct effect 2‐step estimators, provide simulation results, and illustrate with an application study. Our simulation results indicate that the sampling uncertainty in the estimated weights should not be ignored. The standard error estimation using the stacking procedure offers a viable alternative to bootstrap standard error estimation. We discuss broad implications of this approach for causal analysis involving propensity score‐based weighting.