Invited commentary: Decomposing with a lot of supposing

In this issue of the Journal, VanderWeele and Vansteelandt (Am J Epidemiol. 2010;172(12):1339-1348) provide simple formulae for estimation of direct and indirect effects using standard logistic regression when the exposure and outcome are binary, the mediator is continuous, and the odds ratio is the chosen effect measure. They also provide concisely stated lists of assumptions necessary for estimation of these effects, including various conditional independencies and homogeneity of exposure and mediator effects over covariate strata. They further suggest that this will allow effect decomposition in case-control studies if the sampling fractions and population outcome prevalence are known with certainty. In this invited commentary, the author argues that, in a well-designed case-control study in which the sampling fraction is known, it should not be necessary to rely on the odds ratio. The odds ratio has well-known deficiencies as a causal parameter, and its use severely complicates evaluation of confounding and effect homogeneity. Although VanderWeele and Vansteelandt propose that a rare disease assumption is not necessary for estimation of controlled direct effects using their approach, collapsibility concerns suggest otherwise when the goal is causal inference rather than merely measuring association. Moreover, their clear statement of assumptions necessary for the estimation of natural/pure effects suggests that these quantities will rarely be viable estimands in observational epidemiology.     

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.     

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.     

A sufficient cause based approach to the assessment of mediation

The minimal sufficient cause (MSC) model, also known as the sufficient component cause model, has been used to facilitate understanding of several key concepts in epidemiology. To improve the understanding of mediation, we introduce a causal model for mediation that is grounded in the MSC approach. First, we describe an unbiased model for mediation, to clarify the causal meaning of previously described indirect effects. Through the use of potential outcomes and response types, we express each indirect (and direct) effect in terms of component causes within the MSC model. Second, we use an MSC-based model to illustrate a common cause of the mediator and outcome, i.e. a confounder of the mediator–outcome relationship. By describing this potential source of bias within the MSC-based model, important complexities are noted that impact the magnitude of plausible confounding. In conclusion, an MSC-based approach leads to several important insights concerning the interpretation of indirect and direct effects, as well as the potential sources of bias in mediation analysis.     

Evaluating and Rejecting True Mediation Models: A Cautionary Note

This research note describes an overlooked problem in understanding whether a given variable in a model truly acts as a mediator between some exogenous variable(s) and some final dependent factor. Demonstrations of mediation and the rules for identifying have relied on simple 3-variable models with an explicit direct effects alternative model as the competing explanation. Incorporating a 4th variable demonstrates that it is quite simple to reject mediation when a true form of mediation exists. In the presence of an unobserved relation, correlated error, between mediator variable and outcome variable, the 3-variable model will consistently show direct effects when, in fact, there is no direct effect of the exogenous variable. Applying well-established rules to test for mediation in this circumstance cannot distinguish a model in which pure mediation is rejected from a model in which true mediation is correct. This poses a fundamental problem for the typical assessment of mediation offered by the Baron and Kenny procedures.     

Odds ratios for mediation analysis for a dichotomous outcome

For dichotomous outcomes, the authors discuss when the standard approaches to mediation analysis used in epidemiology and the social sciences are valid, and they provide alternative mediation analysis techniques when the standard approaches will not work. They extend definitions of controlled direct effects and natural direct and indirect effects from the risk difference scale to the odds ratio scale. A simple technique to estimate direct and indirect effect odds ratios by combining logistic and linear regressions is described that applies when the outcome is rare and the mediator continuous. Further discussion is given as to how this mediation analysis technique can be extended to settings in which data come from a case-control study design. For the standard mediation analysis techniques used in the epidemiologic and social science literatures to be valid, an assumption of no interaction between the effects of the exposure and the mediator on the outcome is needed. The approach presented here, however, will apply even when there are interactions between the effect of the exposure and the mediator on the outcome.     

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.     

Sensitivity analysis for unobserved confounding of direct and indirect effects using uncertainty intervals

To estimate direct and indirect effects of an exposure on an outcome from observed data, strong assumptions about unconfoundedness are required. Since these assumptions cannot be tested using the observed data, a mediation analysis should always be accompanied by a sensitivity analysis of the resulting estimates. In this article, we propose a sensitivity analysis method for parametric estimation of direct and indirect effects when the exposure, mediator, and outcome are all binary. The sensitivity parameters consist of the correlations between the error terms of the exposure, mediator, and outcome models. These correlations are incorporated into the estimation of the model parameters and identification sets are then obtained for the direct and indirect effects for a range of plausible correlation values. We take the sampling variability into account through the construction of uncertainty intervals. The proposed method is able to assess sensitivity to both mediator‐outcome confounding and confounding involving the exposure. To illustrate the method, we apply it to a mediation study based on the data from the Swedish Stroke Register (Riksstroke). An R package that implements the proposed method is available.     

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.     

Effect decomposition through multiple causally nonordered mediators in the presence of exposure-induced mediator-outcome confounding

Avin et al (2005) showed that, in the presence of exposure-induced mediator-outcome confounding, decomposing the total causal effect (TCE) using standard conditional exchangeability assumptions is not possible even under a nonparametric structural equation model with all confounders observed. Subsequent research has investigated the assumptions required for such a decomposition to be identifiable and estimable from observed data. One approach was proposed by VanderWeele et al (2014). They decomposed the TCE under three different scenarios: (1) treating the mediator and the exposure-induced confounder as joint mediators; (2) generating path-specific effects albeit without distinguishing between multiple distinct paths through the exposure-induced confounder; and (3) using so-called randomised interventional analogues where sampling values from the distribution of the mediator within the levels of the exposure effectively marginalises over the exposure-induced confounder. In this paper, we extend their approach to the case where there are multiple mediators that do not influence each other directly but which are all influenced by an exposure-induced mediator-outcome confounder. We provide a motivating example and results from a simulation study based on from our work in dental epidemiology featuring the 1982 Pelotas Birth Cohort in Brazil.     

Mediation analysis with multiple versions of the mediator

The causal inference literature has provided definitions of direct and indirect effects based on counterfactuals that generalize the approach found in the social science literature. However, these definitions presuppose well-defined hypothetical interventions on the mediator. In many settings, there may be multiple ways to fix the mediator to a particular value, and these various hypothetical interventions may have very different implications for the outcome of interest. In this paper, we consider mediation analysis when multiple versions of the mediator are present. Specifically, we consider the problem of attempting to decompose a total effect of an exposure on an outcome into the portion through the intermediate and the portion through other pathways. We consider the setting in which there are multiple versions of the mediator but the investigator has access only to data on the particular measurement, not information on which version of the mediator may have brought that value about. We show that the quantity that is estimated as a natural indirect effect using only the available data does indeed have an interpretation as a particular type of mediated effect; however, the quantity estimated as a natural direct effect, in fact, captures both a true direct effect and an effect of the exposure on the outcome mediated through the effect of the version of the mediator that is not captured by the mediator measurement. The results are illustrated using 2 examples from the literature, one in which the versions of the mediator are unknown and another in which the mediator itself has been dichotomized.     

Testing for the indirect effect under the null for genome‐wide mediation analyses

Mediation analysis helps researchers assess whether part or all of an exposure's effect on an outcome is due to an intermediate variable. The indirect effect can help in designing interventions on the mediator as opposed to the exposure and better understanding the outcome's mechanisms. Mediation analysis has seen increased use in genome‐wide epidemiological studies to test for an exposure of interest being mediated through a genomic measure such as gene expression or DNA methylation (DNAm). Testing for the indirect effect is challenged by the fact that the null hypothesis is composite. We examined the performance of commonly used mediation testing methods for the indirect effect in genome‐wide mediation studies. When there is no association between the exposure and the mediator and no association between the mediator and the outcome, we show that these common tests are overly conservative. This is a case that will arise frequently in genome‐wide mediation studies. Caution is hence needed when applying the commonly used mediation tests in genome‐wide mediation studies. We evaluated the performance of these methods using simulation studies, and performed an epigenome‐wide mediation association study in the Normative Aging Study, analyzing DNAm as a mediator of the effect of pack‐years on FEV₁.     

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.     

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.     

A simple unified approach for estimating natural direct and indirect effects

An important problem within both epidemiology and many social sciences is to break down the effect of a given treatment into different causal pathways and to quantify the importance of each pathway. Formal mediation analysis based on counterfactuals is a key tool when addressing this problem. During the last decade, the theoretical framework for mediation analysis has been greatly extended to enable the use of arbitrary statistical models for outcome and mediator. However, the researcher attempting to use these techniques in practice will often find implementation a daunting task, as it tends to require special statistical programming. In this paper, the authors introduce a simple procedure based on marginal structural models that directly parameterize the natural direct and indirect effects of interest. It tends to produce more parsimonious results than current techniques, greatly simplifies testing for the presence of a direct or an indirect effect, and has the advantage that it can be conducted in standard software. However, its simplicity comes at the price of relying on correct specification of models for the distribution of mediator (and exposure) and accepting some loss of precision compared with more complex methods. Web Appendixes 1 and 2, which are posted on the Journal's Web site (http://aje.oupjournals.org/), contain implementation examples in SAS software (SAS Institute, Inc., Cary, North Carolina) and R language (R Foundation for Statistical Computing, Vienna, Austria).     

Mediation analysis with principal stratification

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. Copyright © 2009 John Wiley & Sons, Ltd.     

Can DAGs clarify effect modification?

The system proposed by VanderWeele and Robins for categorization of effect modifiers that are causal nodes in a directed acyclic graph (DAG) was not intended to empower DAGs to fully represent complex interactions among causes. However, once one has algebraically identified effect modifiers, the DAG implies a role for them. The limitations of epidemiologic definitions of "effect modification" are discussed, along with the implications of scale dependency for assessing interactions, where the scale can be either absolute risk, relative risk, or odds. My view is that probabilistic independence leads to the log-complement as a natural scale for interaction, but even that scale does not necessarily admit unambiguous inference. Any 2 direct causes of D are effect modifiers for each other on at least 2 scales, which can make a reasonable person question the utility of the concept. Still, etiologic models for joint effects are important, because most diseases arise through pathways involving multiple factors. I suggest an enhancement in construction of DAGs in epidemiology that includes arrow-on-arrow representations for effect modification. Examples are given, some of which depend on scale and some of which do not. An example illustrates possible biologic implications for such an effect modification DAG.     

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.     

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.