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.     

Causal mediation analysis with multiple mediators in the presence of treatment noncompliance

Randomized experiments are often complicated because of treatment noncompliance. This challenge prevents researchers from identifying the mediated portion of the intention‐to‐treated (ITT) effect, which is the effect of the assigned treatment that is attributed to a mediator. One solution suggests identifying the mediated ITT effect on the basis of the average causal mediation effect among compliers when there is a single mediator. However, considering the complex nature of the mediating mechanisms, it is natural to assume that there are multiple variables that mediate through the causal path. Motivated by an empirical analysis of a data set collected in a randomized interventional study, we develop a method to estimate the mediated portion of the ITT effect when both multiple dependent mediators and treatment noncompliance exist. This enables researchers to make an informed decision on how to strengthen the intervention effect by identifying relevant mediators despite treatment noncompliance. We propose a nonparametric estimation procedure and provide a sensitivity analysis for key assumptions. We conduct a Monte Carlo simulation study to assess the finite sample performance of the proposed approach. The proposed method is illustrated by an empirical analysis of JOBS II data, in which a job training intervention was used to prevent mental health deterioration among unemployed individuals.     

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 mediation effects for zero‐inflated regression models

The goal of mediation analysis is to identify and explicate the mechanism that underlies a relationship between a risk factor and an outcome via an intermediate variable (mediator). In this paper, we consider the estimation of mediation effects in zero‐inflated (ZI) models intended to accommodate ‘extra’ zeros in count data. Focusing on the ZI negative binomial models, we provide a mediation formula approach to estimate the (overall) mediation effect in the standard two‐stage mediation framework under a key sequential ignorability assumption. We also consider a novel decomposition of the overall mediation effect for the ZI context using a three‐stage mediation model. Estimation of the components of the overall mediation effect requires an assumption involving the joint distribution of two counterfactuals. Simulation study results demonstrate low bias of mediation effect estimators and close‐to‐nominal coverage probability of confidence intervals. We also modify the mediation formula method by replacing ‘exact’ integration with a Monte Carlo integration method. The method is applied to a cohort study of dental caries in very low birth weight adolescents. For overall mediation effect estimation, sensitivity analysis was conducted to quantify the degree to which key assumption must be violated to reverse the original conclusion. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation of the mediation effect with a binary mediator

A mediator acts as a third variable in the causal pathway between a risk factor and an outcome. In this paper, we consider the estimation of the mediation effect when the mediator is a binary variable. We give a precise definition of the mediation effect and examine asymptotic properties of five different estimators of the mediation effect. Our theoretical developments, which are supported by a Monte Carlo study, show that the estimators that account for the binary nature of the mediator are consistent for the mediation effect defined in this paper while other estimators are inconsistent. We use these estimators to study the mediation effect of chronic cerebral infarction in the causal relationship between the apolipoprotein E epsilon4 allele and cognitive function among 233 deceased participants from the Religious Orders Study, a longitudinal, clinical-pathologic study of aging and Alzheimer's disease.     

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.     

Quantile causal mediation analysis allowing longitudinal data

Mediation analysis has mostly been conducted with mean regression models. With this approach modeling means, formulae for direct and indirect effects are based on changes in means, which may not capture effects that occur in units at the tails of mediator and outcome distributions. Individuals with extreme values of medical endpoints are often more susceptible to disease and can be missed if one investigates mean changes only. We derive the controlled direct and indirect effects of an exposure along percentiles of the mediator and outcome using quantile regression models and a causal framework. The quantile regression models can accommodate an exposure‐mediator interaction and random intercepts to allow for longitudinal mediator and outcome. Because DNA methylation acts as a complex “switch” to control gene expression and fibrinogen is a cardiovascular factor, individuals with extreme levels of these markers may be more susceptible to air pollution. We therefore apply this methodology to environmental data to estimate the effect of air pollution, as measured by particle number, on fibrinogen levels through a change in interferon‐gamma (IFN‐γ) methylation. We estimate the controlled direct effect of air pollution on the qth percentile of fibrinogen and its indirect effect through a change in the pth percentile of IFN‐γ methylation. We found evidence of a direct effect of particle number on the upper tail of the fibrinogen distribution. We observed a suggestive indirect effect of particle number on the upper tail of the fibrinogen distribution through a change in the lower percentiles of the IFN‐γ methylation distribution.     

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.     

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.     

The mediation proportion: a structural equation approach for estimating the proportion of exposure effect on outcome explained by an intermediate variable

It is often of interest to assess how much of the effect of an exposure on a response is mediated through an intermediate variable. However, systematic approaches are lacking, other than assessment of a surrogate marker for the endpoint of a clinical trial. We review a measure of "proportion explained" in the context of observational epidemiologic studies. The measure has been much debated; we show how several of the drawbacks are alleviated when exposures, mediators, and responses are continuous and are embedded in a structural equation framework. These conditions also allow for consideration of several intermediate variables. Binary or categorical variables can be included directly through threshold models. We call this measure the mediation proportion, that is, the part of an exposure effect on outcome explained by a third, intermediate variable. Two examples illustrate the approach. The first example is a randomized clinical trial of the effects of interferon-alpha on visual acuity in patients with age-related macular degeneration. In this example, the exposure, mediator and response are all binary. The second example is a common problem in social epidemiology-to find the proportion of a social class effect on a health outcome that is mediated by psychologic variables. Both the mediator and the response are composed of several ordered categorical variables, with confounders present. Finally, we extend the example to more than one mediator.     

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.     

多重并行中介分析方法的比较研究

目的:介绍4种多重并行中介模型的分析方法,包括纯回归法、逆概率加权法、扩展的自然效应模型和基于权重的填补法,并对其进行探讨和比较。方法:针对多重并行中介模型,通过3种情境的模拟试验比较不同方法在不同情境下估计直接效应和间接效应的表现,并应用英国生物样本库的数据集进行实例分析。结果:模拟试验和实例分析结果显示纯回归法和逆...     

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.     

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.     

一种新的针对多个中介变量的中介分析方法

中介分析主要用于探究自变量X与因变量Y之间的因果关系机制,将自变量X与因变量Y之间的因果路径进行分解,判断中介变量M是否在其因果路径中起作用及其作用大小。经典的中介分析方法一般仅针对单一中介变量。本文介绍了一种新的针对多个中介变量的中介分析方法。     

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.     

A mediation analysis for a nonrare dichotomous outcome with sequentially ordered multiple mediators

Mediation analyses can help us to understand the biological mechanism in which an exposure or treatment affects an outcome. Single mediator analyses have been used in various applications, but may not be appropriate for analyzing intricate mechanisms involving multiple mediators that affect each other. Thus, in this article, we studied multiple sequentially ordered mediators for a dichotomous outcome and presented the identifiability assumptions for the path-specific effects on the outcome, that is, the effect of an exposure on the outcome mediated by a specific set of mediators. We proposed a closed-form estimator for the path-specific effects by modeling the dichotomous outcome using a probit model. Asymptotic variance of the proposed estimator is derived and can be approximated via delta method or bootstrapping. Simulations under a finite sample showed the validity of our method in capturing the path-specific effects when the probability of each potential counterfactual outcome is not small and demonstrated the utility of a computationally efficient alternative to bootstrapping for calculating variance. The method is applied to investigate the effects of polycystic ovarian syndrome on live birth rates mediated by estradiol levels and the number of oocytes retrieved in a large electronic in vitro fertilization database. We implemented the method into an R package SOMM , which is available at https://github.com/roqe/SOMM .     

Invited commentary: pushing the mediation envelope

The very insightful and clear paper by VanderWeele and Vansteelandt in this issue of the Journal (Am J Epidemiol. 2010;172(12):1339-1348) bridges the gap between biostatistics methodologists focusing on causal methods for mediation analyses and the practitioners of mediational analyses to the benefit of both groups. In an effort to continue the bridging of this gap, this invited commentary relates the important issue of "natural direct effects" to the well-known epidemiologic method of direct standardization. Additionally, attention is paid to the importance of temporal sequencing to help substantiate the mediation relations among the exposure, mediation, and outcome. A crucial mathematical distortion under the logistics model, called "absence of collapsibility," is noted in motivating VanderWeele and Vansteelandt's use of the log-linear model for comparing the effect of exposure adjusted for the mediator with the effect of exposure unadjusted for the mediator. It is also noted that this issue applies to one approach to assessing confounding. Finally, some issues are raised for consideration when testing the interaction between the exposure and mediator before assessing mediation.     

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.