Counterfactual Graphical Models for Longitudinal Mediation Analysis With Unobserved Confounding

Authors


Abstract

Questions concerning mediated causal effects are of great interest in psychology, cognitive science, medicine, social science, public health, and many other disciplines. For instance, about 60% of recent papers published in leading journals in social psychology contain at least one mediation test (Rucker, Preacher, Tormala, & Petty, 2011). Standard parametric approaches to mediation analysis employ regression models, and either the “difference method” (Judd & Kenny, 1981), more common in epidemiology, or the “product method” (Baron & Kenny, 1986), more common in the social sciences. In this article, we first discuss a known, but perhaps often unappreciated, fact that these parametric approaches are a special case of a general counterfactual framework for reasoning about causality first described by Neyman (1923) and Rubin (1924) and linked to causal graphical models by Robins (1986) and Pearl (2006). We then show a number of advantages of this framework. First, it makes the strong assumptions underlying mediation analysis explicit. Second, it avoids a number of problems present in the product and difference methods, such as biased estimates of effects in certain cases. Finally, we show the generality of this framework by proving a novel result which allows mediation analysis to be applied to longitudinal settings with unobserved confounders.

The aim of empirical research in many disciplines is establishing the presence of effects by means of either randomized trials or observational studies if randomization is not possible. For example, a celebrated success of empirical research in epidemiology is the discovery of a causal connection between smoking and lung cancer (Doll & Hill, 1950).

Once the presence of an effect is established, the precise mechanism of the effect becomes a topic of interest as well. A particularly popular type of mechanism analysis concerns questions of mediation, that is, to what extent a given effect of one variable on another is direct and to what extent it is mediated by a third variable. For example, it is known that genetic variants on chromosome 15q25.1 increase both smoking behavior and the risk of lung cancer (VanderWeele et al., 2012). A public health mediation question of interest here is whether these variants increase lung cancer risk by directly making the patients susceptible in some way, or whether the risk increase is driven by the increase in smoking.

In psychology, interest in mediation analysis began partly due to the influential S-O-R model (Woodworth, 1928), where causal relationships between stimulus and response are mediated by mechanisms internal to an organism, and partly due to the multistage causality present in many theories in psychology (such as attitude causing intentions, which in turn cause behavior in social psychology). Today, mediation questions are ubiquitous in psychology. Mediation analysis is used to explicate theories of persuasion (Tormala, Briñol, & Petty, 2007a), ease of retrieval (Schwarz et al., 1991; Tormala, Falces, Briñol, & Petty, 2007b), cognitive priming (Eagly & Chaiken, 1993), developmental psychology (Conger et al., 1990), and to explore many other areas. In fact, about 60% of recent papers published in leading journals in social psychology contain at least one mediation test (Rucker et al., 2011).

A standard approach for mediation analysis involves the use of (linear) structural equation models and the so-called difference method (Judd & Kenny, 1981) and product method (Baron & Kenny, 1986). The first method, more common in epidemiology, considers an outcome model both with and without the mediator and takes the difference in the coefficients for the exposure as the measure of the indirect effect. The second method, more common in the social sciences, takes as a measure of the indirect effect the product of (a) the coefficient for the exposure in the model for the mediator and (b) the coefficient for the mediator in the model for the outcome. These methods suffer from a number of problems. First, interpreting linear regression parameters as causal parameters is not appropriate when non-linearities or interactions are present in the underlying causal mechanism and can lead to bias (Kaufman, MacLehose, & Kaufman, 2004; MacKinnon & Dwyer, 1993). Second, it is not always the case that a regression parameter is interpretable as a causal parameter, even if the parametric structural assumptions of linearity and no interaction hold (Robins, 1986). Finally, these methods are not directly applicable to longitudinal settings (where multiple treatments happen over time) and assume no unmeasured confounding.

The aim of this article is twofold. First, we describe recent developments in the causal inference literature which address the limitations of the approaches based on linear structural equations (Baron & Kenny, 1986; Judd & Kenny, 1981). In particular, we show that the linear structural equation approach to mediation analysis is a special case of a more general framework based on potential outcome counterfactuals, developed by Neyman (1923) and Rubin (1974), and extended and linked to non-parametric structural equations and graphical models by Robins (1986) and Pearl (2000). We show how this more general framework avoids the difficulties of the linear structural equations approach and has additional advantages in making strong causal assumptions necessary for mediation analysis explicit. Second, we use the counterfactual framework to develop novel results which extend existing mediation analysis techniques to longitudinal settings with some degree of unmeasured confounding.

Our argument is that to handle increasingly complex mediation questions in psychology and cognitive science, scientists must necessarily move beyond the linear structural equation approach and embrace more general frameworks for mediation analysis. The linear structural equation approach is simply not applicable in complex data analysis settings, and careless generalizations of this approach will lead to biased conclusions.

The article is organized as follows. In section 'Mediation analysis using linear structural equations models', we describe mediation analysis based on linear structural equations in more detail and describe situations where the use of this method leads to problems. In section 'Potential outcomes and mediation', we introduce a causal inference framework based on potential outcome counterfactuals and graphical models, and show how this framework generalizes the linear structural equation framework and correctly handles the problems described in section 'Mediation analysis using linear structural equations models'. In section 'Path-specific longitudinal mediation with unobserved confounding', we describe two motivating examples involving three complications: unobserved confounding, longitudinal treatments, and path-specific effects, and show how the counterfactual framework is able to handle these complications with ease. Section 'Conclusions' contains the discussion and our conclusions. The general theory necessary to solve examples of the type shown in section 'Path-specific longitudinal mediation with unobserved confounding' is contained in the supplementary materials.

1. Mediation analysis using linear structural equations models

The standard mediation setting contains three variables, the cause or treatment variable, which we will denote by A, the effect or outcome variable, which we will denote by Y, and the mediator variable, which we will denote by M. The treatment A is assumed to have an effect on both mediator M and outcome Y, while the mediator M has an effect on the outcome Y. A typical goal of causal inference is establishing the presence of the total effect, or just the causal effect, of A on Y. The goal of mediation analysis is to decompose the total effect into the direct effect of the treatment A on the outcome Y, with the indirect or mediated effect of the treatment A on the outcome Y through the mediator M.

Causal relationships in mediation analysis are often displayed by means of causal diagrams. A causal diagram is a directed graph where nodes represent variables of interest, in our case the treatment A, the mediator M, and the outcome Y, and directed arrows represent, loosely, “direct causation.” The mediation setting is typically represented by means of a causal diagram shown in Fig. 1A.

The situation represented by this picture contains a treatment that is either randomly assigned by the experimenter or randomized naturally. For example, genetic variants on chromosome 15q25.1 which are linked with smoking behavior and lung cancer (VanderWeele et al., 2012) can generally (modulo possibly some confounding due to population genetics) be assumed to be naturally randomized. In psychology, a randomized treatment is often a treatment or prevention program, such as drug prevention.

Another common situation assumes that the treatment is not randomized, but all causes of the treatment are observed. One example of this situation is shown in Fig. 1B, which contains a single observed confounder C. Extending methods described in this section to this case is straightforward.

Given the causal structure shown in Fig. 1, the statistical analysis proceeds as follows. First, the causal relationships between treatment, mediator, and outcome are assumed to take the form of a causal regression model, or linear structural equation:

display math(1)
display math(2)

where math formula are intercepts, math formula are regression coefficients, math formula are mean zero noise terms, and the covariance of the noise terms for Y and M is assumed to equal 0: math formula.

For the “difference method,” a regression model for the outcome where the mediator is omitted is also included in the analysis:

display math(3)

For binary outcomes, it is straightforward to specify alternative regression models, such as logistic regression models. However, as we shall soon see, even this simple modeling change requires care.

The total effect under these models is taken to equal to math formula and can be derived using Sewall Wright's rules of path analysis (Wright, 1921). The direct effect under these models is taken to equal to the regression coefficient math formula of the treatment in the outcome model (Eq. (1)). The “product method” (Baron & Kenny, 1986) and the “difference method” (Judd & Kenny, 1981) both aim to express the indirect effect of A on Y in terms of statistical parameters of these regression models. The product method takes as a measure of the indirect effect the product of (a) the coefficient for the treatment in the model for the mediator (math formula in Eq. (2)) and (b) the coefficient for the mediator in the model for the outcome math formula in Eq. (1)). The difference method considers the outcome model with (Eq. (1)) and without the mediator (Eq. (3)) and takes the difference in the coefficients for the treatment in these two models (math formula and math formula) as the measure of the indirect effect. If the outcome and mediator are continuous and there are no interaction terms in the regression model for the outcome, the two methods produce identical answers for the indirect effect (MacKinnon, Warsi, & Dwyer, 1995).

An important property in mediation analysis is the decomposition property:

display math(4)

This property allows the investigator to quantify how much of an existing total effect of treatment on outcome is due to the direct influence on the outcome, and how much is due to the influenced mediated by a third variable. Note that it is possible for the total effect to be weak or non-existent, and direct and indirect effects to both be strong. This situation can occur due to cancellation of effects. For instance, there may be a strong positive direct effect, but an equally strong negative mediated effect, resulting in a weak total effect. The decomposition property holds for linear structural equation models with continuous outcomes, for indirect effects defined by both the product and difference methods.

The advantage of the product and difference methods is their simplicity—they rely on standard software for fitting regression models. The disadvantage is their lack of flexibility. In order to work, these methods require assumptions of linearity, no unmeasured confounding between mediator and outcome, and continuous outcomes. As we will see in the next section, careless application of these methods in settings where one or more of these assumptions are violated will result in bias and counterintuitive conclusions.

1.1. Problems with the product and difference methods

With binary outcomes (MacKinnon & Dwyer, 1993), or interaction terms in the outcome (VanderWeele & Vansteelandt, 2009), the two methods above no longer agree on the estimate of the indirect effect. In addition, there is evidence that in the case of non-linearities or interactions in the outcome model, neither method gives a satisfactory measure of the indirect effect (VanderWeele & Vansteelandt, 2009).

Furthermore, even for the case of continuous outcome models with no interaction terms, certain underlying causal structures can make it impossible to associate any standard regression parameter with direct and indirect effects. Consider the causal diagram shown in Fig. 2. This diagram represents a situation where we have a randomized treatment A and the outcome Y, but instead of a single mediator, we have two mediating variables L and M. Furthermore, we have reasons to believe there is a strong source of unobserved confounding (which we call U) between one of the mediators L and the outcome Y. For instance, if A represents a primary prevention program (say drug prevention), and M represents a secondary prevention program (say a program designed to increase screening rates for serious illness), then L might represent some observable intermediate outcome of people enrolled in the primary program, perhaps linked to eventual outcome Y via some unobserved measure of conscientiousness or health consciousness. Assume for the moment that all variables are continuous, and we can model their relationships using linear regression models:

display math(5)
display math(6)
display math(7)

We model the presence of the U confounder by allowing that math formula, while assuming math formula. We still assume mean zero error terms. Note that though the directed arrows in the graph in Fig. 2 are causal, not all of the regression coefficients in above equations have causal interpretations. In particular, math formula, and math formula do not have causal interpretations, while math formula does (as the direct effect of M on Y).

We are interested in quantifying the direct effect of A on Y, and the effect of A on Y mediated by M. The question is, what (combination of) parameters of the regression models we specified correspond to these effects. A naive approach would be to consider a regression model in Eq. (5) and take the regression parameter math formula associated with A as the measure of the direct effect. This approach is wrong and will lead to bias. The difficulty with this example is that a regression coefficient of a particular independent variable X represents the extent to which the dependent variable Y depends on X given that we condition on all other independent variables. In our example, the regression coefficient for A represents dependence of Y on A given that we conditioned on L and M (we do not condition on U since U is not observed). Unfortunately, conditioning on L makes U and A dependent due to the phenomenon known as “explaining away.”

Consider a toy causal system: A light in a hallway is wired to two toggle light switches on the opposite ends of the hallway. If either of the light switches is flipped, the light turns on. Two people, Alice and Uma, stand at opposite ends of the hallway, each near a switch. Alice sees the light turn on and knows she did not toggle the switch. She can then conclude (“explain away” the light turning on) that Uma toggled the switch. In our graph, Alice's switch is A, Uma's switch is U, and the light itself is L. Conditional on L, we can learn information about U if we know something about A. In other words, conditional on L, A and U become dependent. Of course, U is a direct cause of Y. This means that some of the variation of Y due to A, represented by the regression coefficient of A in Eq. (5), is actually due to the “explaining away” effect correlating A and U, which in turn correlates A and Y in a non-causal way. In particular, even if there is no direct effect of A on Y, the regression coefficient of A will not vanish in most models.

In fact, it can be shown that in examples of this sort, the presence of unobserved confounders, coupled with the “explaining away” effect, will prevent us from associating any standard function of regression coefficients with causal parameters in a way that avoids bias. Furthermore, even if the correct expression for the direct effect is used (as derived in a subsequent section, and shown in Eq. 17), using standard statistical models in that expression can result in cases where the absence of direct effect is not possible given the model. In particular, if we use a linear regression model with no interaction terms for a continuous outcome Y, and a logistic regression model with no interaction terms for a binary mediator L, then the absence of direct effect is impossible given those models in the sense that the expression in Eq. 17 will never equal 0. This difficulty, which applies not only to regression models but to almost any standard parametric statistical model associated with causal diagrams such as Fig. 2, is known as the “null paradox” (Robins & Wasserman, 1997).

Finally, even if assumptions of linearity, no interaction, and no unobserved confounding hold, no function of the observed data will equal to either direct or indirect effect in general. In order for this equality to hold, it must be the case that error terms of the outcome and mediator model remain uncorrelated for any possible set of assignments of independent variables to the model. This assumption is also necessary in order to derive mediation effects from double randomization studies (Imai, Tingley, & Yamamoto, 2013; Word, Zanna, & Cooper, 1974). The assumption cannot easily be tested and can be viewed as ruling out unobserved confounding between variables in different counterfactual situations. It will be described in more detail later. Deriving analogs of this crucial assumption in more complex settings, for the purposes of sensitivity analysis, can be challenging.

A way out of many of these difficulties involves generalizing from linear regression models to a general non-parametric framework based on potential outcome counterfactuals. This framework will be described in great detail in the next section. We will show how this framework gives a more general representation of direct and indirect effects that will happen to coincide with the results of the product and difference methods in the special case of linear regression models. We will also show how assumptions underlying mediation analysis can be clearly explained as independence statements among random counterfactual variables, displayed graphically by a causal diagram. We will discuss possible solutions to the null paradox that can be derived in this framework. Finally, the flexibility of the framework will allow us to pose more complex questions of mediation, such as “what is the effect of A on Y along the path AMY in the graph in Fig. 2?” and answer these questions in complex settings involving multiple time-dependent treatments, and unobserved confounding.

2. Potential outcomes and mediation

Typically, the notion of causal effect of treatment A on outcome Y refers to change in the outcome between the control group and the test group in a randomized control trial. A general representation of causality, divorced from a particular statistical model such as a regression model, must capture this notion in some way. An idealized, mathematical representation of a randomized control trial captures the notion of controlling a variable by means of an intervention. An intervention on A, denoted by do(a) by Pearl (2000), refers to an operation that fixes the value of A to a regardless of the natural variability of A. An intervention represents an assignment of treatment to the test group, or a decision to set A to a. The variation in the outcome after an intervention is captured by means of an interventional distribution, sometimes denoted by p(y|do(a)).

Crucially, intervening to force A to value a is not the same as observing that A attains the value a, that is: p(y|a)≠p(y|do(a)). As an example: “only Olympic sprinters that can run quickly win gold medals (observation), therefore I should wear a gold medal to run faster (intervention).”1 This is the essence of the common refrain that correlation (statistical dependence) does not imply causation.

A potential outcome counterfactual refers to the value of a random variable under a particular intervention do(a) for a particular unit (individual) u and is denoted by Y(a, u). If we wish to average over units in a particular study, we would obtain a random variable Y(a), representing variation in the outcome after the intervention do(a) was performed. In other words, Y(a) is a random variable with a distribution p(y|do(a)).

Assume for the moment the simplest mediation setting with variables A, M, Y, shown in Fig. 1, and assume the causal relationships between A, M, and Y can be captured by structural equations shown in Eqs. (1) and (2). The intervention do(a) in these systems of equations is represented by replacing the random variable A in each equation with the intervened value a. Alternatively, if we augment Eqs. (1) and (2) with another equation for A itself, such as:

display math(8)

then the intervention on A can be represented by replacing Eq. (8) by another equation that sets A to a constant a. If interventions are represented in this way, then the total effect of A on Y, equal to math formula, can be viewed as

display math(9)

In other words, the total effect is the expected difference of outcomes under two hypothetical interventions. In one intervention, A is set to 1, and in another A is set to 0. Note that this definition is non-parametric in that it does not rely on the model for Y being a linear regression model. In fact, the definition remains sensible even if we replace the models for Y and M by arbitrary functions:

display math(10)
display math(11)
display math(12)

These models can be viewed as (non-parametric) structural equations and are discussed in great detail by Pearl (2000). The key idea is that we assume the causal relationship between a variable, say Y, and its direct causes is by means of some unrestricted causal mechanism function math formula. These structural models can still be modeled by means of causal diagrams, but they are no longer bound by linearity, lack of interactions, or other parametric assumptions.

2.1. Direct and indirect effects as potential outcomes

Representing direct and indirect effects using potential outcomes is slightly more involved. In the case of total effects, the intuition was that A being set to 0 represents “no treatment,” while A being set to 1 represents “treatment,” and we want to subtract off the expected outcome under no treatment (the baseline effect) from the expected outcome under treatment. In the case of direct effects we still would like to subtract off the baseline, but from the effect that considers only the direct influence of A on Y in some way.

One approach that preserves the attractive property of decomposition of total effects into direct and indirect effects proceeds as follows. We consider a two-stage potential outcome. In the first stage, we consider for a particular unit u, the value the mediator would take under baseline treatment a = 0: M(0, u). We then consider the outcome value of that same unit if the treatment was set to 1, and mediator was set to M(0, u): Y(1, M(0, u), u). In other words, the direct influence of A on Y for this unit is quantified by the value of the outcome in a hypothetical situation where we give the individual the treatment but also force the mediator variable to behave as if we did not give the individual treatment. In graphical terms, this is the outcome value if active treatment a = 1 is only active along the direct path AY, but not active along the path AMY, since we force M to behave as if treatment was set to 0 for the purposes of that path. If we average over units, we get a nested potential outcome random variable: Y(1, M(0)). We define the direct effect as the difference in expectation between this random variable and the baseline outcome:

display math(13)

Note that E[Y(0)] = E[Y(0,M(0))]. The indirect effect is defined similarly, except we now subtract off the direct influence of A on Y from the total effect of setting A to 1:

display math(14)

It is not difficult to show that these definition reduce to definitions in terms of regression coefficients given in the previous sections in the special case where Y is continuous, math formula, and math formula are linear functions with no interactions, and all ε noise terms are Gaussian. However, these effect definitions, known as natural (Pearl, 2001) or pure (Robins & Greenland, 1992) are the only sensible definitions of direct and indirect effects currently known that simultaneously maintain the decomposition property (4), and apply to arbitrary functions in structural Eqs. (10), (11), and (12).

2.2. Assumptions underlying mediation analysis

Defining the influence of A on Y for a particular unit u as Y(1, M(0, u), u) involved a seemingly impossible hypothetical situation, where the treatment given to u was 0 for the purposes of the mediator M, and 1 for the purposes of the outcome Y. In other words, this situation is a function of multiple, conflicting hypothetical worlds. In general, no experimental design is capable of representing this situation unless it is possible to bring the unit by some means to the pre-intervention state (perhaps by means of a “washout period,” or some other method). In order to express direct and indirect effects defined in the previous section as functions of the observed data, such as regression coefficients, we must be willing to make certain assumptions that make our impossible hypothetical situation amenable to statistical analysis.

A typical assumption that makes our situation tractable is expressed in terms of conditional independence statements on potential outcome counterfactuals:

display math(15)

where math formula stands for “X is marginally independent of Y,” and math formula stands for “X is conditionally independent of Y given Z.”

This assumption states that if we happen to have some information on how the mediator varies after treatment is set to 0, this does not give us any information about how the outcome varies if we set the treatment to 1 and the mediator to (arbitrary) m. Note that this assumption immediately follows if we assume independent error terms in a non-parametric structural equation model defined by Eqs. (10), (11), and (12). This assumption allows us to perform the following derivation:

display math(16)

This derivation expressed our potential outcome as a product of two terms, with each of these terms representing variation in a random variable after a well-defined intervention. This represents progress, since we were able to express a random variable not typically representable by any experimental design in terms of results of two well-defined randomized trials, one involving Y as the outcome and A,M as treatments, and one involving M as the outcome, and A as treatment.

Unfortunately, even a single randomized study can be expensive or possibly illegal to perform on people (if the treatment is harmful), let alone two. For this reason, a common goal in causal inference is to find ways of expressing interventional distributions as functions of observed data. In the causal inference literature, this problem is known as the identification problem of causal effects.

As mentioned earlier, the interventional distribution, such as that corresponding to Y(a), namely p(y|do(a)), is not necessarily equal to a conditional distribution p(y|a). Nevertheless, such an equality holds if there are no unobserved confounders or common causes between A and Y. This happens to be the case in our example. In terms of potential outcomes, the lack of unobserved confounding is expressed in terms of the ignorability assumption

display math

In words, this assumption states that if we happen to have information on the treatment variable, it does not give us any information about the outcome Y after the intervention do(a) was performed. A graphical way of describing ignorability is to say that there does not exist certain kinds of paths between A and Y, called back-door paths (Pearl, 2000), in the causal diagram. Such paths are called “back-door” because they start with an arrow pointing into A. It can be shown that if ignorability holds for Y(a) and A (alternatively if there are no back-door paths from A to Y in the corresponding causal diagram), then p(y|do(a)) = p(y|a).

If there exist common causes of A and Y but they are observed, as is the case of node C in Fig. 1B, it is possible to express a more general assumption known as the conditional ignorability assumption

display math

In words, this assumption states that if we happen to have information on the treatment variable, then conditional on the observed confounder C, this information gives us no information about the outcome Y after the intervention do(a) was performed. In graphical terms, this assumption is equivalent to stating that C “blocks” all back-door paths from A to Y.2 It can be shown that if conditional ignorability math formula holds, then math formula. This formula is known as the back-door formula, or the adjustment formula.

Figure 1.

(A) Typical mediation setting: a is the treatment, m is the mediator, y is the outcome. (B) Mediation setting with observed confounding: c is a confounder due to being a common cause of treatment, Mediator, and outcome variables.

Sometimes identification of causal effects is possible even in the presence of unobserved confounding. See the work of Tian and Pearl (2002a), Huang and Valtorta (2006), and Shpitser and Pearl (2006a, b, 2008) for a general treatment of the causal effect identification problem.

In our case, the ignorability assumption for M(a) and A, as well as for Y(1, m) and A, M allows us to further express each of the terms in the product in Eq. (16) in terms of observed data as follows:

display math

Plugging this last expression into the formula (13) for direct effects gives us

display math

This expression is known as the mediation formula (Pearl, 2011). Note that the mediation formula does not require a particular functional form for causal mechanisms relating Y, M, and A.

Note also that assumption (15) is untestable, since it is positing a marginal independence between two potential outcomes, one of which involves the treatment being set to 1, and another involves the treatment being set to 0. A form of this assumption is still necessary in order to equate direct and indirect effects with functions of regression coefficients in the simple linear regression setting described in previous sections, in the sense that violations of the assumption will generally prevent us from uniquely expressing a given direct or indirect effect as a function of observed data (e.g., the effect becomes non-identifiable). For this reason, even in the simplest mediation problems, care must be taken to either justify assumption (15) on strong substantive grounds, perform a reasonable sensitivity analysis (Tchetgen & Shpitser, 2012b), or reduce the mediation problem to a testable problem involving interventions without conflicts (Robins & Richardson, 2010).

2.3. Mediation with unobserved confounding

One of the advantages of the potential outcome framework is its flexibility. Since it does not rely on parametric assumptions, it can be readily extended to handle modeling complications. Consider again our two mediator example in Fig. 2. We mentioned in the previous section that product and difference methods will result in biased estimates of direct effects of A on Y not through M, due to a combination of unobserved confounding and the “explaining away” effect in that example. A non-parametric definition of direct effect based on potential outcomes avoids these difficulties. Our expression for direct effect is E[Y(1, M(0))]−E[Y(0)]. Since A is randomized (there is no unobserved confounding between A and the outcome Y), the second term can be shown to equal E[Y|A = 0]. The first term can be shown, given assumption (15), and a general theory of identification of causal effects (Shpitser & Pearl, 2006b, 2008; Tian & Pearl, 2002a) to equal

display math

The direct effect is then equal to

display math(17)

while the indirect effect is equal to

display math

It can be shown that not only do the direct and indirect effects add up to the total effect in this case, but the quantity 17 equals 0 precisely when the effect of A on Y along the arrow AY is in some sense absent.3 However, even though we used simple linear regression models in this example, neither of these effects reduces to any straightforward function of the regression coefficients. It is possible to express these kinds of functional as functions of regression coefficients in an appropriately adjusted model (such as the marginal structural model, which is estimated by fitting weighted regression models (Robins, Hernan, & Brumback, 2000), or as functions of parameters in a non-standard parameterization of causal models, where statistical parameters correspond to causal parameters directly (Richardson, Robins, & Shpitser, 2012; Shpitser, Richardson, & Robins, 2011).

3. Path-specific longitudinal mediation with unobserved confounding

In the previous section, we saw how the presence of unobserved confounders and multiple mediators can easily result in situations where regression coefficients cannot be meaningfully associated with direct and indirect causal effects. In this section, we consider even more complex mediation settings, which can nevertheless be handled appropriately using the potential outcome counterfactual framework representing (possibly non-linear) structural equations. We motivate the discussion with two examples, one from HIV research and one from psychology.

The human immunodeficiency virus causes AIDS by attacking and destroying helper T cells. If the concentration of these cells falls below a critical threshold, cell-mediated immunity is lost, and the patient eventually dies to an opportunistic infection. Patients infected with HIV with reduced T cell counts are typically put on courses of antiretroviral therapy (ART), as a first-line therapy. Unfortunately, side effects of many types of ART medication may cause poor adherence to the therapy (i.e., patients do not always take the medication on time or stop taking it altogether). Side effects are often caused by toxicity of the medication or patient's adverse reaction to the medication. Severity of the side effects is often linked to the patient's “overall health level” (ill defined, and thus not measured), which also affects the eventual outcome of the therapy (survival or death). If the ART happens to not be very effective at viral suppression and results in patient deaths, this could be because the ART itself is not very good, or it could be due to poor patient adherence. In other words, poor outcomes of ART result in a natural mediation question in HIV research—is the poor total effect possibly due to cancelation of a strong direct effect of the medication on survival by an equally strong indirect effect of poor adherence?

The situation is shown graphically in Fig. 3. Here, we show ART taken over the course of 2 months, represented by two time slices. In practical studies, ART is taken over a period of years, and the number of time slices is quite large. In this graph, the ART is represented by nodes math formula and math formula, the patient outcome by Y, patient adherence at each time slice by math formula and math formula, toxicity of the medication by math formula, math formula, and finally the unobserved state of patient's health affecting reaction to the medication and the outcome by U. Since we are interested in the indirect effect of ART on survival mediated by adherence, we are only interested in the effect along the paths from math formula on Y which pass through math formula. These paths are shown in green in the graph.

Our second example, which is isomorphic to the HIV example above, concerns the use of prevention programs to promote positive outcomes in vulnerable populations. Assume the primary interventions math formula involve attending a drug prevention program. Like ART in the previous example, the program is an ongoing intervention (say a monthly meeting). However, there is also a secondary intervention math formula which is meant to increase the rates of screening for serious illness such as cancer. Although the secondary intervention is not directly related to drug prevention, it is conceivable that there is a synergistic effect between the primary and secondary intervention to promote positive outcomes (say staying drug free), perhaps due to the fact that both interventions promote good habits and health consciousness. In this example, math formula are, loosely speaking, “the participant's responsiveness,” which may be affected by unobserved factors involving family, friends, socioeconomic background (U), and so on. These unobserved factors also influence the outcome. The mediation question here is quantifying the extent to which the outcome is influenced by the primary intervention itself, versus an indirect effect via the secondary intervention. The indirect effect mediated by the secondary intervention is, again, shown in green in Fig. 3.

Aside from unobserved confounding represented by U, a complication also present in the example in Fig. 2 in the previous section, what makes these examples difficult is first the longitudinal setting where treatments recur over multiple time slices, and second that we are interested in effects along a particular bundle of causal paths. In previous sections, we were interested in effects either only along the direct path AY, or only along all paths other than the direct one. In these cases, we are interested in some indirect paths (through math formula), but not others (through math formula).

In the case of multiple treatments, causal effect from these treatments to the outcome Y is transmitted along special paths called proper causal paths of the form math formula, where math formula is one of the treatments, and where this path cannot intersect any other treatment other than math formula (otherwise it is really a causal path from that second treatment to the outcome). We are interested in quantifying the effect along a subset of such paths, displayed graphically as those paths consisting entirely of green arrows. Algebraically, we will denote this set of paths of interest as π. The proper causal paths along which the causal effect is transmitted but which do not lie in π is displayed graphically as those paths which contain at least one blue arrow.

3.1. Formalization of path-specific effects

Naturally, even if the statistical model associated with the causal diagram shown in Fig. 3 is given in terms of linear regressions, it is not possible to express effects of interest as simple functions of regression coefficients. However, it is possible to express these path-specific effects (Avin, Shpitser, & Pearl, 2005; Pearl, 2001) in terms of potential outcome counterfactuals.

We will use an inductive rule to construct a potential outcome representing the effect of math formula on Y only along green paths. For the purposes of this rule, we will represent values math formula of math formula to represent “baseline treatment” or “no treatment” (in the previous section we used 0), and values math formula to represent “active treatment” (in the previous section we used 1). This potential outcome will involve Y, and interventions on all direct causes of Y (that is all nodes X such that XY exists in the graph). These interventions are defined as follows.

If the arrow XY is blue, this means we are not interested in the effect transmitted along this arrow. In previous sections, we represented this by considering the value of X “as if treatment was baseline,” or X(0). In our case, we will do the same, except since we have two treatments we set them both to the baseline values math formula. For example, math formula and math formula are both direct causes of Y along blue arrows. This means we intervene on whatever values they would have had if math formula were set to baseline, or math formula, math formula. If the direct cause of Y is one of the treatments math formula or math formula then we intervene to set their value to “active” if the arrow from treatment to outcome is green (e.g. we are interested in the effect) or “baseline” if the arrow is blue (e.g., we are not interested in the effect). Finally, if the direct cause of Y is not a treatment but is a direct cause along a green arrow, we inductively set the value of that cause to whatever value it would have had under a path-specific effect of math formula on that cause. For instance, math formula is a direct cause of Y along a green arrow, which means we set the value of math formula to whatever value is dictated by the path-specific effect of math formula on math formula. To figure out what that value is, we simply apply our rule inductively from the beginning, except to math formula as the outcome, rather than Y.

Applying the first stage of our rule gives us the potential outcome math formula, where math formula and math formula are path-specific effects of math formula on math formula and math formula, respectively, along the green paths only. If we apply the rule inductively math formula, and math formula, and plug in and simplify, we get that our path-specific effect is the following complex potential outcome

display math(18)

While this expression looks algebraically complex, what it is actually expressing is a rather simple idea. We have two treatment levels: “baseline” and “active.” For the purposes of green paths, the causal paths we are interested in, we pretend treatment levels are active. For the purposes of all other paths, we pretend treatment levels are baseline. In this way, the treatment is active only along the paths we are interested in, and all other paths are “turned off.” We use this rule to select what values we intervene on, and then use these interventions in a nested way, following the causal paths of the graph. For the special case of single treatments and no unobserved confounding, an equivalent definition of path-specific effects phrased in terms of replacing structural equations is given by Pearl (2001).

3.2. The total effect decomposition property for path-specific effects

In the previous sections, we defined the direct and indirect effects by taking a difference of expectations (see Eqs. (13) and (14)). We can generalize such definitions to path-specific effects to obtain a decomposition of the total effect into a sum of two terms, one representing the effect along proper causal paths in π, and one representing the effect along proper causal paths not in π.

First, assume the distribution for the nested potential outcome defining the path-specific effect along proper causal paths in π of A on Y is given by math formula. Then we have

display math(19)

and

display math(20)

Since the total effect can be defined as math formula, we have

display math

which is an intuitive additivity property stating that the total effect can be decomposed into a sum of two terms, where one term quantifies the effect operating along a given bundle of proper causal paths π, and another term quantifies the effect operating along all proper causal paths other than those in π. Note that this property generalizes the additivity property for direct and indirect effects, where π was taken to mean a single arrow from A to Y.

3.3. Path-specific effects as functions of observed data

In the previous section, we showed that a path-specific effect can be defined in terms of a nested potential outcome after we “turn off” causal paths we are not interested in. Regardless of how sensible such a definition may be, it is not very useful unless this potential outcome can be expressed as a function of the observed data, and thus become amenable to statistical analysis.

In a previous section, we showed that in order to express direct and indirect effects in terms of observed data, we needed to make an untestable independence assumption (shown in Eq. (15)). Path-specific effects generalize direct and indirect effects, and thus require even more assumptions.

In fact, for the path-specific effect along green paths in Fig. 3, it suffices to believe the following independence claim for any value assignments math formula:

display math(21)

If we believe this assumption, we can express the path-specific effect in Eq. 18 as

display math(22)
display math

This expression is a product of terms, where each term is a well-defined interventional density (i.e., there are no conflicts involving different hypothetical worlds). If we further make use of the general theory of identification of interventional densities from observed data (Shpitser & Pearl, 2006b, 2008; Tian & Pearl, 2002b), we can express the above as follows

display math(23)

This expression is a function of the observed data.

What is left is finding an expression for the total effect of math formula on Y. It is not difficult to show that math formula. By analogy with a previous section, we can express the path-specific effect via fully green paths in π through math formula as a difference of expectations

display math(24)

while the effect via all paths that are not fully green (that is, proper causal paths not in π) as another difference

display math(25)

These quantities can be estimated with standard statistical methods by simply positing a model for each term, for instance a regression model, estimating the models from data, and computing the estimated functional. This is the so-called parametric g-formula approach (Robins, 1986). With this method, care must be taken to avoid the “null paradox” issue, as was the case with direct and indirect effects. A less straightforward approach which only relies on modeling the probability of the treatment in each times lice given the past, is to generalize marginal structural models (Robins et al., 2000), which were originally developed in the context of estimating total effects in longitudinal settings with confounding. Another alternative is to extend existing multiply robust methods for point treatment mediation settings (Tchetgen & Shpitser, 2012b) based on semi-parametric statistics to the longitudinal setting.

3.4. Expressing arbitrary path-specific effects in terms of observed data

One difficulty with path-specific effects is that the corresponding potential outcome counterfactual is nested and therefore complicated. On the other hand, the graphical representation of path-specific effects on a causal diagram is fairly intuitive (effect along green paths only). For this reason, it would be desirable to obtain a result which says, for a particular bundle of green paths on a particular causal diagram, whether the corresponding counterfactual can be expressed as a function of the observed data, without going into the details of the counterfactual itself. In this section, we give just such a result, which generalizes existing results on path-specific effects in cases with a single treatment and no unobserved confounding (Avin et al., 2005).

We first start with a few preliminaries on graphs. We will display causal diagrams with unobserved confounding, such that those in Figs. 2 and 3 by means of a special kind of mixed graph containing two kinds of edges, directed edges (→), either blue or green depending on whether we are interested in the corresponding causal path, and red bidirected edges (↔). The former represent direct causation edges, as before. The latter represent the presence of some unspecified unobserved common cause. For example, we represent the causal diagram in Fig. 3 by means of the mixed graph shown in Fig. 4. Note that since U links three nodes, math formula, each pair of these three is joined by a bidirected arrow. We call this type of mixed graph an acyclic directed mixed graph (ADMG) (Richardson, 2009). Verma and Pearl (1990) called these types of graphs latent projections. The reason we use bidirected arrows is both to avoid cluttering the graph with potentially many possible unobserved confounders, and because certain crucial definitions involving confounders are easier to state in terms of bidirected arrows.

Figure 2.

Mediation setting with an unobserved confounder U, two mediators L and M, and no direct effect of A on Y.

A sequence of distinct edges such that the first edge connects to X, the last edge connects to Y, and each kth and k+1th edge shares a single node in common is called a path from X to Y. If vertices X,Y are connected by a path of the form X→…→Y, then we say X is an ancestor of Y and Y is a descendant of X. Such a path is called directed. A path is called bidirected if it consists exclusively of bidirected edges. For an ADMG math formula with a set of nodes V, we define a subgraph math formula over a subset AV of nodes to consist only of nodes in A and edges in math formula with both endpoints in A. If an edge XY exists, X is called a parent of Y, and Y a child of X.

Definition 1 (district). Let math formula be an ADMG. Then for any node a, the set of nodes in math formula reachable from a by bidirected paths is called the district of a, written math formula. For example, in the graph in Fig. 4, math formula.

Definition 2 (recanting district). Let math formula be an ADMG, A, Y sets of nodes in math formula, and π a subset of proper causal paths which start with a node in A and end in a node in Y in math formula. Let math formula be the set of nodes not in A which are ancestral of Y via a directed path which does not intersect A. Then a district D in an ADMG math formula is called a recanting district for the π-specific effect of A on Y if there exist nodes math formula (possibly math formula), math formula, and math formula (possibly math formula) such that there is a proper causal path math formula in π, and a proper causal path math formula not in π.

It turns out that the recanting district criterion characterizes situations when a potential outcome counterfactual can be expressed in terms of well-defined interventions, without conflicts, as long as we assume that the causal diagram represents a set of (arbitrary) structural equations.

Theorem 3 (recanting district criterion). Let math formula be an ADMG representing a causal diagram with unobserved confounders corresponding to a structural causal model. Let A, Y sets of nodes in math formula, and π a subset of proper causal paths which start with a node in A and end in a node in Y in math formula. Then the π-specific effect of A on Y is expressible as a functional of interventional densities if and only if there does not exists a recanting district for this effect.

The functional referenced in the theorem is equal to

display math(26)

where D ranges over all districts in the graph math formula, math formula refers to nodes with directed arrows pointing into D but which are themselves not in D, and value assignments math formula are assigned as follows. If any element a in A occurs in math formula in a term math formula, then it is assigned a baseline value if the arrows from a to elements in D are all blue, and an active value if the arrows from a to elements in D are all green. All other elements in math formula and D are assigned values consistent with the values indexed in the summation.

The proof of this theorem is given in the supplementary materials. As an example, assume we are interested in the effect of math formula on Y along only the green paths in the graph in Fig. 3. The set of nodes math formula in this case is just math formula. There are three districts in the corresponding graph math formula, which is just the graph obtained from Fig. 3 by removing math formula and all edges adjacent to these nodes. These three districts are math formula, math formula, and math formula. It is never the case that both a green and a blue arrow from math formula or math formula points to nodes in the same district such that these nodes are ancestors of Y. This means there is no recanting district for this effect, which in turn means the effect is expressible as a functional of interventional densities. We have already verified this fact in the previous section where this functional was given as Eq. (17), and which can be rephrased in the do(.) notation as follows:

display math

which can be shown equals to

display math(27)

which is easily verified to be an example of Eq. (20).

On the other hand, in either of the graphs shown in Fig. 5, the recanting district exists. In Fig. 5a, the district math formula is recanting, since the path math formula is not fully green (e.g., we are not interested in the effect along this path), while the path math formula is fully green, and math formula and math formula lie in the same district. Similarly, in Fig. 5b, the district math formula is recanting, since the path math formula is not fully green, while the path math formula is fully green, and math formula is its own district.

The recanting district criterion generalizes an earlier result for static treatments and no unobserved confounding known as a recanting witness, where the “witness” is a singleton node (Avin et al., 2005). The term is “recanting” because for the purposes of one path from a particular treatment math formula the witness (or district in our case) pretends the treatment should be active, while for the purposes of another path from that same treatment math formula the witness (or district) “changes the story” and pretends the treatment should be baseline. Identification of path-specific effects in terms of interventional distributions must always avoid this “recanting” phenomenon. Note that even in the case of multiple longitudinal treatments, the “recanting” phenomenon still involves a single treatment but spoils the identification of the whole effect of multiple treatments.

The presence of the recanting district prevents the expression of path-specific effects in terms of either observed or interventional data in the sense that it is possible to construct two distinct causal models which are observationally and interventionally indistinguishable, which are represented by the same causal diagram, but which disagree on the value of the path-specific effect. Furthermore, if the recanting district criterion does not exist, it is possible to characterize cases in which the expression for the path-specific effect in terms of interventional densities can be further expressed in terms of observational data.

Theorem 4. Let math formula be an ADMG with nodes V representing a causal diagram with unobserved confounders corresponding to a structural causal model. Let A, Y sets of nodes in math formula, and π a subset of proper causal paths which start with a node in A and end in a node in Y in math formula. Assume there does not exist a recanting district for the π-specific effect of A on Y. Then the counterfactual representing the π-specific effect of A on Y is expressible in terms of the observed data p(V) if and only if the total effect p(y|do(a)) is expressible in terms of p(V). Moreover, the functional of p(V) equal to the counterfactual is obtained from Eq. (20) by replacing each interventional term in Eq. (20) by a functional of the observed data identifying that term given by Tian's identification algorithm (Shpitser & Pearl, 2006b, 2008; Tian & Pearl, 2002a).

General theory of identification of causal effects states that if p(y|do(a)) is identifiable in terms of observed data, then it can be expressed as the functional very similar to that in Eq. (20), except all variables in A are assigned “active values” a. If p(y|do(a)) is identifiable from observed data, then each of the interventional terms in the functional is expressible in terms of observed data. This theorem simply states that to obtain our path-specific effect all we have to do is obtain the functional of the observed data expressing p(y|do(a)) and replace the appropriate “active values” a by “baseline values” math formula in those terms of the functional which correspond to districts containing children of treatment A via blue arrows. For example, it can be shown that the total effect math formula in Fig. 3 is equal to

display math(28)

Replacing math formula and math formula by math formula and math formula in expressions for Y, math formula and math formula (which are the only terms where the node before the conditioning bar is a child of math formula or math formula along blue arrows) yields precisely Eq. 23, which is the function of the observed data equal to the path-specific effect. We conclude by giving a form of the mediation formula generalized for path-specific effects on the difference scale.

Corollary 5 (generalized mediation formula for path-specific effects). Let math formula be an ADMG with nodes V representing a causal diagram with unobserved confounders corresponding to a structural causal model. Let A be a set of nodes, Y a single node in math formula, and π a subset of proper causal paths which start with a node in A and end in Y in math formula. Assume there does not exist a recanting district for the π-specific effect of A on Y. Assume p(y|do(a)) is expressible as a functional math formula of the observed data, and the path-specific effect is equal to the functional math formula obtained in Theorem 2. Then the path-specific effect along the set of paths π on the mean difference scale for active value a and baseline value math formula is equal to

display math

while the path-specific effect along all paths not in π on the mean difference scale is equal to

display math

An example of the generalized mediation formula applied to the longitudinal mediation setting shown in Fig. 3 is shown in Eq. (18) for the direct effect, and Eq. (19) for the indirect effect. The proof of these assertions is also given in the supplementary materials.

Figure 3.

A longitudinal mediation setting with an unobserved confounder u, where we are interested in effects along all paths from math formula and math formula to Y only through math formula or math formula. Paths we are interested in are shown in green.

Figure 4.

A mixed graph representing unobserved confounding in Fig. 3.

Figure 5.

Two variations on the graph in Fig. 4 where the path-specific effect of math formula on Y along the green paths is not identifiable from observational (or even interventional) data. (A) The presence of an unobserved parent of math formula and math formula spoils identification. (B) If math formula is a direct cause of math formula, the effect along green paths is not identifiable.

4. Conclusions

In this article, we have shown that existing methods for mediation analysis in epidemiology and psychology literature based on the product and difference methods (Baron & Kenny, 1986; Judd & Kenny, 1981) and linear regression models suffer from problems in the presence of interactions, non-linearities, binary outcomes, unobserved confounders, and other modeling complications. We have described a general framework developed in the causal inference literature based on potential outcome counterfactuals, non-parametric structural equations, and causal diagrams which recovers the product and difference methods as a special case, but which is flexible enough to handle multiple types of difficulties which arise in practical mediation analysis situations.

Our article serves two aims. We first wish to caution against careless use of mediation methodology based on linear regressions in situations where such methodology is not suitable. Such careless use may invalidate any conclusions about mediation that are drawn. Second, we want to show that appropriate use of functional models and potential outcomes is a very flexible strategy for tackling complex questions in causal inference, including mediation questions in longitudinal settings with unobserved confounding. We demonstrate this flexibility by developing a complete characterization of situations when path-specific effects are expressible as functionals of the observed data. This result paves the way for using statistical tools for answering general mediation questions in longitudinal observational studies. We argue that methods based on functional models and potential outcomes are often a more appropriate methodology in complex mediation setting than simpler methods based on linear structural equations.

Notes

  1. 1

    We want to thank the lesswrong.com community for this example.

  2. 2

    Pearl (1988, 2000) gives a more detailed discussion of the notion of “blocking” that has to be employed here.

  3. 3

    As long as the parametric models for the functionals in the formula are general enough to avoid the “null paradox” issue. Linear regressions for all terms suffice, but a no-interaction linear regression for a continuous outcome Y, and a no-interaction logistic regression for a binary mediator L does not suffice. The general rule of thumb is the models must be general enough to permit the above mean differences to equal zero for some parameter settings.

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