Mediation and mechanism

Abstract

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

Eur J Epidemiol (2009) 24:217224 DOI 10.1007/s10654-009-9331-1

COMMENTARY

Mediation and mechanism

Tyler J. VanderWeele

Received: 17 November 2008 / Accepted: 5 March 2009 / Published online: 28 March 2009 Springer Science+Business Media B.V. 2009

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

Keywords Causal inference Counterfactual

Direct and indirect effects Mechanism Mediation

Sufcient cause

Introduction

A recent paper by Hafeman [1] approaches the question of mediation from a mechanistic perspective. In particular, Hafeman relates formal denitions of direct and indirect effects, mediational quantities, to the sufcient cause framework which concerns different mechanisms for a particular outcome. This commentary explores the distinctions and relationships between the concepts of mediation and mechanism more generally. The commentary will be structured as follows. First, discussion will be

devoted to making explicit various distinctions between the concepts of mediation and mechanism. Second, we will show that the concepts of mediation and mechanism can both be illustrated in a single causal diagram. Third, we will discuss various logical implications that hold relating mediation and mechanism. Fourth, we will use the counterfactual denitions of direct and indirect effects to describe how a researcher can examine the portion of an effect mediated through a particular mechanism. Finally, we will discuss a distinction that is sometimes drawn between statistical or experimental causality and mechanistic causality and we will describe how the methods in causal inference recently proposed to address questions of mediation and mechanism blur this distinction between statistical causality and mechanistic causality.

The concepts of mediation and mechanism

The concept of mediation is generally framed within context of the estimation of the direct and indirect effects of some exposure, cause or treatment. Some of the effects of an exposure X on outcome Y may be thought to be mediated by some intermediate M and it will sometimes be of interest to determine the proportion of the effect of X on Y that is mediated by M. Such questions arise with considerable regularity within the social sciences; though, as noted by Hafeman, until recently these questions have been posed with only limited frequency by epidemiologists. The structural equation methods typically employed in the social sciences to address questions of mediation [26] have been subject to a number of limitations and criticisms concerning the inapplicability of the approach in the presence of interactions and non-linearities [79]. Recent

T. J. VanderWeele (&)

Department of Health Studies, University of Chicago, 5841 S Maryland Ave, MC 2007, Chicago, IL 60637, USA e-mail: [email protected]

123

218 T. J. VanderWeele

literature in causal inference has sought to address some of these limitations and to supply denitions of direct and indirect effects that allow for effect decomposition even in settings with interactions and non-linearities [711]; this causal inference literature also claried the no-unmeasured confounding assumptions necessary to identify direct and indirect effects and in particular noted the need to control for confounders of the relationships between the mediator and the outcome when conditioning on the mediator in the analysis.

Within the counterfactual (or potential outcomes) framework used by Hafeman, we may dene Yx as the potential outcome Y if, possibly contrary to fact, X were set to X. Potential outcomes such as Yx are sometimes also referred to as counterfactual outcomes. The total effect of

X on Y is then given by Y1-Y0. As in the paper by Hafeman, for simplicity, we will generally restrict our attention to the case in which X, M and Y are all binary. We also may dene Yxm as the potential outcome Y if, possibly contrary to fact, X were set to x and M were set to m.

Similarly we may dene Mx as the potential outcome M if, possibly contrary to fact, X were set to x.

Robins and Greenland [7] and Pearl [8] have given formal denitions for direct and indirect effects. As described by Hafeman, a controlled direct effect comparing X = 1 and X = 0 for some level of M = m can be dened as Y1m-Y0m; natural direct effects are dened by Y1M0

Y0M0 (the pure direct effect) and Y1M1 Y0M1 (the total

direct effect); natural indirect effects are dened by Y0M1

Y0M0 (the pure indirect effect) and Y1M1 Y1M0 (the total

indirect effect). Note that Pearl refers to pure and total direct effects as types of natural direct effects and refers to pure and total indirect effects as types of natural indirect effects. Note that the pure and total effects are reversed by relabeling the exposure levels of X so that 1 becomes 0 and 0 becomes 1. As discussed below, Hafeman makes certain monotonicity assumptions and thus the distinction between pure and total is relevant.

Alternative counterfactual-based denitions of direct and indirect effects based on principal stratication are also possible [1214]. The conceptual relationships between the denitions given above and those based on principal stratication are described elsewhere [14]; it can be shown for example that the presence of a principal strata direct effect necessarily implies the presence of a controlled direct effect and a natural direct effect but that similar implications do not hold for principal strata indirect effects and natural indirect effects [14].

In a highly inuential paper in the psychology literature, Baron and Kenny [3] considered the distinction between mediation and moderation (also known as modication in the epidemiologic literature). In this commentary we will consider the relationship not between mediation and

moderation but between mediation and mechanism and for this we turn to the conceptualization of mechanism provided by the sufcient cause framework.

Within epidemiology, mechanisms are typically conceptualized by Rothmans sufcient cause framework [15 17]. A sufcient cause model for a particular outcome posits a collection of different mechanisms each of which is capable of bringing about the outcome under consideration. A particular mechanism operates when some minimal set of actions, events or states of nature is obtained; when all components required for the mechanism are present, the outcome inevitably occurs. These mechanisms are thus referred to as sufcient causes since the conjunction of all the components required for a particular mechanism to operate is sufcient for that outcome; the individual components required for particular mechanisms are then each referred to as component causes. There are several antecedents to Rothmans formulation in the statistics, philosophical and epidemiologic literatures [1820]; the sufcient cause model has subsequently been formulated both formally [17] and graphically [21, 22].

Hafeman considers a situation in which the exposure X never prevents the intermediate M or the outcome Y and in which the intermediate M never prevents the outcome Y. Such assumptions are sometimes referred to as monotonicity assumptions and they may or may not be reasonable assumptions given a particular context. In any case, under such monotonicity assumptions, there are two possible sufcient causes for the M: one sufcient that requires X and possibly some other factors, denoted by Hafeman by A, to operate; and a second sufcient cause that may operate irrespective of whether X is present, provided some other factors, denoted by K, are present. For the outcome Y there are four sufcient causes: one involving both X and M and possibly some other factors F; one involving just X and possibly some other factors C; one involving just M and possibly some other factors B; and one requiring neither X nor M but simply some other factors L. The two sufcient causes for M are thus K and AX; the four sufcient causes for Y are L, BM, CX, and FXM. Hafeman [1] graphically depicts the causal pies in Fig. 1 of her paper. Hafemans particular contribution, however, is relating controlled direct effect and natural direct and indirect effects to the background components of the sufcient cause model, namely K, A, L, B, C and F.

In general, we would say that M mediates the effect of X on Y if for some individual one of the natural indirect effects, Y0M1 Y0M0 (pure) or Y1M1 Y1M0 (total), is non

zero. If no mediation is present (i.e. if the pure indirect effect and the total indirect effect are both zero for all individuals) then the pure and total direct effects will both coincide with the total effect of X on Y (see the technical appendix for details). We propose that the condition of a

123

Mediation and mechanism 219

Fig. 1 A causal diagram depicting both mediation and mechanism: Y is the outcome; M is the mediator; X is the exposure; K, A, L, B, C, and F are causal partners involved in various mechanisms

non-zero natural indirect effect for some individual is a sufcient condition for saying that M mediates the effect of X on Y. Below we consider whether it is also necessary. In general controlled direct effects cannot be used to characterize mediation because, if there is interaction between the effects of X and M on Y, then controlled direct effects may differ from total effect even if X is not a cause of M so that none of the effect of X on Y is mediated by M. See the technical appendix for further discussion.

The condition given above for mediation in terms of natural indirect effects is fairly subtle in that the denition of natural indirect effects involves composite potential outcomes such as those of the form Y1M0 which considers the potential outcome of Y if X were set to 1 and M were set to what it would have been if X had been 0. In contrast to mediation, a characterization of mechanism is relatively straightforward, though, as we will see, some of the implications of this characterization are subtle.

Within the sufcient cause framework we would say that M participates in a mechanism for the outcome Y whenever there is a sufcient cause that involves M. The characterization arises immediately and directly from the sufcient cause framework. However, a subtlety of this characterization is that it cannot be entirely reduced to potential outcomes. Certainly if there is some individual for whom Y would take the value 1 if M were 1 and for whom Y would take the value 0 if M were 0 then M must participate in some sufcient cause for Y. However, it may be that interventions on M leave Y unchanged for all individuals in the population but that M still participates in some mechanism for Y. This may happen because a mechanism is present that involves M and another mechanism is present that does not involve M, so that an individual may have Y = 1 regardless of whether M is

removed, but if M is present the individual may have Y = 1 from the mechanism involving M.

For example, in the setting described by Hafeman, there were four sufcient causes for outcome Y: L, BM, CX, and FXM. For simplicity, suppose that the sufcient cause FXM were absent i.e., F = 0 so that there would be just three sufcient causes L, BM and CX. Now suppose that Y denotes some congenital abnormality and that M and X denote some environmental exposures. Suppose that L, B and C denoted certain proteins such that whenever protein B and environmental exposure M are both present Y occurs; similarly whenever protein C and environmental exposure X are both present Y occurs; and further suppose that the protein L on its own is also sufcient for outcomeY. Suppose that protein B is only ever present as a consequence of some genetic variant; suppose further that whenever this genetic variant is present it would also give rise to protein L; thus whenever B is present, L is also present. Suppose that L alone gives rise to Y through some mechanism; and B and M together give rise to Y through some other mechanism; suppose that whenever L, B and M are all present the mechanism BM leading to Y takes precedence over the mechanism by which L leads to Y so that when L, B and M are all present, it is the BM mechanism that leads to Y.

In this case, M participates in a mechanism for Y which sometimes operates to cause Y. However, in this example, interventions on M will not change the value of Y for any individual. This is because for the mechanism for M to operate, B must also be present and whenever B is present L is also. So even when B is present, interventions on M will not change the value of Y because even under interventions to set M to 0, Y will still be 1 because L is 1. Thus in this example, interventions on M do not change the value of Y but M yet participates in a mechanism for the outcome Y.

We see then that the characterization of mechanism given in the sufcient cause framework cannot be reduced to potential outcomes. There may be more than one mechanism operating that can potentially bring about the outcome so that even if one mechanism were blocked another would still cause the outcome. In the literature on the philosophy of causation this issue is sometimes referred to as that of preemption or overdetermination [23]; the issue is also discussed in the epidemiologic literature [24]. Similar points also pertain to synergism within the sufcient cause framework [16, 17, 25].

With these concepts in place, in the next three sections we will explore in greater detail the relationship between mediation and mechanism. We will show how both can be represented on a single causal diagram, we will consider various implications that hold between the concepts of mediation and mechanism, and we will consider how a

123

220 T. J. VanderWeele

researcher can examine the portion of an effect mediated through a particular mechanism.

Causal diagrams depicting both mediation and mechanism

The paper by Hafeman gave a graphical representation of the sufcient causes for M; a second diagram in the same gure was given for the sufcient causes for Y (see Hafeman [1] Fig. 1b). In fact, the diagram for the sufcient causes for M and the diagram for the sufcient causes for Y can be combined to form a single causal directed acyclic graph (DAG) by using rules developed by VanderWeele and Robins [21, 22] for incorporating sufcient causes into causal DAGs. This graph is given in Fig. 1.

Ellipses are placed around the collection of the sufcient cause nodes for M and around the collection of those for Y to indicate that these are the only sufcient causes for M and Y, respectively. This thus graphically indicates that M is 1 if and only if one of K or AX is 1, and Y is 1 if and only one of L, BM, CX, or FXM is 1. In this graph, there are mechanisms for Y in which M participates, namely in the mechanisms represented by the sufcient cause nodes BM and FXM. It can also be seen from the graph in Fig. 1 that M is a mediator of the effect of X on Y. The effect of X on Y is mediated through M on the pathways X-AX-M-BM-Y and X-AX-M-FXM-Y. Thus in the graph on Fig. 1, we have mediation and mechanism depicted in a single causal diagram.

Hafeman assumed that the components K, A, L, B, C and F were all independent. In fact this assumption is not necessary to produce a causal DAG depicting both mediation and mechanism. Even if K, A, L, B, C and F were not all independent, Fig. 1 could still be made into a causal DAG by adding a node U1 with arrows into K and A and a node U2 with arrows into L, B, C and F. When the effects of X and M are monotonic, VanderWeele and Robins [22] showed that such a construction is always possible; they furthermore showed that even without the monotonicity assumption one could construct and graphically represent on a causal DAG all of the sufcient causes for the DAGs various nodes. Essentially, if the exposure, mediator and outcome are all binary, then it is always possible to construct a single causal directed acyclic graph to represent both mediation and mechanism in cases in which they are present.

Mediation implies mechanism but mechanism need not imply mediation

With the concepts and graphical tools of the previous section in place, we can now consider the implications that hold between mediation and mechanism. First, we can see

that mediation implies mechanism; specically the statement M mediates the effect of X on Y implies the statement M participates in a mechanism for Y. In particular, if one of the natural indirect effects, Y0M1 Y0M0

or Y1M1 Y1M0, is non-zero, then we must have for some

individual that M1 = M0 for if we had M1 = M0 then we would have Y0M1 Y0M0 and Y1M1 Y1M0 and thus no

natural indirect effects. If M1 = M0 and one of Y0M1

Y0M0 or Y1M1 Y1M0 is non-zero then a change of M from

M0 to M1 will change the value of Y and from the discussion above concerning mechanism we know would that there is a mechanism for Y in which M participates. We thus see that mediation implies mechanism. Often when questions of mediation and direct and indirect effects are of interest, researchers will speak of identifying the mechanisms through which a particular exposure acts. The implication that mediation implies mechanism provides justication for this language. If an analysis implies that some portion of an effect of an exposure is mediated through a particular intermediate then that intermediate must be present in some mechanism for the outcome, even if the precise set of conditions or biological pathways for this mechanism are not entirely identied.

Although mediation implies mechanism, we will now argue that mechanism need not imply mediation. Specically, M may participate in a mechanism for Y but X may not affect M. If X does not affect M, then M cannot mediate the effect of X on Y. In the sufcient cause model described by Hafeman we might have mechanism without mediation for the variable M if the sufcient cause AX were absent as depicted in Fig. 2. In Fig. 2, M participates in mechanisms for Y but X has no effect on M and thus M does not mediate the effect of X on Y.

Fig. 2 A causal diagram in which M participates in mechanisms for the outcome Y but M is not a mediator for the effect of the exposure X on the outcome Y

123

Mediation and mechanism 221

We thus see that the statement M participates in a mechanism for Y need not imply the statement M mediates the effect of X on Y. In fact, we might even have a situation in which X affects M and M participates in a mechanism for Y but M may still not be a mediator for the effect of X on Y. This could occur if the individuals for whom X affects M were such that, for these individuals, M had no effect on Y. In other words, there may be some individuals for whom X affects M and some individuals for whom M affects Y but these two groups of individuals may not overlap. Alternatively, if M is not binary, it may be the case that X affects M for certain values of the variable M but those are not the values for which M affects Y. In any case, we have now seen that mechanism needs not imply mediation.

A further issue remains to be examined. We noted above that the presence of a natural indirect effect for some individual was a sufcient condition for the statement that M mediates the effect of X on Y. It remains to determine whether it is also necessary. Suppose that there were two sufcient causes for M, namely K and AX as before, in Fig. 1.

Suppose that whenever K were present, A were also present. In this case we would have M1 = M0 and thus

Y0M1 Y0M0 and Y1M1 Y1M0, the natural indirect effects,

would be zero. But suppose that whenever K, A and X were all present the AX mechanism would take precedence over the K mechanism. Suppose further that the components L, C, and F in the sufcient causes for Y were all absent and that only B was present. Thus the only way Y could come about would be through the BM mechanism. Thus we would be in a setting in which, when X were present, X would affect M through the AX mechanism and M would affect Y through the BM mechanism and thus the effect of X on Y, at a mechanistic level, would be mediated by M even though the natural indirect effects are zero and even though the potential outcomes Yx and Yxm would not vary with x.

Similarly, we might have the natural indirect effects Y0M1 Y0M0 and Y1M1 Y1M0 being zero but the effect of X

on Y, at a mechanistic level, being mediated by M if L were 1 whenever B or F were 1. We thus have examples in which we would likely want to say that M mediates the effect of X on Y even though the natural indirect effects are zero. This is, once again, a problem of overdetermination [23]. There may be multiple mechanisms operating that can potentially bring about the outcome so that even if one mechanism were blocked another would still cause the outcome.

In any case it appears that the condition that one of the natural indirect effects is non-zero for some individual is only a sufcient condition, not a necessary condition, for the statement M mediates the effect of X on Y. In the context of the sufcient cause model with K and AX as sufcient causes for M and with L, BM, CX, and FXM as sufcient causes for Y, we would likely say that M

mediates the effect of X on Y when M occurs because of the AX mechanism and when subsequently Y occurs because of either the BM or FXM mechanisms. In these settings, X causes the outcome Y through its effect on M. More generally, we might characterize the statement M mediates the effect of X on Y as follows: we would say that this statement holds if there is some individual for whom M occurs because of a mechanisms involving X and for whom Y occurs because of a mechanism involving M. Within the sufcient cause framework this characterization becomes M mediates the effect of X on Y if there is some individual for whom M occurs because of a sufcient cause involving X and for whom Y occurs because of a sufcient cause involving M. It is proposed that these conditions characterize (that is, they are both necessary and sufcient) for the statement M mediates the effect of X onY. This characterization of mediation, like that of mechanism above, is in some cases not determined by a complete knowledge of the potential outcomes. In some cases, even if a complete knowledge of the potential outcomes were available, some further knowledge of the biology of the situation under consideration may be necessary to evaluate the statement M mediates the effect of X on Y.

Of course, in practice, we rarely have this knowledge. We thus rely on estimating causal effects to see if there is, on average, a causal effect or a natural indirect effect. If there is an effect on average there must be some individuals for whom there is an effect. Thus if, on average, M has a causal effect on Y then there must be a mechanism for Y in which M participates. If, on average, there is an indirect effect of X on Y through M then there must be individuals for whom M mediates the effect of X on Y. In practice we rely on the sufcient conditions for mediation and mechanism, rather than on the necessary and sufcient characterizations of these concepts.

Mediation by mechanism

We saw in the previous section that mediation implies mechanism, that if M mediates the effect of X on Y then there is a sufcient cause for Y which requires M to operate. This gives rise to the question of the extent to which the effect of X on Y is mediated by particular sufcient causes. Hafeman expressed various direct and indirect effects, with some intermediate M taken as the mediator, in terms of the background components of sufcient causes. One could also express various direct and indirect effects, with different sufcient causes (that is, different mechanisms) taken as the mediator, in terms of the background components of sufcient causes. In the general setting depicted in Fig. 1, we thus might consider

123

222 T. J. VanderWeele

the extent to which the effect of X on Y is mediated by one of the sufcient causes BM, CX, or FXM. We could let S denote one of these three sufcient causes. We might then consider potential outcomes of the form Sx and Yxs which, respectively, denote the potential outcome of S if X were set to X and the potential outcome of Y if X were set to x and S were set to s. We could then consider various controlled direct effects and natural direct and indirect effects with S taken as the mediator.

For example, we could consider average controlled direct effects when one of the sufcient causes BM, CX, or FXM is taken as the mediator and we could express the controlled direct effect in terms of the background causes K, A, L, B, C, and F. Let Lc denote the complement of L (i.e., not L), let Kc denote the complement of K, etc. For controlled direct effects, it is shown in the technical appendix that under the general setting described above and denoted in Fig. 1, if we take S to be the sufcient cause BM then the average controlled direct effect E[Yx=1,s=0-Yx=0,s=0] is given by P[LcC

or LcF(A or K)]. Similarly, if we take S to be the sufcient cause CX then the average controlled direct effect E[Yx=1,s=0-Yx=0,s=0] is given by P[LcKcBA or LcKcFA or

LcBcF(A or K)]. Finally, if we take S to be the sufcient cause FXM then the average controlled direct effect E[Yx=1,s=0-

Yx=0,s=0] is given by P[LcKcBA or Lc(Bc or Kc)C]. Similar

expressions could be derived for expressing the pure direct effect, E[Yx1;s0 Yx0;s0], the total direct effect, E[Yx1;s1

Yx0;s1], the pure indirect effect, E[Yx0;s1 Yx0;s0], and the

total indirect effect, E[Yx1;s1 Yx1;s0], in terms of the

probabilities of the background component causes.

Discussion of the identication of these direct and indirect effects is beyond the scope of this commentary but we note that when these effects are identied the results of Hoffmann et al. [26] could be used to estimate controlled direct effects of the form Yx=1,s=0-Yx=0,s=0 where S is

some sufcient cause for Y.

Mechanistic causality and statistical causality

In a recent overview paper on causality in the statistics literature, Aalen and Frigessi [27] distinguish between what they call experience-based or counterfactual-based causality and mechanistic causality. Counterfactual-based causality is essentially concerned with the effects of a particular intervention or exposure without regard to the mechanisms by which these effects arise. Conclusions about causal effects are drawn either through randomized trials or through the careful design and analysis of observational data in which the researcher attempts to control for all the variables that confound the exposure-outcome relationship. Aalen and Frigessi also describe this approach as black box causality since the methods used for the

estimation of causal effects can be valid irrespective of how the exposure produces its effect.

Mechanistic causality, on the other hand, attempts to understand the mechanisms governing the various processes which give rise to particular outcomes. Assessing mechanistic causality requires closer observations and a good deal of scientic knowledge. The model for mechanistic causality is the natural sciences in which attempts are made to identify the natural laws and precise workings behind the phenomena we observe. In the mechanistic approach, one attempts to look inside the black box. A similar distinction to that made by Aalen and Frigessi is also made by Heckman [28, 29]. Heckman calls the counterfactual-based causality described above statistical causality. He contrasts this with what he calls the scientic model of causality [28] or econometric causality [29]. Heckman criticizes the statistical literature on causality for not making use of theory and for not taking into account agent choice, equilibrium processes and feedback which he argues are the mechanisms by which outcomes are generated.

Note that in their discussions, these authors are using the word mechanism in a much broader sense than the sense of mechanism which is formalized by the sufcient cause framework. Very roughly, this broader sense of mechanism refers to an explanation concerning how certain initial states lead to particular nal states through a process or a series of processes involving different intermediate stages. Such explanation can be given at more or less detailed levels. See the work of Machamer et al. [30] for a fuller attempt at describing the notion of mechanism as employed in various scientic disciplines.

The distinction between a statistical-experimentalist approach to causality and a mechanistic approach to causality may be useful, but it is perhaps best viewed as a spectrum rather than as an absolute distinction. As noted by Aalen and Frigessi [27], and as is clear to any practicing epidemiologist, even under a statistical-experimentalist approach to causality, some scientic knowledge is still necessary in order to determine which confounding variables need to be controlled for in order to interpret estimates causally. The use of causal direct acyclic graphs [3133] to represent and reason about control for confounding also makes clear the relationship between the statistical-experimentalist approach to causality in observational studies and scientic theory or knowledge.

Moreover, the recent theory and methods in the statistics and epidemiology literature concerning mediation and mechanism particularly blur the distinction between statistical-experimentalist causality and mechanistic causality. The recent work in causal inference on the identication and estimation of direct and indirect effects [711, 3436] comes out of a statistical-experimentalist approach to causality based on the careful control of confounding

123

Mediation and mechanism 223

variables and various identication assumptions. However, the types of conclusions this work allows the researcher to draw concern whether the effect of one variable on another is mediated by an intermediate, a mechanistic conclusion. Similarly, recent work on the sufcient cause framework concerning tests for whether two causes ever participate in a single causal mechanism [16, 17, 37] arises from a statistical-experimentalist approach to causality. However, once again, the type of conclusion that the researcher can draw from these methods concerns mechanism.

This recent work on methods to draw conclusions about mediation and mechanism may thus be seen to lie somewhere in the middle of the statistical causalitymechanistic causality spectrum. It may thus make sense to follow Cox and Wermuth [38] who speak of different levels of causality. A particular study may only identify the average causal effects of an intervention; another study may uncover something of the mechanistic process by which outcomes are generated but may leave other aspects of the mechanism unknown. Only gradually, and with increasingly detailed and extensive data, will additional knowledge of mechanisms be possible.

Technical appendix

Mediation, controlled direct effects, and natural direct and indirect effects

Note that the total effect Y1-Y0 decomposes as the sum of a total direct effect and a pure indirect effect, Y1-

Y0 = Y1M1 Y0M1 Y0M1 Y0M0, or as the sum of a

total indirect effect and a pure direct effect, Y1-

Y0 = Y1M1 Y1M0 Y1M0 Y0M0. If both the pure

indirect effect and the total indirect effect are zero, i.e., if both Y0M1 Y0M0 0 and Y1M1 Y0M1 0, then we will

have that Y1-Y0 = Y1M1 Y0M1 and that Y1-

Y0 = Y1M0 Y0M0 so that the total effect, the total direct

effect and the pure direct effect all coincide. We thus say that M mediates the effect of X on Y whenever one of the natural indirect effects is non-zero since, if they are both zero then the total effect and the natural direct effects coincide.

Note that it is more difcult to use the controlled direct effect to draw conclusions about mediation. The controlled direct effect takes the form Y1m-Y0m. Even if X has no

effect on M so that there is no mediation, the controlled direct effect Y1m-Y0m may differ from the total effect. For example, suppose that X has no effect on M so that there is no mediation of the effect of X on Y by M but suppose there is interaction between X and M. If there is interaction between the effects of X and M on Y, then Y1m-Y0m will differ for

different values of m and thus one of the controlled direct

effects Y11-Y01 or Y10-Y00 will differ from Y1-Y0.

Obtaining a controlled direct effect that is different than the

total effect is thus not evidence that mediation is present. If there is no interaction between the effects of X and M on Y then Robins [35] has shown that the controlled direct, the total direct effect and the pure direct effect all coincide; furthermore all of these quantities will be equal to the total effect if there are no natural indirect effects; in this case one can compare total effects and controlled direct effects to assess mediation. Thus only under the assumption of no interaction can one use controlled direct effects to assess mediation. See the work of Robins and Greenland [7] and Kaufman et al. [9] for further critique of using controlled direct effects to assess mediation and indirect effects.

Relationship between direct and indirect effects, with a sufcient cause taken as the mediator, and the probabilities of the background component causes

Here we express controlled direct effects, with some sufcient cause S taken as the mediator, in terms of the probabilities of the background components of the sufcient causes in Fig. 1. Note that if S were set to 1 then Y would be 1 since S is a sufcient cause for Y; it thus does not make sense to discuss controlled direct effects of the form Yx=1,s=1-Yx=0,s=1 since

this controlled direct effect is zero for any sufcient cause and we report only the controlled direct effects of the form Yx=1,s=0-Yx=0,s=0. Note that a binary variable Z can be treated

as an event and so we will let P(Z) denote P(Z = 1).For S = BM:

E Yx1;s 0 Yx

0;s 0

PL or C or FMx

P LcC or LcFA or K

For S = CX:

E Yx1;s0 Yx0;s0

PL or BMx1 or FMx1

PL or BMx0

P L or BA or K or FA or K

PL or BK P LcKcBA or LcKcFA

or LcBcFA or K

For S = FXM:

E Yx1;s0 Yx0;s0

PL or BMx1 or C

PL or BMx0

P L or BA or K or C

PL or BK P LcKcBA or LcBc or KcC

1 PL P LcC or FMx

1

123

224 T. J. VanderWeele

References

1. Hafeman D. A sufcient cause based approach to the assessment of mediation. Eur J Epidemiol. 2008;23:71121. doi:http://dx.doi.org/10.1007/s10654-008-9286-7

Web End =10.1007/ http://dx.doi.org/10.1007/s10654-008-9286-7

Web End =s10654-008-9286-7 .

2. Judd CM, Kenny DA. Estimating mediation in treatment evaluations. Process Anal. 1981;5:60219.

3. Baron RM, Kenny DA. The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical considerations. J Pers Soc Psychol. 1986;51:117382. doi:http://dx.doi.org/10.1037/0022-3514.51.6.1173

Web End =10.1037/0022-3514.51.6.1173 .4. Bollen KA. Total, direct and indirect effects in structural equation models. In: Clogg CC, editor. Sociological methodology. Washington, DC: American Sociological Association; 1987. p. 3769.

5. Holland PW. Causal inference, path analysis, and recursive structural equations models. In: Clogg CC, editor. Sociological methodology. Washington, DC: American Sociological Association; 1988. p. 44984.

6. Sobel ME. Effect analysis and causation in linear structural equation models. Psychometrika. 1990;55:495515. doi:http://dx.doi.org/10.1007/BF02294763

Web End =10.1007/ http://dx.doi.org/10.1007/BF02294763

Web End =BF02294763 .

7. Robins JM, Greenland S. Identiability and exchangeability for direct and indirect effects. Epidemiology. 1992;3:14355. doi:http://dx.doi.org/10.1097/00001648-199203000-00013

Web End =10.1097/00001648-199203000-00013 .

8. Pearl J. Direct and indirect effects. In: Proceedings of the seventeenth conference on uncertainty and articial intelligence. San Francisco: Morgan Kaufmann; 2001:411-420.

9. Kaufman JS, MacLehose RF, Kaufman S. A further critique of the analytic strategy of adjusting for covariates to identify biologic mediation. Epidemiol Perspect Innov. 2004;1:4. doi:http://dx.doi.org/10.1186/1742-5573-1-4

Web End =10.1186/1742-5573-1-4 .

10. Peterson ML, Sinisi SE, van der Laan MJ. Estimation of direct causal effects. Epidemiology. 2006;17:27684. doi:http://dx.doi.org/10.1097/01.ede.0000208475.99429.2d

Web End =10.1097/ http://dx.doi.org/10.1097/01.ede.0000208475.99429.2d

Web End =01.ede.0000208475.99429.2d .11. VanderWeele TJ. Marginal structural models for the estimation of direct and indirect effects. Epidemiology. 2009;20:1826. doi:http://dx.doi.org/10.1097/EDE.0b013e31818f69ce

Web End =10.1097/EDE.0b013e31818f69ce .

12. Frangakis CE, Rubin DB. Principal stratication in causal inference. Biometrics. 2002;58:219. doi:http://dx.doi.org/10.1111/j.0006-341X.2002.00021.x

Web End =10.1111/j.0006- http://dx.doi.org/10.1111/j.0006-341X.2002.00021.x

Web End =341X.2002.00021.x .

13. Rubin DB. Direct and indirect effects via potential outcomes. Scand J Stat. 2004;31:16170. doi:http://dx.doi.org/10.1111/j.1467-9469.2004.02-123.x

Web End =10.1111/j.1467-9469.2004.02- http://dx.doi.org/10.1111/j.1467-9469.2004.02-123.x

Web End =123.x .

14. VanderWeele TJ. Simple relations between principal stratication and direct and indirect effects. Stat Probab Lett. 2008;78:295762. doi:http://dx.doi.org/10.1016/j.spl.2008.05.029

Web End =10.1016/j.spl.2008.05.029 .

15. Rothman KJ. Causes. Am J Epidemiol. 1976;104:58792.16. VanderWeele TJ, Robins JM. The identication of synergism in the sufcient-component cause framework. Epidemiol. 2007;18:32939. doi:http://dx.doi.org/10.1097/01.ede.0000260218.66432.88

Web End =10.1097/01.ede.0000260218.66432.88 .17. VanderWeele TJ, Robins JM. Empirical and counterfactual conditions for sufcient cause interactions. Biometrika. 2008;95:4961. doi:http://dx.doi.org/10.1093/biomet/asm090

Web End =10.1093/biomet/asm090 .

18. Cayley A. On a question in the theory of probabilities. Philos Mag. 1853;6:259.

19. Mackie JL. Causes and conditions. Am Philos Q. 1965;2:24555.

20. MacMahon B, Pugh TF. Causes and entities of disease. In: Clark DW, MacMahon B, editors. Preventive medicine. Boston: Little, Brown, and Company; 1967. p. 118.

21. VanderWeele TJ, Robins JM. Directed acyclic graphs, sufcient causes and the properties of conditioning on a common effect. Am J Epidemiol. 2007;166:1096104. doi:http://dx.doi.org/10.1093/aje/kwm179

Web End =10.1093/aje/kwm179 .

22. VanderWeele TJ, Robins JM. Minimal sufcient causation and directed acyclic graphs. Ann Stat. 2009. (in press).

23. Hall N, Paul LA. Causation and preemption. In: Clark P, Hawley K, editors. Philosophy of science today. Oxford: Oxford University Press; 2003. p. 10029.

24. Greenland S, Robins JM. Conceptual problems in the denition and interpretation of attributable fractions. Am J Epidemiol. 1988;128:118597.

25. Greenland S, Brumback B. An overview of relations among causal modeling methods. Int J Epidemiol. 2002;31:10307. doi:http://dx.doi.org/10.1093/ije/31.5.1030

Web End =10.1093/ije/31.5.1030 .

26. Hoffmann K, Heidemann C, Weikert C, Schulze MB, Boeing H. Estimating the proportion of disease due to classes of sufcient causes. Am J Epidemiol. 2006;163:7683. doi:http://dx.doi.org/10.1093/aje/kwj011

Web End =10.1093/aje/ http://dx.doi.org/10.1093/aje/kwj011

Web End =kwj011 .

27. Aalen OO, Frigessi A. What can statistics contribute to a causal understanding? Scand J Stat. 2007;34:15568. doi:http://dx.doi.org/10.1111/j.1467-9469.2006.00549.x

Web End =10.1111/ http://dx.doi.org/10.1111/j.1467-9469.2006.00549.x

Web End =j.1467-9469.2006.00549.x .28. Heckman JJ. The scientic model of causality. Sociol Methodol. 2005;1:198. doi:http://dx.doi.org/10.1111/j.0081-1750.2006.00163.x

Web End =10.1111/j.0081-1750.2006.00163.x .

29. Heckman JJ. Econometric causality. Int Stat Rev. 2007;76:127. doi:http://dx.doi.org/10.1111/j.1751-5823.2007.00024.x

Web End =10.1111/j.1751-5823.2007.00024.x .

30. Machamer P, Darden L, Craver CF. Thinking about mechanisms. Philos Sci. 2000;67:125. doi:http://dx.doi.org/10.1086/392759

Web End =10.1086/392759 .

31. Pearl J. Casual diagrams for empirical research (with discussion). Biometrika. 1995;82:669710. doi:http://dx.doi.org/10.1093/biomet/82.4.669

Web End =10.1093/biomet/82.4.669 .32. Greenland S, Pearl J, Robins JM. Causal diagrams for epidemiologic research. Epidemiology. 1999;10:3748. doi:http://dx.doi.org/10.1097/00001648-199901000-00008

Web End =10.1097/ http://dx.doi.org/10.1097/00001648-199901000-00008

Web End =00001648-199901000-00008 .

33. Glymour MM, Greenland S. Causal diagrams. In: Rothman KJ, Greenland S, Lash TL, editors. Modern epidemiology. 3rd ed. Philadelphia: Lippincott Williams and Wilkins; 2008. p. 183 209.

34. Robins JM. Testing and estimation of direct effects by reparameterizing directed acyclic graphs with structural nested models. In: Glymour C, Cooper GF, editors. Computation, causation, and discovery. Menlo Park, CA, Cambridge, MA: AAAI Press/The MIT Press; 1999. p. 349405.

35. Robins JM. Semantics of causal DAG models and the identication of direct and indirect effects. In: Green P, Hjort NL, Richardson S, editors. Highly structured stochastic systems. New York: Oxford University Press; 2003. p. 7081.

36. Joffe M, Small D, Hsu CY. Dening and estimating intervention effects for groups that will develop an auxiliary outcome. Stat Sci. 2007;22:7497. doi:http://dx.doi.org/10.1214/088342306000000655

Web End =10.1214/088342306000000655 .

37. VanderWeele TJ. Sufcient cause interactions and statistical interactions. Epidemiology. 2009;20:613. doi:http://dx.doi.org/10.1097/EDE.0b013e31818f69e7

Web End =10.1097/ http://dx.doi.org/10.1097/EDE.0b013e31818f69e7

Web End =EDE.0b013e31818f69e7 .

38. Cox DR, Wenmuth N. Causality: a statistical view. Int Stat Rev. 2004;72:285305.

123