A unification of mediation and

interaction: a four-way

decomposition

Tyler J. VanderWeele

Departments of Epidemiology and Biostatistics

Harvard School of Public Health

1

Plan of Presentation

(1) Questions of Mediation and Interaction

(2) A Unification of Mediation and Interaction

(3) Regression Approaches and Ratio Scales

(4) Application to Genetic Epidemiology

(5) Relation to Prior Decompositions

(6) Concluding Remarks

2

Mediation

In some research contexts we might be interested in the extent to

which the effect of some exposure A on some outcome Y is

mediated by an intermediate variable M and to what extent it is

direct

Stated another way, we are interested in the direct and indirect

effects of the exposure

In other research contexts we may be interested in whether A and

M interact in their effects, and how much of their effects are due

to interaction

A M Y

3

MediationIn some cases, we may be interested in both mediation and interaction

In 2008, GWAS studies found variants 15q25.1 associated with lung cancer

(Thorgeirsson et al., 2008; Hung et al., 2008; Amos et al., 2008)

These same variant were known to be associated with smoking (average

cigarettes per day) (Saccone et al., 2007; Spitz et al., 2008)

The variants also increased vulnerability to the harmful effect of smoking, a

gene-environment interaction e.g. carriers of the variant allele extract more

nicotine and toxins from each cigarette (Le Marchand, 2008)

The causal inference literature has developed methods that can assess

mediation in the presence of interaction to get direct and indirect effects

In this example from genetic epidemiology, most of the effect seemed

“direct” (94%) with respect to cigarettes per day (VanderWeele et al. 2012)

But this does not clarify the role of interaction itself 4

NotationLet Y denote some outcome of interest for each individual

Let A denote some exposure or treatment of interest for

each individual

Let M denote some post-treatment intermediate(s) for each

individual (potentially on the pathway between A and Y)

Let C denote a set of covariates for each individual

Let Ya be the counterfactual outcome (or potential outcome)

Y for each individual when intervening to set A to a

Let Ma be the counterfactual outcome M for each individual

when intervening to set A to a

Let Yam be the counterfactual outcome Y for each individual

when intervening to set A to a and M to m5

A Unification of Mediation and

Interaction

We can in fact decompose a total effect, TE = Y1 - Y0, into four components

(VanderWeele, 2014) under the “composition” assumption that Ya =YaMa

(1)A controlled direct effect (CDE): the effect of A in the absence of M

(2)A reference interaction (INTref): The interaction that operates only if the

mediator is present in the absence of exposure

(3)A mediated interaction (INTmed): The interaction that operates only if the

exposure changes the mediator

(4)A pure indirect effect (PIE): The effect of the mediator in the absence of

the exposure times the effect of the exposure on the mediator 6

A Unification of Mediation and

Interaction

We can summarize the four components as:

(1)CDE: Neither mediation nor interaction

(2)INTref: Interaction but not mediation

(3)INTmed: Both mediation and interaction

(4)PIE: Mediation but not interaction

7

A Unification of Mediation and

Interaction

We cannot identify these effects for an individual but, under certain

confounding assumptions (next slides), we can identify them on average for

a population. If so, we let pam = P(Y=1|A=a,M=m) then we have:

We could calculate the proportions due to each of the components:

8

A Unification of Mediation and

InteractionThe four components are:

We could add E[INTref] and E[INTmed] for the overall proportion due to interaction:

We could add E[PIE] and E[INTmed] for the overall proportion due to mediation:

9

Identification

The confounding assumptions are the same as those generally used in the

causal inference literature to identify direct and indirect effects:

(1) There are no unmeasured exposure-outcome confounders given C

(2) There are no unmeasured mediator-outcome confounders given (C,A)

(3) There are no unmeasured exposure-mediator confounders given C

(4) None of the mediator-outcome confounders are affected by exposure

For controlled direct effects,

only assumptions (1) and (2)

are needed

Note (1) and (3) are guaranteed

when treatment is randomized

A M YC1

C3 C2 10

Identification

More formally, in counterfactual notation, these assumptions are:

(1)is Yam | | A | C

(2) is Yam | | M | C,A

(3) is Ma | | A | C

(4) is Yam | | Ma* | C

For controlled direct effects,

only assumptions (1) and (2)

are needed

Note (1) and (3) are guaranteed

when treatment is randomized

A M YC1

C3 C2 11

Regression ApproachSimilar results hold if one or both of A or M are binary

Under the confounding assumptions we can estimate each of the four

components in a straightforward way using regression models for Y and M:

Under these models if our confounding assumptions, then the effects for a

change in the exposure from reference level a* to level a are given by:

12

Relation to Mediation Decompositions

Our basic four-way decomposition was:

If we combine the CDE and INTref we obtain what is sometimes called the

“natura/pure direct effect”If we combine the PIE and INTmed we obtain what is some times called the

“natural/total indirect effect” (Robins and Greenland1992;Pearl 2001)

PDE = Pure direct effect (natural direct effect) =

TIE = Total indirect effect (natural indirect effect =

These are also sometimes called natural direct and indirect effects

This is the decomposition of Robins and Greenland (1992) and Pearl (2001)

This is essentially the decomposition used in epidemiology and the social

sciences when interaction is absent 13

Relation to Prior DecompositionsVanderWeele and Tchetgen Tchetgen (2014) also showed the total effect could be

divided into CDE, PIE and proportion attributable to interaction; the 4-way

decomposition unites all other; We can summarize in a figure:

14

Ratio ScaleA similar four-way decomposition also holds using a ratio scale

Where RRam = pam /p00 and where κ = p00 / pa=0 is a scaling factor

If we divide each component by the sum, then κ drops out:

We can estimate the components using logistic regression (w/SAS code)

We can also proceed with case-control data under a rare outcome assumption 15

Genetic EpidemiologyIn 2008, GWAS studies found variants 15q25.1 associated with lung cancer

(Thorgeirsson et al., 2008; Hung et al., 2008; Amos et al., 2008)

These same variant were known to be associated with smoking (average

cigarettes per day) (Saccone et al., 2007; Spitz et al., 2008)

The variants also increased vulnerability to the harmful effect of smoking, a

gene-environment interaction e.g. carriers of the variant allele extract more

nicotine and toxins from each cigarette (Le Marchand, 2008)

When methods for direct and indirect effects were employed most of the

effect seemed “direct” with respect to cigarettes per day (VanderWeele et al.

2012)

But this did not fully capture the role of interaction; there was evidence for

such interaction (Li et al, 2010; Truong et al, 2010; VanderWeele et al, 2012)

Now we will examine what proportion of the effect is due (i) to just mediation,

(ii) to just interaction, (iii) to both and (iv) to neither 16

Genetic Epidemiology

The study sample consists of 1836 cases and 1452 controls is

from a case control study (cf. Miller et al., 2002) assessing the

molecular epidemiology of lung cancer, which began in 1992 at

the Massachusetts General Hospital (MGH)

Eligible cases included any person over the age of 18 years, with

a diagnosis of primary lung cancer that was further confirmed by

an MGH lung pathologist.

The controls were recruited from among the friends or spouses of

cancer patients or the friends or spouses of other surgery

patients in the same hospital.

Potential controls that carried a previous diagnosis of any cancer

(other than non-melanoma skin cancer) were excluded from

participation. 17

Genetic Epidemiology

Sample characteristics of cases and controls

_________________________________________________________________

Cases (N=1836) Controls (N=1452)

_________________________________________________________________

Average Cigarettes per Day 25.42 13.97

Smoking Duration 38.50 18.93

Age 64.86 58.58

College Education 31.3% 33.5%

Sex Male 50.1% 56.1%

Female 49.9% 43.9%

rs8034191 C alleles

0 33.8% 43.3%

1 48.5% 43.7%

2 17.7% 13.0%

18

Assumptions About Confounding

To use our approach with the genetic variants we need to assume no

unmeasured confounding for the (1) exposure-outcome, (2) mediator-

outcome, and (3) exposure-mediator relationships

Assumptions (1) and (3) are probably plausible for the exposure (the

genetic variant) subject to no population stratification (the analysis was

restricted to Caucasians)

*(2)* No confounding may be less plausible for the smoking – lung

cancer association (e.g. SES / neighborhood)

We consider sensitivity analysis later

(4) Smoking duration may affect

cigarettes/day and lung cancer and

may affected by the variant (though not

much evidence) and results are similar

when duration is omitted

A M YC

C U

Genetic EpidemiologyWhen we apply the four-way decomposition using logistic regression for lung

cancer and linear regression for square root of average cigarettes per day

(this measure is more normally distributed) comparing 2 to 0 variant alleles

we obtain a total effect risk ratio of RR=1.77 and:

Most of the direct effect (which is 94%) appears to be due to INTref i.e. to be

due to interaction but not mediation; mediation is only about 6%

As suspected, the proportion due to interaction is substantial, but now we

can quantify this

20

Genetic EpidemiologyWith the two-way (Robins and Greenland, 1992; Pearl, 2001; VanderWeele et al.,

2012) decomposition we obscure the role of interaction here because these

combine the CDE and INTref into the PDE

21

Study Summary

(1) Most of the effect seemed to be due to interaction, in the absence

of mediation (at least with respect to cigarettes per day)

(2) Both the mediated effect and the reference interaction may be

underestimated due to measurement error in the self-reported

cigarettes per day measure (Valeri et al., 2014)

(3) Other aspects of smoking (e.g. depth of inhalation) may mediate

more of the relationship

(4) The strong interaction might likewise depend on the smoking

variable used (cigarettes per day versus depth of inhalation)

(5) At least with respect to cigarettes per day, however, most of the

effect is not by increasing cigarettes per day (it does this only by

1 CPD for smokers) but rather because of the interaction

22

APOE and MemoryThe same technique was applied to a similar genetic example with APOE

e4 alleles (Sajeev et al., 2015)

APOE e4 is associated with Alzheimer’s, cognitive decline, memory, etc.

To what extent is the effect of e4 alleles on memory mediated by

cerebrovascular disease markers e.g. microbleeds?

To what extent is it due to interaction?

Data come from 4121 participants in the population-based Age-

Gene/Environment Susceptibility (AGES) Study in Reykjavik, Iceland

We use techniques for the 4-way decomposition

All models adjusted for age, sex, education, diabetes, smoking status,

and midlife measures of physical activity, body mass index, systolic blood

pressure, and total cholesterol 23

APOE and Memory

24

Concluding RemarksOne further theoretical point is of interest

Sometimes a portion eliminated measure is proposed as being of more policy

relevance (Robins and Greenland, 1992; VanderWeele, 2013):

E[PE] = E[TE] – E[CDE] = E(Y1 - Y0) - E(Y10 - Y00)i.e. what portion (or proportion) of the effect could we eliminate if we set M=0

The four-way decomposition gives a further causal interpretation of PE:

i.e. it is the proportion due to mediation or interaction or both

When we fix the mediator to 0, we eliminate both mediation and interaction

This is different from the portion mediated (PM) in that it includes INTref

In the example PM=6% but PE=61% because of the interaction

The CDE (and thus also the portion eliminated) are easier to identify from the

data (only confounding assumptions 1 and 2, not 3 and 4, are required) 25

Concluding Remarks

(1) The four-way decomposition makes clear what proportion of an

effect is due (i) to just mediation, (ii) to just interaction, (iii) to both

and (iv) to neither

(2) It unites, within a single framework, prior decompositions for

mediation and prior decompositions for interaction

(3) It gives the most insight into both phenomena of mediation and

interaction (cf. VanderWeele, 2015)

(4) It is relatively straightforward to implement with SAS code

(5) Sensitivity analysis for measurement error and unmeasured

confounding are available for some mediation and interaction

measures; it would be good to extend these to cover each of the

four components

26

27

OXFORD UNIVERSITY PRESS

Explanation in Causal

Inference

Methods for Mediation and

Interaction

2015 │ Hardcover│ ISBN:

9780199325870

ReferencesAmos, C.I., et al. Genome-wide association scan of tag SNPs identifies a

susceptibility locus for lung cancer at 15q25.1. Nat. Genet. 40, 616-622 (2008).

Baron RM, Kenny DA. The moderator-mediator variable distinction in social psycho-

logical research: conceptual, strategic, and statistical considerations. Journal of

Personality and Social Psychology. 1986; 51:1173-1182.

Chanock, S.J. & Hunter, D.J. When the smoke clears… Nat. 452, 537-538 (2008).

Hosmer, D.W., Lemeshow, S. (1992). Confidence interval estimation of interaction.

Epidemiology 3:452-56.

Hung, R., et al. A susceptibility locus for lung cancer maps to nicotinic acetylcholine

receptor subunit genes on 15q25. Nat. 452, 633-637 (2008).

Imai, K., Keele, L., Yamamoto, T. (2010a). Identification, inference, and sensitivity

analysis for causal mediation effects. Statistical Science, 25:51-71.

28

ReferencesImai, K., Keele, L., Tingley, D. (2010b). A general approach to causal mediation

analysis. Psychological Methods, 15:309-334.

Imai, K., Keele, L., Tingley, D., Yamamoto, T. (2010c). Causal mediation analysis

using R. In: H.D. Vinod (ed.), Advances in Social Science Research Using R. New

York: Springer (Lecture Notes in Statistics), p.129-154.

Judd CM, Kenny DA. Process analysis: estimating mediation in treatment

evaluations. Eval Rev, 1981;5:602-619.

Lange, T., Vansteelandt, S., and Bekaert, M. (2012). A simple unified approach for

estimating natural direct and indirect effects. Am J Epidemiol, 176:190-195.

Le Marchand, L., et al. Smokers with the CHRNA lung cancer-associated variants

are exposed to higher levels of nicotine equivalents and a carcinogenic tobacco-

specific nitrosamine. Cancer Res. 68, 9137-9140 (2008).

Pearl J. (2001). Direct and indirect effects. In Proceedings of the Seventeenth

Conference on Uncertainty and Artificial Intelligence, 411-20. Morgan Kaufmann,

San Francisco.

29

ReferencesRobins JM, Greenland S. (1992). Identifiability and exchangeability for direct and

indirect effects. Epidemiology 3, 143-155.

Rothman, K. J. Modern Epidemiology. 1st ed. Little, Brown and Company, Boston,

MA (1986).

Tchetgen Tchetgen, E.J. (2011). On causal mediation analysis with a survival

outcome. International Journal of Biostatistics, 7:Article 33, 1-38.

Tchetgen Tchetgen, E.J. and Shpitser, I. (2012). Semiparametric theory for causal

mediation analysis: efficiency bounds, multiple robustness, and sensitivity analysis.

Annals of Statistics, 40:1816-1845.

Valeri, L., Lin, X., and VanderWeele, T.J. (2014). Mediation analysis when a

continuous mediator is measured with error and the outcome follows a generalized

linear model. Statistics in Medicine.

Valeri, L. and VanderWeele, T.J., Mediation analysis allowing for exposure-mediator

interactions and causal interpretation: theoretical assumptions and implementation

with SAS and SPSS macros. Psychological Methods, 18:137-150.30

References

VanderWeele, T.J. (2010). Bias formulas for sensitivity analysis for direct and indirect

effects. Epidemiology, 21:540-551.

VanderWeele, T.J. (2011). Causal mediation analysis with survival data. Epidemiol,

22:582-585.

VanderWeele, T.J. (2013). Policy-relevant proportions for direct effects.

Epidemiology, 24:175-176.

VanderWeele, T.J. (2013). A three-way decomposition of a total effect into direct,

indirect, and interactive effects. Epidemiology, 24: 24:224-232.

VanderWeel, T.J. (2014). A unification of mediation and interaction: a four-way

decomposition. Epidemiology, 25:749-761.

VanderWeel, T.J. (2015). Explanation in Causal Inference: Methods for Mediation

and Interaction. Oxford University Press: New York, in press.

31

References

VanderWeele, T.J., Asomaning, K., Tchetgen Tchetgen, E.J., Han, Y., Spitz, M.R.,

Shete, S., Wu, X., Gaborieau, V., Wang, Y., McLaughlin, J., Hung, R.J., Brennan, P.,

Amos, C.I., Christiani, D.C. and Lin, X. (2012). Genetic variants on 15q25.1, smoking

and lung cancer: an assessment of mediation and interaction. American Journal of

Epidemiology, 75:1013-1020.

VanderWeele, T.J. and Tchetgen Tchetgen, E.J. (2014). Attributing effects to

interactions. Epidemiology, 25:711-722.

VanderWeele, T.J. and Vansteelandt, S. (2009). Conceptual issues concerning

mediation, interventions and composition. Statistics and Its Interface 2:457-468.

VanderWeele, T.J. and Vansteelandt, S. (2010). Odds ratios for mediation analysis

with a dichotomous outcome. American Journal of Epidemiology, 172:1339-1348.

VanderWeele, T.J. and Vansteelandt, S. (2013). Mediation analysis with multiple

mediators. Epidemiologic Methods, 2:95-115.

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General Decomposition

33

General Decomposition

34