A unification of mediation and interaction: a four-way - - PowerPoint PPT Presentation
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
Tyler J. VanderWeele Departments of Epidemiology and Biostatistics Harvard School of Public Health
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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
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In 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
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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
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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
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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:
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The 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:
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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,
are needed Note (1) and (3) are guaranteed when treatment is randomized A M Y C1 C3 C2
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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,
are needed Note (1) and (3) are guaranteed when treatment is randomized A M Y C1 C3 C2
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Similar 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:
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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
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VanderWeele 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:
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A 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
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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) 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
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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.
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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 33.8% 43.3% 1 48.5% 43.7% 2 17.7% 13.0%
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To use our approach with the genetic variants we need to assume no unmeasured confounding for the (1) exposure-outcome, (2) mediator-
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 Y C C U
When 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
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With 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
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(1) Most of the effect seemed to be due to interaction, in the absence
(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
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The 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
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One 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)
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(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
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Explanation in Causal Inference Methods for Mediation and Interaction
2015 │ Hardcover│ ISBN: 9780199325870
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