Mediation: The Causal Inference Approach

David A. Kenny

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WarningI claim minimal expertise in

this area.Welcome suggestions or

corrections.I include this because this

approach is important.

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OriginsPaper by Robins &

Greenland (1992)Also key papers by Pearl,

Imai, and VanderWeele

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Starting PointBasic mediation model of X, M, and YThe variables need not be interval but

can be any level of measurement.X and M are presumed to interact when

causing Y.Often, though not always, X is presumed

to be manipulated and be a dichotomy (0 = control; 1 = treatment).

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DAGsUse directed acyclic graphs or DAGS, not

path diagrams.

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Confounders• All assumptions concern “omitted

variables” but are called confounders.

• There would be a XY confounder if there exists any variable that causes both X and Y but it is not included in the model.

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Assumptions• Condition 1: No unmeasured confounding

of the XY relationship. • Condition 2: No unmeasured confounding

of the MY relationship.• Condition 3: No unmeasured confounding

of the XM relationship. • Condition 4: Variable X must not cause

any confounder of the MY relationship.

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Condition 2

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Note that …• Condition 4 would be satisfied if

Condition 2 is satisfied.• Added to the list because conclusions are a

bit different if Condition 4 does or does not hold and Condition 2 also holds.

• Although not obvious, these conditions imply no measurement error in M and X and no reverse causation.

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Potential OutcomesImagine someone in the control condition; the person’s

score on Y would be denoted as Y(0).What would have someone scored on Y if their score on

X was 1 and not 0 or Y(1)?Also referred to as a counterfactual.Note that a potential outcome is not different from a

predicted value of a properly specified structural equation.

The causal effect of X on Y is the value of the difference between E[Y(1)] – E[Y(0)] across individuals.

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Causal EffectLet E[Y(1)] be the expected value for members in

the population when X = 1 and E[Y(0)] be the expected value for members in the population when X = 0.

The causal effect of X on Y equals: E[Y(1)] – E[Y(0)]

This looks stranger than it is. In a randomized study it is nothing more than the difference between the population means of experimental (X = 1) and control (X = 0) groups.

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Potential Outcomes with Mediation

To refer to a potential outcome of Y as function of X and M, the following convention is adopted: E[Y(i, j)] where i refers to the score on X and j to the score on M.

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How Do We Measure a Potential Outcome?

With randomization can use the expected value of the other group.

If the Four Conditions hold then can use a model predicted score.

In some case, propensity scores can be used to estimate what the score would be in the other condition.

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Measuring EffectsIn the classical model, the effect of X is a regression coefficient.A regression coefficient, say c, expresses the difference in Y between those varying by one unit on X.We could then denote the effect as

E[Y(1)] – E[Y(0)]Note because of linearity and lack of interaction this would not be the same value for all one unit differences in X.

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Controlled Direct EffectThe Controlled Direct Effect for M equal to M

CDE(M) = E[Y(1,M)] –E[Y(0,M)]

If X and Y interact, this difference is going to be different for different values of M.

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Natural Direct EffectTo obtain a single measure of the DE, several different suggestions have been made. One idea is to determine the Natural Direct Effect which is defined as:

NDE = E[Y(1,M0)] - E[Y(0,M0)]

where M0 is the potential outcome for M when X is 0.

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M0

Note that if M is continuous variable, e.g., how much coping that you do, M0 is simply the predicted mean of M for those whose score on X is zero.If however, M is a 0-1 dichotomy, e.g., whether you cope (1) or not (0), the M0 term is a probability. In that case, we would have:

Y(1,M0) = (1 - M0) E[Y(1,0)] + (M0) E[Y(1,1)]

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Natural Indirect and Total Effects

Natural indirect effect:NIE = E[Y(1,M1)] –E[Y(1,M0)]

Total Effect: TE = E[Y(1)] –E[Y(0)] = E[Y(1,M1)] – E[Y(1,M0)] + E[Y(1,M0)] – E[Y(0,M0) ] = E[Y(1,M1)] – E[Y(0,M0)]

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Practicality

The NDE and NIE are relatively straightforward when X is dichotomy and there is a single mediator. But if X has many levels or there is a second mediator, there are many of these effects and it is not at all clear which of these effects one should use.

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EstimationIn some cases, the estimate simple reduces

to classical mediation and multiple regression or SEM can be used.

Can also use G estimation or Marginal Structural Equation Modeling.

ProgramsTingley, Yamamoto, Keele, & Imai, RVanderWeele, SAS and SPSS

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Sensitivity AnalysesEmphasized strongly in this

tradition.Each of the Four Conditions is

examined.

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The FutureThe Causal Inference Approach is evolving and is sure to change.

Stay tuned.

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Additional Presentation

• Sensitivity Analysis

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ThanksJudea Pearl