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 Y

The 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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DAGs

Use 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.

Programs

Tingley, Yamamoto, Keele, & Imai, R

VanderWeele, SAS and SPSS

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Sensitivity Analyses

Emphasized 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