Mediation: The Causal Inference Approach David A. Kenny

2 Warning I claim minimal expertise in this area. Welcome suggestions or corrections. I include this because this approach is important.

3 Origins Paper by Robins & Greenland (1992) Also key papers by Pearl, Imai, and VanderWeele

4 Starting Point Basic 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).

5 DAGs Use directed acyclic graphs or DAGS, not path diagrams.

6 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.

7 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.

8 Condition 2

9 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.

10 Potential Outcomes Imagine 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.

11 Causal Effect Let 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.

12 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.

13 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.

14 Measuring Effects In 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.

15 Controlled Direct Effect The 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.

16 Natural Direct Effect To 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,M 0 )] - E[Y(0,M 0 )] where M 0 is the potential outcome for M when X is 0.

17 M0M0 Note that if M is continuous variable, e.g., how much coping that you do, M 0 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 M 0 term is a probability. In that case, we would have: Y(1,M 0 ) = (1 - M 0 ) E[Y(1,0)] + (M 0 ) E[Y(1,1)]

18 Natural Indirect and Total Effects Natural indirect effect: NIE = E[Y(1,M 1 )] –E[Y(1,M 0 )] Total Effect: TE = E[Y(1)] –E[Y(0)] = E[Y(1,M 1 )] – E[Y(1,M 0 )] + E[Y(1,M 0 )] – E[Y(0,M 0 ) ] = E[Y(1,M 1 )] – E[Y(0,M 0 )]

19 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.

20 Estimation In 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

21 Sensitivity Analyses Emphasized strongly in this tradition. Each of the Four Conditions is examined.

22 The Future The Causal Inference Approach is evolving and is sure to change. Stay tuned.

23 Additional Presentation Sensitivity Analysis

24 Thanks Judea Pearl