Mediation Analysis With Intermediate Confounding: Structural Equation Modeling Viewed Through the Causal Inference Lens

The study of mediation has a long tradition in the social sciences and a relatively more recent one in epidemiology. The first school is linked to path analysis and structural equation models (SEMs), while the second is related mostly to methods developed within the potential outcomes approach to causal inference. By giving model-free definitions of direct and indirect effects and clear assumptions for their identification, the latter school has formalized notions intuitively developed in the former and has greatly increased the flexibility of the models involved. However, through its predominant focus on nonparametric identification, the causal inference approach to effect decomposition via natural effects is limited to settings that exclude intermediate confounders. Such confounders are naturally dealt with (albeit with the caveats of informality and modeling inflexibility) in the SEM framework. Therefore, it seems pertinent to revisit SEMs with intermediate confounders, armed with the formal definitions and (parametric) identification assumptions from causal inference. Here we investigate: 1) how identification assumptions affect the specification of SEMs, 2) whether the more restrictive SEM assumptions can be relaxed, and 3) whether existing sensitivity analyses can be extended to this setting. Data from the Avon Longitudinal Study of Parents and Children (1990–2005) are used for illustration.


Part C: L-Y confounding
Consider the setting where there is an additional variable U that is a common cause of L and Y with the usual assumption of uncorrelated errors. Also assume for simplicity that there are no background confounders C, no interactions, and that all variables have zero mean and unit variance: where for completeness we have added an equation for X.
We are interested in what would happen if we omitted U from these models, i.e. if we wrongly assumed the model to be: In this web appendix, we use the theory described by Wermuth and Cox (1) to express the parameters of model (2) in terms of those of model (1). Even though they do not all coincide, we will show that when combined into the causal mediation estimands CDE(m), PNDE and TNIE, the bias is compensated, so that both models, (1) and (2), lead to identical mediation estimands.
Model (1) can be re-written as: Or equivalently in matrix form AY = where Y = (Y, M, L, U, X) T , = ( y , m , l , u , x ) T , and Σ, a diagonal matrix, is the variancecovariance matrix of . A is then given by: After marginalizing over U , our equations are written in matrix form as: where Y = (Y, M, L, X) T , η = (η y , η m , η l , η x ) T , and K is the variance-covariance matrix of η. A is given by: Re-arranging the rows and columns of A so that U appears first, we re-write equation 3 as The first step in Wermuth and Cox is to form a matrix B by partial inversion ofÃ wrt U .
HereŪ refers to "everything but U ", and we use this notation for partitioning matrices throughout.
Evaluating equation 4, we obtain: Next, since we wish to know the coefficients of the regression of Y on M , L and X when U is ignored, we form the matrix C, which is the partial inversion ofBŪŪ wrt Y . Applying the partial inversion formula: Wermuth and Cox (1) show that the coefficients of the regression of Y on M , L and X when U is ignored are given by: where Q = invȲW and W is the variance-covariance matrix of the vector andW is this matrix rearranged so that the row and column corresponding to Y appears last (so that the order of the variables is M, L, X, Y ). In our setting, CȲ ,Y = (0, 0, 0) T and so W is equal to K, the variance-covariance matrix of η. Wermuth and Cox show that this can be derived as Returning to expression (5), we obtain that the coefficients of the regression of Y on M , L and X when U is ignored are given by: It remains to write σ 2 l and σ 2 u in terms of the elements of A.
since Var (U ) = Var (X) = 1 and Cov (X, u ) = 0 by the uncorrelated errors assumption. This gives us Then, for σ 2 l :

This gives us
Putting these back into equation (6), we obtain: Going through the same calculations for the regression coefficients for L and X in the equation for M (after marginalising over U ), and for the regression coefficient for X in the equation for L (after marginalising over U ), we similarly obtain: These results are intuitive, since U does not appear in the true data generating equation for M , and since marginalising over U apportions the path from X to U to L to the path from X to L.
Putting all this into the expressions for the PNDE, TNIE and CDE, we obtain: Thus, when a (background or intermediate) confounder U of the L-Y relationship is ignored, even though some of the individual SEM parameters are biased (i.e. cannot be given a causal interpretation as data generating parameters), this bias is compensated when the mediation estimands are calculated, so that the PNDE, TNIE, and CDE can be identified without data on U .

WEB TABLE 1
Estimated coefficients of the SEMs corresponding to the models reported in Table 3