Analytic bounds on causal risk differences in directed acyclic graphs involving three observed binary variables

Abstract

We apply a linear programming approach which uses the causal risk difference (RDC) as the objective function and provides minimum and maximum values that RDC can achieve under any set of linear constraints on the potential response type distribution. We consider two scenarios involving binary exposure X, covariate Z and outcome Y. In the first, Z is not affected by X, and is a potential confounder of the causal effect of X on Y. In the second, Z is affected by X and intermediate in the causal pathway between X and Y. For each scenario we consider various linear constraints corresponding to the presence or absence of arcs in the associated directed acyclic graph (DAG), monotonicity assumptions, and presence or absence of additive-scale interactions. We also estimate Z-stratum-specific bounds when Z is a potential effect measure modifier and bounds for both controlled and natural direct effects when Z is affected by X. In the absence of any additional constraints deriving from background knowledge, the well-known bounds on RDc are duplicated: −Pr(Y≠X) ≤ RDC ≤ Pr(Y=X). These bounds have unit width, but can be narrowed by background knowledge-based assumptions. We provide and compare bounds and bound widths for various combinations of assumptions in the two scenarios and apply these bounds to real data from two studies.

An Introduction to Causal Inference

“…Chickering and Pearl (1997) further used Bayesian techniques (with Gibbs sampling) to investigate the sharpness of these bounds as a function of sample size. Kaufman, Kaufman, and MacLenose (2009) used this technique to bound direct and indirect effects (see Section 6.1).…”

Interpretable Conditions for Identifying Direct and Indirect Effects

“…Interest in mediation analysis stems from both scientific and practical considerations. Scientifically, mediation tells us "how nature works," and practically, it 1992, resulted in nine years of abandonment, during which natural effects were considered void of empirical content and were not investigated (Kaufman et al, 2009).…”

Alternative Monotonicity Assumptions for Improving Bounds on Natural Direct Effects

“…Since CDEs and NDEs cannot generally be identified in the presence of unmeasured confounders between a mediator and an outcome, several researchers have discussed their bounds [7][8][9][10][11], which is a strategy for inference on non-identified (causal) parameters. Recently, bounds on both direct effects were developed using a linear programming technique [7][8][9].…”

“…Recently, bounds on both direct effects were developed using a linear programming technique [7][8][9]. Although they have the advantage that they have the tightest width under the given assumptions, they also have the disadvantages that their use is limited to cases with a binary outcome and the given assumptions are very strict, that is, they must hold for all individuals.…”