Causal directed acyclic graphs and the direction of unmeasured confounding bias

Abstract

We present results that allow the researcher in certain cases to determine the direction of the bias that arises when control for confounding is inadequate. The results are given within the context of the directed acyclic graph causal framework and are stated in terms of signed edges. Rigorous definitions for signed edges are provided. We describe cases in which intuition concerning signed edges fails and we characterize the directed acyclic graphs that researchers can use to draw conclusions about the sign of the bias of unmeasured confounding. If there is only one unmeasured confounding variable on the graph, then nonincreasing or nondecreasing average causal effects suffice to draw conclusions about the direction of the bias. When there are more than one unmeasured confounding variable, nonincreasing and nondecreasing average causal effects can be used to draw conclusions only if the various unmeasured confounding variables are independent of one another conditional on the measured covariates. When this conditional independence property does not hold, stronger notions of monotonicity are needed to draw conclusions about the direction of the bias.

Figure 1

Example illustrating confounding by health status: Y - disease; A - exposure; U - unmeasured health status.

Figure 2

Graph that is not minimal with respect to A, Y and X: Y indicates outcome; A, indicates exposure; X, indicates measured confounding variable; V, W, indicate unmeasured confounding variables; Q, Z, indicate intermediate variables.

Figure 3

Graph that is minimal with respect to A, Y and X: Y indicates outcome; A, indicates exposure; X, indicates measured confounding variable; V, W, indicate unmeasured confounding variables.

Figure 4

Example illustrating the use of Result 1: Y indicates asthma; A, indicates antihistamine treatment; S, indicates sex; V, indicates air pollution; W, indicates bronchial reactivity.

Figure 5

Example illustrating that Result 1 may fail for graphs on which the conditional independence condition is not satisfied: Y indicates outcome; A, indicates exposure; V, W, indicate unmeasured confounding variables.

Figure 6

Example illustrating that positive average monotonic effects are not transitive: A indicates a variable with a positive average monotonic effect on B; B, indicates a varaible with a positive average monotonic effect on C; C, indicates outcome.