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 non-increasing or non-decreasing average causal effects suffice to draw conclusions about the direction of the bias. When there are more than one unmeasured confounding variable, non-increasing and non-decreasing 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.

Obesity Paradox and Stroke

“…The direction and degree of the bias is difficult to determine, especially if there are numerous factors and confounding variables, which are difficult or impossible to measure directly. More advanced statistical techniques such as causal-directed acyclic graphs 9 and counterfactuals 10 to accurately tally these interrelationships may be needed. Of note, a recent study using propensity score matching design in patients with chronic heart failure and diabetes mellitus has not encountered the "obesity paradox" at all.…”

Directed acyclic graphs with edge-specific bounds

“…We present an example adapted from Greenland et al (1999) and discussed by VanderWeele et al (2008). Using bounds for edges rather than signed edges allows us to derive bounds for the causal effect under fewer assumptions.…”

“…As is clear from Example 2, Theorem 2 in this paper allows the researcher to draw conclusions about bounds on causal effects in the presence of unmeasured confounding by making assumptions concerning fewer edges than were previously required. VanderWeele et al (2008) showed that to draw conclusions about the sign of the bias in the presence of unmeasured confounding using signed edges the treatment had to be binary or comparison had to be made between the minimum and maximum levels of treatment. Intuition about the sign of the bias could fail if an intermediate level of treatment was considered.…”

“…We could then conclude from this study that antihistamine use truly had a beneficial effect on asthma. To draw conclusions about bounds for the causal effect using signed edges, VanderWeele et al (2008) had to assume that the W → U , W → A, U → A and U → Y edges all had positive sign. From these assumptions, they concluded that…”

Epidemiological Approaches to Metal Toxicology

“…If exposures occur both prenatally and postnatally, the decision whether to adjust one for the other becomes difficult, as both may be causally related to the same household characteristics (Figure 2). Directed acyclic graphs (VanderWeele et al, 2008), such as those illustrated in the figure, can be helpful when determining whether confounding is present and how unmeasured confounding variables affect the findings.…”