A directed acyclic graph for interactions

Abstract

Background: Directed acyclic graphs (DAGs) are of great help when researchers try to understand the nature of causal relationships and the consequences of conditioning on different variables. One fundamental feature of causal relations that has not been incorporated into the standard DAG framework is interaction, i.e. when the effect of one variable (on a chosen scale) depends on the value that another variable is set to. In this paper, we propose a new type of DAG-the interaction DAG (IDAG), which can be used to understand this phenomenon.

Methods: The IDAG works like any DAG but instead of including a node for the outcome, it includes a node for a causal effect. We introduce concepts such as confounded interaction and total, direct and indirect interaction, showing that these can be depicted in ways analogous to how similar concepts are depicted in standard DAGs. This also allows for conclusions on which treatment interactions to account for empirically. Moreover, since generalizability can be compromised in the presence of underlying interactions, the framework can be used to illustrate threats to generalizability and to identify variables to account for in order to make results valid for the target population.

Conclusions: The IDAG allows for a both intuitive and stringent way of illustrating interactions. It helps to distinguish between causal and non-causal mechanisms behind effect variation. Conclusions about how to empirically estimate interactions can be drawn-as well as conclusions about how to achieve generalizability in contexts where interest lies in estimating an overall effect.

Keywords: Causal inference; external validity; generalizability; interaction; internal validity; mediation.

Figure 1

An example of a standard directed acyclic graph (DAG) (panel A) and two possible interaction DAGs (IDAGs) (panels B and C). Variables A (warfarin) and Q (smoking) influence Y (ischaemic stroke). Panel B suggests that Q also influences the effect of A on Y, whereas panel C suggests that this is not the case

Figure 2

Confounded interaction or ‘effect modification by proxy’. A standard directed acyclic graph (DAG) is given in panel A and an interaction DAG (IDAG) in panel B. Variables X (genotype) and A (bariatric surgery) influence Y (weight loss), with an interaction present. The effect of A is modified by Q (hair colour), but there is no interaction between A and Q

Figure 3

Two examples of standard directed acyclic graphs (DAGs) (left) and two interaction DAGs (IDAGs) (right). The variable Y (a disease) is directly influenced by A (treatment), Q (smoking) and potentially also X (education). The DAG in panel A is compatible with the IDAG in panel C, whereas the DAG in panel B is compatible with either of the IDAGs in panels C and D

Figure 4

Sample selection potentially compromising generalizability. Individuals are selected based on S. X may represent socioeconomic status, A some treatment, and Y a disease. The standard directed acyclic graph (DAG) in panel A is compatible either with the interaction DAG (IDAG) in panel B or the one in panel C, where generalizability is only compromised in the scenario depicted in panel B