One way to understand DAGs is as a collection of causal paths. A causal path can be ‘direct’ such as E → O, or ‘indirect’ such as E → M → O. Crucially, all causal paths must be acyclical – a path started at any variable may never return to that variable (thus directed acyclic graphs). This is how
24 DAGs encode the linearity of time. This collection of direct and indirect acyclic causal paths can be used to identify the causal structures of DAGs described above (i.e. confounders, mediators, and colliders) and their corresponding biases (i.e. confounding bias, ‘overcontrol’ for mediators, and collider/selection bias). DAGs achieve this using the d-separation algorithm, sometimes known as the backdoor criterion, which can be understood as a method for reliably identifying these forms of bias given the causal paths between variables in any DAG. Or in other words, d-separation determines the ‘adjustment set’ by identifying variables that cause bias and should be adjusted for, and variables which would cause bias if they were adjusted.
There is only one type of variable that must be adjusted for – a variable that ‘opens a backdoor path’. A backdoor path starts from the outcome and ends at the exposure, and can therefore be confusing because it requires the researcher to ignore the direction of edges in the DAG. Take Figure 2-11 below. Using DAGitty, the E node has been set to the exposure, and the O node has been set to the outcome. DAGitty automatically performs the d-separation algorithm and has determined that C must be adjusted to estimate the effect of E on O. From the above description of a confounder, we know that is because C is a mutual cause of the exposure E and the outcome O. In DAG terminology, however, we adjust for C because it opens a backdoor path from O to E. Adjusting for the confounder ‘blocks’ it as a variable, and thus closes the open backdoor path. That path, ignoring the direction of directed edges, is O ← C → E → O (note there are other versions via M as well). This exercise would have to be performed for every possible backdoor path in the DAG.
Figure 2-11: Simple DAG for demonstrating an open backdoor path
There are three ways to open a backdoor path – failing to adjust for a confounder, adjusting for a mediator, and adjusting for a collider. Conversely, adjusting for a confounder, and not adjusting for mediators or colliders closes a backdoor path. For example, a closed backdoor path in the above Figure is O ← M ← E → O, because it contains no unadjusted confounders and the mediator is not adjusted. To understand why confounders, mediators and colliders can open backdoor paths, consider each through the lens of bias:
1. Failing to adjust for a confounder opens a backdoor path because the relationship between the exposure and outcome is biased by the confounder’s relationship to both.
25 2. Adjusting for a mediator opens a backdoor path because it systematically removes part of the
effect of interest.
3. Adjusting for a collider opens a backdoor path because it creates a spurious association between its mutual causes (a detailed example is included below).
Thus, even if C is adjusted for Figure 2-11, and the backdoor path O ← C → E → O is then closed, the exposure-outcome relationship would still be biased if M were also adjusted for. This is because the previously closed backdoor path O ← M ← E → O has now been opened by adjusting for M. Adjusting for confounders and mediators in the same statistical model thus induces bias. This is a common problem sometimes referred to as the Table 2 fallacy (Westreich and Greenland, 2013). In this thesis, the more specific term ‘overcontrol for mediators’ is used instead. Note that this is not to say that adjusting for anything that is not a confounder automatically biases effect estimates. For example, the common practice to adjust for ‘other risk factors’ to improve the precision of estimates does not induce bias as a long as that risk factor’s effect is independent of the exposure of interest given the outcome (i.e. it influences the outcome, but not the exposure in any way).
Colliders (i.e. variables caused by two other variables) are likely to be very common in more complex DAGs. However, this is not problematic as the spurious associations induced by adjusting for a collider can often be accounted for by further adjustment. Figure 2-12 below introduces what is sometimes referred to as the M-bias graph, which demonstrates this effect. In Figure 2-12(a), we can see that C1, C2, and C3 each confound the relationship of interest, E → O. The confounding bias of both C1 and C3 is mediated by C2, meaning that C2 is mutually caused by C1 and C3 – it is a collider. However, Figure 2-12(b) shows that, by adjusting for C1 and C2, we can get an unbiased estimate of E → O (i.e. the red/pink arrows are now black). To see why consider first that adjusting for C2 ‘blocks’ the path from C1 to the outcome and the path from C3 to the exposure (such that neither can confound E → O because they do not influence both the exposure and outcome).
However, the spurious association that is induced between C1 and C3 by controlling only for the collider C2 would still bias E → O. Think of this spurious association as a latent, if artificial, construct (i.e. analogous to a variable). This is sometimes represented in DAGs by a curved bi- directional arrow, as per Figure 2-12(c). This construct, induced by controlling only for C2 in this example, biases E → O because it has opened the backdoor path O ← C3 ↔ C1 → E → O. In Figure 2-12(d), however, we can see that controlling for C3 in addition to C2 closes this backdoor path because the collider bias is now independent of the outcome. Thus, the spurious association induced by adjusting for a collider can be accommodated using d-separation. The M-bias graph also
comments on another aspect of DAGs – there can be several adjustment sets which can satisfy the backdoor criterion (i.e. in this example there are three viable adjustment sets: C1 + C2; C2 + C3; or
26 C1 + C2 + C3). In an applied setting, the adjustment set that is preferred depends on practical issues (e.g. which has the fewest variables, which variables have the fewest missing data, etc.).
(a) (b)
(c) (d)
Figure 2-12: The M-bias example of adjusting for a collider
Thus, once researchers have set down their causal assumptions in the form of a DAG, and assuming that there are no cycles in their diagram, d-separation will indicate set of variables that can be adjusted for, thus directly informing adjustment strategies. For example, if researchers wanted to use multiple regression to estimate the effect of an exposure on an outcome, the DAG would indicate which baseline confounders should be added to the model, and which mediators and colliders should not.