Since their mainstream introduction in the 1990s, causal diagrams, including directed acyclic graphs (DAGs), are increasingly used to depict our causal knowledge of the world we study and to guide study design, analysis, and interpretation for causal inference [1, 2]. Beyond describing the data-generating mechanism, DAGs are typically used to select variables for confounding control [2]. However, to do more, researchers have had to modify or augment their DAGs with additional features reflecting theoretical what-if scenarios or study-dependent processes affecting the data, analysis, and interpretation. In this journal, Mansournia et al. [3] show how to depict balancing scores on DAGs, a welcome addition to the growing use of augmented graphs. This commentary will first place this work in the broader context of how researchers have tried to do more with augmented causal diagrams, including augmented directed acyclic graphs (ADAGs) which we use more expansively to include all such graphs in this commentary. Then, we highlight some useful features of ADAGs that depict balancing scores, focusing on the propensity score for illustration. We conclude with some remarks on the future of graphical augmentation.

Over the years, DAGs have been augmented to depict counterfactual variables and study design, measurement, analytical, and interventional features [1, 4–10], yielding what we call augmented DAGs (ADAGs). Consider the nonaugmented DAG in Figure 1a with an exposure A, an outcome Y, a mediator M, and two confounders L1 and L2, which could be used to study the effect of A on Y and the joint effect of A and L1 and Y. Suppose we are interested in visualizing non-differential independent measurement error in A, effect-measurement modification of the effect of A on Y by L1, interaction between A and M in causing Y, that the confounder L2 is unmeasured but its proxies E2 and O2 are observed, and the consequences of selection on unmeasured L2 which affects whether Y is observed.

These study-dependent processes or features can be added to DAG 1a to obtain the ADAG in Figure 1b. For example, the measurement error in exposure is added as a new node A* with an arrow from A [4, 5]. The effect-measure modification of A→Y by L1 is visualized using a product node AL1 with arrows from A and L1 and an outgoing arrow into Y [6]. Similar augmentation draws the interaction between A and M in affecting Y. The selection mechanism that results from selection on S = 1 and how S defines whose Y is observed in the study are shown, respectively, with an arrow from unobserved L2 (a parent of Y) into S and arrows from S and unobserved Y into observed Y^obs in the ADAG 1b [7, 8]. A signed ADAG depicting the direction of effects with + and – symbols can be used for reasoning about uncontrolled confounding (due to L2 here) (Figure 1b). Others, such as single-world intervention graphs (as in ADAG 1c) and twin networks (as in ADAG 1d), have been developed to show interventions and thus potential outcomes [chapter 41 in reference 1]. Although standard d-separation can be used to read some key independencies (e.g., conditional exchangeability) for using the PS in these ADAGs to estimate A→Y, other independence implications of deterministic augmentations need the (modified) D-separation criterion [9].

Given the growing use of ADAGs, it is unsurprising to see ones depicting balancing scores, namely propensity scores (PS) and disease risk scores (DRS), used in confounding control [3]. Like others [10], Mansournia et al. [3] propose inserting the PS between the exposure A and the confounders L1 and L2, and the DRS between the confounders and the (potential) outcome. [By the way, we believe Mansournia et al’s arrow from A to the computational PS in their Figure 8c should be omitted.] Is it possible to anticipate or design the theoretical balancing scores from the unmodified DAG, like the one in Figure 1e? Is there a general principle for augmenting DAGs with the balancing scores? Which variables should be included? Can we evaluate variable balance using the resulting ADAGs? What else can we infer from them?

Consider the DAG in Figure 1e, adapted from Mansournia et al. by including L0 (which affects exposure A directly and Y only through A), L4 (which affects Y only), and a collider L3 (which forms an M-bias structure between A and Y). Under Pearl’s Structural Causal Model [1] that would have given rise to DAG 1e, A is given by the function f_A(L0, L1, L2, U₁, U_A) with U_A reflecting the unknown parents of A that are independent of the unknown parents of the other variables in DAG 1e. Then, imagine inventing the PS from DAG 1e by partitioning A into two independent parts which would have arrows from them into A: a deterministic part we call the PS which is formed by the Ls in f_A (and represented with arrows from the Ls to the PS in ADAGs 1f to 1j), and a variable ‘error’ part containing at least U_A. Therefore, a general principle for representing the PS as a function of selected Ls on the ADAG involves moving any existing d-connection between the Ls and A from Ls to PS and an arrow from PS to A, as shown in ADAGs 1f to 1j. Nodes like L3 or L4, which do not cause A but are used to specify PS, can be connected using dashed arrows to distinguish their non-causal role in the partitioning of A. We could conduct a similar visual thought experiment to ‘invent’ the DRS on an ADAG using the structural model for Y, f_Y(A, L1, L2, L4, U₂, U_Y).

If the Ls used in forming the PS suffice to close the backdoor between A and Y, then the PS could also close the same backdoor (see ADAGs 1f to 1j). This answers the question of which variables to include in the PS: at a minimum, those Ls (L1 and L2) that would suffice to ‘deconfound’ A→Y. Mansournia et al. use a SWIG (a causal diagram augmented with interventional features) to illustrate how conditioning on the DRS satisfies the traditional conditional ignorability/exchangeability assumptions, which could also be shown for the PS. Table 1 presents five PS models using various combinations of L0 through L4, corresponding to ADAGS 1f to 1j. Only the first four PS models that exclude the collider L3 can be expected to result in no open backdoor between A and Y. From Table 1, we also see that a PS model can only be used to achieve variable balance (or independence) between any L and A if that L is included in the PS. Any L not included in a PS (e.g., L0 or L4 for PS1 in ADAG 1f) retains its original (im)balance across levels of A. Introducing a collider L3 into a PS induces collider-bias (ADAG 1j and PS5 in Table 1). ADAGs also give intuition about implications of variable selection for statistical efficiency based on the relationship of selected variables to the exposure and outcome (Table 1).

Finally, from the ADAGs, we can also infer that we could regress Y on the residuals (U_A) from the exposure model (A = PS + U_A) used to create the PS to get unconfounded causal estimates. For example, by using L1 and L2 to specify PS1 (in ADAG 1f), the error term would contain only information from the variables that are instruments for A (i.e. L0, U_A1, and the unspecified unmeasured common cause(s) of A and L3 represented by the dashed bidirected arc).

Balancing scores, such as PS and DRS, can be depicted on ADAGs [3, 10]. Due to space constraints, we focused on PS in DAGs, although similar results can be developed for the DRS. Drawing ADAGs should be used after drawing and using the unmodified DAG for identification. In this commentary, we highlighted only some implications of PS-augmented graphs, but we hope more work will be done on extending them. We argue that ADAGs can be more useful than plain DAGs in empirical causal inference work and deserve further research to help us understand what we can and cannot do with them.