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Towards identifiability of micro total effects in summary causal graphs with latent confounding: extension of the front-door criterion

Charles K. Assaad

TL;DR

This paper addresses identifying total causal effects from observational data when only a partially specified, dynamic causal graph—an SCG—is available, potentially containing cycles and latent confounding. It shows that standard adjustment and existing front-door criteria fail in this setting and introduces the SCG-front-door criterion, plus a corresponding do-free identification formula, to recover $P(y_t \mid do(x_{t-\gamma}))$ under a broad class of SCGs. The main contribution is a graphical condition and constructive formula that enable identifiability from SCGs without enumerating all compatible FT-ADMGs, along with careful discussion of cases where identifiability fails. This advances causal inference in time-evolving systems where full temporal graphs are unavailable, with practical implications for observational studies in epidemiology and related fields.

Abstract

Conducting experiments to estimate total effects can be challenging due to cost, ethical concerns, or practical limitations. As an alternative, researchers often rely on causal graphs to determine whether these effects can be identified from observational data. Identifying total effects in fully specified causal graphs has received considerable attention, with Pearl's front-door criterion enabling the identification of total effects in the presence of latent confounding even when no variable set is sufficient for adjustment. However, specifying a complete causal graph is challenging in many domains. Extending these identifiability results to partially specified graphs is crucial, particularly in dynamic systems where causal relationships evolve over time. This paper addresses the challenge of identifying total effects using a specific and well-known partially specified graph in dynamic systems called a summary causal graph, which does not specify the temporal lag between causal relations and can contain cycles. In particular, this paper presents sufficient graphical conditions for identifying total effects from observational data, even in the presence of cycles and latent confounding, and when no variable set is sufficient for adjustment.

Towards identifiability of micro total effects in summary causal graphs with latent confounding: extension of the front-door criterion

TL;DR

This paper addresses identifying total causal effects from observational data when only a partially specified, dynamic causal graph—an SCG—is available, potentially containing cycles and latent confounding. It shows that standard adjustment and existing front-door criteria fail in this setting and introduces the SCG-front-door criterion, plus a corresponding do-free identification formula, to recover under a broad class of SCGs. The main contribution is a graphical condition and constructive formula that enable identifiability from SCGs without enumerating all compatible FT-ADMGs, along with careful discussion of cases where identifiability fails. This advances causal inference in time-evolving systems where full temporal graphs are unavailable, with practical implications for observational studies in epidemiology and related fields.

Abstract

Conducting experiments to estimate total effects can be challenging due to cost, ethical concerns, or practical limitations. As an alternative, researchers often rely on causal graphs to determine whether these effects can be identified from observational data. Identifying total effects in fully specified causal graphs has received considerable attention, with Pearl's front-door criterion enabling the identification of total effects in the presence of latent confounding even when no variable set is sufficient for adjustment. However, specifying a complete causal graph is challenging in many domains. Extending these identifiability results to partially specified graphs is crucial, particularly in dynamic systems where causal relationships evolve over time. This paper addresses the challenge of identifying total effects using a specific and well-known partially specified graph in dynamic systems called a summary causal graph, which does not specify the temporal lag between causal relations and can contain cycles. In particular, this paper presents sufficient graphical conditions for identifying total effects from observational data, even in the presence of cycles and latent confounding, and when no variable set is sufficient for adjustment.
Paper Structure (8 sections, 11 theorems, 6 equations, 5 figures)

This paper contains 8 sections, 11 theorems, 6 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and problem setup
  3. FT-ADMG notions
  4. Unsuitability of the standard front-door criterion when applied to SCGs
  5. The identifiability of total effects in SCGs with latent confounding
  6. Towards non identifiability results
  7. Conclusion
  8. Appendix

Key Result

Lemma 4.1

Consider an SCG $\mathcal{G}^s$. If a set of macro vertices $\mathbb{W}$ intercepts all directed paths from $X$ to $Y$ in $\mathcal{G}^s$ then $\{ (\mathbb{W}_{t-\gamma+\ell})_{0\leq \ell \le \gamma} \}$sub-intercepts all directed paths from $X_{t-\gamma}$ to $Y_t$ in any candidate FT-ADMG in $\math

Figures (5)

  • Figure 1: Three SCGs and six FT-ADMGs, where $\gamma_{\max}=1$, such that FT-ADMGs 1 and 2 are compatible with SCG 1, FT-ADMGs 3 and 4 are compatible with SCG 2, and FT-ADMGs 5 and 6 are compatible with SCG 3. Each pair of red and blue vertices represents the cause and the effect of interest and green vertices are those that interpect all paths from the cause to the effect of interest. In SCG 1 and SCG 2, $\{W\}$ satisfies Definition \ref{['def:front_door_SCG']} for the total effect $\Pr(y_t|do(x_{t-1}))$. However, $\{W\}$ does not satisfy Definition \ref{['def:front_door_SCG']} in SCG 3 for the total effect $\Pr(y_t|do(x_{t-1}))$ since $Cycles(X, \mathcal{G}^s)\ne \emptyset$ and $\gamma\ne0$.
  • Figure 2: Ten SCGs satisfying Definition \ref{['def:front_door_SCG']} for $W$ relative to the pair of micro vertices $(X_{t-\gamma}, Y_t)$, $\forall \gamma \in \{0, \cdots, \gamma_{\max}\}$, i.e., for the first half of these SCGs, Conditions \ref{['item:front_door_SCG:1']}, Conditions \ref{['item:front_door_SCG:2']}, Conditions \ref{['item:front_door_SCG:3']}, and Conditions \ref{['item:front_door_SCG:a']} in Definition \ref{['def:front_door_SCG']} are satisfied and for the second half, Conditions \ref{['item:front_door_SCG:1']}, Conditions \ref{['item:front_door_SCG:2']}, Conditions \ref{['item:front_door_SCG:3']}, and Conditions \ref{['item:front_door_SCG:c']} in Definition \ref{['def:front_door_SCG']} are satisfied. Each pair of red and blue vertices represents the total effect of interest and green vertices are those that intercept all directed paths from the cause to the effect of interest.
  • Figure 3: Five SCGs satisfying Definition \ref{['def:front_door_SCG']} for $W$ relative only to the pair of micro vertices $(X_{t}, Y_t)$, i.e., Conditions \ref{['item:front_door_SCG:1']}, Conditions \ref{['item:front_door_SCG:2']}, Conditions \ref{['item:front_door_SCG:3']}, and Conditions \ref{['item:front_door_SCG:b']} in Definition \ref{['def:front_door_SCG']} are satisfied. Each pair of red and blue vertices represents the total effect of interest and green vertices are those that intercept all directed paths from the cause to the effect of interest.
  • Figure 4: Five SCGs not satisfying Definition \ref{['def:front_door_SCG']} for any vertex relative to the micro of vertices $(X_{t-\gamma}, Y_t)$. The SCG in (a) and (d) does not satisfy Definition \ref{['def:front_door_SCG']} because Condition \ref{['item:front_door_SCG:2']} is not satisfied. The SCG in (b) and (e) does not satisfy Definition \ref{['def:front_door_SCG']} because Condition \ref{['item:front_door_SCG:3']} is not satisfied. The SCG in (c) does not satisfy Definition \ref{['def:front_door_SCG']} because Condition \ref{['item:front_door_ADMG:2']} and \ref{['item:front_door_ADMG:3']} are not satisfied. Each pair of red and blue vertices represents the total effect of interest and green vertices are those that intercepts all directed paths from the cause to the effect of interest.
  • Figure 5: Four FT-ADMGs that correspond respectively to the SCGs in Figures \ref{['fig:satisfying:1.1']}, \ref{['fig:satisfying:6']}\ref{['fig:satisfying_inst:1.1']}, and \ref{['fig:satisfying:2']}. In each graph, the red and blue vertices represent the total effect of interest, while the green vertices are those that intercept all directed paths from the cause to the effect of interest. In FT-ADMGs (a), (b), and (c), all back-door paths from the red vertex (in bold) to the green vertices (in bold) are blocked by the brown, pink, and gray vertices. In FT-ADMG (d), all back-door paths from each green vertex (in bold)to the blue vertex (in bold) are blocked by the orange, purple, and other green vertices that are temporally prior to the selected green vertex.

Theorems & Definitions (26)

  • Definition 1: Discrete-time dynamic structural causal model (DTDSCM)
  • Definition 2: Full-Time Acyclic Directed Mixed Graph
  • Definition 3: Identifiability of total effects in FT-ADMGs
  • Definition 4: Summary Causal Graph with possible latent confounding
  • Definition 5: Identifiability of total effects in SCGs
  • Definition 6: Standard front-door criterion naively applied to SCGs
  • Definition 7: SCG-front-door criterion
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 16 more