Average Controlled and Average Natural Micro Direct Effects in Summary Causal Graphs

TL;DR

The paper addresses the identifiability of average controlled micro direct effects and average natural micro direct effects from summary causal graphs that abstract time-series dynamics, allowing cycles and hidden confounding in a non-parametric setting. It develops graphical conditions using the do-calculus and the notion of possible parents PP(Y_t) to establish identifiability results for ACMDE and ANMDE, showing sufficiency and, in some cases, necessity under no hidden confounding. A key contribution is a framework that yields interventional expressions for these direct effects solely in terms of observed quantities, enabling identification despite incomplete temporal detail. The results have practical relevance for epidemiology and other dynamic systems where full causal graphs are impractical, and they include a real-world example illustrating identifiable and non-identifiable scenarios.

Abstract

In this paper, we investigate the identifiability of average controlled direct effects and average natural direct effects in causal systems represented by summary causal graphs, which are abstractions of full causal graphs, often used in dynamic systems where cycles and omitted temporal information complicate causal inference. Unlike in the traditional linear setting, where direct effects are typically easier to identify and estimate, non-parametric direct effects, which are crucial for handling real-world complexities, particularly in epidemiological contexts where relationships between variables (e.g, genetic, environmental, and behavioral factors) are often non-linear, are much harder to define and identify. In particular, we give sufficient conditions for identifying average controlled micro direct effect and average natural micro direct effect from summary causal graphs in the presence of hidden confounding. Furthermore, we show that the conditions given for the average controlled micro direct effect become also necessary in the setting where there is no hidden confounding and where we are only interested in identifiability by adjustment.

Figures (6)

• Figure 1: An SCG in (a) with two compatible FT-ADMGs in (b) and (c). Each pair of red and blue vertices represents the micro direct effect we are interested in. Both the controlled direct effect and the natural direct effect are not identifiable using the conditions given in this paper.
• Figure 2: An SCG in (a) with a compatible FT-ADMG in (b). Each pair of red and blue vertices represents the micro direct effect we are interested in. Both the controlled direct effect and the natural direct effect are not identifiable using the conditions given in this paper.
• Figure 3: An SCG in (a) with two compatible FT-ADMGs in (b) and (c). Each pair of red and blue vertices represents the micro direct effect we are interested in. The controlled direct effect is identifiable according to our condition but the natural direct effect is not.
• Figure 4: An SCG in (a) with a compatible FT-ADMG in (b). Each pair of red and blue vertices represents the micro direct effect we are interested in. The controlled direct effect is identifiable according to our condition but the natural direct effect is not.
• Figure 5: Three SCGs. Each pair of red and blue vertices represents the micro direct effect we are interested in. The controlled direct effect is identifiable in all these SCGs using the conditions given in this paper. The natural direct effect is not identifiable in (a), identifiable if $\gamma = \gamma_{max}$ in (b) and identifiable for all $\gamma$ in (c).

Theorems & Definitions (15)

• Definition 1: Discrete-time dynamic structural causal model (DTDSCM)
• Definition 3: Full-Time Acyclic Directed Mixed Graph
• Definition 4: Average Controlled Micro Direct Effect Pearl_2001
• Definition 5: Average Natural Micro Direct Effect Pearl_2001
• Definition 6: Summary Causal Graph with possible latent confounding
• Definition 7: Possible Parents
• Theorem 9
• Proposition 10
• Lemma 11
• Theorem 12