Identifiability of Direct Effects from Summary Causal Graphs

TL;DR

The paper tackles the problem of identifying direct causal effects from summary causal graphs (SCGs) when the full-time causal graph is not available, within linear dynamic SCMs under no hidden confounding. It introduces a complete graphical identifiability criterion that characterizes exactly when the direct effect $\alpha_{X_{t-\gamma_{xy}},Y_t}$ is estimable from an SCG, accounting for cycles and temporal lags. When identifiability holds, it furnishes two finite adjustment sets that yield unbiased estimators of the direct effect from data, with a more conservative but larger set and a smaller, efficient alternative. These results extend prior work to general cyclic SCGs, provide practical estimation tools, and open avenues for future work on completeness of adjustment sets and extensions to nonlinear or cyclic full-time graphs.

Abstract

Dynamic structural causal models (SCMs) are a powerful framework for reasoning in dynamic systems about direct effects which measure how a change in one variable affects another variable while holding all other variables constant. The causal relations in a dynamic structural causal model can be qualitatively represented with an acyclic full-time causal graph. Assuming linearity and no hidden confounding and given the full-time causal graph, the direct causal effect is always identifiable. However, in many application such a graph is not available for various reasons but nevertheless experts have access to the summary causal graph of the full-time causal graph which represents causal relations between time series while omitting temporal information and allowing cycles. This paper presents a complete identifiability result which characterizes all cases for which the direct effect is graphically identifiable from a summary causal graph and gives two sound finite adjustment sets that can be used to estimate the direct effect whenever it is identifiable.

Figures (4)

• Figure 1: An example of an SCG in (a and b) with four of its compatible FTCGs in (b), (c), (e) and (f). The pair of red and blue vertices in the SCG and in the FTCGs represents the direct effect of interest, i.e., $\alpha_{X_t,Y_t}$ and $\alpha_{X_{t-1},Y_t}$. In this example, $\alpha_{X_{t-1},Y_t}$ is non-identifiable given the SCG because for this direct effect, in the FTCG in (b), $X_t$ should be in every valid adjustment set but $X_t$ should not be in any valid adjustment set in the FTCG in (c). Similarly, $\alpha_{X_t,Y_t}$ is non-identifiable given the SCG because for this direct effect (equal to zero), in the FTCG in (e), at least one vertex in $\{U_t, Z_t, W_t\}$ should be in every valid adjustment set but none of the vertices in $\{U_t, Z_t, W_t\}$ should be in any valid adjustment set in the FTCG in (f).
• Figure 2: An example of an SCG in (a) with two of its compatible FTCGs in (b) and (c). The pair of red and blue vertices in the SCG and in the FTCGs represents the direct effect of interest, i.e., $\alpha_{X_t,Y_t}$. In this example, $\alpha_{X_t,Y_t}$ is non-identifiable given the SCG because for this direct effect at least one vertex in $\{U_t, Z_t, W_t\}$ should be in every valid adjustment set for the first FTCG but none of the vertices in $\{U_t, Z_t, W_t\}$ should be in any valid adjustment set for the second FTCG.
• Figure 3: An example of an SCG in (a) with two of its compatible FTCGs in (b) and (c). The pair of red and blue vertices in the SCG and in the FTCGs represents the direct effect of interest, i.e., $\alpha_{X_{t-1},Y_t}$. In this example, $\alpha_{X_{t-1},Y_t}$ is non-identifiable given the SCG because for this direct effect at least one vertex in $\{U_t, Z_t, W_t\}$ should be in every valid adjustment set for the first FTCG but none of the vertices in $\{U_t, Z_t, W_t\}$ should be in any valid adjustment set for the second FTCG.
• Figure 4: Examples of SCGs with $5$ vertices where the direct effect $\alpha_{X_{t-\gamma_{xy}},Y_t}$ is identifiable for all $\gamma_{xy}$. Red and blue vertices respectively represent the cause and the effect we are interested in and the thick edge corresponds to the the edge between them. All SCGs share the same skeleton, the edges $X\rightarrow Y$, $Y\leftrightarrows W$, and $Z\leftrightarrows W$ and the cycles of size $2$ on $Y, W,Z$ and $U$.

Theorems & Definitions (44)

• Definition 1: Linear Dynamic SCM
• Definition 2: Direct Effect, Pearl_2000
• Definition 3: Full-Time Causal Graph
• Definition 4: Summary Causal Graph
• Definition 5: Graphical Identifiability of a Direct Effect from an SCG
• Definition 6: Paths in FTCGs
• Definition 7: Paths in SCGs
• Definition 8: Cycles in SCGs
• Definition 9: Blocked Path in FTCGs
• Definition 10: Blocked Path in SCGs