Assessing Identifiability with Unmeasured Confounders/Mediators

1. General Graphical Criteria for Identifiability

The backdoor and front-door criteria are sufficient for identification, but they are not the only ways. A causal effect might be identifiable even if neither criterion is met. The rules of do-calculus are complete, meaning they can identify any effect that is identifiable. However, applying them manually can be complex. Fortunately, we can rely on more general graphical criteria to build our intuition.

The video below from Brady Neal's "Causal Inference" course introduces a more general sufficient condition for identifiability and also a necessary one, helping us understand the boundaries of what's possible.

The main takeaway is that identifiability hinges on our ability to isolate the causal flow from treatment to outcome by blocking all confounding pathways. This can be more complex than blocking a single backdoor path, as the set of variables needed for adjustment must work simultaneously without opening new paths.

2. Handling Unmeasured Network Confounding with Double Negative Controls

When confounding is unmeasured, especially in a network, we need more advanced tools. The method of Double Negative Controls (DNC), part of a broader field called Proximal Causal Learning, offers a path forward by using proxies to learn about the unmeasured confounder.

This is particularly relevant for your triadic model, where unobserved factors like "pre-existing group dynamics" or "latent compatibility" can act as network confounders, influencing both the choices people make (treatment) and their eventual satisfaction (outcome). This is the problem of homophily and contextual confounding that plagues social network analysis.

The following image illustrates the core challenge in a networked setting. An unmeasured confounder $U$ can create spurious correlations between individuals. In panel A, it causes homophily (people with similar $U$ connect), and conditioning on the connection $S=1$ creates a confounding path. In panel B, $U$ is a shared context that affects all parties.

The paper "Identification and estimation of causal peer effects using double negative controls" by Naoki Egami provides a rigorous yet applicable framework for this problem. Given your background in econometrics and causal analysis, you will find its logic compelling.

The DNC Logic in a Nutshell

The DNC method hinges on finding two special variables, $W$ and $Z$, that act as proxies for the unmeasured confounder $U$.

1. Negative Control Outcome (W): This variable is a proxy for the outcome. It is assumed to be affected by the unmeasured confounder $U$ but is not affected by the treatment $A$.
2. Negative Control Exposure (Z): This variable is a proxy for the exposure to the confounder. It is related to $U$ but does not directly affect the outcome $Y$ or the NCO $W$.

The relationship between how $U$ affects $W$ and how $U$ affects $Y$ is captured by a confounding bridge function. The NCE, $Z$, provides the information needed to identify this bridge function, effectively allowing you to "adjust" for $U$ without ever measuring it.

Applying DNC to the Vee-Triad Experiment

Let's translate this to your experiment on the vee-triad.

• Ego ($i$): The person whose outcome we are measuring (e.g., the pivot woman).
• Treatment ($A_i$): The ego's exposure under the experimental condition (e.g., their living arrangement, a communication protocol). This is part of the full treatment vector $\mathbf{A}$ for the triad.
• Outcome ($Y_i$): The ego's satisfaction.
• Unmeasured Confounder ($U$): A latent characteristic of the triad, like 'pre-existing emotional cohesion' or 'conflict avoidance tendency'.

To use DNC, we need to select a plausible NCO and NCE based on domain knowledge.

• Potential NCO ($W_i$): We need a variable for the pivot woman ($i$) that is affected by the triad's 'emotional cohesion' ($U$) but not by the specific living arrangement ($A$). A candidate could be a measure of 'sleep quality' or 'frequency of non-relationship-related rumination'.

  • Justification: It's plausible that low underlying cohesion ($U$) might cause stress that degrades sleep quality ($W_i$), regardless of the specific living arrangement ($A$). You must argue that the arrangement itself (e.g., living together vs. apart) does not have a direct causal arrow to sleep quality, except through satisfaction.

• Potential NCE ($Z$): This is where the network structure of the triad is powerful. As Egami's paper suggests, we can use variables from peers. The NCE for the pivot woman could be a variable measured on her partners.

  • Justification: Let's choose one of the partners, $j$. A candidate NCE could be partner $j$'s baseline score on a 'locus of control' personality trait, measured before the experiment begins. The argument would be that this stable trait is correlated with the formation of the triad and its latent cohesion ($U$), but it does not have a direct causal effect on the pivot woman's satisfaction ($Y_i$) or her sleep quality ($W_i$). Its influence is mediated through the system dynamics captured by $U$.

Finding and justifying these negative controls is the most challenging part of this method, but it provides a rigorous path to identification where simpler methods fail. The dissertation by Kao also emphasizes the theoretical importance of accounting for network properties to achieve unconfoundedness, which is precisely what the DNC method helps you do in practice. The "Unconfounded Influence Network Assumption" discussed by Kao is the formal condition that DNCs help satisfy by using proxies for latent network-related attributes.

3. When Identification Fails: An Introduction to Sensitivity Analysis

What if you cannot find plausible negative controls? Or what if, even after applying all these methods, you conclude that your causal effect of interest is simply not identifiable from the available data? This does not mean your research is invalid. The next step is to perform a sensitivity analysis.

Instead of a binary "yes/no" on identification, sensitivity analysis asks:

How strong would an unobserved confounder need to be, in terms of its effect on both the treatment and the outcome, to change my conclusion?

If a very weak confounder could flip your result from significant to non-significant, your finding is considered "sensitive" or fragile. If it would take an implausibly large confounding effect to overturn your result, your finding is "robust."

The image below shows a contour plot of a "bias factor." This is a typical output of a sensitivity analysis. It illustrates how the bias in an estimate changes as a function of the unmeasured confounder's properties (e.g., its mean in the treatment vs. control group, and its effect on the outcome).

We will not go into the calculations today, but it is critical to know that this is the principled approach to handling situations where identification is not possible. We will return to this and learn how to compute and interpret these measures (like E-values) in Module 8.

Conclusion

In this lesson, we have moved beyond simple identification recipes to a more robust framework for assessing and achieving identifiability in the face of unmeasured variables, particularly within your complex triadic system.

Key Takeaways:

• Identifiability can be assessed with general graphical criteria that are more powerful than the basic backdoor and front-door rules.
• When unmeasured confounding is present, especially in networks, Double Negative Controls (DNC) offer a powerful identification strategy by using proxy variables for the confounder.
• The selection of an NCO (unaffected by treatment, affected by confounder) and an NCE (related to the confounder, but not the outcome) is a critical design step that relies on deep domain knowledge.
• If an effect is ultimately non-identifiable, the next step is not to abandon the analysis, but to conduct a sensitivity analysis to quantify the robustness of your findings to potential confounding.

In our next lesson, we will explore another subtle threat to valid causal inference: selection bias. We will specifically examine how Berkson's paradox can arise from conditioning on a collider, such as the triad's joint decision to continue participating in your study, which can induce spurious associations even in a well-designed experiment.