A proxy-based approach for unmeasured confounding in electronic health records research

1. Problem Statement

Electronic Health Records (EHR) are a primary resource for observational clinical research, but they suffer significantly from unmeasured confounding. While clinicians record many variables (e.g., vital signs, triage acuity) that reflect a patient's underlying health status, these are often treated merely as measured confounders or ignored.

• The Challenge: Standard causal inference methods (like Inverse Probability Weighting or Instrumental Variables) often rely on assumptions that are difficult to verify or hold in practice.
• The Gap: Existing proxy-based methods, such as Proximal Causal Inference (PCI), require specific data structures (Negative Control Exposures and Outcomes) and involve solving complex, ill-posed integral equations (bridge functions), making them difficult to implement and prone to instability.
• The Goal: To develop a practical, robust, and easily implementable method to use a vector of proxy variables to adjust for unmeasured confounding in EHR data without requiring the strict structural assumptions of PCI.

2. Methodology

The authors propose a two-step framework that leverages Factor Analysis to construct covariates representing the unmeasured confounder, which are then used in a standard regression model.

A. Causal Framework

The authors define a Non-Parametric Structural Equation Model (NPSEM) where:

• $X$: Measured confounders.
• $U$: Unmeasured confounders (latent variables).
• $Z$: Proxy variables (observed measurements influenced by $U$).
• $A$: Treatment.
• $Y$: Outcome.
  Crucially, the method relaxes the strict requirement that proxies must be independent of treatment and outcome given $U$ (as required by simple adjustment). It allows for scenarios where proxies might be influenced by treatment or outcome, provided specific independence conditions in the potential outcomes framework hold.

B. Identification Strategy

The core theoretical contribution is a two-step identification strategy based on two assumptions:

1. Recoverability: There exists a function $g(U, X)$ such that its conditional expectation $E[g(U, X) | Z, A, X]$ can be identified from observed data.
2. Separability: The potential outcome $E[Y(a) | U, X]$ can be written as $\tau(a, X)'g(U, X)$.

Theorem 3.1 establishes that if these assumptions hold, the Conditional Average Treatment Effect (CATE) can be identified as:
$$\beta(Z, X) = (\tau(a_1, X) - \tau(a_0, X))' E[g(U, X) | Z, X]$$

C. Practical Implementation (The "Factor-Regression" Approach)

The authors propose a concrete parametric implementation using the Multiple Indicators Multiple Causes (MIMIC) model:

1. Step 1: Latent Variable Estimation (Factor Analysis):
   • Fit a MIMIC model to the observed proxies $Z$, treatment $A$, and covariates $X$.
   • This model assumes $Z$ are noisy measurements of the latent factor $U$.
   • The model estimates the conditional expectation $\hat{E}[U | Z, A, X]$ for each patient.
2. Step 2: Outcome Regression:
   • Construct a working model for the outcome: $E[Y | Z, A, X] = \nu_0 + \nu_1' \hat{E}[U | Z, A, X] + \nu_2' \hat{E}[U | Z, A, X]A + \nu_3 A$.
   • Perform linear regression of $Y$ on $A$, $X$, and the estimated latent scores $\hat{E}[U | Z, A, X]$.
3. Step 3: CATE Calculation:
   • Compute the treatment effect using the recovered coefficients and the estimated latent scores for the counterfactual treatment state.

This approach avoids solving integral equations, relying instead on standard factor analysis (e.g., via R's lavaan or MPlus) and linear regression.

3. Key Contributions

1. New Identification Route: The paper introduces a method that treats a vector of proxies as a single source of information for a latent confounder, rather than requiring the specific NCE/NCO pairing mandated by PCI.
2. Robustness to Misspecification: The authors demonstrate that their method is surprisingly robust to violations of standard assumptions, including:
   • Non-linear outcome models (e.g., quadratic in $U$) when a linear model is used.
   • Non-normal distributions of the latent variable (e.g., skew-normal).
   • Binary treatments estimated as continuous.
3. Handling Structural Ambiguity: The method remains valid even if the causal role of a proxy is ambiguous (e.g., if a variable is a direct confounder rather than a proxy, or an instrument), provided the model is not severely misspecified.
4. Practical Accessibility: By utilizing familiar tools (Factor Analysis + Regression), the method lowers the barrier to entry for applied researchers compared to the complex mathematical requirements of PCI.

4. Results

Simulation Studies

The authors compared their method against Inverse Probability Weighting (IPW), standard linear regression, Instrumental Variables (IV), and Proximal Causal Inference (PCI).

• Accuracy: Under correct model specifications, the proposed method outperformed IPW and standard linear regression. It rivaled PCI in accuracy but with lower variance.
• Proxy-to-Latent Ratio: Accuracy improved significantly when the ratio of proxies ($p$) to latent dimensions ($k$) was $\geq 3$, aligning with factor analysis requirements.
• Misspecification Robustness:
  • When the true outcome was quadratic but modeled as linear, the proposed method remained nearly unbiased, while competitors showed significant bias.
  • When the latent variable was skew-normal (violating normality assumptions), the method maintained lower bias than most competitors.
  • When a direct confounder was incorrectly treated as a proxy, the method produced estimates with the correct sign and order of magnitude, though slightly biased.
• Coverage: Bootstrap confidence intervals achieved coverage rates near 95% (0.94–0.97) across various scenarios.

Case Study: Chest Pain in Older Adults

The method was applied to MIMIC-IV data (7,081 ED visits) to estimate the effect of hospital admission vs. discharge on 30-day readmission/mortality for patients aged 65+ with chest pain.

• Context: Admission poses risks (deconditioning, infection), but discharge risks missing serious conditions. Unmeasured confounding (severity of illness) is a major concern.
• Comparison Results:
  • Unadjusted/IPW/Linear: All suggested a positive effect of admission (increased risk of readmission/death), with estimates ranging from 0.074 to 0.648.
  • PCI: Produced a range of positive estimates (median 0.089).
  • Proposed Method: Estimated a negative effect ($-0.021$), suggesting hospital admission might reduce the risk of readmission.
• Interpretation: The negative estimate aligns better with clinical expectations and prior literature, suggesting that the other methods were likely biased by unmeasured confounding (sicker patients were admitted, inflating the apparent risk of admission).

5. Significance

This work provides a practical, robust, and accessible alternative for addressing unmeasured confounding in EHR research.

• Bridging Theory and Practice: It moves proxy-based causal inference from theoretical complexity (solving bridge functions) to practical application using standard statistical software.
• Clinical Impact: The case study demonstrates that ignoring unmeasured confounding or using less robust methods can lead to clinically misleading conclusions (e.g., concluding admission is harmful when it may be beneficial).
• Future Direction: The authors suggest this approach could encourage broader adoption of proxy-based methods in healthcare research, provided practitioners can correctly identify proxy variables. Future work aims to automate the distinction between proxies and confounders.

Limitations: The method assumes a specific parametric form for the latent variable (normality) and linearity in the outcome model (though robust to some deviations). It also relies on bootstrapping for confidence intervals, which can be computationally intensive and conservative. Validation via Randomized Controlled Trials (RCTs) is still necessary for definitive clinical validation.