Orthogonal machine learning for conditional odds and risk ratios

1. Problem Statement

Conditional treatment effects are crucial for precision medicine, allowing researchers to understand how treatment efficacy varies across different subgroups defined by covariates ($X$). While the Conditional Average Treatment Effect (CATE) on the additive scale (Risk Difference) has been extensively studied using modern machine learning (ML) techniques, estimation on multiplicative scales—specifically the Conditional Odds Ratio (OR) and Conditional Risk Ratio (RR)—lags behind.

• Limitations of Current Methods: Traditional approaches rely on parametric models (e.g., logistic regression). These are prone to bias under model misspecification, a common issue in complex real-world data.
• Gap in Orthogonal Methods: State-of-the-art "orthogonal" estimators (like the DR-learner and R-learner), which use efficient influence functions (EIFs) to create pseudo-outcomes robust to nuisance parameter estimation errors, have primarily been developed for the additive scale (CATE). They have not been generalized to ratio-based parameters (OR and RR).
• The Challenge: Directly plugging in estimated probabilities into OR/RR formulas (plug-in estimators) can lead to high variance and bias if the underlying outcome models are misspecified or if the target parameter (the ratio) is smooth while the nuisance parameters (probabilities) are not.

2. Methodology

The authors propose a framework to extend Orthogonal Statistical Learning to conditional OR and RR. The core idea is to derive orthogonal pseudo-outcomes based on the Efficient Influence Functions (EIFs) of the target parameters, ensuring that the estimation error depends on nuisance parameters only at a second order.

A. Theoretical Foundations

The authors define the target parameters:

• $OR(X) = \frac{p_1(X)/(1-p_1(X))}{p_0(X)/(1-p_0(X))}$
• $RR(X) = \frac{p_1(X)}{p_0(X)}$
  where $p_t(X) = P(Y=1|T=t, X)$.

They derive uncentered EIFs for $E[OR(X)]$, $E[\log OR(X)]$, $E[RR(X)]$, and $E[\log RR(X)]$. These EIFs serve as the basis for constructing pseudo-outcomes ($\varphi$) that satisfy the Neyman orthogonality condition.

B. Proposed Estimators

The paper introduces two main families of estimators for each target parameter:

1. DR-Learners (Double Robust):

   • Mechanism: Instead of estimating $p_1(X)$ and $p_0(X)$ separately and plugging them in (DR-P), the DR-learner constructs a pseudo-outcome $\hat{\varphi}(Z)$ using the EIF derived from the nuisance estimates ($\hat{p}_1, \hat{p}_0, \hat{e}$).
   • Second Stage: A flexible machine learning model (e.g., SuperLearner) regresses the pseudo-outcome $\hat{\varphi}(Z)$ directly on the covariates $X$ to estimate the conditional effect.
   • Variants:
     • DR-OR / DR-LOR: For Odds Ratio (and log-OR).
     • DR-RR / DR-LRR: For Risk Ratio (and log-RR).
   • Theoretical Guarantee: The authors prove that the conditional expectation of these pseudo-outcomes equals the true parameter up to a second-order remainder term involving the product of nuisance estimation errors. This ensures that if the nuisance models converge at a rate faster than $n^{-1/4}$, the final estimator achieves oracle efficiency.

2. R-Learners (Residual-based):

   • Mechanism: Similar to the DR-learner but uses a propensity-overlap weighted loss function in the second stage. The observations are reweighted by $w(X) = \hat{e}(X)(1-\hat{e}(X))$.
   • Variants: R-OR, R-LOR, R-RR, R-LRR.
   • Distinction: While theoretically related, the weighting can alter the bias-variance trade-off, particularly in regions of low overlap.

C. Simulation Framework

To evaluate performance, the authors adopted a universal Monte Carlo simulation approach (following Rudolph et al., 2023):

• Data Generating Processes (DGPs): 600 distinct DGPs were generated using a Bayesian-inspired approach with random hyperparameters (interaction orders, nonlinearity, confounding bias).
• Metrics: Integrated Mean Squared Error (iMSE), Integrated Squared Bias (iBias2), and Integrated Variance (iVariance).
• Comparators: The proposed DR/R-learners were compared against:
  • Parametric: Logistic Regression (LR, LR-T).
  • Flexible Plug-in: SuperLearner (SL, SL-T).
  • Existing Orthogonal: DR-P (plug-in using DR-learner probabilities).

3. Key Contributions

1. Novel Estimators: First derivation of orthogonal pseudo-outcomes and corresponding DR-learner/R-learner estimators specifically for conditional OR and RR.
2. Theoretical Proofs: Rigorous proof that these new pseudo-outcomes satisfy second-order conditional-mean identities and that the associated loss functions satisfy Neyman orthogonality, guaranteeing robustness to first-stage nuisance estimation errors.
3. Comprehensive Empirical Evidence: A large-scale simulation study covering hundreds of data-generating mechanisms, providing the first systematic comparison of orthogonal methods for ratio-scale parameters.
4. Real-World Application: Demonstration using NHANES data to analyze the effect of physical activity on sleep trouble, showing how these methods uncover heterogeneity obscured by traditional regression.

4. Key Results

Simulation Findings

• Complex, Data-Rich Settings (High Heterogeneity, Large $n$):
  • The proposed DR-learners (specifically DR-LOR and DR-LRR) significantly outperformed traditional logistic regression and often outperformed SuperLearner in terms of median iMSE.
  • Bias-Variance Trade-off: Direct-targeting DR-learners (DR-LOR/DR-LRR) generally achieved lower variance than DR-P (plug-in), while maintaining low bias.
  • R-Learners: Consistently exhibited high variance, making them less attractive than DR-learners in these settings.
• Simple or Data-Limited Settings (Low Heterogeneity, Small $n$):
  • SuperLearner (SL) and Logistic Regression (LR) often performed best.
  • In small samples ($n=200$), the variance penalty of the orthogonal methods (DR-learners) outweighed their bias-reduction benefits. The "oracle" advantage of orthogonalization only materialized with sufficient sample size.
• Scale Sensitivity: Estimators on the log scale (DR-LOR, DR-LRR) generally performed better than those on the raw ratio scale, likely due to the stabilization of the variance of ratios.

Real-World Application (NHANES)

• Context: Analyzing the effect of physical activity on sleep trouble.
• Finding: Traditional logistic regression suggested a constant (homogeneous) effect. In contrast, the flexible DR-learners revealed substantial heterogeneity.
• Impact: The treatment rules derived from DR-learners led to significantly better counterfactual outcomes (lower mean sleep trouble) compared to rules derived from logistic regression or "treat-all" strategies. This highlights the potential for improved clinical decision-making using data-adaptive methods.

5. Significance and Practical Recommendations

This work bridges a critical gap in causal inference, enabling the application of robust, double-robust machine learning techniques to ratio-based effect measures, which are often preferred in epidemiology and clinical research for their interpretability and invariance properties.

Practical Guidelines derived from the study:

• Complex Scenarios (High Heterogeneity, Large $N$): Use DR-learner estimators (specifically the log-scale variants: DR-LOR, DR-LRR). They offer the best balance of bias reduction and variance control. Avoid R-learners due to high variance.
• Simple or Small Sample Scenarios: Use SuperLearner (SL) or Logistic Regression (LR). The complexity of orthogonal methods is not justified when the signal is simple or data is scarce; simpler models often yield lower MSE due to lower variance.
• Uncertain Settings: SuperLearner serves as a robust default choice.

Conclusion: The paper demonstrates that while parametric models are sufficient for simple problems, the proposed orthogonal nonparametric estimators are essential for uncovering complex treatment effect heterogeneity in ratio scales, leading to more precise and effective personalized interventions.