Causal Stability Selection

Technical Summary: Causal Stability Selection

Problem Formulation
The central problem addressed is the identification of effect modifiers—covariates that explain heterogeneity in treatment effects. While many studies report null average treatment effects (ATE), treatment responses often vary systematically with pre-treatment covariates ($X$). Formally, effect modifiers are defined via the Conditional Average Treatment Effect (CATE), $\tau(x) = E[Y(1) - Y(0) | X=x]$. A covariate $X_j$ is an effect modifier if $\tau(X) \not\perp\!\!\perp X_j | X_{-j}$.

Existing data-adaptive procedures for variable selection often fail to provide finite-sample control over the expected number of false discoveries. As illustrated in the paper's motivating simulation, standard approaches like honest splitting or K-fold cross-fitting combined with Lasso or stability selection often result in severely inflated false discovery rates (FDR), undermining the reliability of downstream targeted interventions. The challenge is to discover a parsimonious set of genuine effect modifiers while maintaining rigorous error rate guarantees in finite samples, despite the fact that the true CATE $\tau(X)$ is unobserved and must be estimated.

Methodology: Causal Stability Selection (CausalStabSel)
The authors propose Causal Stability Selection, an algorithm that integrates cross-fitted CATE estimation with Integrated Path Stability Selection (IPSS). The method operates in two primary stages:

1. Cross-Fitted Estimation of Selection Probabilities:
   Unlike standard stability selection which assumes the outcome is directly observed, CausalStabSel acknowledges that the "response" for variable selection is the unobserved CATE. The algorithm proceeds via $B$ iterations:

   • In each iteration, the data is split into two disjoint subsets, $A$ and $A^c$.
   • A CATE estimator $\hat{\tau}$ is trained on the held-out set $A^c$.
   • The trained model predicts CATE values for the samples in $A$.
   • A base variable selector $\hat{S}_\lambda$ (e.g., Lasso or XGBoost importance scores) is applied to the pairs $(x_i, \hat{\tau}(x_i))$ for $i \in A$.
   • This process is repeated with roles reversed ($A^c$ for training, $A$ for selection) to generate $2B$ subsamples.
   • The selection probability $\hat{\pi}_{j,m}(\lambda)$ for covariate $j$ is calculated as the proportion of times $j$ is selected across all $2B$ subsamples.

2. Construction of the Selection Set:
   The final set of effect modifiers is constructed using the IPSS criterion, which integrates the selection probabilities over a parameter grid $\Lambda$ rather than simply maximizing them. A variable $j$ is selected if:
   $$\int_{\Lambda} f(\hat{\pi}_{j,m}(\lambda)) \mu(d\lambda) \geq \gamma$$
   where $f(x) = (2x-1)^3 \mathbb{1}(x \geq 1/2)$ and $\mu$ is a probability measure. This approach yields a tighter upper bound on the expected number of false positives (EFP) compared to traditional stability selection. The threshold is determined by a target EFP level $t$ or a nominal FDR level $\alpha$.

Key Contributions
The paper makes three primary contributions:

1. Unified Framework with Finite-Sample Guarantees: The authors provide the first framework for effect modifier discovery that offers a non-asymptotic, finite-sample upper bound on the expected number of false positives. This guarantee holds without restricting the form of the base selector or the CATE estimator.
2. Theoretical Connection between Estimation and Discovery: The paper proves that the estimated selection probabilities converge to their "oracle" counterparts (where true $\tau$ is known) at a rate governed directly by the convergence rate of the underlying CATE estimator. Specifically, if the CATE estimator achieves a parametric rate ($n^{-1/2}$), the selection probabilities are asymptotically unbiased.
3. Computational Tractability and Stability: By aggregating results across repeated subsamples, the method yields stable discoveries even in high-dimensional settings, remaining computationally feasible while controlling false discoveries.

Results
The method was evaluated through extensive simulation studies and two real-world applications:

• Simulations: Across 34 experiments covering linear and nonlinear settings with varying sample sizes, dimensions, and confounding structures, CausalStabSel was the only method that maintained empirical FDR at or below the nominal level uniformly across all regimes. Competing methods, including IPSS, Knockoffs, and Benjamini-Hochberg, exhibited anti-conservative behavior (FDR significantly exceeding nominal levels) when applied to estimated CATEs. CausalStabSel achieved high True Positive Rates (TPR) comparable to the best-performing alternatives without sacrificing power.
• Real-World Applications:
  • PRIME Randomized Trial (Oncology): Applied to a trial of anti-EGFR therapy in metastatic colorectal cancer. CausalStabSel identified KRAS mutation status (BMMTR1, BMMTR3) as effect modifiers, consistent with known clinical literature, alongside platelets, age, and lactate dehydrogenase.
  • Maternal Smoking and Birthweight (Observational): Applied to Pennsylvania birth records. The method identified maternal age and first-born status as effect modifiers, consistent with epidemiological findings regarding smoking-induced birthweight reduction.
  • In both cases, held-out inference using BLP-on-CATE specifications provided corroboration for the selected modifiers.

Significance and Claims
The paper claims that CausalStabSel establishes a direct, quantitative link between treatment effect estimation and effect modifier discovery. By demonstrating that the bias in selection probabilities is controlled by the CATE estimation error, the authors show that improvements in causal inference directly translate to more reliable variable selection.

The significance of the work lies in its ability to solve the "false discovery" problem inherent in current data-adaptive causal inference. The authors emphasize that without finite-sample control, spurious findings risk failing to replicate, which is particularly dangerous in precision medicine and policy targeting. CausalStabSel provides a robust, theoretically grounded solution that accommodates arbitrary treatment effect estimators and base selectors, making it applicable to both randomized trials and observational studies under standard identifying assumptions (SUTVA and strong ignorability).