Index: Treatment Effect Estimation

Treatment Effect Estimation

Routing Summary

This folder covers metalearner algorithms for estimating heterogeneous treatment effects (CATE) using machine learning base learners. Based on Künzel, Sekhon, Bickel & Yu (PNAS 2019). Contains 6 notes.

• Need the general CATE estimation framework and notation? → Metalearners for CATE
• Need the single-model approach (CATE = predicted difference from one model)? → S-Learner
• Need the two-model approach + minimax rate theorem? → T-Learner and Minimax Rate
• Need the main contribution: X-learner for unbalanced groups? → X-Learner
• Need empirical results (voter turnout, transphobia)? → Metalearner Simulation Results

Concept Map

Concept	Note	Type	Depends On	Key Result
Metalearner framework	Metalearners for CATE	definition	Causal Estimands	${\hat{\tau }}^S$, ${\hat{\tau }}^T$, ${\hat{\tau }}^X$ as wrappers around base ML
S-learner	S-Learner	concept	Metalearners for CATE	Single model; treatment may be regularized to zero
T-learner	T-Learner and Minimax Rate	theorem	Metalearners for CATE	Rate: $\min (m,n{)}^{-a_0}$; optimal when balanced
X-learner	X-Learner	theorem	T-Learner and Minimax Rate	Rate: $m^{-a_{\tau }}+n^{-a_0}$; optimal for unbalanced groups
Empirical validation	Metalearner Simulation Results	example	X-Learner	X-RF best on voter turnout (38K treated, 191K control)

Concept Dependency Chain

Potential Outcomes Framework + Causal Estimands (existing)
  └─► Metalearners for CATE (framework)
        ├── S-Learner (single model; simple but regularization risk)
        ├── T-Learner (separate models; limited by small group)
        │     └─► Theorem 1: minimax rate = min(m,n)^{-a_0}
        └── X-Learner (cross-imputation; exploits large group)
              └─► Theorem 2: rate = m^{-a_τ} + n^{-a_0} (adapts to CATE smoothness)
                    └─► Metalearner Simulation Results (voter turnout, transphobia)

Notes

• Künzel 2019 - Overview — CONTAINS: paper overview, PNAS 2019; metalearner concept, key results summary
• Metalearners for CATE — CONTAINS: potential outcomes notation; EMSE definition; families $S(a)$; three metalearner comparison table
• S-Learner — CONTAINS: definition ${\hat{\tau }}^S=\hat{\mu }(x,1)-\hat{\mu }(x,0)$; regularization failure mode; when S-learner works
• T-Learner and Minimax Rate — CONTAINS: definition of two separate models; Theorem 1 (minimax rate $n^{-a_0}$); unbalanced failure mode
• X-Learner — CONTAINS: full 3-step algorithm; imputed ITEs ${\tilde{D}}_i^1$, ${\tilde{D}}_i^0$; propensity score weighting; Theorem 2 (adaptive rate)
• Metalearner Simulation Results — CONTAINS: simulation study; voter turnout application (ATE = 8.1%, unbalanced); transphobia canvassing (ATE = 0.22)

Sources

• Künzel et al. - 2017 - Metalearners for estimating heterogeneous treatment effects using machine learning.pdf — Künzel SR, Sekhon JS, Bickel PJ, Yu B. 2019. PNAS 116(10): 4156–4165.

Cross-Links to Existing Vault Notes

• Causal Estimands — CATE is the target quantity throughout (ITE, SATE, CATE, PATE defined)
• Potential Outcomes Framework — foundational setup (potential outcomes, ignorability)
• Propensity Score in Bayesian CI — propensity score $e(x)$ used as weight in X-learner
• Nonparametric Causal Inference — BART is a common base learner for metalearners
• The Experimental Ideal — randomization justifies metalearner assumptions (ignorability)