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When treatment assignment is partially missing, estimators that use the missing-at-random assumption are never less efficient—and usually more efficient—than complete-case estimators that discard partial observations, even when the stronger

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MAR-based doubly robust estimators dominate complete-case estimators for CATE and policy value under MCCAR missing treatment data, with proven efficiency gains and empirical validation.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection MAR beats complete-case under MCCAR is a real, non-obvious result, but the abstract oversells it and the key n^{-1/4} nuisance rate is unchecked in the main experiments. the 3 major comments →

arxiv 2607.14346 v1 pith:AMA7LPNI submitted 2026-07-15 cs.LG stat.ME

Learning Who to Treat When Treatment is Missing

Johnna Sundberg , Rayid Ghani , Eli Ben-Michael , Edward Kennedy This is my paper
classification cs.LG stat.ME MSC 62D1062G0562G20
keywords treatment missingnesspolicy learningconditional average treatment effectinfluence functionsmissing at randommissing completely conditionally at randomasymptotic relative efficiencycomplete-case analysis
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks how to learn which individuals to treat when treatment assignment is missing for some units. It extends efficient influence-function estimators for average treatment effects to policy value and conditional average treatment effects, under two missingness assumptions. The central result is that the MAR-based estimator—which borrows partially observed units—is never less efficient than the complete-case MCCAR estimator, and is strictly more efficient except in knife-edge cases. This gives a formal argument against default complete-case analysis. Experiments on synthetic and semi-synthetic data confirm that MAR-based estimators remain competitive under MCCAR and strongly outperform when missingness depends on outcomes.

Core claim

The paper proves that when treatment is missing in a way that depends only on observed covariates (MCCAR), the estimator derived under the weaker missing-at-random (MAR) assumption—which reuses partially observed units through an outcome-dependent treatment propensity—has asymptotic variance no larger than, and usually strictly smaller than, the complete-case estimator. The asymptotic relative efficiency is (1 − g(P)/Var(φ_MCCAR))^{-1}, where g(P) ≥ 0 is the missingness-weighted signal that the complete-case estimator throws away. Hence defaulting to complete-case analysis is wasteful even when the stronger assumptions that justify it hold.

What carries the argument

The central objects are two influence functions—φ_MAR and φ_MCCAR—for the value of a treatment policy under missing treatment data. Influence functions are bias-corrected scores that yield √n-consistent, asymptotically normal estimators when nuisance functions are estimated well. The proof shows Var(φ_MAR) = Var(φ_MCCAR) − g(P), isolating g(P) ≥ 0 as the efficiency gap that grows with the missingness rate, the augmented propensity score, and their correlation.

Load-bearing premise

The efficiency gain requires the treatment-assignment-matching probability γd(X) = P(A=d(X)|X) to be estimated at nearly n^{-1/4} rate; if nuisance estimators converge more slowly in high-dimensional or complex settings, the MAR estimator may not reach √n-consistency and the advantage over complete cases can vanish.

What would settle it

Simulate the synthetic MCCAR setting with known νd and ηd, plug in the true nuisance functions, and compare the empirical variance ratio of the MAR and MCCAR estimators to the formula (1 − g(P)/Var(φ_MCCAR))^{-1}; if the ratio differs by more than simulation error, the variance derivation is wrong.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Under both MAR and MCCAR missingness, the MAR-based estimator is valid; the MCCAR complete-case estimator is only valid under MCCAR.
  • Even when MCCAR holds, the MAR estimator has asymptotic relative efficiency (1 − g(P)/Var(φ_MCCAR))^{-1} ≥ 1, so discarding partial observations is never the efficient choice.
  • The efficiency advantage grows with the missingness rate, the probability that treatment matches the policy, and the correlation between these two quantities.
  • Misspecified missingness mechanisms produce bias that does not disappear with sample size; correctly modeling the missingness mechanism is necessary for unbiased CATE estimates.
  • In semi-synthetic policy experiments, MAR-based estimators maintain positive gains over random assignment with only 10–30% of treatments observed, while complete-case and propensity-based alternatives frequently fail.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A data-driven version of the ARE formula could serve as a pre-analysis check: estimate g(P) and Var(φ_MCCAR) from the sample, and use the predicted efficiency gain to decide whether MAR-based estimation is worth the extra nuisance modeling.
  • The same influence-function decomposition suggests that when γd is hard to estimate, an adaptive estimator that falls back on the complete-case influence function for high-dimensional strata might preserve some of the efficiency gain; the paper does not propose this.
  • For threshold-based budget policies, the asymptotic normality under estimated policies implies that the MAR pseudo-outcomes could be plugged into ranking objectives rather than generic CATE regression; the paper leaves this as future work.
  • The MCCAR-versus-MAR efficiency gap is a special case of a more general principle: discarding data that is only partially informative is wasteful whenever a valid imputation model exists; the same ARE argument may carry over to missing outcomes or missing confounders.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies policy learning when treatment indicators are missing. It proposes two families of estimators for the value of a fixed treatment policy and for the CATE: a complete-case MCCAR estimator based on the outcome regression νd(X) and a MAR estimator based on the nuisance functions λd, π, βd, γd from Kennedy (2020). The paper derives influence functions, proves √n-consistency and asymptotic normality under product-rate conditions (Propositions 3.2 and 3.3), and then proves an asymptotic relative efficiency result (Theorem 3.4) claiming that, under MCCAR, the MAR estimator is never less efficient than the MCCAR estimator, with a closed-form ARE. The methods are instantiated as DR-Learner variants for CATE estimation and evaluated on synthetic, high-dimensional, and semi-synthetic RCT datasets, with code provided.

Significance. If the efficiency theorem holds, the paper makes a practically important point: when treatment missingness is MCCAR, discarding partially observed units (complete-case analysis) is asymptotically suboptimal, and an MAR estimator that uses all units can be strictly more efficient. This is a substantive extension of Kennedy (2020) from ATE to policy value, and the paper supplies detailed von Mises expansions and variance derivations, as well as reproducible code. The empirical section is extensive and includes both synthetic rate studies and two realistic semi-synthetic benchmarks. The main caveats are that the efficiency claim is conditional on nuisance estimation rates and is proven for fixed policies rather than for the learned optimal policy; these caveats are not fully reflected in the abstract and contribution statements.

major comments (3)
  1. [Theorem 3.4 and Section 5] The abstract and Section 1 (contribution 2) state that the MAR estimator is 'both valid and more efficient than the MCCAR estimator when MCCAR assumptions hold.' However, Theorem 3.4 and Proposition 3.3 make the efficiency gain conditional on ∥bγd−γd∥ = o_P(n^{−1/4}) (Proposition 3.3, condition 3). In the high-dimensional experiment (Section 4.1.2), γd is estimated with Random Forests with d=50 covariates and no rate verification is provided. If γd converges slower than n^{−1/4}, bψMAR can carry first-order bias, its variance is not Var(φMAR), and the ARE formula does not apply. The Discussion (Section 5) acknowledges the condition, but the abstract and contribution statements do not. Please qualify the claims and, if the recommendation is to prefer MAR-based estimation broadly, provide empirical evidence or a formal argument that the rate condition is met in the regimes covered by the e
  2. [Section 3.2 and Theorems A.5–A.6] Theorem 3.4 establishes an efficiency comparison for the value of a fixed policy d. The paper's title and Section 3.2, however, frame the contribution as 'learning who to treat.' The asymptotic results for the data-dependent policy bd^∗ (Theorems A.5 and A.6) give only asymptotic normality, not a relative efficiency comparison between the MAR and MCCAR estimators of V(bd^∗). Thus the formal justification for preferring MAR-based estimation in policy learning is incomplete: the proven efficiency gain applies to fixed-policy evaluation, while the learned-policy setting is left without an efficiency statement. Either prove an efficiency result for the estimated policy under a margin condition, or temper the policy-learning claims to fixed-policy evaluation.
  3. [Remark 3.5] The convergence rates of the CATE estimators bτMAR and bτMCCAR are disposed of in Remark 3.5 by citing Kennedy (2020), Nie and Wager (2021), and Foster and Syrgkanis (2023). Since the CATE estimators are a main contribution (Section 3.2 and all of Section 4), the paper should state the precise assumptions under which these rates hold, including the smoothness/sparsity conditions on the CATE function and the rate conditions on the first-stage nuisances. As written, the empirical claims about CATE performance are not tied to a formal theorem, which weakens the theoretical support for the decision-rule part of the paper.
minor comments (6)
  1. [Abstract and Section 1] The phrase 'more efficient than the MCCAR estimator when MCCAR assumptions hold' should be qualified with 'under the regularity conditions of Proposition 3.3.'
  2. [Figure 1 caption] The right panel caption says MAR 'allows for dependence on the observed outcome Y either directly or through unobserved U.' Under the stated MAR assumption R⊥⊥A|(X,Y), dependence through an unobserved U is not automatically allowed unless U is conditionally independent of A given (X,Y); the phrase is confusing and should be clarified or the DAG adjusted.
  3. [Section 4.1.1] The text says the noise ε∼N(−n^{−α}, n^{−2α}) 'precisely control[s] the rate of convergence,' but the reported RMSE curves are median over 500 iterations; it would help to state whether the median is taken of the L2 error or of RMSE, and to give the standard error bands (or at least a mention of variability).
  4. [Appendix C, Algorithm 3] The nested cross-fitting loop is described in words but not with formal notation over the two inner folds; adding equations for bβ(j), bγ(j), and the averaging step would improve reproducibility.
  5. [Various] There are repeated typos: 'V oting' should be 'Voting'; 'R-Positivity' is sometimes written with a dash and sometimes without; 'Kennedy [2020]' is used both as a method and a reference, which can confuse the provenance of the MAR IF.
  6. [Section 2.2, Table 1] The asterisk footnote in Table 1 is helpful but the table layout makes the statement 'MAR is strictly weaker' hard to parse; consider adding a sentence explaining that the R-positivity under MAR is implied by MCCAR's R-ignorability and R-positivity, so the overall MAR assumption remains weaker.

Circularity Check

0 steps flagged

No significant circularity: the efficiency comparison is derived from the variance formulas, not assumed.

full rationale

The derivation chain is self-contained on the points that matter for circularity. The MAR estimand ψMAR=E[βd/γd] is identified from Assumptions 2.1/2.3 in Appendix A.1, and the policy-value influence function φMAR is adapted and its variance is computed in Appendix A.2.2. The remainder bound for bψMAR uses Lemma 1 of Kennedy (2020), a prior theorem by a co-author; that is independent theorem support with stated assumptions, not a definitional identity, and the paper separately derives the policy-value extension and the variance expressions. The central efficiency claim (Theorem 3.4) is proved by deriving Var(φMAR)=Var(ϕMCCAR)−g(P) with g(P)≥0, which is a direct algebraic comparison of the two variance formulas, not a fitted parameter renamed as a prediction. The n^{-1/4} condition on γd is a hypothesis of Proposition 3.3, and the Discussion explicitly acknowledges that the efficiency gain is asymptotic and requires nuisance convergence conditions; the abstract's unconditional phrasing is an overstatement of scope but not a circular step. The experiments in Fig. 2 are simulations that vary λd, π, and their correlation and evaluate the variance formulas; they do not fit the theorem's conclusion. No circular step is present.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claim rests on standard causal and missingness assumptions plus a rate condition on nuisance estimators. No novel entities are introduced; the paper uses standard potential outcomes and influence-function machinery.

axioms (6)
  • domain assumption Standard causal assumptions: consistency, A-ignorability, A-positivity (Assumption 2.1)
    Needed to identify potential outcome means from observed data. Standard in causal inference.
  • domain assumption MCCAR R-ignorability and R-positivity (Assumption 2.2)
    Defines the complete-case estimand and is used in Theorem 3.4 to simplify the MAR variance.
  • domain assumption MAR R-ignorability and R-positivity (Assumption 2.3)
    Defines the MAR estimand and the validity of the MAR estimator.
  • domain assumption Bounded outcome Y (Remark 3.1)
    Used to guarantee finite variance of the estimators.
  • standard math Nuisance functions estimated on independent sample (sample splitting) (Remark 3.1)
    Standard device to avoid empirical process conditions; used in the proofs of Propositions 3.2 and 3.3.
  • domain assumption Margin condition and no-ties assumption for estimated policies (Assumption A.4 and Appendix A.3.1)
    Used only for the secondary results on policy value at the estimated optimal policy, not for the central efficiency theorem.

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of Learning Who to Treat When Treatment is Missing." pith.science (2026). https://pith.science/paper/AMA7LPNI

@misc{pith2026260714346,
  author       = {Pith},
  title        = {Pith review of: Learning Who to Treat When Treatment is Missing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AMA7LPNI}},
  note         = {Machine review of arXiv:2607.14346}
}
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read the original abstract

Policy learning methods are increasingly used to inform treatment allocation under budget constraints. Most proposed methods assume complete treatment data, yet applications frequently suffer from missingness that can bias estimates and lead to suboptimal policies. We address this gap by extending efficient estimators for average treatment effect (ATE) estimation to policy value and conditional average treatment effect (CATE) estimation under missing at random (MAR) and missing completely conditionally at random (MCCAR) treatment data. Through asymptotic efficiency analysis, we prove that the MAR estimator, which leverages partially-observed units, is both valid and more efficient than the MCCAR estimator when MCCAR assumptions hold. This result provides formal justification for preferring MAR-based estimation in policy learning under both missing data settings. Our comprehensive experiments using synthetic and semi-synthetic datasets confirm that correctly specifying the missingness mechanism is crucial: misspecified estimators remain biased regardless of sample size, while our estimators achieve near-oracle performance when assumptions are satisfied. Our work provides practitioners with theoretically grounded, empirically validated tools for robust policy learning in the presence of missing treatment data.

Figures

Figures reproduced from arXiv: 2607.14346 by Edward Kennedy, Eli Ben-Michael, Johnna Sundberg, Rayid Ghani.

Figure 1
Figure 1. Figure 1: Left: Under MCCAR Assumption 2.2, missing [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Asymptotic Relative Efficiency gains of φMAR over ϕMCCAR  ARE(ϕMCCAR, φMAR) := Var(ϕMCCAR) Var(φMAR)  as a function of the average of λd (left), the average of 1 − π (center), and the correlation between λd and 1 − π (right). When not varied, the average values of λd, 1 − π, and their correlation are held fixed at 0.5. ARE increases with the augmented propensity score λd, missingness probability 1 − π, a… view at source ↗
Figure 3
Figure 3. Figure 3: Median RMSE of CATE estimates comparing τbMAR (blue), τbMCCAR (orange), and τbOracle (green) in the MAR (top) and MCCAR (bottom) settings over 500 itera￾tions against the rate of noise (ϵ ∼ N(−n −α, n−2α)) added to all nuisance functions. In the MAR setting, τbMCCAR re￾mains biased even after more treatments are observed, while in the MCCAR setting both estimators are unbiased and τbMAR achieves slightly f… view at source ↗
Figure 5
Figure 5. Figure 5: Average RMSE of the MAR CATE estimator under misspecification, where ˜f denotes a misspecified f. Under Proposition 3.3, the model is robust to misspecifica￾tion of λd or π, but not both λd and π or γd. 4.1.2 High-Dimensional Experiment Our second experiment evaluates performance in a sparse setting with different CATE structure. In addition to oracle models, we compare our estimators to outcome regression… view at source ↗
Figure 6
Figure 6. Figure 6: Average Area-Under-the-Prescriptive-Effect [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Median RMSE of CATE estimates comparing τbMAR (blue), τbMCCAR (orange), and τbOracle (green) in the MAR (a) and MCCAR (b) settings over 500 iterations against the rate of noise added to all nuisance functions. In the MAR setting (top), τbMCCAR remains biased even after more treatments are observed, while in the MCCAR setting (bottom) both estimators are unbiased and τbMAR achieves slightly faster convergen… view at source ↗
Figure 8
Figure 8. Figure 8: Average AUPEC of CATE estimators (zero is no gain over a random policy), averaged over 10 seeds (STAR) and [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: AUPEC with 95% confidence intervals, where zero indicates no gain over random policy. Results show average [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Example policy values across budgets and treatment observation rates. At low treatment observation rates, [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Example policy values over different budgets and treatment observation rates. In the MCCAR setting, all methods [PITH_FULL_IMAGE:figures/full_fig_p032_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Example policy values across budgets and treatment observation rates. Even at low treatment observation rates, [PITH_FULL_IMAGE:figures/full_fig_p033_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Example policy values over different budgets and treatment observation rates. At treatment observation rates [PITH_FULL_IMAGE:figures/full_fig_p034_13.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 2, 2026.