{"id":"e97d9c8c-a872-4f61-a377-d049329a6de9","arxiv_id":"1908.10448","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"New projection-based estimators for optimal joint testing and treatment regimes leverage the no-direct-effect-of-testing assumption to achieve lower asymptotic variance than standard opt-SNMM g-estimators while retaining double robustness.","lead":"This paper derives new estimators for optimal testing-and-treatment regimes that exploit the fact that a diagnostic test usually changes health only through the treatment it triggers. The estimators have smaller statistical uncertainty than previous methods, which matters for deciding whether expensive tests are worth their cost.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Near-optimality for continuous outcomes rests on an unproved convergence assertion (Remark 4); the recommended finite-basis estimator is only proven to improve on g-estimation, not to attain the claimed efficiency bound.","rationale":"The reader's weakest_assumption is the NDE(Yd) condition itself. That is a substantive limitation and the paper is explicit that all gains disappear if testing has a direct effect on the health outcome. I do not dispute this, but it is a stated premise rather than an internal gap: under the assumption, Theorem 3's proof and variance reduction argument appear sound. The more load-bearing internal problem is Remark 4, where the recommended continuous-outcome estimator's near-optimality is asserted with 'should' but not proved. The practical estimator \\hat\\Psi(q,bsub) is the one the paper recommends, and without a proof that \\cup_\\xi \\Omega_\\xi is dense in \\Lambda^\\perp_{NDE} (and suitable rates for \\xi(n) preserving RALness), the claim that its efficiency can be made arbitrarily close to that of the intractable optimal estimator is unverified. This does not invalidate the core Theorem 3 result, so the existing CONDITIONAL verdict remains appropriate: accept only after the convergence claim is proved or explicitly downgraded to a conjecture. The reader's rationale mentions Remark 4, so my concern partially overlaps, but the reader's formal weakest_assumption is the NDE premise, which I would not rank as the single most load-bearing issue for the paper's practical recommendation.","tokens_in":62577,"tokens_out":14063,"duration_ms":151204,"concrete_test":"Use the Section 9.1 DGP (K=1, continuous Y) with the same saturated opt-SNMM. For \\xi = 6, 12, 24, 48, compute the oracle variance V_{oracle}(q,bsub) via the closed-form projection in Corollary 1 with true (or very large-sample-estimated) nuisance functions, and compare it with a numerical solution of the integral equations in Appendix A.3 for \\Pi[U(q,\\Psi^*)|\\Lambda^\\perp_{NDE}]. If the variance does not decrease monotonically to the optimal value as \\xi grows, or if the gap remains bounded away from zero, Remark 4's convergence claim is false or at least unsubstantiated, and the near-optimality assertion should be withdrawn.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5.2 and Remark 4. For continuous Y, the recommended estimator \\hat\\Psi(q,bsub) projects onto the finite-basis subspace \\Omega = \\sum_t \\Gamma_t with T_t = T_{b*_t}. The paper asserts (Remark 4) that as \\xi\\to\\infty, \\Pi[U|\\Omega] 'should converge' to \\Pi[U|\\Lambda^\\perp_{NDE}] and that choosing \\xi\\to\\infty slowly with n makes \\hat\\Psi(q,bsub) and \\hat\\Psi(q,bopt) 'asymptotically equivalent.' No proof is given, no rates for \\xi(n) are supplied, and no conditions are stated under which the map b \\mapsto T_b preserves L2 density. The claimed near-optimality of the practical estimator depends exactly on this unproved step. If the closure of \\cup_\\xi \\Omega_\\xi is a proper subspace of \\Lambda^\\perp_{NDE}—for example, because the conditional expectations in T_b smooth out high-frequency components of b—then the asymptotic variance of \\hat\\Psi(q,bsub) has a strictly positive gap from V_{oracle}(q,bopt). The paper's central practical message that relative efficiency can be made arbitrarily close to optimal would then be unsupported, even though Theorem 3's fixed-b variance reduction over standard g-estimation would remain valid. The text itself flags the gap with the word 'should', so this is an omitted proof, not a merely technical detail.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops semiparametric estimators of the parameters of an optimal regime structural nested mean model (opt-SNMM) under the assumption that a diagnostic test has no direct effect on the health outcome except through treatment choice (the NDE assumption). The main construction subtracts from a standard g-estimating function its projection onto the ortho-complement of the tangent space of the NDE model, thereby producing estimating equations with smaller asymptotic variance. The paper characterizes that ortho-complement (Theorem 1), proves a variance-reduction result for any user-specified projection subspace (Theorem 3), and provides a closed-form projection formula for a finite-dimensional subspace spanned by treatment-history indicators and basis functions of the outcome (Theorem 4 and Corollary 1). For discrete outcomes the subspace can be taken to be the full ortho-complement, while for continuous outcomes the authors propose a finite-basis subspace and assert in Remark 4 that the resulting estimator is asymptotically equivalent to the optimal estimator as the basis dimension grows. The paper also develops a cross-fitted doubly robust feasible estimator and studies the NDE-IPW estimator as a special case, with simulations illustrating large efficiency gains in an HIV monitoring setting.","tokens_in":62905,"tokens_out":4513,"duration_ms":52360,"significance":"If the central results stand, the paper makes a substantive contribution to causal inference for dynamic treatment and testing regimes. Theorem 3 gives a clean projection argument that, for fixed b, any estimator solving the projected estimating equation is RAL and dominates the standard opt-SNMM g-estimator whenever the projection is nonzero, while retaining double robustness. Theorem 1 and Theorem 4 are proved in sufficient detail and provide a workable route to constructing the projection, including a closed form for discrete outcomes. The explicit connection to NDE-IPW estimation and the simulation studies, which show large efficiency gains and improved regime selection, are valuable and are based on reproducible code in the supplementary materials. The main weakness is that the recommended near-optimal estimator for continuous outcomes relies on an unproved convergence assertion in Remark 4; as written, the paper proves improved efficiency over g-estimation for any finite basis but does not prove the advertised approach to the efficiency bound.","major_comments":[{"comment":"The claim that \\hat\\Psi(q,bsub) is asymptotically equivalent to \\hat\\Psi(q,bopt) for continuous outcomes is supported only by the sentence that the projection 'should converge' to \\Pi[U|\\Lambda^\\perp_NDE]. No theorem establishes L2(P)-density of \\cup_\\xi \\Omega_\\xi in \\Lambda^\\perp_NDE, no rates for \\xi(n) are given, and no conditions are stated under which the map b \\mapsto T_b preserves L2 density. This statement is load-bearing: it is the basis for recommending \\hat\\Psi(q,bsub) as a near-optimal estimator. If the closure of \\cup_\\xi \\Omega_\\xi is a proper subspace of \\Lambda^\\perp_NDE, the asymptotic variance of \\hat\\Psi(q,bsub) has a strictly positive gap from V_oracle(q,bopt), and the paper's central practical message that relative efficiency can be made arbitrarily close to optimal would be unsupported. The fixed-b variance reduction of Theorem 3 would remain valid, but the near-optimality claim requires a proof or explicit sufficient conditions.","section":"Sec. 5.2, Remark 4"},{"comment":"The entire projection construction uses that E[T_b]=0, which follows from Eq. (8), a consequence of the NDE(Yd) assumption. The manuscript acknowledges in Section 7 that NDE(Yd) can fail in realistic settings (for example, through ancillary care), but it does not quantify the bias of the adjusted estimators under such violations or provide a sensitivity analysis. This is not an internal inconsistency, but it is a substantive limitation of the practical recommendation: the efficiency gains are conditional on an untestable assumption, and the paper would be strengthened by an explicit statement of the resulting bias-variance trade-off or a small sensitivity analysis.","section":"Sec. 4, Eq. (8) and Sec. 7"},{"comment":"The feasible estimator is shown to be RAL under high-level conditions E[\\hat U(q,\\Psi^*)|Nu] = op(n^{-1/2}) and \\sum_t E[\\hat T_{b,t}|Nu] = op(n^{-1/2}). These conditions are stated as sufficient and are standard in the double/debiased machine learning literature, but for the recommended near-optimal estimator with estimated bsub they are combined with the unproved density claim of Remark 4. The paper should make explicit that the practical guarantee for \\hat\\Psi(q,\\hat bsub) is therefore only the fixed-b variance reduction of Theorem 3 unless Remark 4 is upgraded to a theorem.","section":"Sec. 6, Theorem 5"}],"minor_comments":[{"comment":"The sentence 'We let \\bar H_m be the sample space of the random vector \\bar H_m' uses the same symbol for the random vector and its sample space; this is confusing and should be rephrased, for example by using a script or calligraphic letter for the sample space.","section":"Sec. 2, paragraph after notation"},{"comment":"Remark 4 refers to \\phi(Y) while Corollary 1 defines b*_t in terms of \\phi(Y_d). Since the total utility Y includes the known testing cost and the NDE assumption concerns Y_d, the paper should clarify which variable is used in the basis functions and why the cost-adjusted version is appropriate or not.","section":"Sec. 5.2, Remark 4"},{"comment":"The 50-fold efficiency gain attributed to Caniglia et al. [3] concerns an NDE-IPW estimator in a dyn-MSM analysis, not the opt-SNMM estimators developed here; the text should make this distinction explicit to avoid overstating the simulation evidence for the proposed estimators.","section":"Introduction, Section 1"},{"comment":"The paper excludes exceptional laws but does not define them in the main text; a one-sentence definition or a more precise pointer to Robins [27] would make the exclusion self-contained.","section":"Sec. 3.2, paragraph after Eq. (4)"},{"comment":"Theorem 3 assumes \\hat\\Psi(q,b) is RAL rather than stating conditions under which the estimator solving 0 = \\hat U(q,b,\\Psi) is RAL; the regularity conditions in Section 6 and Appendix A.5 are stated for the cross-fitted version, and it would help the reader if the theorem explicitly noted that these conditions are being assumed.","section":"Sec. 5.1, Theorem 3"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong semiparametric contribution with correct-looking projection algebra and useful closed-form results for discrete outcomes. The main issue is Remark 4: the near-optimality of the recommended continuous-outcome estimator is asserted without proof, and this is exactly the part that supports the paper's strongest practical claim. I would not reject, because the missing convergence proof seems plausibly within reach and the fixed-b variance reduction remains valid. The NDE assumption is a substantive modeling choice, not an internal flaw, but the paper would benefit from a sensitivity discussion."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, what to know: this paper earns its keep on Theorem 3. The idea is simple and clever: under the no direct effect assumption, the estimating function from opt-SNMM g-estimation can be projected onto a space of mean-zero variables implied by NDE, and the leftover estimating function is more efficient. That theorem is proved, and the algebra is sound. The simulation results back it up, and the authors ship R code. The NDE-IPW estimator from Robins et al. 2008 is contextualized nicely; the paper clearly shows what is new: no one had used the NDE assumption inside opt-SNMM estimation via influence-function projection before. For discrete outcomes, the closed-form projection in Theorem 4 is a genuine contribution, and the double robustness preservation is real. The HIV example and the value-of-information framing are useful for decision-science readers, not just methodologists.\n\nNow the soft spots, in proportion. The main one is Remark 4, and the stress-test note lands. For continuous Y, the practical estimator projects onto a finite-dimensional basis space of dimension xi. The paper asserts, without proof, that as xi grows, the projection converges to the projection onto the full ortho-complement, and that xi can be chosen to grow slowly with n so the estimators become asymptotically equivalent. The text literally says 'should converge' and 'should converge' again. That is an omitted proof, not a technical aside. If the closure of the union of these finite-dimensional subspaces is a proper subspace of the ortho-complement—say, because the conditional expectations in T_b smooth out high-frequency components—then the practical estimator has a positive variance gap from the oracle, and the 'nearly optimal' selling point is unsupported. The fixed-b variance reduction of Theorem 3 still holds; the paper is just overclaiming the asymptotic equivalence part. This should be fixed by either providing conditions and a proof, or softly rephrasing the claim to 'conjectured to be nearly optimal.'\n\nA second, smaller point: the exceptional-law exclusion is explicit and reasonable, but it means the method does not cover the full optimal-regime space. The NDE assumption itself is substantive and, as the paper acknowledges in Section 7, can fail in ways that bias the projected estimators; that is a modeling assumption, not a math error.\n\nWho should read this: anyone doing causal inference for dynamic treatment regimes, especially with testing-and-treatment decisions. It deserves serious peer review: the central variance-reduction result is correct, the discrete-outcome case is clean, and the simulation evidence is transparent. My clear recommendation: send it to a referee, but instruct them to focus on Remark 4. If that step is not provable, the near-optimality language must be softened before publication.","headline":"The core projection-based variance reduction is real and well proven, but the paper's headline near-optimality claim for continuous outcomes rests on an unproved convergence step that should be flagged in the review.","tokens_in":63416,"tokens_out":1683,"would_cite":true,"duration_ms":22498,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G05","62G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under the no-direct-effect-of-testing assumption, projecting g-estimating functions onto mean-zero testing-treatment residuals yields more efficient, doubly robust estimators of optimal testing and treatment regimes.","keywords":["optimal dynamic treatment regimes","structural nested mean models","g-estimation","no direct effect assumption","semiparametric efficiency","doubly robust estimation","value of information","screening tests"],"falsifier":"Estimate $\\mathrm{E}[D_{b,t}]$ from data with variation in testing and treatment, where $D_{b,t} = b_t(\\bar H_t,\\bar S_t,Y^d)\\,W_{t+1}^{-1}(A_t - \\mathrm{E}[A_t\\mid \\bar H_t,S_t])$; a nonzero mean for some $b_t$, or a difference in mean outcome between randomized testing arms with treatment held fixed, refutes the NDE assumption and implies the efficiency gains rest on a false premise.","tokens_in":62390,"feed_emoji":"🩺","tokens_out":9439,"duration_ms":93420,"temperature":0.7,"pith_summary":"This paper establishes that the no-direct-effect (NDE) assumption—that a diagnostic test affects outcomes only through the treatment it triggers—can be converted into a source of statistical precision, not just an identifying restriction. The authors construct estimators for optimal joint testing-and-treatment regimes by taking the usual g-estimating function and subtracting its projection onto a space of mean-zero test-treatment residual variables implied by NDE. The resulting estimators are regular asymptotically linear, never less efficient than standard g-estimators, and strictly more efficient whenever the residual correlation is nonzero, while remaining doubly robust. This matters for cost-benefit analyses and for estimating the value of information of expensive tests, because the efficiency gain can translate into dramatically smaller required sample sizes.","feed_headline":"Projection onto test-effect residuals sharpens regime estimates","feed_subtitle":"They stay valid when one nuisance model is wrong, with HIV monitoring gains up to 50-fold.","key_machinery":"The central object is the ortho-complement $\\Lambda_{\\mathrm{NDE}}^{\\perp}$ of the tangent space of the NDE model. The paper represents it as $\\{T_b = \\sum_{t=0}^{K} T_{b,t}\\}$, where $T_{b,t}$ is the residual from projecting $D_{b,t} = b_t(\\bar H_t,\\bar S_t,Y^d)\\,W_{t+1}^{-1}(A_t - \\mathrm{E}[A_t\\mid \\bar H_t,S_t])$ onto the space of testing and treatment scores. Here $W_{t+1}$ is the product of treatment probabilities, and the NDE restriction makes every $T_b$ have mean zero. Subtracting the population least-squares projection $c_{\\mathrm{OLS}}T_b$ of an influence function $U$ onto this space yields a new influence function, and the Pythagorean theorem guarantees the variance reduction. For feasibility, the paper projects onto a large subspace $\\Omega$ built from $b_t^\\ast = (\\phi_1(Y^d)I_t^{\\top},\\ldots,\\phi_\\xi(Y^d)I_t^{\\top})^{\\top}$, with $I_t$ the vector of indicators of full treatment histories; Theorem 4 supplies recursive least-squares coefficients for the closed-form projection.","core_discovery":"Under the NDE assumption, the paper constructs estimators $\\tilde\\Psi(q,b)$ that solve $0 = \\hat U(q,b,\\Psi)$, where $\\hat U(q,b,\\Psi)$ is the residual from projecting the influence function of the usual opt-SNMM estimating function $\\hat U(q,\\Psi)$ onto the space $T_b$ of mean-zero random variables implied by NDE. Theorem 3 shows each such estimator is regular asymptotically linear with asymptotic variance $V^{\\mathrm{oracle}}(q,b) = J^{-1}\\{\\operatorname{var}[U(q,\\Psi)] - c_{\\mathrm{OLS}}\\mathrm{E}[T_b T_b^\\top]c_{\\mathrm{OLS}}^\\top\\}J^{-\\top}$, which is no larger, and strictly smaller whenever $\\mathrm{E}[U(q,\\Psi^\\ast)T_b] \\neq 0$, than the variance of the standard g-estimator. The projected estimators remain doubly robust, and the paper provides a closed-form feasible construction based on a large subspace spanned by basis functions of the health outcome and treatment-history indicators, whose efficiency approaches the intractable optimal projection as the basis grows. Simulations and an HIV monitoring application indicate gains that can amount to a roughly 50-fold reduction in variance.","pith_inferences":["An immediate extension is to use the same projected quantities as a specification test: the NDE assumption implies infinitely many mean-zero restrictions, so a data-driven check of whether empirical projections are near zero could precede the efficiency-gaining analysis.","The construction suggests a general principle for causal inference: any domain assumption that generates extra mean-zero variables can be converted into precision by projecting influence functions onto the ortho-complement of the implied tangent space, not just the NDE assumption considered here.","In cost-benefit decisions near a value-of-information threshold, the variance reduction could change conclusions: the paper's simulation shows the projected estimator raising the empirical rejection rate of 'screening is not cost-effective' from around 25 percent to 100 percent, so policy conclusions may flip.","At the boundary where everyone is tested, the NDE assumption identifies parameters that are otherwise unidentified; whether the optimal regime can still be computed there is left open, so a natural extension is to study computation and estimation under such positivity failures."],"forward_implications":["The projected estimators are never less efficient than standard opt-SNMM g-estimators, and strictly more efficient whenever the initial estimating function correlates with the NDE-implied residuals.","The efficiency gain translates into a sample-size reduction: in the HIV monitoring application cited by the paper, exploiting NDE was associated with a roughly 50-fold variance reduction.","The same projection recipe applies to other doubly robust RAL estimators, including estimators of dynamic marginal structural models, so the improvement is not tied to opt-SNMMs.","A feasible, closed-form implementation exists by projecting onto a basis-expanded subspace; its efficiency approaches the semiparametrically optimal projected estimator as the basis dimension grows.","Under stronger NDE variants, such as no direct effect on covariates or on latent test results, larger projection spaces are available, so the estimators are at least as efficient and typically more efficient than under NDE on the health outcome alone."],"supporting_citations":[{"why":"Defines opt-SNMMs and the g-estimator whose efficiency the paper improves.","marker":"[27]"},{"why":"Derives the observed-data NDE restrictions that give the residual variables T_b mean zero.","marker":"[25]"},{"why":"Documents the 50-fold efficiency gain in an HIV monitoring analysis that motivates the projection construction.","marker":"[3]"},{"why":"Introduces the NDE-IPW estimator that the paper compares and refines by projection.","marker":"[23]"},{"why":"Establishes the double-robustness property that the projected estimators inherit.","marker":"[1]"},{"why":"Provides the cross-fitted machine-learning framework used for feasible nuisance estimation.","marker":"[4]"}],"fun_headline_variants":["Projection onto NDE residuals sharpens optimal regime estimates","NDE projection yields efficient regime estimation","Sharper optimal regimes via test-effect residual projection","NDE projection cuts variance up to 50-fold in regimes","Efficient joint testing-treatment regimes via NDE projection"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that testing has no direct effect on the health outcome, meaning a test changes outcomes only through the treatment it triggers; if testing affects outcomes directly, the mean-zero property of the residual variables fails and the projected estimators become biased.","fun_headline_variants_meta":{"raw":{"variants":["Projection onto NDE residuals sharpens optimal regime estimates","NDE projection yields efficient regime estimation","Sharper optimal regimes via test-effect residual projection","NDE projection cuts variance up to 50-fold in regimes","Efficient joint testing-treatment regimes via NDE projection"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000824,"raw_usage":{"total_tokens":3591,"prompt_tokens":920,"completion_tokens":2671,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":2596}},"tokens_in":536,"tokens_out":2671,"duration_ms":22009,"temperature":1.0,"reasoning_tokens":2596,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:43:27.820077+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Estimate $\\mathrm{E}[D_{b,t}]$ from data with variation in testing and treatment, where $D_{b,t} = b_t(\\bar H_t,\\bar S_t,Y^d)\\,W_{t+1}^{-1}(A_t - \\mathrm{E}[A_t\\mid \\bar H_t,S_t])$; a nonzero mean for some $b_t$, or a difference in mean outcome between randomized testing arms with treatment held fixed, refutes the NDE assumption and implies the efficiency gains rest on a false premise.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines opt-SNMMs and the g-estimator whose efficiency the paper improves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the observed-data NDE restrictions that give the residual variables T_b mean zero."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the 50-fold efficiency gain in an HIV monitoring analysis that motivates the projection construction."},{"cited_title":"Orellana, and A","cited_arxiv_id":null,"evidence_quote":"Introduces the NDE-IPW estimator that the paper compares and refines by projection."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the double-robustness property that the projected estimators inherit."},{"cited_title":"Chetverikov, M","cited_arxiv_id":null,"evidence_quote":"Provides the cross-fitted machine-learning framework used for feasible nuisance estimation."}],"review_version":1}