{"id":"1eb4ca03-c946-4c48-8a54-3a6bcd3606f9","arxiv_id":"2506.01219","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A selective inference method produces valid p-values for interaction effects after a sparse additive model is fit to the same data, using external randomization and full-data statistics.","lead":"This paper develops a way to test whether adding a feature interaction improves an additive model, while accounting for the fact that the additive model was chosen using the same data. The method uses randomized selective inference to produce p-values that correct for this data reuse, and it shows higher power than data splitting in simulations.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unverified Laplace approximation undermines the 'valid p-values' claim; near-zero selected group norms are a concrete failure mode not covered by the paper's simulations.","rationale":"The reader's weakest_assumption correctly identifies the Laplace approximation. In stress-testing, I find no internal inconsistency in Theorem 1; the exact conditional likelihood derivation appears coherent. However, the step from Theorem 1 to the p-values in Section 3.5 involves two unquantified approximations: the Laplace approximation to c(theta,beta) and the Wald Gaussian approximation to the MLE's distribution. The Laplace approximation is more fundamental because the Wald step is downstream of it; if the normalizing constant is inaccurate, the likelihood surface itself is wrong. The boundary regime (small selected-group norms) is a natural failure mode because the integrand in c(theta,beta) includes a positive-orthant constraint and a log-determinant that varies with gamma; near gamma=0 the Laplace approximation to a truncated integral with a non-quadratic log-determinant is not justified by the cited previous work, which assumes a stronger curvature condition. The simulations in Section 4 cover a range of noise, correlation, interaction strength, and number of interactions, but not weakly identified main effects. The real-data application is a useful demonstration but not a calibration check. Theorem 2 is stated without proof; while the formula is plausible (it matches a Schur-complement derivation of the observed Fisher information of the approximate likelihood), the lack of proof is secondary given the simulations. Overall, the paper's contribution is promising but the central 'valid p-values' claim is stronger than what is proven; conditional acceptance is appropriate, pending an error analysis or boundary-regime simulations.","tokens_in":21649,"tokens_out":19160,"duration_ms":193950,"concrete_test":"Add a simulation setting to Section 4 where one true main effect has a tiny coefficient (e.g., scale 0.1) so SPAM selects it with small gamma_j, and test the ECDF of the proposed pivots for interactions involving that feature against Uniform(0,1) over 1000 replications. Additionally, for one dataset, compare p-values from the Laplace-approximated likelihood to p-values obtained by high-accuracy Monte Carlo integration of c(theta,beta); if the two differ beyond Monte Carlo error, the approximation is the bottleneck.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of valid p-values depends on the Laplace approximation in Section 3.4 replacing the normalizing integral c(theta_M_jk, beta_M_jk) with a pointwise maximum plus a barrier. The paper provides no finite-sample error bound; the reference to Huang et al. (2023) covers a different construction. The risk is concrete when a selected group has small norm gamma_j: the integrand then has mass near the boundary g'_j=0, where the log(1+1/g) barrier and the log-determinant term make it strongly non-quadratic, so a single Laplace mode does not approximate the integral. Because the Wald p-values in Section 3.5 are derived from this approximate likelihood, the pivot Phi(z_jk) need not be Uniform(0,1) under the null in that regime, and the simulations in Section 4 do not include such weak-boundary settings.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a two-step procedure for testing whether linear interaction terms should be added to a sparse additive model (SPAM). In Step 1, a SPAM is fitted via a randomized group lasso with Gaussian external randomization; in Step 2, for each interaction in a data-adaptive set T^M, the paper constructs p-values for H_0: θ^M_jk = 0 under the model y ~ N(Ψ_M β + I_jk θ, σ² I). The p-values are derived from the conditional distribution of the key statistics given the observed SPAM selection, using a change of variables from the randomization (Sections 3.1–3.3). Theorem 1 gives an exact selective log-likelihood; Section 3.4 replaces its normalizing constant by a Laplace approximation with a barrier; Section 3.5 obtains an MLE and observed Fisher information and forms Wald p-values. Simulations and a JFK flight delay analysis compare the method with naive z-tests and data splitting.","tokens_in":21808,"tokens_out":6064,"duration_ms":65990,"significance":"If the approximation is reliable, the paper fills a gap by providing selective inference for interactions after additive-model selection, with the advantage of using the full data rather than a holdout. The exact conditional likelihood derivation in Theorem 1 is coherent and builds sensibly on the randomized group-lasso machinery of Panigrahi et al. (2023) and Huang et al. (2023). The empirical results show nearly uniform pivots in many settings and improved F1 over data splitting. However, the central validity claim is approximate, and the Laplace approximation is not quantified; this is the main obstacle to accepting the paper in its current form.","major_comments":[{"comment":"The validity of the proposed p-values rests on the Laplace approximation of the normalizing constant c(θ^M_jk, β^M_jk). The manuscript provides no finite-sample error bound for replacing the integral by its supremum plus a barrier, and the reference to Huang et al. (2023) is not directly transferable because the integrand here includes log(1 + 1/g_k) and -log det D_ΠS, which are strongly non-quadratic when any selected group has small norm γ_j. In that regime the Laplace mode need not represent the integral, and the Wald pivot from Section 3.5 need not be Uniform(0,1) under H0. Since the simulations in Section 4 do not include such weak-boundary selected groups, the central claim of valid p-values is not established for this regime. Please add a quantitative approximation bound under interpretable conditions, or at minimum a simulation study that deliberately generates selected groups with small γ_j and reports the empirical distribution of the pivots.","section":"Section 3.4, Eq. (12)"},{"comment":"Theorem 2 is the computational core of the proposed inference, but its proof is omitted with the statement that it is similar to Theorem 4.1 in Huang et al. (2023). Because the expressions for the MLE and observed Fisher information in the approximate selective likelihood are not identical to that theorem, and because the Wald p-values are built on them, a proof or a detailed derivation should be included in the paper or appendix.","section":"Section 3.5, Theorem 2"},{"comment":"The theory treats σ as known: Lemma 2 defines Σ^M_jk in terms of σ², and the selective likelihood in Theorem 1 inherits this. In the numerical experiments and JFK application, σ is replaced by a plug-in estimator bσ, but no adjustment or theory is provided for the effect of this estimation on the selective likelihood or the Wald pivot. This is a second source of approximation error beyond Laplace that is not reflected in the stated validity claims.","section":"Section 4.2 vs. Section 3"}],"minor_comments":[{"comment":"The text in Section 2 and Section 3 refers to 'Algorithm 2' and 'Step 2.1 of Algorithm 2', but the displayed algorithm is labeled Algorithm 1; the numbering should be made consistent.","section":"Section 2, Algorithm 1"},{"comment":"In Section 3.5 the pivot is written as z_jk = (bθ^M_jk − θ^M_jk)/sqrt(I^{-1}), while Section 4.2 defines it via bθ^M_mle; the notation should be aligned so that the reader can see that the OLS estimator is replaced by the selective MLE.","section":"Section 3.5 vs. Section 4.2"},{"comment":"The caption of Table 2 says 'by both naive inference and data splitting', but the columns are headed 'Naive' and 'Proposed'; the caption should be corrected to refer to the proposed method.","section":"Table 2 caption"},{"comment":"The heading 'Precison' appears in Table 1 and Figure 7; it should read 'Precision'.","section":"Table 1 and Figure 7"},{"comment":"The notation σ_jk is used for both the naive pivot and the selective pivot; different symbols would avoid confusion.","section":"Section 4.2"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the authors' own prior framework, and the Laplace approximation is the main technical risk. The paper is likely publishable if the authors can provide a concrete error bound or boundary-focused simulations for the approximate likelihood. The manuscript is within scope for a selective-inference-oriented statistics journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper takes the authors' own randomized group-lasso selective-inference machinery and points it at a new and practical target: testing linear interaction effects after fitting a sparse additive model. The key-statistic decomposition in Section 3.2 is genuinely new, and the simulations make a credible case that the method beats data splitting in power while keeping Type I error near nominal. The repeated-subsampling real-data analysis is a nice touch and gives more confidence than a single split.\n\nThe soft spots are real but not fatal. The abstract says the p-values are \"valid,\" but they come from a Laplace approximation to the selective likelihood and a Wald approximation, with no finite-sample error bound. The stress-test worry about small selected group norms is plausible: near g=0 the log-barrier and log-determinant terms are strongly non-quadratic, so a single Laplace mode could miss a substantial part of the integral. The simulations do not cover that regime, so we are left not knowing how bad the miscalibration gets. That is the main question a referee should push on. Theorem 2 is also stated without proof, which is tolerable given it closely follows earlier work, but it deserves a citation-to-proof rather than a hand-wave.\n\nNone of this undercuts the core approach. The derivation is coherent, the problem is well motivated, and the empirical work is fairly thorough. The authors should soften \"valid\" to \"approximately valid\" and add either a diagnostic or a simulation in the small-norm regime. If the Laplace approximation breaks down there, the method needs a fix or a caveat; if it does not, that is worth showing.\n\nThis deserves a serious referee. I would send it to review without hesitation.","headline":"A useful extension of randomized group-lasso selective inference to interaction testing, with a solid empirical case but an under-scrutinized Laplace approximation near small selected group norms.","tokens_in":22361,"tokens_out":2074,"would_cite":true,"duration_ms":25446,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62J07","62F03"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs valid p-values for data-adaptive hypotheses about linear interaction effects by conditioning on a randomized sparse additive model fit.","keywords":["selective inference","post-selection inference","sparse additive models","group lasso","interaction effects","data-adaptive hypotheses","external randomization","Wald p-values"],"falsifier":"Generate many null datasets with $\\theta^{M}_{jk}=0$ and a large selected main-effect set, compute the proposed Wald p-values, and compare the empirical CDF of the pivots with $\\mathrm{Uniform}(0,1)$; in a small-$|M|$ design, also compute the normalizing constant exactly by numerical integration and substitute it for the Laplace approximation. If the approximate-pivot ECDF deviates beyond Monte Carlo tolerance while the exact version stays uniform, the Laplace step—not the conditioning argument—is what fails.","tokens_in":21419,"feed_emoji":"📊","tokens_out":8928,"duration_ms":78075,"temperature":0.7,"pith_summary":"The paper's aim is to let an analyst test whether a linear interaction between two features is needed, after a sparse additive model (SPAM) has been fitted to the same data, without the p-values being corrupted by that first-stage fit. Naive z-tests ignore the fact that the SPAM's selected main effects were chosen by looking at the data, so they overstate evidence and admit spurious interactions. The authors show that adding Gaussian randomization to the group-lasso objective used to fit the SPAM makes the selection event tractable, and they derive the exact selective log-likelihood of the interaction coefficient conditional on that event. From this likelihood they build Wald p-values and confidence intervals that control Type I error while using the full dataset, which gives more power than data splitting. A sympathetic reader cares because this turns the reluctance principle—do not move beyond an additive model unless the data warrant it—into a statistically honest workflow.","feed_headline":"Conditioning fixes interaction p-values after additive fits","feed_subtitle":"External randomization lets you test interactions with full data and no selection-bias inflation.","key_machinery":"The engine is external Gaussian randomization added to the group-lasso objective in (1). Because the randomization vector $\\omega$ is independent of $y$, a change of variables from $\\omega$ to the group-lasso estimators $(\\hat\\gamma, \\hat U, \\hat Z)$ turns the opaque event $\\{\\hat M = M\\}$ into simple sign constraints $\\hat\\gamma \\succ 0$ together with fixed unit-norm and subgradient constraints (Lemma 1). The key statistics $\\hat S^{M}_{jk} = (\\hat\\theta^{M}_{jk}, (\\hat\\beta^{M}_{jk})^\\top, (\\hat A^{M}_{jk})^\\top)^\\top$ have a Gaussian law for fixed $M$, and Proposition 1 gives their density conditional on the selection event as a product of that Gaussian, the randomization density evaluated through a mapping $\\Pi_S(\\gamma,U,Z)$, and a Jacobian determinant. Theorem 1 marginalizes $\\gamma$ out, yielding a selective log-likelihood whose normalizing constant $c(\\theta^{M}_{jk}, \\beta^{M}_{jk})$ is approximated by Laplace's method with a barrier function; Theorem 2 then solves a low-dimensional convex problem for the approximate MLE and observed Fisher information, from which Wald pivots $\\Phi(z_{jk})$ are formed.","core_discovery":"On the paper's own terms, the central claim is that after solving the randomized group-lasso problem (1) to fit a SPAM and observing the selected main-effect set $M$, valid inference for the linear interaction coefficient $\\theta^{M}_{jk}$ in the model $y \\sim N(\\Psi_M \\beta^{M}_{jk} + I_{jk}\\,\\theta^{M}_{jk}, \\sigma^2 I_n)$ can be obtained from the conditional distribution of the OLS statistics given the event that the SPAM selected $M$. Theorem 1 gives the resulting selective log-likelihood, and Theorem 2 gives the MLE and observed Fisher information that turn it into approximate Gaussian pivots and Wald confidence intervals. The paper claims these p-values properly account for the data-adaptive selection of $M$, are valid where the naive z-test is not, and are more powerful than data splitting because inference uses the full dataset rather than a holdout. With small randomization variance (chosen as $r=0.9$), the randomized SPAM fit closely matches the nonrandomized fit, so the validity guarantee costs almost nothing in main-effect recovery.","pith_inferences":["The paper fixes $r=0.9$ throughout; a natural extension is to tune $r$ as a deliberate trade-off between main-effect selection fidelity and inferential power, or to choose it data-adaptively.","The selective likelihood should extend to joint tests of several interactions at once, which the paper names as future work; if the Laplace approximation holds there, an analyst could test an entire interaction set rather than one pair at a time.","Because the conditioning argument relies only on the Gaussian law of $y$ and $\\omega$, the same derivation could be carried over to non-Gaussian responses through an asymptotic selective likelihood, an extension the authors flag."],"forward_implications":["Analysts can screen interactions under a weak-hierarchy rule after a SPAM fit and trust the reported p-values, because the selection of main effects is explicitly conditioned on.","Using the full data for inference yields shorter confidence intervals and better F1 recovery of true interactions than data splitting, whose holdout can become rank-deficient.","A small randomization variance keeps the randomized SPAM almost identical to the plain SPAM, so the validity guarantee does not require a visibly different first-stage model.","The same selective likelihood also produces valid p-values and confidence intervals for main-effect coefficients in the fitted SPAM, a byproduct noted in the paper."],"supporting_citations":[{"why":"Supplies the randomized group-lasso selective inference machinery and the Jacobian/change-of-variables computation that Proposition 1 builds on.","marker":"Panigrahi et al. (2023)"},{"why":"Develops the maximum-likelihood selective inference framework for general models and the Laplace approximation to the selective likelihood; Theorem 2's proof is said to mirror its Theorem 4.1.","marker":"Huang et al. (2023)"},{"why":"Provides the randomized maximum-likelihood approach that the paper adopts for inference on interaction effects after the SPAM fit.","marker":"Panigrahi and Taylor (2023)"},{"why":"Introduces the reluctance principle that motivates testing interactions only after main effects have been fitted.","marker":"Yu et al. (2019)"},{"why":"Establishes polyhedral post-selection inference for the lasso, the baseline selective-inference approach that does not extend to SPAM selection.","marker":"Lee et al. (2016)"},{"why":"Provides the data-splitting strategy that the paper compares against and that it aims to improve on by using the full data.","marker":"Rasines and Young (2023)"},{"why":"Introduced the group-lasso-style SPAM formulation whose randomized version is the first-stage model in this paper.","marker":"Ravikumar et al. (2009)"}],"fun_headline_variants":["Randomized SPAM enables valid interaction p-values","Selective inference for interactions after additive fit","Full-data interaction tests without selection bias","Testing interactions post-SPAM with valid p-values"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Valid p-values rest on the assumption that the Laplace approximation to the normalizing constant $c(\\theta^{M}_{jk}, \\beta^{M}_{jk})$ in Section 3.4 is accurate for the data at hand; the paper gives no finite-sample error bound for this approximation, only a reference to prior work and simulation evidence.","fun_headline_variants_meta":{"raw":{"variants":["Randomized SPAM enables valid interaction p-values","Selective inference for interactions after additive fit","Full-data interaction tests without selection bias","Testing interactions post-SPAM with valid p-values"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00012,"raw_usage":{"total_tokens":1097,"prompt_tokens":959,"completion_tokens":138,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":81}},"tokens_in":575,"tokens_out":138,"duration_ms":2157,"temperature":1.0,"reasoning_tokens":81,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:47:26.993045+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate many null datasets with $\\theta^{M}_{jk}=0$ and a large selected main-effect set, compute the proposed Wald p-values, and compare the empirical CDF of the pivots with $\\mathrm{Uniform}(0,1)$; in a small-$|M|$ design, also compute the normalizing constant exactly by numerical integration and substitute it for the Laplace approximation. If the approximate-pivot ECDF deviates beyond Monte Carlo tolerance while the exact version stays uniform, the Laplace step—not the conditioning argument—is what fails.","supporting_citations":[{"cited_title":"Selective inference using randomized group lasso estimators for general models","cited_arxiv_id":"2306.13829","evidence_quote":"Develops the maximum-likelihood selective inference framework for general models and the Laplace approximation to the selective likelihood; Theorem 2's proof is said to mirror its Theorem 4.1."},{"cited_title":"D., Sun, D","cited_arxiv_id":null,"evidence_quote":"Establishes polyhedral post-selection inference for the lasso, the baseline selective-inference approach that does not extend to SPAM selection."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the data-splitting strategy that the paper compares against and that it aims to improve on by using the full data."},{"cited_title":"(2009), Sparse additive models, Journal of the Royal Statistical Society Series B: Statistical Methodology\\/ , 71, 1009--1030","cited_arxiv_id":null,"evidence_quote":"Introduced the group-lasso-style SPAM formulation whose randomized version is the first-stage model in this paper."}],"review_version":1}