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REVIEW 3 major objections 5 minor 38 references

Reluctant Interaction Inference after Additive Modeling

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper constructs valid p-values for data-adaptive hypotheses about linear interaction effects by conditioning on a randomized sparse additive model fit.

desk verdict 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. read the letter →

arxiv 2506.01219 v1 pith:IMOWCGHZ submitted 2025-06-02 stat.ME

Yiling Huang , Snigdha Panigrahi , Guo Yu , Jacob Bien This is my paper
classification stat.ME MSC 62J0762F03
keywords selectiveinferencepost-selectionsparseadditivemodelsgrouplassointeractioneffectsdata-adaptivehypothesesexternalrandomizationWaldp-values
verification ladder T0 review T1 audit T2 compute T3 formal

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'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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Section 3.4, Eq. (12)] 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.
  2. [Section 3.5, Theorem 2] 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.
  3. [Section 4.2 vs. Section 3] 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.
minor comments (5)
  1. [Section 2, Algorithm 1] 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.
  2. [Section 3.5 vs. Section 4.2] 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.
  3. [Table 2 caption] 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.
  4. [Table 1 and Figure 7] The heading 'Precison' appears in Table 1 and Figure 7; it should read 'Precision'.
  5. [Section 4.2] The notation σ_jk is used for both the naive pivot and the selective pivot; different symbols would avoid confusion.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the selective likelihood is derived from the joint distribution of data and external randomization; the approximate p-values are not fitted to force uniformity. Main caveat is a load-bearing self-citation for the Laplace approximation's formal justification, which is a correctness risk rather than a circular reduction.

full rationale

The paper's central derivation is not circular. Step 1 defines a randomized group lasso SPAM fit (Eq. 1). Lemma 1 characterizes the selection event via KKT conditions; Lemma 2 gives the marginal distribution of the key statistics; Lemma 3 and Proposition 1 build the joint conditional density; Theorem 1 then gives an exact selective log-likelihood for the interaction coefficient. The p-value in Section 3.5 is obtained by plugging an MLE and observed Fisher information from an approximate version of this likelihood into a Wald pivot. At no point is a parameter fitted to force the resulting pivot to be uniform, and the likelihood is not defined in terms of the target null hypothesis. The paper is also transparent that the pivot is 'approximate Uniform(0,1)', not exactly uniform. The one legitimate concern is that the formal justification for the Laplace approximation replacing the normalizing constant is outsourced to Huang et al. (2023), a previous paper with overlapping authorship, and no finite-sample error bound is supplied. This is a load-bearing reliance on self-citation for the approximation's accuracy, and the weak-boundary regime of small selected group norms is not covered by the simulations. However, this is an unsupported approximation and a correctness risk, not a circular derivation: the exact selective likelihood is proven in the appendix, and the approximate pivot is not constructed by design to equal its own input. The paper is therefore essentially self-contained against external benchmarks, with only minor self-citation in the approximation justification, supporting a low circularity score of 2.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim depends on several assumptions inherited from prior work on randomized group lasso selective inference, plus the accuracy of the Laplace approximation. The free parameters are standard tuning choices rather than fitted parameters of the inference itself.

free parameters (5)
  • r (randomization proportion) = 0.9
    Controls the variance of the external randomization, Omega = sigma^2 (1-r)/r Psi^T Psi. Chosen in Section 4.2 to mimic data splitting with 90% selection and to keep the SPAM fit close to the unrandomized fit. Not fitted to data but user-set.
  • lambda_j (group lasso tuning) = 0.5 sigma sqrt(n) sqrt(B_j) sqrt(2 log q)
    Set by a formula in Section 4.2, not data-driven. Influences which main effects are selected and hence the conditioning event.
  • epsilon (ridge penalty in group lasso) = not specified
    Small ridge penalty added in (1) to ensure strict convexity and uniqueness of the solution. No specific value is given in the paper, so a user must choose it.
  • B-spline basis degree and knots = degree 2, 6 knots
    Modeling choices for the additive feature expansions, used in all simulations and the real data analysis. They affect the design matrix and the group structure.
  • t0 (true interaction threshold in F1 score) = 0.1
    Threshold used only in the evaluation metric to decide which interaction effects are true. Not part of the inferential method.
assumptions (5)
  • standard math The selection event {Mhat = M} has the equivalent characterization {bgamma > 0, bU = U, bZ = Z} from Lemma 1, borrowed from Panigrahi et al. (2023) and Huang et al. (2023).
    This is a proven result in the cited prior work. The paper relies on it to make the conditioning event tractable.
  • standard math The change-of-variables mapping Pi from the randomization omega to (bgamma, bU, bZ) is one-to-one with the Jacobian determinant given in Proposition 1.
    The Jacobian formula is taken from prior work by Panigrahi et al. (2023). The paper does not re-derive it fully.
  • domain assumption The Laplace approximation to the normalizing constant c(theta, beta) in Section 3.4 is accurate enough for the resulting p-values to be approximately valid.
    The paper cites Huang et al. (2023) for formal justification but provides no finite-sample error bound. The simulations are the only evidence of accuracy in finite samples.
  • domain assumption The response y follows the normal linear model y ~ N(Psi_M beta + I_jk theta, sigma^2 I) when testing the interaction I_jk.
    This model is stated in (3) and underlies the distribution of the key statistics. It is a simplifying assumption that the test is performed in a linear model with one interaction added to the selected main effects.
  • domain assumption The external randomization omega is independent of y and has a known Gaussian distribution N(0, Omega).
    This independence is used in Proposition 1 to write the joint density. In practice Omega is estimated with a plug-in sigma, which is an approximation.

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Cite this review

Pith. "Pith review of Reluctant Interaction Inference after Additive Modeling." pith.science (2026). https://pith.science/paper/IMOWCGHZ

@misc{pith2026250601219,
  author       = {Pith},
  title        = {Pith review of: Reluctant Interaction Inference after Additive Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IMOWCGHZ}},
  note         = {Machine review of arXiv:2506.01219}
}
read the original abstract

Additive models enjoy the flexibility of nonlinear models while still being readily understandable to humans. By contrast, other nonlinear models, which involve interactions between features, are not only harder to fit but also substantially more complicated to explain. Guided by the principle of parsimony, a data analyst therefore may naturally be reluctant to move beyond an additive model unless it is truly warranted. To put this principle of interaction reluctance into practice, we formulate the problem as a hypothesis test with a fitted sparse additive model (SPAM) serving as the null. Because our hypotheses on interaction effects are formed after fitting a SPAM to the data, we adopt a selective inference approach to construct p-values that properly account for this data adaptivity. Our approach makes use of external randomization to obtain the distribution of test statistics conditional on the SPAM fit, allowing us to derive valid p-values, corrected for the over-optimism introduced by the data-adaptive process prior to the test. Through experiments on simulated and real data, we illustrate that--even with small amounts of external randomization--this rigorous modeling approach enjoys considerable advantages over naive methods and data splitting.

Figures

Figures reproduced from arXiv: 2506.01219 by the authors.

Figure 1
Figure 1. ECDF of the uniform pivots obtained by naive inference, data splitting, and the [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. F1 scores for true main effects selection (left) and average confidence interval [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. ECDF of the uniform pivots Φ(zjk) obtained by naive inference, data splitting, and the proposed method in all simulation settings 26 [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Average confidence interval lengths for θ M jk obtained by naive inference, data splitting, and the proposed method in all simulation settings treat the remaining features linearly, indexed by L ⊂ [p]. A summary of the linear and nonlinear features is provided in Table…
Figure 5
Figure 5. Figure 5: F1 scores of hypothesis testings of θ M jk obtained by naive inference, data splitting, and the proposed method in all simulation settings with varying α 28 [PITH_FULL_IMAGE:figures/full_fig_p028_5.png]
Figure 6
Figure 6. Figure 6: Fitted main effects using the proposed method and naive inference [PITH_FULL_IMAGE:figures/full_fig_p029_6.png]
Figure 7
Figure 7. Figure 7: Precison, recall, and F1 scores of naive inference and the proposed method [PITH_FULL_IMAGE:figures/full_fig_p030_7.png]
Figure 8
Figure 8. Figure 8: Frequency of significance of interaction signals that were most frequently deemed [PITH_FULL_IMAGE:figures/full_fig_p031_8.png]

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