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Selective inference is easier with p-values

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Pith's one-line read A p-value with non-decreasing null density can be adjusted for any known selection procedure by one integral, making selective inference a plug-in operation.

desk verdict A genuinely new and mostly correct unification of selective inference via p-values, with an honest mapping of its limitations; deserves serious peer review. read the letter →

arxiv 2411.13764 v1 pith:ZLV7SR7E submitted 2024-11-21 stat.ME math.STstat.TH

Anav Sood This is my paper · ORCID
classification stat.MEmath.STstat.TH MSC 62F0362J05
keywords selectiveinferencepost-selectionp-valuesdominanceonwinnersrankverificationdatacarvingFishercombinationtest
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

Selective inference is the problem of doing valid tests after the data have already picked the question. This paper proposes a general solution: any p-value whose null density is non-decreasing, which it calls selectively dominant, can be corrected for an arbitrary known selection procedure by a single integral transform. If true, standard p-values—from two-sided parametric tests, one-sided tests in monotone likelihood-ratio and exponential families, F-tests, and permutation tests—can all be used for selective inference, and several known selective methods become simple consequences rather than bespoke derivations. The paper demonstrates this by re-deriving conditional inference on winners, hybrid inference, rank verification, and new selective versions of Fisher's combination test.

What carries the argument

The central object is the selection function $s(x,z)=P(S=1\mid p=x,Z=z)$, assumed known, which governs how the analyst decides to test a null after seeing the p-value. For a selectively dominant p-value—defined as one whose post-selection distribution stochastically dominates that of a uniform p-value subjected to the same selection—Theorem 1 says the selectively adjusted p-value $p_{\mathrm{sel}} = \int_0^p s(x,Z)\,dx/\int_0^1 s(x,Z)\,dx$ is stochastically uniform after selection, so rejecting when $p_{\mathrm{sel}}\le\alpha$ controls selective error. Theorem 2 identifies selective dominance with the condition that the null conditional density of $p$ given $Z$ is non-decreasing, which is verified for UMP and UMPU p-values, permutation p-values, and $F$-test p-values.

What would settle it

A decisive check: simulate any standard test statistic under its null, transform to a p-value, and look at the histogram conditional on the auxiliary variable $Z$. If the density decreases on $[0,1]$ for a set of $Z$ with positive probability, Theorem 2 says the p-value is not selectively dominant, so a single such example among commonly used tests would break the paper's blanket claim. A toy calculation shows the failure mode is real: with $p=1-\sqrt{1-U}$ (a valid decreasing-density p-value) and selection $s(x)=\mathbf{1}\{x\le0.1\}$, the corrected test rejects with probability $P(p\le0.01\mid p\le0.1)\approx0.105$, above the nominal $0.1$.

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Extended reading notes

Core claim

The central discovery is that selective dominance holds exactly when the null conditional density of the p-value $p$ given $Z$ is non-decreasing, and that this condition turns the selection-adjusted p-value $p_{\mathrm{sel}} = \int_0^p s(x,Z)\,dx \,/\, \int_0^1 s(x,Z)\,dx$ into a valid p-value after selection, where $s(x,z)=P(S=1\mid p=x,Z=z)$ is the known selection function. The paper shows that all the p-values practitioners commonly use satisfy this condition, and then uses the corrected p-value to give short derivations of conditional and hybrid inference on winners, rank verification in exponential families, data carving, and selective variants of Fisher's combination test. It also shows that the correction is tight when the original p-value is exactly uniform under the null.

Load-bearing premise

The load-bearing assumption is that the selection function $s(x,z)=P(S=1\mid p=x,Z=z)$ is known exactly and depends on no unknown parameters; whenever selection probabilities must be estimated or depend on the effect size being tested, the adjusted p-value cannot be computed from the formula.

Editorial extensions

If this is right

  • A researcher who sees a p-value only when $p\le\alpha$ can correct for this publication bias by rejecting when $p\le\alpha^2$, for any selectively dominant p-value.
  • For independent selectively dominant p-values, the winning null is rejected when $p_{(1)}\le\alpha p_{(2)}$, and the closed version of this test makes sequential discoveries while controlling family-wise error.
  • In the Gaussian rank-verification problem, a two-sided rejection at level $\alpha$ justifies the statement that the winner is strictly larger, while verifying that the winner is at least as large requires no selection correction at all.
  • Fisher's top-$k$ and truncated combination tests remain valid even when some null p-values are conservative (super-uniform), whereas earlier versions required exact uniformity.
  • Data carving becomes a general tool: when data fission or thinning makes the conditional distribution of the selection statistic given the p-value parameter-free, the selection function is known and the integral correction can be computed, at least numerically.

Reading between the lines

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

  • The paper treats the selection function as known; a natural extension, not pursued in the paper, is a plugin or conservative-envelope version that uses an estimated or upper-bounded $s(x,z)$ and studies how misspecification degrades selective error.
  • The explicit tie-handling correction in rank verification suggests that other selective methods that ignore ties in discrete or grouped data may be either conservative or anticonservative, and the same $1/N$ tie-breaking adjustment could be ported to them.
  • Practically, the framework turns common heuristics like 'we only publish $p\le0.05$' into a transparent correction $p/0.05$ that applies to almost any standard p-value, without deriving a truncated-normal distribution for each new test statistic.
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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

0 major / 5 minor

Summary. The paper introduces selectively dominant p-values, defined as p-values whose conditional density under the null is non-decreasing on [0,1]. It proves (Theorem 2) that this density condition is equivalent to selective dominance: after any selection process with known selection probability s(x,z), the conditional distribution of the p-value given selection stochastically dominates that of a uniform random variable subjected to the same selection. Theorem 1 then shows that the adjusted p-value p_sel = ∫_0^p s(x,Z)dx / ∫_0^1 s(x,Z)dx is uniformly valid post-selection. The theory is applied to inference on winners (Section 3), rank verification in exponential families (Section 4), and combination tests including data carving and publication-bias-aware meta-analysis (Section 5). The paper also corrects a tie-handling error in Hung and Fithian (2019) and gives a counterexample to their claimed level inflation.

Significance. If the results hold, this is a substantial unification of selective inference: instead of deriving conditional procedures case by case, one only needs a valid p-value, a non-decreasing null density, and a known selection function. The appendices provide careful measure-theoretic proofs of the main theorems, a verified counterexample to the rank-verification level claim, and a transparent coupling argument for the data-carving example. The paper also makes its code available. The central assumption—that the selection function is known and parameter-free—is explicitly stated and scoped; it limits applicability but does not affect the internal validity of the theorems. My reading confirms the soundness of the main derivations.

minor comments (5)
  1. [Abstract and Section 1.2] The phrase 'all commonly used p-values' overstates the support provided by Examples 2–6, which cover important parametric families, permutation tests, and F-tests but not a mathematically defined class of 'commonly used' p-values; moreover, Theorem 2 requires a conditional density, so discrete p-values are excluded unless randomized. Suggest qualifying the wording.
  2. [Section 2.3, Example 9] The statement that the selection function is 'always accessible to us via extensive simulations' deserves a caveat: a Monte Carlo estimate of s(x,z) does not by itself provide exact finite-sample Type I error control unless the estimation error is incorporated; the paper currently treats s as known.
  3. [Section 5.1] The statement that pO/α remains valid under 'p-hacking' is heuristic, based on modeling null p-hacked p-values as having increasing density on [0,α]; this is a reasonable empirical model but should be labeled as a heuristic rather than a theorem.
  4. [Appendix A.5] The LASSO post-selection inference example assumes the noise level σ is known; the paper could note that unknown σ requires additional treatment (e.g., via the square-root LASSO) so that readers do not infer that the framework resolves all LASSO settings without further conditions.
  5. [Appendix A.12] In the displayed re-expression of psel, the denominator is written as q+(Z) + (1/N(Z))(q+(Z) - q(Z)), which has the sign reversed relative to the correct expression q+(Z) + (1/N(Z))(q(Z) - q+(Z)) shown in the main text around equation (23); the surrounding narrative and Example 12 use the correct form, so this appears to be a typographical error.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central theorems are proved from definitions and standard testing theory; applications are checked against external benchmarks.

full rationale

The paper's load-bearing claims are Theorem 1 and Theorem 2. Theorem 1 is a direct consequence of the definition of selective dominance: if p's post-selection distribution stochastically dominates that of a uniform under the same selection, then applying the conditional CDF of that uniform, psel = F_{U|Z,S'}(p), yields a post-selection p-value. This is a proof, not an assumption of the conclusion; the substantive content is the characterization in Theorem 2 and the demonstration in Appendix B that UMP/MLR and UMPU p-values satisfy the non-decreasing-density condition. The proof of Theorem 2 constructs the relevant density inequalities for arbitrary selection functions and gives an explicit violating selection function for the converse, so the characterization is not circular. The examples and corollaries are derived from these theorems rather than fitted to data, and the reproductions of conditional LCB, LASSO post-selection inference, and hybrid inference are validated against external references (Lee et al., Fithian et al., Andrews et al.). The correction to Hung and Fithian's tie handling is supported by an explicit selection-function computation and simulation. The known-selection-function assumption is stated openly as a scope condition, not smuggled in, and the paper contains no load-bearing self-citations or fitted parameters renamed as predictions. Overall the derivation chain is self-contained.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the tuning parameters in hybrid inference (beta) and Fisher variants (k, tau) are user-chosen, not fitted to data. The axioms are standard selective inference assumptions: known selection and existence of conditional densities. No new entities are posited; the selectively dominant class is a property of existing p-values.

assumptions (4)
  • domain assumption Under the null, p-values considered have a conditional density given Z.
    Definition 1 requires a conditional PDF; Section 2.1 says randomization ensures this is without loss, and the paper restricts to p-values with densities.
  • domain assumption The selection function s(x,z) is known to the analyst and independent of unknown parameters.
    Theorem 1's adjusted p-value uses s(x,z); Section 2 assumes the selection process is decided by the practitioner, and Example 9 stresses the parameter-free property.
  • domain assumption All p-values in the main applications are independent and selectively dominant, verified for UMP, MLR, exponential, F, and permutation cases.
    The framework requires each input p-value to be selectively dominant; Appendix B establishes this for MLR and exponential families, and Examples 2, 5, and 6 cover the others.
  • standard math Standard Neyman-Pearson testing theory: UMP and UMPU tests exist and have the stated form.
    Used throughout (Lehmann et al. 1986), e.g., Lemmas B.1-B.3 and Examples 3-4.

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Pith. "Pith review of Selective inference is easier with p-values." pith.science (2026). https://pith.science/paper/ZLV7SR7E

@misc{pith2026241113764,
  author       = {Pith},
  title        = {Pith review of: Selective inference is easier with p-values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZLV7SR7E}},
  note         = {Machine review of arXiv:2411.13764}
}
abstract

Selective inference is a subfield of statistics that enables valid inference after selection of a data-dependent question. In this paper, we introduce selectively dominant p-values, a class of p-values that allow practitioners to easily perform inference after arbitrary selection procedures. Unlike a traditional p-value, whose distribution must stochastically dominate the uniform distribution under the null, a selectively dominant p-value must have a post-selection distribution that stochastically dominates that of a uniform having undergone the same selection process; moreover, this property must hold simultaneously for all possible selection processes. Despite the strength of this condition, we show that all commonly used p-values (e.g., p-values from two-sided testing in parametric families, one-sided testing in monotone likelihood ratio and exponential families, $F$-tests for linear regression, and permutation tests) are selectively dominant. By recasting two canonical selective inference problems-inference on winners and rank verification-in our selective dominance framework, we provide simpler derivations, a deeper conceptual understanding, and new generalizations and variations of these methods. Additionally, we use our insights to introduce selective variants of methods that combine p-values, such as Fisher's combination test.

Figures

Figures reproduced from arXiv: 2411.13764 by the authors.

Figure 1
Figure 1. The first panel (left) shows the growth of the quantile [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The first panel (left) depicts an example distribution of a p-value [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. We plot the level α = 0.1 conditional LCB ˆµ for different gaps between the winning value XW and the runner up value XR and highlight the distance between ˆµ and XW in red. The LCB ˆµ is chosen exactly so that the tail probability P(N(ˆµ, 1) > XR), shaded in blue, is 1/α = 10 times the tail probability P(N(ˆµ, 1) > XW ), shaded in red (the overlap appears purple). As XW and XR get closer, we need to take µˆ further … view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Over B = 1000 different sub-samples of n = 2 failure times from the dataset Molotaliev [2024], the distribution of the conditional and simultaneous LCB for the “winning” parameter. The conditional LCB is always vacuous. 3.2 Hybrid inference Hybrid inference, originally…
Figure 5
Figure 5. Figure 5: For the n = 20 dimensional Gaussian problem Xi ∼ N(µi , σ2 ) with largest observation XW and second largest observation XR, the standardized distance (XW − µˆ)/σ from XW to the level α = 0.1 hybrid, conditional, and simultaneous LCB ˆµ as a function of the standardized…
Figure 6
Figure 6. Figure 6: For µ2 = 0 and different µ1, the Type I error of rejecting H0,W R : µW −µR < 0 and H0,W R : µW − µR ≤ 0 when the level α one-sided test comparing the winner of X1 ∼ N(µ1, 1/ √ 2) and X2 ∼ N(µ2, 1/ √ 2) to the runner-up rejects. 4.2 Rank verification in exponential fami…
Figure 7
Figure 7. Figure 7: Considering three independent binomials Xi ∼ Bin(b, si) with b = 4 and si = 1/2, the first panel (left) depicts N = 106 draws from the conditional distribution of the p-value p12 (used for testing H0,12 : s1 ≤ s2) given W = 1 and the nuisance statistics (X1 + X2)/2 = 2…
Figure 8
Figure 8. Figure 8: Scatter plot of the original pO and replication pR p-values for 92 psychology studies from the open science collaboration’s replication analysis Collaboration [2015]. Note that the x-axis, which ranges from [0, 0.05] is on a different scale than the y-axis, which range…
Figure 9
Figure 9. Figure 9: For ℓ = 3 (left), ℓ = 5 (middle), and ℓ = 10 (right), power of the k = 3 conditional, τ = 0.5 truncated, and original Fisher’s combination test for data drawn from N(µ, I10) with µ1 = · · · = µℓ varying according to the x-axis and µℓ+1 = · · · = µn = −2. Power results …
Figure 10
Figure 10. Figure 10: For n = 10 (left), n = 100 (middle), and n = 1000 (right) the distance between the hybrid LCB to the winner (dash-dot line) and union bound LCB to the winner (dotted and solid line) with α = 0.05 and β = 0.005 plotted as a function of the gap between the winning and r…
Figure 11
Figure 11. Figure 11: For µ2 = 0 (left), µ2 = −0.5 (middle), and µ2 = −1 (right) the empirical power over N = 104 trials of the hybrid inference approach versus the union bound approach for the needle in the haystack alternative. One standard error bands are also plotted. For the most part…

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