Minimally Adjusted P-Values

• Minimally adjusted p-values are minimally transformed raw p-values designed to restore validity under specific inferential constraints, such as order-statistic combination or multiplicity corrections.
• They play a critical role in controlling family-wise error rates in multiple testing, sequential analyses, and discrete statistical tests by applying the least conservative correction.
• By balancing error control and statistical power, these methods ensure reliable inference while making the smallest necessary modification to the original p-value.

Searching arXiv for recent and relevant papers on minimally adjusted p-values and related notions. A minimally adjusted p-value is an adjusted significance measure obtained by altering a raw or intermediate p-value only as much as required by a specified inferential constraint. Across the literature, that constraint varies: validity of a combined order-statistic p-variable, family-wise error control under multiplicity, compatibility with repeated analyses in group sequential designs, accommodation of discreteness, calibration of permutation tails, or consistency with auxiliary control information. The common theme is extremal optimization: among all admissible adjustments satisfying the target validity property, the preferred construction is the smallest, closest, or least conservative one in an appropriate order, metric, or testing framework (Mattner, 2010, Zhao et al., 2023, Habiger et al., 20 Feb 2026).

1. Formal meanings of minimal adjustment

The phrase does not denote a single universal object. In the cited work, it appears in at least two recurrent formal senses. First, a minimally adjusted p-value may be the smallest transformation of a statistic that restores validity. In "Combining individually valid and conditionally i.i.d. P-variables" (Mattner, 2010), the problem is to combine individually valid, conditionally i.i.d. P-variables $U_1,\ldots,U_n$ through an order statistic $U_{k:n}$. The paper studies increasing functions $f$ such that $f(U_{k:n})$ is again a valid P-variable and proves that validity for every hypothesis and every P-kernel is equivalent to the pointwise inequality $f \ge f_{n,k}$, where

$f_{n,k}(u) = u\,\psi_{n,k}(p_{n,k}\vee u) = \begin{cases} c_{n,k}u, & u\in[0,p_{n,k}],\[0.4em] \mathrm{B}_{n,u}(\{k,\ldots,n\}), & u\in[p_{n,k},1]. \end{cases}$

Thus $f_{n,k}(U_{k:n})$ is the smallest increasing valid summary P-variable within that class.

Second, a minimally adjusted p-value may be the least multiplicity correction compatible with a designated error criterion. In such settings, the adjustment is defined relative to a family of nulls, a resampling distribution, or a closure principle, and the resulting value is interpreted as the smallest level at which rejection is justified.

Setting	Minimal object	Validity target
Order-statistic combination	$f_{n,k}(U_{k:n})$	Valid P-variable (Mattner, 2010)
Multiple analyses / minP	Null distribution of $\min p_i$	Weak FWER (Mandl et al., 2024)
Group sequential multiplicity	Adjusted-sequential p-values	FWER across looks and hypotheses (Zhao et al., 2023)
Functional pointwise inference	GET-adjusted $\tilde p(s)$ and refinements	FWER over the domain (Xu et al., 2019)
Discrete testing	MD/MR p-values	Stochastic, convex-order, or exact uniform validity (Habiger et al., 20 Feb 2026)

This variety suggests that minimality is always relative to a class of admissible procedures and a chosen notion of validity, not an intrinsic property of a number alone.

2. Multiple testing, closure, and minP constructions

In multiple testing, minimally adjusted p-values are closely tied to closed testing and minP methodology. For elementary hypotheses $U_{k:n}$0, closed testing introduces every nonempty intersection hypothesis

$U_{k:n}$1

The adjusted p-value for an elementary hypothesis is then the maximum of the valid intersection-hypothesis p-values over all supersets containing that hypothesis. In the group sequential setting, "Adjusted inference for multiple testing procedure in group sequential designs" (Zhao et al., 2023) extends this logic to repeated analyses, proposing adjusted-sequential p-values for elementary hypotheses together with sequential p-values for intersection hypotheses. The inferential target is strong control of the family-wise Type I error rate across all hypotheses and all looks.

A different minimality principle appears in minP procedures. When many candidate analyses are entertained and the smallest observed p-value is selected, the relevant statistic is

$U_{k:n}$2

"Addressing researcher degrees of freedom through minP adjustment" (Mandl et al., 2024) formalizes analytic flexibility as multiplicity and defines the minP-adjusted p-value through the null distribution of the minimum p-value, approximated by permutation: $U_{k:n}$3 Because the dependence structure among analyses is preserved by permutation, this adjustment is less conservative than Bonferroni or Šidák when tests are highly correlated, while ensuring weak FWER control under the global null.

The same strategy is developed for score tests with inequality constraints in "MinP Score Tests with an Inequality Constrained Parameter Space" (Cavaliere et al., 2021). There the minimum is taken over one-sided score-test p-values, optionally augmented by a global cone or $U_{k:n}$4-type score-test p-value. Bootstrap calibration of the joint null distribution yields global MinP tests and a stepdown multiple-testing procedure with asymptotic FWER control. In this usage, the minimally adjusted quantity is not a closed-form correction but the bootstrap-calibrated distribution of the minimum itself.

3. Sequential, functional, and control-informed variants

Minimal adjustment also arises when multiplicity is indexed by time or by a continuum. In group sequential designs, repeated interim analyses induce multiplicity even for a single hypothesis; multiple hypotheses induce a second layer. The adjusted-sequential p-values of (Zhao et al., 2023) are designed precisely for the joint problem: simultaneous adjustment for multiple hypotheses and multiple looks, with rejection of an elementary hypothesis when its adjusted-sequential p-value is less than or equal to the desired FWER.

For functional hypotheses, "Distribution-Free Pointwise Adjusted P-Values for Functional Hypotheses" (Xu et al., 2019) constructs pointwise adjusted p-values on a grid $U_{k:n}$5 from global envelope tests. If $U_{k:n}$6 is the pointwise rank of the observed curve and $U_{k:n}$7 is the min-rank depth of permutation curve $U_{k:n}$8, the single-step GET-adjusted p-value is

$U_{k:n}$9

These pointwise adjusted p-values control the FWER, and the paper proposes step-down and ERL-based variants that are always less than or equal to $f$0 while preserving FWER control. In this context, minimal adjustment is explicitly tied to exploiting the empirical joint permutation null rather than generic worst-case inequalities.

A more localized notion appears in paired-control calibration for intracellular cytokine staining assays. "Determining vaccine responders in the presence of baseline immunity using single-cell assays and paired control samples" (Chen et al., 8 Jul 2025) defines a family of p-values $f$1 indexed by misclassification parameters $f$2, restricted by a control-based confidence set $f$3. The minimally adjusted p-value is

$f$4

where $f$5 is the unadjusted p-value. The corresponding theorem states that, under pathwise connectedness of $f$6, the minimally adjusted p-value is, with probability at least $f$7 asymptotically, at least as close to the true p-value as the unadjusted p-value. Unlike the maximally adjusted p-value in the same paper, it is not designed for strict type I error control; it is a closest-compatible correction.

4. Discreteness, mid-p values, and optimal-transport adjustments

Discrete tests supply a major source of minimally adjusted p-values because ordinary exact p-values are conservative. For a discrete test statistic $f$8, the ordinary p-value is stochastically larger than $f$9, whereas the mid-p-value

$f(U_{k:n})$0

splits the atom at the observed statistic. "Meta-analysis of mid-p-values: some new results based on the convex order" (Rubin-Delanchy et al., 2015) shows that the mid-p-value is not conservative in the usual stochastic order but satisfies $f(U_{k:n})$1, that is, it is dominated by the uniform distribution in convex order. This weaker structure suffices to derive conservative combination rules for independent mid-p-values.

A broader discrete framework is given in "Multiple Test Functions and Adjusted p-Values for Test Statistics with Discrete Distributions" (Habiger, 2014), which unifies randomized p-values, mid-p-values, abstract randomized p-values, and multiplicity-adjusted versions thereof. The paper emphasizes that, whenever randomized and nonrandomized decisions may differ, the adjusted abstract randomized p-value and the associated test function should be reported, especially when the number of tests is large.

Optimal-transport formulations sharpen the notion of minimality. "A minimum Wasserstein distance approach to Fisher's combination of independent discrete p-values" (Contador et al., 2023) defines the minimally adjusted discrete statistic as the one minimizing the Wasserstein distance to a target continuous null law. In that framework, Lancaster’s mid-p-value and mean-value $f(U_{k:n})$2 statistic arise as special cases. The follow-up "Optimal Adjustment and Combination of Independent Discrete $f(U_{k:n})$3-Values" (Contador et al., 4 Aug 2025) extends the construction to Fisher’s, Pearson’s, George’s, Stouffer’s, and Edgington’s statistics by replacing each discrete transform with the conditional mean of the continuous target transform over the corresponding p-value atom. The paper further shows that accurate Type I error control is achieved when the variance of the adjusted discrete statistic closely matches that of the continuous case.

The most explicit discrete minimality terminology appears in "Minimally Discrete and Minimally Randomized p-Values" (Habiger et al., 20 Feb 2026). There, minimally discrete natural and mid-p-values refine the support by ranking the entire sample space rather than only the original test statistic, and they dominate their non-MD counterparts in the stochastic and convex order. Minimally randomized p-values remain exactly uniform under the null but reduce the conditional variance attributable to the auxiliary uniform variate. These constructions make the discreteness effect as small as possible while preserving the target validity property.

5. Calibration, generalized p-like quantities, and indirect-information shifts

Some papers broaden minimal adjustment beyond ordinary multiplicity or discreteness. "Testing with p*-values: Between p-values, mid p-values, and e-values" (Wang, 2020) introduces p*-variables through the stochastic order $f(U_{k:n})$4, weaker than the p-variable condition $f(U_{k:n})$5. A central result is that a random variable is a p*-variable if and only if it is a convex combination of p-variables; posterior predictive p-values are a principal example, and mid-p-values fall inside this class. For deterministic calibration back to ordinary p-values, the paper shows that

$f(U_{k:n})$6

dominates all other p*-to-p calibrators. This establishes a sharp factor-of-two bridge between minimally adjusted p-like quantities and ordinary valid p-values.

A different use of minimal adjustment appears in indirect-information testing. "Smaller $f(U_{k:n})$7-values via indirect information" (Hoff, 2019) develops FAB p-values by replacing the classical two-sided statistic $f(U_{k:n})$8 with a shifted statistic $f(U_{k:n})$9, where the scalar $f \ge f_{n,k}$0 is learned from indirect information through a linking model. The resulting p-value

$f \ge f_{n,k}$1

remains exactly uniform under the null for any fixed $f \ge f_{n,k}$2, and still remains uniform when $f \ge f_{n,k}$3 is random but independent of the direct statistic. In that sense, the adjustment is minimal with respect to null calibration: the shift can make p-values smaller on average under informative alternatives without altering the null law.

These two lines of work show that minimal adjustment need not mean “closest to the raw p-value” numerically. It may instead mean “smallest relaxation of the p-value axioms” or “smallest structural modification of the test statistic consistent with exact null validity.”

6. Interpretation, applications, and limitations

Several misconceptions recur in this literature. A minimally adjusted p-value is not inherently a posterior probability that the null is true. "Simple estimators of false discovery rates given as few as one or two p-values without strong parametric assumptions" (Bickel, 2011) treats local false discovery rates as conservative posterior-probability-like quantities, but explicitly contrasts them with adjusted p-values from FWER or FDR procedures. Likewise, "From $f \ge f_{n,k}$4-Values to Posterior Probabilities of Hypothesis" (Vélez et al., 2022) calibrates p-values to Bayes factors and posterior probabilities; those quantities are probability-like transformations, not multiplicity-adjusted p-values in the classical sense.

The validity target also matters. Some minimally adjusted procedures guarantee exact or asymptotic p-value validity under a null model, such as $f \ge f_{n,k}$5 (Mattner, 2010), FAB p-values (Hoff, 2019), or minimally randomized p-values (Habiger et al., 20 Feb 2026). Others target FWER under closure or resampling, such as adjusted-sequential p-values (Zhao et al., 2023), GET-adjusted pointwise p-values (Xu et al., 2019), or minP procedures (Mandl et al., 2024). Still others, such as the paired-control minimally adjusted p-value of (Chen et al., 8 Jul 2025), are explicitly exploratory and may be closer to the true p-value without providing strict type I error control.

In practical terms, selection among minimally adjusted p-values is therefore problem-specific. If the core issue is hidden multiplicity across analyses, a dependence-aware minP adjustment is appropriate. If the issue is repeated looks in a trial, a sequentially closed testing construction is required. If discreteness dominates, mid-p, MD, MR, or Wasserstein-optimal adjustments become relevant. If the goal is to exploit side information without sacrificing null calibration, FAB-style shifts are natural. The encyclopedia-level lesson is that minimal adjustment is best understood as a design principle: impose the weakest modification that attains the inferential guarantee actually needed, and no stronger one.