Signed-Knockoff Procedure for FDR Control

• The paper introduces the signed-knockoff procedure, which leverages sign-adaptive p-values and knockoff reflections to achieve finite-sample false discovery rate control.
• It extends classical knockoff methods by using asymmetric rejection regions based on signed test statistics to enhance power and directional accuracy.
• Extensions such as multi-knockoffs, aggregation techniques, and calibrated wrappers improve stability, power, and interpretability in applications like genomics and finance.

The signed-knockoff procedure denotes a line of FDR-controlling methodology in which sign information is built into knockoff inference. In the classical knockoff literature, sign enters through antisymmetric statistics $W_j$: $|W_j|$ measures evidence against the null, while $\operatorname{sign}(W_j)$ records whether the original variable appears more important than its knockoff. In a later and more explicit formulation, the “signed-knockoff procedure” is a direction-adaptive multiple-testing method that operates on signed $p$-values and their knockoff reflections, allowing asymmetric rejection boundaries while retaining finite-sample FDR control (Luo et al., 2022, Tian et al., 21 Jul 2025).

1. Historical emergence and scope

Earlier knockoff work already treated directional information as a first-class inferential target, even when the exact phrase “signed-knockoff procedure” was not used. In high-dimensional selective inference for linear models, one framework split the observations into a screening part and an inference part, then applied the knockoff filter on the reduced model and proved control of the directional false discovery rate. In that formulation, the procedure “discovers” important variables as well as the directions, or signs, of their effects, and the expected proportion of wrongly chosen signs is controlled below the target level; the guarantee is non-asymptotic and holds for any distribution of the original features and any values of the unknown regression coefficients (Barber et al., 2016).

A later development made the terminology explicit. “A powerful procedure that controls the false discovery rate with directional information” introduces a method named the signed-knockoff procedure (SK) for large-scale multiple testing with directional information. Its stated goal is to use the signs of test statistics, such as up- or down-regulation in genomics, to improve power while controlling the ordinary FDR in finite samples (Tian et al., 21 Jul 2025).

The literature represented here suggests two closely related uses of the term. One use is broad: any knockoff procedure based on a signed antisymmetric comparison between an original variable and its knockoff. The other is narrow: the 2025 SK procedure based on signed $p$-values $q_i$ and knockoff reflections $\tilde q_i$ (Tian et al., 21 Jul 2025).

2. Classical signed knockoff mechanism

In fixed-$X$ and model-$X$ knockoffs, the central object is a feature statistic $W_j\in\mathbb R$ constructed from the original feature and its knockoff. The standard interpretation is that $|W_j|$0 measures the overall importance of the pair, while $|W_j|$1 records whether the original variable appears more important than its knockoff. This is the fundamental signed structure of knockoff inference (Luo et al., 2022).

A common construction begins with paired importance scores $|W_j|$2 and $|W_j|$3 and forms a signed contrast

$|W_j|$4

Other standard choices include the lasso coefficient-difference statistic

$|W_j|$5

and the lasso signed-max statistic

$|W_j|$6

These satisfy antisymmetry: swapping the original and knockoff flips the sign of $|W_j|$7 (Luo et al., 2022, Gimenez et al., 2018).

Under the null, this antisymmetry yields the sign-flip law

$|W_j|$8

which is the calibration device behind knockoff FDR control. The standard knockoff+ filter defines

$|W_j|$9

with FDP estimator

$\operatorname{sign}(W_j)$0

and threshold

$\operatorname{sign}(W_j)$1

The rejection set is then

$\operatorname{sign}(W_j)$2

In this classical formulation, “signed” refers to original-versus-knockoff dominance, not necessarily to the sign of a regression coefficient or scientific effect (Luo et al., 2022).

3. The explicit signed-knockoff procedure based on signed $\operatorname{sign}(W_j)$3-values

The 2025 SK procedure begins with test statistics $\operatorname{sign}(W_j)$4 and two-sided $\operatorname{sign}(W_j)$5-values $\operatorname{sign}(W_j)$6, and defines the signed $\operatorname{sign}(W_j)$7-value

$\operatorname{sign}(W_j)$8

Under the null, if $\operatorname{sign}(W_j)$9 and $p$0 is independent of $p$1, then $p$2. The associated knockoff is

$p$3

so that $p$4 and $p$5 are reflected around $p$6 on the positive side and around $p$7 on the negative side (Tian et al., 21 Jul 2025).

The procedure splits the signed $p$8-values into positive and negative groups, orders them by closeness to $p$9 or $p$0, and iteratively shrinks a rejection region from the middle toward the extremes. After $p$1 shrinkage steps, the rejection region is

$p$2

At each step, the algorithm removes exactly one pair from one side, with the side choice required to be measurable with respect to the masking filtration

$p$3

The estimated FDR is

$p$4

and the algorithm stops when $p$5, or when both sides are exhausted (Tian et al., 21 Jul 2025).

For power, the paper proposes a side-selection rule based on estimated local FDR under a parametric mixture

$p$6

with alternative density

$p$7

The next side is chosen by comparing the estimated local FDRs of the next positive-side and negative-side masked pairs (Tian et al., 21 Jul 2025).

4. Error criteria and theoretical guarantees

The 2025 SK paper controls the ordinary false discovery rate,

$p$8

not a separate directional FDR. Its main theorem states that, under the null independence condition, if null $p$9-values are $q_i$0 and independent of the signs of the test statistics, then the signed-knockoff procedure controls the FDR at level $q_i$1 (Tian et al., 21 Jul 2025).

This differs from earlier directional knockoff work in high-dimensional linear models, where the inferential target was explicitly the directional false discovery rate in a reduced model after screening. There the guarantee concerned wrongly chosen signs among selected variables, and the result was non-asymptotic (Barber et al., 2016).

A separate asymptotic strand studied sign errors for standard model-$q_i$2 knockoffs augmented with sign decisions. In that setting, after selecting

$q_i$3

one reports

$q_i$4

and analyzes the false sign proportion

$q_i$5

That work treats signed inference as an analytical extension of standard knockoff selection, rather than as a new finite-sample directional filter (Weinstein et al., 2020).

The resulting picture is technically important. Some signed-knockoff procedures control ordinary FDR while exploiting directional information in the ranking or rejection geometry; other knockoff procedures target directional or sign-error criteria directly. The distinction is substantive, not terminological (Tian et al., 21 Jul 2025, Barber et al., 2016).

5. Variants, generalizations, and power-oriented refinements

Several later developments preserve the signed core of knockoff inference while altering its stability, power, or geometric resolution. One extension is simultaneous multi-knockoffs, where the binary sign of a single $q_i$6 is replaced by a winner label $q_i$7 and a margin statistic $q_i$8. In the single-knockoff case, $q_i$9 and $\tilde q_i$0 corresponds to $\tilde q_i$1; under the null, the binary sign symmetry becomes uniform label symmetry on $\tilde q_i$2. This lowers the effective detection threshold and improves stability and power in sparse-signal regimes (Gimenez et al., 2018).

Another extension is Aggregation of Multiple Knockoffs (AKO), which keeps the signed statistic $\tilde q_i$3 within each run but repeats knockoff generation, converts each run into an intermediate $\tilde q_i$4-value

$\tilde q_i$5

and aggregates across runs. AKO therefore stabilizes signed-knockoff evidence rather than replacing it (Nguyen et al., 2020).

Power-improving wrappers preserve the same signed/antisymmetric core. The calibrated knockoff procedure begins from any valid fixed-$\tilde q_i$6 or model-$\tilde q_i$7 knockoff rejection set $\tilde q_i$8, uses the usual signed statistics $\tilde q_i$9, and augments the rejection set by conditionally calibrated fallback tests while retaining FDR control. Its purpose is to improve power without altering the underlying sign-flip mechanism (Luo et al., 2022).

The quality of signed comparisons also depends on knockoff construction. “Powerful Knockoffs via Minimizing Reconstructability” argues that minimizing mean absolute correlation can make original-versus-knockoff signs unreliable, because machine-learning procedures may reconstruct signal through the knockoffs. It proposes minimizing reconstructability instead, with the aim of producing stronger and more reliable antisymmetric comparisons such as

$X$0

This places knockoff construction, not only downstream statistics, at the center of signed-knockoff performance (Spector et al., 2020).

6. Interpretation, applications, and recurrent misconceptions

A persistent interpretive issue is the meaning of “sign.” In standard knockoff methodology, the sign of $X$1 indicates whether the original feature defeats its knockoff, not whether the underlying scientific effect is positive or negative. A finance application makes this distinction explicit: the relevant sign is induced by comparing the importance score of each original factor with that of its knockoff, and it is distinct from the sign of returns and also distinct from the sign of regression coefficients (Challet et al., 2021).

By contrast, the 2025 SK procedure uses the sign of the original test statistic $X$2 to define

$X$3

so its signed structure is directly directional in the scientific sense. This is why the method can use asymmetric rejection regions of the form

$X$4

and adapt differently to the positive and negative sides (Tian et al., 21 Jul 2025).

Applications reflect both meanings. Earlier high-dimensional knockoff work applied directional sign control to genome-wide association analysis with a continuous phenotype (Barber et al., 2016). The explicit SK procedure is motivated by genetics and illustrated on gene-expression-style settings where the sign of the test statistic corresponds to up- or down-regulation (Tian et al., 21 Jul 2025). Other knockoff-based applications, including finance and multi-resolution localization, rely on the signed antisymmetry of original-versus-knockoff comparisons even when no effect-direction claim is made (Challet et al., 2021, Gablenz et al., 2023).

Construction validity is another recurring theme. In case-control studies, knockoff variables can be built using controls only, cases only, or arbitrary mixtures, and the resulting exchangeability validates the construction step for any downstream signed-knockoff pipeline that depends on the usual swap symmetry (Barber et al., 2018). Robustness work on approximate knockoffs shows, in a different way, that the standard signed/antisymmetric logic can survive misspecified feature models asymptotically when approximate statistics can be coupled closely to ideal model-$X$5 statistics (Fan et al., 2023).

Taken together, these developments establish the signed-knockoff procedure as both a specific direction-adaptive testing algorithm and a broader inferential principle. In the narrow sense, it is the signed-$X$6-value procedure with knockoff reflections and finite-sample ordinary FDR control (Tian et al., 21 Jul 2025). In the broader sense, it is the use of signed antisymmetric knockoff statistics—together with null sign symmetry, positive-versus-negative thresholding, and knockoff construction—to perform controlled variable selection or multiple testing with sign-aware evidence (Luo et al., 2022, Barber et al., 2016).