Beyond False Discovery Rate: A Stepdown Group SLOPE Approach for Grouped Variable Selection

Xuelin Zhang; Jingxuan Liang; Xinyue Liu; Hong Chen; Biqin Song

stat.MEMar 1, 2026

Beyond False Discovery Rate: A Stepdown Group SLOPE Approach for Grouped Variable Selection

Authors: Xuelin Zhang, Jingxuan Liang, Xinyue Liu, Hong Chen, Biqin Song

Organizations: College of Informatics, Huazhong Agricultural University, Wuhan, 430070, Hubei Province, China.

Abstract

High-dimensional feature selection is routinely required to balance statistical power with strict control of multiple-error metrics such as the k-Family-Wise Error Rate (k-FWER) and the False Discovery Proportion (FDP), yet some existing frameworks are confined to the narrower goal of controlling the expected False Discovery Rate (FDR) and can not exploit the group-structure of the covariates, such as Sorted L-One Penalized Estimation (SLOPE). We introduce the Group Stepdown SLOPE, a unified optimization procedure which is capable of embedding the Lehmann-Romano stepdown rules into SLOPE to achieve finite-sample guarantees under k-FWER and FDP thresholds. Specifically, we derive closed-form regularization sequences under orthogonal designs that provably bound k-FWER and FDP at user-specified levels, and extend these results to grouped settings via gk-SLOPE and gF-SLOPE, which control the analogous group-level errors gk-FWER and gFDP. For non-orthogonal general designs, we provide a calibrated data-driven sequence inspired by Gaussian approximation and Monte-Carlo correction, preserving convexity and scalability. Extensive simulations are conducted across sparse, correlated, and group-structured regimes. Empirical results corroborate our theoretical findings that the proposed methods achieve nominal error control, while yielding markedly higher power than competing stepdown procedures, thereby confirming the practical value of the theoretical advances.

Figures & tables

	FDR 1 Control	FDP 1 Control	$k$ -FWER 1 Control	Group-wise
SLOPE ( Bogdan et al., 2015 ; Su and Candès, 2016 ; Larsson et al., 2020a ; Dupuis and Tardivel, 2024 ; Minsker et al., 2024 )	$\surd$	$\times$	$\times$	$\times$
g-SLOPE ( Brzyski et al., 2019 ; Wei et al., 2021 ; Chen and Luo, 2023 ; Feser and Evangelou, 2023 )	$\surd$	$\times$	$\times$	$\surd$
Stepdown (FDP) ( Lehmann and Romano, 2005 )	$\times$	$\surd$	$\times$	$\times$
Stepdown ( $k$ -FWER) ( Lehmann and Romano, 2005 )	$\times$	$\times$	$\surd$	$\times$
F-SLOPE (Ours)	$\surd$	$\surd$	$\times$	$\times$
$k$ -SLOPE (Ours)	$\surd$	$\times$	$\surd$	$\times$

Table 1: Algorithmic properties (

\surd

- has the given information,

\times

- hasn’t the given information). 1 1 footnotetext: The group-level definitions of these three criteria (gFDR, gFDP and g

k

-FWER) are considered with respect to g-SLOPE, gF-SLOPE and g

k

-SLOPE.

t	$k$ -SLOPE			F-SLOPE
t	$\mathrm{Prob}(\mathrm{FDP}>\gamma)$	FDR	Power	$\mathrm{Prob}(\mathrm{FDP}>\gamma)$	FDR	Power
50	0.001	0.006	1.000	0.003	0.007	1.000
100	0.000	0.002	0.998	0.002	0.005	1.000
200	0.001	0.001	1.000	0.000	0.007	1.000
300	0.002	0.001	1.000	0.000	0.005	0.995
400	0.000	0.001	0.995	0.001	0.004	0.994

Table 2: Results for controlled feature selection under the orthogonal design (different

t

and fixed

k=5

Figure 1:

k

-FWER provided by different approaches for controlled feature selection under orthogonal design (with different

k

and

t

). The value in the small square is the size of

k

-FWER. The darker the color, the larger the

k

-FWER and vice versa.

Figure 2: Result for controlled feature selection on the simulated data. The black dashed lines indicate the target FDR level. Constance for

k

-SLOPE is

k=6

in the second column (from left to right). The value in the small square is the size of

k

-FWER in the third and fourth columns (from left to right). The darker the color, the larger the

k

-FWER and vice versa.

$t$	F-SLOPE	$k$ -SLOPE	Sd (FDP)	Sd ( $k$ -FWER)
10	0.00	0.04	0.02	0.08
20	0.03	0.00	0.00	0.03
30	0.00	0.01	0.01	0.01
40	0.00	0.00	0.00	0.00
50	0.01	0.00	0.00	0.00
60	0.00	0.00	0.00	0.00

Table 3:

\mathrm{Prob(FDP}>\gamma

) results on the simulated data for multiple mean testing (

k=6

)

$k/t$	10	30	40	50	60	70	80
2	0.02	0.02	0.00	0.04	0.04	0.07	0.07
4	0.00	0.00	0.01	0.00	0.00	0.00	0.01
6	0.00	0.00	0.00	0.00	0.00	0.00	0.00
8	0.00	0.00	0.01	0.00	0.00	0.00	0.01

Table 4:

k

-FWER of

k

-SLOPE (

m=2n

) on the simulated data under the weak signals

$k/t$	30	40	50	60	70	80
2	0.05	0.03	0.10	0.12	0.07	0.13
4	0.00	0.01	0.00	0.00	0.01	0.06
6	0.00	0.00	0.00	0.00	0.00	0.01
8	0.00	0.01	0.00	0.02	0.00	0.00

Table 5: FDR of F-SLOPE (

m=2n

) on the simulated data under the weak signals

Figure 3: Power and FDR of F-SLOPE under Gaussian design (different

t

). The black dashed line indicates the target FDR level.

Figure 4: The g

k

-FWER provided by different approaches for controlled feature selection under orthogonal design. The value in the small square is the size of g

k

-FWER. The darker the color, the larger the g

k

-FWER and vice versa.

Figure 5: The gFDR of g-SLOPE, g

k

-SLOPE and gF-SLOPE on the simulated data in orthogonal design when

k=15

$t$	$m=2n$		$m=n/2$
$t$	weak	moder	weak	moder
10	0.03	0.00	0.05	0.00
20	0.07	0.00	0.03	0.01
30	0.01	0.00	0.00	0.00
40	0.00	0.01	0.00	0.00
50	0.00	0.00	0.00	0.00

Table 6: Prob(FDP

>\gamma

) of F-SLOPE on the simulated data under the weak and moderate signals (different

t

the number	50	100	150	200
g-SLOPE	0.51	0.73	0.98	1.00
g $k$ -SLOPE	0.01	0.00	0.00	0.01
gFSLOPE	0.01	0.00	0.00	0.00

Table 7: Prob(FDP

>\gamma

) of g-SLOPE, g

k

-SLOPE and gF-SLOPE on the simulated data.

the number	50	100	150	200	250
g-SLOPE	0.8	0.9	0.93	0.95	0.96
g $k$ -SLOPE	0.99	0.98	1	0.99	1
gF-SLOPE	1	0.99	1	1	0.98

Table 8: Power values of g-SLOPE, g

k

-SLOPE and gF-SLOPE in orthogonal design.

Figure 6: The g

k

-FWER provided by different approaches for controlled feature selection under orthogonal design. The value in the small square is the size of g

k

-FWER. The darker the color, the larger the g

k

-FWER and vice versa.

Figure 7: The gFDR of g-SLOPE, g

k

-SLOPE and gF-SLOPE on the simulated data as

k=6

Size	10	20	30	40	50
g-SLOPE	0.42(0.32)	0.28(0.38)	0.46(0.48)	0.24(0.44)	0.48(0.56)
gk-SLOPE	0.16(0.00)	0.08(0.00)	0.12(0.02)	0.02(0.00)	0.1(0.00)
gF-SLOPE	0.16(0.06)	0.08(0.00)	0.12(0.00)	0.02(0.00)	0.1(0.00)

Table 9: Results of Prob(FDP

>\gamma

) on the simulated data. The values in different group configurations are shown in parentheses.

Size	50	100	150	200	250
g-SLOPE	0.68	0.87	0.9	0.92	0.93
g $k$ -SLOPE	1	1	1	1	1
gF-SLOPE	1	1	1	1	1

Table 10: Results of power values under general design.

Method	MSE ( $\times 10^{-2}$ )	Time (s)	gk-FWER	Prob(gFDP $\geq$ 0.1)
g-SLOPE	2.84 ± 0.21	2.34 ± 0.82	0.82	0.48
Group Lasso	2.79 ± 0.19	0.82 ± 0.02	0.91	0.63
g $k$ -SLOPE	2.61 ± 0.15	1.51 ± 0.43	0.12	0.17
gF-SLOPE	2.60 ± 0.14	1.59 ± 0.47	0.19	0.11

Table 11: Results on real-world ADNI data.

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