How important are the genes to explain the outcome - the asymmetric Shapley value as an honest importance metric for high-dimensional features

Mark A. van de Wiel; Jeroen Goedhart; Martin Jullum; Kjersti Aas

stat.MLMar 5, 2026

How important are the genes to explain the outcome - the asymmetric Shapley value as an honest importance metric for high-dimensional features

Authors: Mark A. van de Wiel, Jeroen Goedhart, Martin Jullum, Kjersti Aas

Organizations: Dept of Epidemiology and Data Science, Amsterdam Public Health research institute, Amsterdam University Medical Centers, Amsterdam, The Netherlands · Norwegian Computing Center, Oslo, Norway

Abstract

In clinical prediction settings the importance of a high-dimensional feature like genomics is often assessed by evaluating the change in predictive performance when adding it to a set of traditional clinical variables. This approach is questionable, because it does not account for collinearity nor known directionality of dependencies between variables. We suggest to use asymmetric Shapley values as a more suitable alternative to quantify feature importance in the context of a mixed-dimensional prediction model. We focus on a setting that is particularly relevant in clinical prediction: disease state as a mediating variable for genomic effects, with additional confounders for which the direction of effects may be unknown. We derive efficient algorithms to compute local and global asymmetric Shapley values for this setting. The former are shown to be very useful for inference, whereas the latter provide interpretation by decomposing any predictive performance metric into contributions of the features. Throughout, we illustrate our framework by a leading example: the prediction of progression-free survival for colorectal cancer patients.

Figures & tables

Shapley value ∗	Local feature importance (explaining individual prediction)
SAGE	Global feature importance (explaining model performance)
Symmetric	No directional effects
Asymmetric	Directional and undirectional effects
SHAP	Estimation by marginalizing over features not in the subset
Refitting	Estimation by refitting models on subsets of features
Marginal approach	SHAP assuming independence for marginalization

Table 1: Concepts related to Shapley values.

Figure 1: Model for information flow for

G

: gene expression profile,

C

: confounders,

D

: disease state,

Y

: outcome. Right arrows depict unidirectional effects, bidirectional arrows depict correlations. Grey arrows indicate prediction, black arrows dependency between variables. Modules may contain multiple features.

	C-index	tAUC
blockForest	0.754	0.789
fusedTree	0.747	0.777
ridge0	0.710	0.715
blockForest (reduced)	0.731	0.737

Table 2: Predictive performances

	Sym	Asym	$\text{LLR}_{\text{Sym}}$	$\text{LLR}_{\text{Asym}}$	$p_{\text{Sym}}$	$p_{\text{Asym}}$
intercept	0.500	0.500
Genes ( $G$ )	0.130	0.173	12.39	14.56	1.72e-05	2.12e-06
– $G$ High ( $G_{1}$ )	0.076	0.082	7.87	7.14	7.27e-05	1.57e-04
– $G$ Low ( $G_{2}+G_{3}$ )	0.053	0.091	2.28	4.91	0.102	7.39e-03
— CMS ( $G_{2}$ )	0.028	0.043	1.59	2.17	0.075	0.037
— Pred $D$ ( $G_{3}$ )	0.025	0.048	0.43	2.36	0.356	0.030

Table 3: SAGE decomposition of C-index of blockForest model for symmetric and asymmetric version. Plus (partial) likelihood-ratio (log; LLR) and

p

-values based on the likelihood-ratio test using Shapley values on log cumulative hazard scale.

Figure 2: Mediation effect. Shapley values for Genes

G

(y-axis) against true disease state

D

(x-axis) for the asymmetric (left box) and symmetric (right box) version.

	$G$	$D$	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$	$C_{5}$	$C_{6}$	$C_{7}$	$C_{8}$	$C_{9}$	$C_{10}$	$C_{11}$	$C_{12}$
$\hat{\phi}^{+}_{h}$	1.01	1.00	0.96	0.98	1.01	1.02	1.01	1.01	1.00	1.01	1.00	0.99	1.03	0.98
$\hat{\phi}^{-}_{h}$	0.99	1.01	1.00	1.00	1.02	1.00	1.00	1.00	1.01	0.97	1.01	1.00	0.97	0.99

Table 4: Estimated asymmetric Shapley values for constant contribution function. Estimation based on importance sampling with

B=10,000

sampled subsets

Variable	Asymmetric	Symmetric	Marginal
$G$	$G\left(\beta_{1}+\beta_{2}\gamma\right)$	$\beta_{1}G+\frac{\gamma}{2}\!\left(\beta_{2}G-\beta_{1}D\right)$	$\beta_{1}G$
$D$	$\beta_{2}\!\left(D-\gamma G\right)$	$\beta_{2}D+\frac{\gamma}{2}\!\left(\beta_{1}D-\beta_{2}G\right)$	$\beta_{2}D$

Table 5: Asymmetric, symmetric, and marginal Shapley values (

\phi

) for the toy example described by ( 7 ).

Figure 3: Analytical Shapley values for variables

G

(bottom panels) and

D

(top panels) as a function of the variable values. Both the symmetric (black) and the asymmetric (orange) versions are depicted. Results are shown for varying correlation strengths

\rho

. Remaining parameter settings:

\beta_{1}=\beta_{2}=1.0

\beta_{3}=5.0

, and

\gamma=0.8

. The plot is produced using the R -package ggplot2 ( Wickham, 2016 ) .

Figure 4: Shapley scatter plots for 200 test observations for the low-dimensional simulation. Variable on the x-axis, its Shapley value on y-axis. All Shapley values are estimated using SHAP, i.e. marginalisation.

	Sym-marg	Sym-cond	Asym
$C_{1}$	0.11	0.11	0.11
$C_{2}$	0.00	0.02	0.04
$D$	0.66	0.50	0.26
$G$	0.11	0.25	0.48
Total	0.88	0.88	0.88

Table 6: SAGE-

R^{2}

on test observations for the low-dimensional simulation.

Figure 5: Asymmetric Shapley values based on refitting. Scatter plots for 200 test observations for the low-dimensional simulation. Variable on the x-axis, its Shapley value on y-axis.

	fusedTree		ridge0
	Sym	Asym	Sym	Asym
Intercept	0.500	0.500	0.500	0.500
Genes ( $G$ )	0.122	0.170	0.075	0.143
Disease state ( $D$ )	0.121	0.071	0.118	0.058
Gender ( $C_{1}$ )	0.003	0.005	0.028	0.008
Age ( $C_{2}$ )	0.005	0.007	0.003	0.005

Table 7: SAGE decomposition of C-index of fusedTree and ridge0 models for asymmetric and symmetric setting

Figure 6: Symmetric and asymmetric Shapley values for

G

and

D

for the blockForest model. Shapley values were computed on the log cumulative hazard scale

	Sym	Asym	$p_{\text{Sym}}^{\text{NP}}$	$p_{\text{Asym}}^{\text{NP}}$
intercept	0.500	0.500
Genes ( $G$ )	0.130	0.173	4.39e-3	5.57e-4
– $G$ High ( $G_{1}$ )	0.076	0.082	0.058	0.039
– $G$ Low ( $G_{2}+G_{3}$ )	0.053	0.091	0.502	0.014
— CMS ( $G_{1}$ )	0.028	0.043	0.398	0.054
— Pred $D$ ( $G_{2}$ )	0.025	0.048	0.717	0.245

Table 8: SAGE decomposition of C-index of blockForest model for symmetric and asymmetric version. Plus significance based on the nonparametric (NP) conditional independence test using Shapley values on log cumulative hazard scale.

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