cs.LGFeb 25, 2026

Discovering Subgroups with Exceptional Survival Characteristics

Authors: Mhd Jawad Al Rahwanji, Sascha Xu, Nils Philipp Walter, Jilles Vreeken

Organizations: CISPA Helmholtz Center for Information Security Saarbrücken, Germany

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Abstract

In many applications, it is important to identify subpopulations that survive longer or shorter than the rest of the population. In medicine, for example, it allows determining which patients benefit from treatment, and in predictive maintenance, which components are more likely to fail. Existing methods for discovering subgroups with exceptional survival characteristics rely on restrictive assumptions about the survival model (e.g. proportional hazards), require pre-discretized features, and, as they compare average statistics, tend to overlook individual heterogeneity. In this paper, we propose Sysurv, a non-parametric, fully differentiable method that discovers human-readable rules selecting subgroups with exceptional survival characteristics. Empirical evaluation on a wide range of datasets and settings, including a case study on cancer data, shows that Sysurv reveals insightful and actionable survival subgroups, outperforming the state of the art.

Figures & tables

Figure 1: Survival subgroups . Sysurv finds patients suffering from a therapy-resistant tumor (Left) , and people with exceptionally long (1) resp. short (2) durations until re-employment (Right) .
Figure 2: (Left) Individual survival functions S^(t∣xi)\hat{S}(t|\mathbf{x}_{i}) are more informative than their group-level estimate S^(t)\hat{S}(t) . (Right) The absolute difference between group and reference survival (Pop.) is effective in the absence of crossing ( S^1\hat{S}_{1} and S^2\hat{S}_{2} ) and in its presence ( S^3\hat{S}_{3} ), unlike the signed difference.
Figure 3: Soft rules. The soft condition approaches a hard interval with decreasing temperature τ→0\tau\to 0 (a) . Multiple soft conditions combine to form a soft rule depicted as a hyper box in the covariate space (b) . Adapted from Xu et al. (2024) .
Figure 4: Synthetic setting. Comparison of Sysurv and each of RuleKit , EsmamDs , and Fibers in terms of F1F_{1} -scores recovering planted subgroups with increasingly large dataset dimensionalities (Left) , increasingly many censored subjects (Center) , and increasingly large planted subgroups (Right) . EsmamDs is the closest competitor to Sysurv closely followed by RuleKit . Higher is better. The shaded areas show ±1 standard error over 10 runs.
 Our objectiveLogrankMean-shift
 OursRKEDFibOursRKEDFibOursRKEDFib
UnempDur6.43.83.83.819.3226.5125.0125.02.02.12.12.1
Nwtco1011.6681.0681.0208.6374.2281.331.40.01162.0688.2688.20.6
Rott2559.2326.2487.6487.6267.093.071.7169.331.510.025.225.2
Rdata259.2213.8213.8158.5113.4107.814.643.31116.6933.5933.5705.5
Aids266.660.554.034.70.21.74.30.022.761.261.40.7
Table 1: Real-world setting. Exceptionality of subgroups discovered by Sysurv ( Ours ), RuleKit ( RK ), EsmamDs ( ED ), and Fibers ( Fib ), measured using 3 metrics (our objective, logrank, and mean-shift). Sysurv ranks first across metrics. Greater is better. Highest values marked in bold.
Figure 5: Real-world setting. Survival subgroups discovered in the unemployment ( a ) and heart attack ( b ) datasets using Sysurv and RuleKit resp. EsmamDs . Sysurv learns more exceptional subsets of the subgroups discovered by baselines. The shaded areas show 95% confidence intervals.
Figure 6: Case study. Sysurv discovers, overall, sets of diverse subgroups in the primary cohort ( a ), and two subgroups of poor responders in the postoperative cohort ( b ) of HNSCC data. The shaded areas show 95% confidence intervals.
Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Histogram of the exceptionality scores obtained from 1000 runs of BuildDFD on the postoperative cohort from our case study, with a fitted normal distribution. The mean and standard deviation of the fitted normal distribution are μ=16.9\mu=16.9 and η=2.77\eta=2.77 , respectively.
Figure 8: Synthetic setting. Comparison of Sysurv and each of RuleKit , EsmamDs , and Fibers in terms of runtime recovering planted subgroups with increasingly large dataset dimensionalities (Left) . Lower is better. Also, in terms of F1F_{1} -score, comparisons between Sysurv and baselines under varying population-subgroup hazard ratios (Center) , where the hazard ratio indicates the rate by which the population outsurvives the subgroup, or vice versa, and increasingly large datasets (Right) . Higher is better. EsmamDs is the closest competitor to Sysurv closely followed by RuleKit . The shaded areas show ±1 standard error over 10 runs.
 nnSysurvRuleKitEsmamDsFibers
UnempDur3241112.0 (3.5%)1620.5 (50.0%)1620.5 (50.0%)1585.6 (48.9%)
Nwtco4028168.0 (4.2%)2014.0 (50.0%)1490.0 (37.0%)4027.0 (100.0%)
Rott22982320.0 (10.7%)1825.7 (61.2%)1007.2 (33.8%)1210.6 (40.6%)
Rdata1040140.0 (13.5%)520.0 (50.0%)268.8 (25.8%)503.2 (48.4%)
Aids22839306.0 (10.8%)1342.6 (47.3%)503.7 (17.7%)2834.0 (99.8%)
Dialysis68052665.0 (39.2%)4095.7 (60.2%)2636.8 (38.7%)2568.6 (37.7%)
Appendix
Table 2: Average subgroup sizes (with coverage percentages) per method across real-world datasets.
Table 11
Original rulePruned ruleChanges
“ ACTN1∈[−3.3,−0.9]∧AKT1∈[−3.6,−1.9]∧ALDH3A1∈[−8.9,−1.4]∧ANLN∈[−4.3,−2.2]∧BCL2L1∈[−7.2,−5.3]∧BNIP3L∈[−3.0,−0.6]∧CAV1∈[−4.4,−1.1]∧CDKN3∈[−4.5,−2.5]∧CXCL12∈[−8.2,−2.6]∧DCBLD1∈[−6.9,−4.7]∧ERBB2∈[−1.9,0.7]∧ERBB3∈[−7.0,−2.9]∧GNAI1∈[−5.2,−0.0]∧IGF1R∈[−3.5,−0.9]∧LOX∈[−6.4,−3.3]∧MTOR∈[−3.2,−2.3]∧NOTCH1∈[−4.9,−2.6]∧PLAU∈[−4.5,−1.2]∧RB1∈[−3.8,−2.4]∧RELA∈[−3.2,−1.8]∧RFC4∈[−4.6,−1.6]∧RPA2∈[−4.0,−2.1]∧SNAI1∈[−6.9,−3.3]∧TCF3∈[−3.2,−1.3]∧TGFB1∈[−2.6,−1.0]∧TPI1∈[−7.2,−5.3]∧XPA∈[−4.7,−1.6]∧XRCC1∈[−4.2,−2.6]\mathit{ACTN1}\in[-3.3,-0.9]\land\mathit{AKT1}\in[-3.6,-1.9]\land\mathit{ALDH3A1}\in[-8.9,-1.4]\land\mathit{ANLN}\in[-4.3,-2.2]\land\mathit{BCL2L1}\in[-7.2,-5.3]\land\mathit{BNIP3L}\in[-3.0,-0.6]\land\mathit{CAV1}\in[-4.4,-1.1]\land\mathit{CDKN3}\in[-4.5,-2.5]\land\mathit{CXCL12}\in[-8.2,-2.6]\land\mathit{DCBLD1}\in[-6.9,-4.7]\land\mathit{ERBB2}\in[-1.9,0.7]\land\mathit{ERBB3}\in[-7.0,-2.9]\land\mathit{GNAI1}\in[-5.2,-0.0]\land\mathit{IGF1R}\in[-3.5,-0.9]\land\mathit{LOX}\in[-6.4,-3.3]\land\mathit{MTOR}\in[-3.2,-2.3]\land\mathit{NOTCH1}\in[-4.9,-2.6]\land\mathit{PLAU}\in[-4.5,-1.2]\land\mathit{RB1}\in[-3.8,-2.4]\land\mathit{RELA}\in[-3.2,-1.8]\land\mathit{RFC4}\in[-4.6,-1.6]\land\mathit{RPA2}\in[-4.0,-2.1]\land\mathit{SNAI1}\in[-6.9,-3.3]\land\mathit{TCF3}\in[-3.2,-1.3]\land\mathit{TGFB1}\in[-2.6,-1.0]\land\mathit{TPI1}\in[-7.2,-5.3]\land\mathit{XPA}\in[-4.7,-1.6]\land\mathit{XRCC1}\in[-4.2,-2.6] ”“ CDKN3∈[−4.5,−2.5]∧ERBB2∈[−1.9,0.7]∧LOX∈[−6.4,−3.3]∧RB1∈[−3.8,−2.4]∧RPA2∈[−4.0,−2.1]∧TGFB1∈[−2.6,−1.0]\mathit{CDKN3}\in[-4.5,-2.5]\land\mathit{ERBB2}\in[-1.9,0.7]\land\mathit{LOX}\in[-6.4,-3.3]\land\mathit{RB1}\in[-3.8,-2.4]\land\mathit{RPA2}\in[-4.0,-2.1]\land\mathit{TGFB1}\in[-2.6,-1.0] ”Subgroup size: 6→66\to 6 Exceptionality: 19.1051→19.103719.1051\to 19.1037
“ AKT1∈[−3.2,−1.2]∧ALDH1A1∈[−8.1,−2.3]∧ASS1∈[−4.4,−0.4]∧CENPK∈[−6.1,−4.4]∧CHEK2∈[−5.5,−3.8]∧CXCR4∈[−6.6,−2.8]∧CYP1B1∈[−11.3,−4.5]∧ERBB3∈[−5.5,−3.5]∧ERBB4∈[−10.6,−7.6]∧FGFR3∈[−7.1,−3.1]∧FOSL1∈[−5.3,−0.7]∧GNAI1∈[−3.2,−0.4]∧GPI∈[−2.0,0.2]∧HIF1A∈[−6.8,−4.7]∧KDR∈[−5.5,−3.5]∧LDHA∈[−6.9,−4.3]∧LGALS1∈[−2.6,1.0]∧LIMD1∈[−6.2,−4.7]∧LOX∈[−4.1,−0.9]∧MCM6∈[−5.9,−3.7]∧MET∈[−5.3,−2.8]∧MMP7∈[−9.2,−3.4]∧MYC∈[−2.8,0.3]∧NBN∈[−3.7,−2.7]∧PDK1∈[−4.9,−3.5]∧PFKFB3∈[−3.9,−0.7]∧RASSF6∈[−7.6,−5.0]∧RB1∈[−4.4,−2.5]∧SYK∈[−6.0,−3.9]∧SYNGR3∈[−8.9,−4.7]∧TPI1∈[−7.2,−4.8]∧XRCC1∈[−5.4,−3.4]∧CD44∈[−0.1,2.1]\mathit{AKT1}\in[-3.2,-1.2]\land\mathit{ALDH1A1}\in[-8.1,-2.3]\land\mathit{ASS1}\in[-4.4,-0.4]\land\mathit{CENPK}\in[-6.1,-4.4]\land\mathit{CHEK2}\in[-5.5,-3.8]\land\mathit{CXCR4}\in[-6.6,-2.8]\land\mathit{CYP1B1}\in[-11.3,-4.5]\land\mathit{ERBB3}\in[-5.5,-3.5]\land\mathit{ERBB4}\in[-10.6,-7.6]\land\mathit{FGFR3}\in[-7.1,-3.1]\land\mathit{FOSL1}\in[-5.3,-0.7]\land\mathit{GNAI1}\in[-3.2,-0.4]\land\mathit{GPI}\in[-2.0,0.2]\land\mathit{HIF1A}\in[-6.8,-4.7]\land\mathit{KDR}\in[-5.5,-3.5]\land\mathit{LDHA}\in[-6.9,-4.3]\land\mathit{LGALS1}\in[-2.6,1.0]\land\mathit{LIMD1}\in[-6.2,-4.7]\land\mathit{LOX}\in[-4.1,-0.9]\land\mathit{MCM6}\in[-5.9,-3.7]\land\mathit{MET}\in[-5.3,-2.8]\land\mathit{MMP7}\in[-9.2,-3.4]\land\mathit{MYC}\in[-2.8,0.3]\land\mathit{NBN}\in[-3.7,-2.7]\land\mathit{PDK1}\in[-4.9,-3.5]\land\mathit{PFKFB3}\in[-3.9,-0.7]\land\mathit{RASSF6}\in[-7.6,-5.0]\land\mathit{RB1}\in[-4.4,-2.5]\land\mathit{SYK}\in[-6.0,-3.9]\land\mathit{SYNGR3}\in[-8.9,-4.7]\land\mathit{TPI1}\in[-7.2,-4.8]\land\mathit{XRCC1}\in[-5.4,-3.4]\land\mathit{CD44}\in[-0.1,2.1] ”“ ALDH1A1∈[−8.1,−2.3]∧ASS1∈[−4.4,−0.4]∧CYP1B1∈[−11.3,−4.5]∧ERBB3∈[−5.5,−3.5]∧ERBB4∈[−10.6,−7.6]∧FOSL1∈[−5.3,−0.7]∧GNAI1∈[−3.2,−0.4]∧GPI∈[−2.0,0.2]∧HIF1A∈[−6.8,−4.7]∧KDR∈[−5.5,−3.5]∧LDHA∈[−6.9,−4.3]∧LGALS1∈[−2.6,1.0]∧LIMD1∈[−6.2,−4.7]∧LOX∈[−4.1,−0.9]∧MCM6∈[−5.9,−3.7]∧MET∈[−5.3,−2.8]∧MMP7∈[−9.2,−3.4]∧MYC∈[−2.8,0.3]∧NBN∈[−3.7,−2.7]∧PDK1∈[−4.9,−3.5]∧PFKFB3∈[−3.9,−0.7]∧RASSF6∈[−7.6,−5.0]∧RB1∈[−4.4,−2.5]∧SYK∈[−6.0,−3.9]∧SYNGR3∈[−8.9,−4.7]∧TPI1∈[−7.2,−4.8]∧XRCC1∈[−5.4,−3.4]∧CD44∈[−0.1,2.1]\mathit{ALDH1A1}\in[-8.1,-2.3]\land\mathit{ASS1}\in[-4.4,-0.4]\land\mathit{CYP1B1}\in[-11.3,-4.5]\land\mathit{ERBB3}\in[-5.5,-3.5]\land\mathit{ERBB4}\in[-10.6,-7.6]\land\mathit{FOSL1}\in[-5.3,-0.7]\land\mathit{GNAI1}\in[-3.2,-0.4]\land\mathit{GPI}\in[-2.0,0.2]\land\mathit{HIF1A}\in[-6.8,-4.7]\land\mathit{KDR}\in[-5.5,-3.5]\land\mathit{LDHA}\in[-6.9,-4.3]\land\mathit{LGALS1}\in[-2.6,1.0]\land\mathit{LIMD1}\in[-6.2,-4.7]\land\mathit{LOX}\in[-4.1,-0.9]\land\mathit{MCM6}\in[-5.9,-3.7]\land\mathit{MET}\in[-5.3,-2.8]\land\mathit{MMP7}\in[-9.2,-3.4]\land\mathit{MYC}\in[-2.8,0.3]\land\mathit{NBN}\in[-3.7,-2.7]\land\mathit{PDK1}\in[-4.9,-3.5]\land\mathit{PFKFB3}\in[-3.9,-0.7]\land\mathit{RASSF6}\in[-7.6,-5.0]\land\mathit{RB1}\in[-4.4,-2.5]\land\mathit{SYK}\in[-6.0,-3.9]\land\mathit{SYNGR3}\in[-8.9,-4.7]\land\mathit{TPI1}\in[-7.2,-4.8]\land\mathit{XRCC1}\in[-5.4,-3.4]\land\mathit{CD44}\in[-0.1,2.1] ”Subgroup size: 12→1212\to 12 Exceptionality: 16.6102→16.607916.6102\to 16.6079
“ ADM∈[−5.3,−0.8]∧AKT1∈[−3.6,−1.7]∧ALDH3A1∈[−8.9,−0.5]∧ANLN∈[−4.3,−2.8]∧ASS1∈[−4.9,−0.0]∧ATP5G3∈[−1.3,1.4]∧ATR∈[−5.9,−3.8]∧BCL2L1∈[−7.8,−4.9]∧BIRC5∈[−4.7,−1.0]∧CAV1∈[−4.4,0.3]∧CDKN3∈[−5.9,−2.5]∧CLDN4∈[−5.0,−0.6]∧CXCL12∈[−8.2,−3.2]∧ERBB3∈[−6.3,−3.2]∧ERCC5∈[−5.2,−3.6]∧FAM83B∈[−7.4,−1.7]∧FLT1∈[−5.7,−3.1]∧GNAI1∈[−5.2,−0.2]∧IGF1R∈[−3.5,−0.6]∧INHBA∈[−4.4,1.9]∧KDR∈[−5.5,−2.8]∧LDHA∈[−7.5,−5.7]∧LGALS1∈[−2.1,1.0]∧LOX∈[−6.4,−1.4]∧MDM2∈[−4.6,−2.2]∧MMP13∈[−9.3,1.3]∧MRE11A∈[−6.1,−3.6]∧MTOR∈[−3.7,−2.3]∧NOTCH1∈[−5.9,−2.6]∧P4HA2∈[−4.7,−1.4]∧PSMD9∈[−4.2,−2.1]∧PTEN∈[−7.3,−5.6]∧RAD23B∈[−0.9,0.8]∧RB1∈[−3.8,−2.3]∧RPA2∈[−4.9,−2.1]∧SERPINB2∈[−6.6,−1.2]∧TCF3∈[−3.5,−1.0]∧TGFB1∈[−2.6,−0.3]∧TPI1∈[−7.2,−5.0]∧XRCC1∈[−5.3,−2.6]∧XRCC5∈[−2.5,−0.8]∧CD44∈[−0.8,2.5]\mathit{ADM}\in[-5.3,-0.8]\land\mathit{AKT1}\in[-3.6,-1.7]\land\mathit{ALDH3A1}\in[-8.9,-0.5]\land\mathit{ANLN}\in[-4.3,-2.8]\land\mathit{ASS1}\in[-4.9,-0.0]\land\mathit{ATP5G3}\in[-1.3,1.4]\land\mathit{ATR}\in[-5.9,-3.8]\land\mathit{BCL2L1}\in[-7.8,-4.9]\land\mathit{BIRC5}\in[-4.7,-1.0]\land\mathit{CAV1}\in[-4.4,0.3]\land\mathit{CDKN3}\in[-5.9,-2.5]\land\mathit{CLDN4}\in[-5.0,-0.6]\land\mathit{CXCL12}\in[-8.2,-3.2]\land\mathit{ERBB3}\in[-6.3,-3.2]\land\mathit{ERCC5}\in[-5.2,-3.6]\land\mathit{FAM83B}\in[-7.4,-1.7]\land\mathit{FLT1}\in[-5.7,-3.1]\land\mathit{GNAI1}\in[-5.2,-0.2]\land\mathit{IGF1R}\in[-3.5,-0.6]\land\mathit{INHBA}\in[-4.4,1.9]\land\mathit{KDR}\in[-5.5,-2.8]\land\mathit{LDHA}\in[-7.5,-5.7]\land\mathit{LGALS1}\in[-2.1,1.0]\land\mathit{LOX}\in[-6.4,-1.4]\land\mathit{MDM2}\in[-4.6,-2.2]\land\mathit{MMP13}\in[-9.3,1.3]\land\mathit{MRE11A}\in[-6.1,-3.6]\land\mathit{MTOR}\in[-3.7,-2.3]\land\mathit{NOTCH1}\in[-5.9,-2.6]\land\mathit{P4HA2}\in[-4.7,-1.4]\land\mathit{PSMD9}\in[-4.2,-2.1]\land\mathit{PTEN}\in[-7.3,-5.6]\land\mathit{RAD23B}\in[-0.9,0.8]\land\mathit{RB1}\in[-3.8,-2.3]\land\mathit{RPA2}\in[-4.9,-2.1]\land\mathit{SERPINB2}\in[-6.6,-1.2]\land\mathit{TCF3}\in[-3.5,-1.0]\land\mathit{TGFB1}\in[-2.6,-0.3]\land\mathit{TPI1}\in[-7.2,-5.0]\land\mathit{XRCC1}\in[-5.3,-2.6]\land\mathit{XRCC5}\in[-2.5,-0.8]\land\mathit{CD44}\in[-0.8,2.5] ”“ ANLN∈[−4.3,−2.8]∧ATR∈[−5.9,−3.8]∧CLDN4∈[−5.0,−0.6]∧CXCL12∈[−8.2,−3.2]∧ERBB3∈[−6.3,−3.2]∧ERCC5∈[−5.2,−3.6]∧LDHA∈[−7.5,−5.7]∧LGALS1∈[−2.1,1.0]∧MDM2∈[−4.6,−2.2]∧MTOR∈[−3.7,−2.3]∧PTEN∈[−7.3,−5.6]∧RAD23B∈[−0.9,0.8]∧RB1∈[−3.8,−2.3]∧SERPINB2∈[−6.6,−1.2]∧TCF3∈[−3.5,−1.0]∧TGFB1∈[−2.6,−0.3]∧TPI1∈[−7.2,−5.0]\mathit{ANLN}\in[-4.3,-2.8]\land\mathit{ATR}\in[-5.9,-3.8]\land\mathit{CLDN4}\in[-5.0,-0.6]\land\mathit{CXCL12}\in[-8.2,-3.2]\land\mathit{ERBB3}\in[-6.3,-3.2]\land\mathit{ERCC5}\in[-5.2,-3.6]\land\mathit{LDHA}\in[-7.5,-5.7]\land\mathit{LGALS1}\in[-2.1,1.0]\land\mathit{MDM2}\in[-4.6,-2.2]\land\mathit{MTOR}\in[-3.7,-2.3]\land\mathit{PTEN}\in[-7.3,-5.6]\land\mathit{RAD23B}\in[-0.9,0.8]\land\mathit{RB1}\in[-3.8,-2.3]\land\mathit{SERPINB2}\in[-6.6,-1.2]\land\mathit{TCF3}\in[-3.5,-1.0]\land\mathit{TGFB1}\in[-2.6,-0.3]\land\mathit{TPI1}\in[-7.2,-5.0] ”Subgroup size: 28→2928\to 29 Exceptionality: 16.0098→15.793416.0098\to 15.7934
“ AKT1∈[−3.6,−1.5]∧ALDH3A1∈[−8.9,−0.5]∧ANLN∈[−4.5,−2.6]∧ASS1∈[−4.9,−0.0]∧ATP5G3∈[−1.3,1.7]∧ATR∈[−5.9,−3.7]∧BCL2L1∈[−7.7,−5.0]∧BIRC5∈[−5.3,−1.3]∧BNIP3L∈[−3.1,−0.6]∧CAV1∈[−4.4,0.4]∧CDKN3∈[−5.8,−2.5]∧CLDN4∈[−5.0,−0.7]∧CXCL12∈[−8.2,−2.2]∧ERBB3∈[−6.2,−3.2]∧ERBB4∈[−10.6,−7.4]∧ERCC5∈[−5.2,−3.6]∧FAM83B∈[−6.8,−1.6]∧GNAI1∈[−5.2,−0.2]∧HIF1A∈[−7.7,−4.7]∧IGF1R∈[−3.5,−0.6]∧INHBA∈[−4.8,1.8]∧KDR∈[−5.5,−3.6]∧LDHA∈[−7.5,−5.3]∧LGALS1∈[−2.6,1.0]∧LOX∈[−6.4,−1.7]∧MCM6∈[−5.9,−3.0]∧MMP13∈[−9.3,1.3]∧MRGBP∈[−5.0,−2.8]∧MTOR∈[−3.8,−2.3]∧P4HA2∈[−4.7,−1.4]∧PDK1∈[−4.9,−2.3]∧PGAM1∈[−2.6,−0.7]∧PSMD9∈[−4.2,−2.0]∧PTEN∈[−7.3,−5.5]∧RAD23B∈[−0.8,0.8]∧RB1∈[−3.9,−2.3]∧TCF3∈[−3.4,−1.4]∧TGFB1∈[−2.6,−0.0]∧TP53∈[−4.3,−1.0]∧TPI1∈[−7.2,−4.9]∧CD44∈[−0.4,2.1]\mathit{AKT1}\in[-3.6,-1.5]\land\mathit{ALDH3A1}\in[-8.9,-0.5]\land\mathit{ANLN}\in[-4.5,-2.6]\land\mathit{ASS1}\in[-4.9,-0.0]\land\mathit{ATP5G3}\in[-1.3,1.7]\land\mathit{ATR}\in[-5.9,-3.7]\land\mathit{BCL2L1}\in[-7.7,-5.0]\land\mathit{BIRC5}\in[-5.3,-1.3]\land\mathit{BNIP3L}\in[-3.1,-0.6]\land\mathit{CAV1}\in[-4.4,0.4]\land\mathit{CDKN3}\in[-5.8,-2.5]\land\mathit{CLDN4}\in[-5.0,-0.7]\land\mathit{CXCL12}\in[-8.2,-2.2]\land\mathit{ERBB3}\in[-6.2,-3.2]\land\mathit{ERBB4}\in[-10.6,-7.4]\land\mathit{ERCC5}\in[-5.2,-3.6]\land\mathit{FAM83B}\in[-6.8,-1.6]\land\mathit{GNAI1}\in[-5.2,-0.2]\land\mathit{HIF1A}\in[-7.7,-4.7]\land\mathit{IGF1R}\in[-3.5,-0.6]\land\mathit{INHBA}\in[-4.8,1.8]\land\mathit{KDR}\in[-5.5,-3.6]\land\mathit{LDHA}\in[-7.5,-5.3]\land\mathit{LGALS1}\in[-2.6,1.0]\land\mathit{LOX}\in[-6.4,-1.7]\land\mathit{MCM6}\in[-5.9,-3.0]\land\mathit{MMP13}\in[-9.3,1.3]\land\mathit{MRGBP}\in[-5.0,-2.8]\land\mathit{MTOR}\in[-3.8,-2.3]\land\mathit{P4HA2}\in[-4.7,-1.4]\land\mathit{PDK1}\in[-4.9,-2.3]\land\mathit{PGAM1}\in[-2.6,-0.7]\land\mathit{PSMD9}\in[-4.2,-2.0]\land\mathit{PTEN}\in[-7.3,-5.5]\land\mathit{RAD23B}\in[-0.8,0.8]\land\mathit{RB1}\in[-3.9,-2.3]\land\mathit{TCF3}\in[-3.4,-1.4]\land\mathit{TGFB1}\in[-2.6,-0.0]\land\mathit{TP53}\in[-4.3,-1.0]\land\mathit{TPI1}\in[-7.2,-4.9]\land\mathit{CD44}\in[-0.4,2.1] ”“ ALDH3A1∈[−8.9,−0.5]∧CLDN4∈[−5.0,−0.7]∧ERBB3∈[−6.2,−3.2]∧ERBB4∈[−10.6,−7.4]∧HIF1A∈[−7.7,−4.7]∧LDHA∈[−7.5,−5.3]∧MMP13∈[−9.3,1.3]∧MRGBP∈[−5.0,−2.8]∧P4HA2∈[−4.7,−1.4]∧PDK1∈[−4.9,−2.3]∧PGAM1∈[−2.6,−0.7]∧PTEN∈[−7.3,−5.5]∧RAD23B∈[−0.8,0.8]∧TP53∈[−4.3,−1.0]∧TPI1∈[−7.2,−4.9]\mathit{ALDH3A1}\in[-8.9,-0.5]\land\mathit{CLDN4}\in[-5.0,-0.7]\land\mathit{ERBB3}\in[-6.2,-3.2]\land\mathit{ERBB4}\in[-10.6,-7.4]\land\mathit{HIF1A}\in[-7.7,-4.7]\land\mathit{LDHA}\in[-7.5,-5.3]\land\mathit{MMP13}\in[-9.3,1.3]\land\mathit{MRGBP}\in[-5.0,-2.8]\land\mathit{P4HA2}\in[-4.7,-1.4]\land\mathit{PDK1}\in[-4.9,-2.3]\land\mathit{PGAM1}\in[-2.6,-0.7]\land\mathit{PTEN}\in[-7.3,-5.5]\land\mathit{RAD23B}\in[-0.8,0.8]\land\mathit{TP53}\in[-4.3,-1.0]\land\mathit{TPI1}\in[-7.2,-4.9] ”Subgroup size: 25→2625\to 26 Exceptionality: 15.8073→15.206915.8073\to 15.2069
Appendix
Table 5: Rules learned on the primary RCT cohort before and after pruning using a Jaccard similarity threshold of 0.950.95 for membership indicators along with the respective changes in subgroup sizes and exceptionality.
Original rulePruned ruleChanges
“ ALDH1A1∈[−7.0,−3.2]∧BCL2L1∈[−6.9,−5.5]∧BIRC5∈[−4.6,−1.6]∧BNIP3L∈[−2.5,−1.1]∧CA9∈[−9.0,−4.4]∧CBX4∈[−5.3,−3.7]∧CLDN4∈[−5.3,−1.3]∧CXCL12∈[−5.2,−1.1]∧EGLN3∈[−4.4,−0.9]∧ENO1∈[−0.2,1.2]∧ENO2∈[−5.8,−2.4]∧EPHA1∈[−7.2,−4.6]∧GPI∈[−1.7,−0.2]∧HSPB1∈[−4.0,−2.0]∧ITGB2∈[−4.7,−2.3]∧KCTD11∈[−2.3,0.1]∧MAP2K2∈[−4.3,−3.3]∧MCM6∈[−5.8,−4.0]∧MMP13∈[−1.9,1.7]∧MMP9∈[−2.7,0.6]∧MPRS17∈[−5.0,−1.2]∧PFKFB3∈[−2.7,−0.2]∧PTEN∈[−6.6,−5.1]∧RAD23B∈[−1.0,0.9]∧RB1∈[−5.0,−2.5]∧RELA∈[−2.9,−1.6]∧SERPINB2∈[−7.6,−2.6]∧SLC5A1∈[−9.9,−3.6]∧TCF3∈[−3.8,−2.3]∧XRCC1∈[−5.2,−3.2]∧XRCC4∈[−6.1,−4.2]∧XRCC5∈[−2.1,−0.6]\mathit{ALDH1A1}\in[-7.0,-3.2]\land\mathit{BCL2L1}\in[-6.9,-5.5]\land\mathit{BIRC5}\in[-4.6,-1.6]\land\mathit{BNIP3L}\in[-2.5,-1.1]\land\mathit{CA9}\in[-9.0,-4.4]\land\mathit{CBX4}\in[-5.3,-3.7]\land\mathit{CLDN4}\in[-5.3,-1.3]\land\mathit{CXCL12}\in[-5.2,-1.1]\land\mathit{EGLN3}\in[-4.4,-0.9]\land\mathit{ENO1}\in[-0.2,1.2]\land\mathit{ENO2}\in[-5.8,-2.4]\land\mathit{EPHA1}\in[-7.2,-4.6]\land\mathit{GPI}\in[-1.7,-0.2]\land\mathit{HSPB1}\in[-4.0,-2.0]\land\mathit{ITGB2}\in[-4.7,-2.3]\land\mathit{KCTD11}\in[-2.3,0.1]\land\mathit{MAP2K2}\in[-4.3,-3.3]\land\mathit{MCM6}\in[-5.8,-4.0]\land\mathit{MMP13}\in[-1.9,1.7]\land\mathit{MMP9}\in[-2.7,0.6]\land\mathit{MPRS17}\in[-5.0,-1.2]\land\mathit{PFKFB3}\in[-2.7,-0.2]\land\mathit{PTEN}\in[-6.6,-5.1]\land\mathit{RAD23B}\in[-1.0,0.9]\land\mathit{RB1}\in[-5.0,-2.5]\land\mathit{RELA}\in[-2.9,-1.6]\land\mathit{SERPINB2}\in[-7.6,-2.6]\land\mathit{SLC5A1}\in[-9.9,-3.6]\land\mathit{TCF3}\in[-3.8,-2.3]\land\mathit{XRCC1}\in[-5.2,-3.2]\land\mathit{XRCC4}\in[-6.1,-4.2]\land\mathit{XRCC5}\in[-2.1,-0.6] ”“ EPHA1∈[−7.2,−4.6]∧GPI∈[−1.7,−0.2]∧KCTD11∈[−2.3,0.1]∧MAP2K2∈[−4.3,−3.3]∧MMP13∈[−1.9,1.7]∧MMP9∈[−2.7,0.6]∧XRCC5∈[−2.1,−0.6]\mathit{EPHA1}\in[-7.2,-4.6]\land\mathit{GPI}\in[-1.7,-0.2]\land\mathit{KCTD11}\in[-2.3,0.1]\land\mathit{MAP2K2}\in[-4.3,-3.3]\land\mathit{MMP13}\in[-1.9,1.7]\land\mathit{MMP9}\in[-2.7,0.6]\land\mathit{XRCC5}\in[-2.1,-0.6] ”Subgroup size: 12→1212\to 12 Exceptionality: 27.8189→27.743527.8189\to 27.7435
“ ANXA5∈[−3.0,−0.9]∧ATM∈[−3.8,−1.6]∧ATP5G3∈[−1.1,1.0]∧BNIP3∈[−6.1,−2.6]∧BSG∈[−9.4,−5.4]∧CD24∈[−8.8,−4.5]∧CXCL12∈[−6.2,−1.3]∧DCBLD1∈[−7.5,−4.5]∧DKK3∈[−5.3,−1.4]∧FANCA∈[−6.7,−3.1]∧FN1∈[−4.6,1.8]∧HIF1A∈[−7.1,−4.6]∧HK2∈[−6.7,−2.6]∧HSPA4∈[−3.9,−2.2]∧INHBA∈[−7.9,−1.3]∧ITGB1∈[−2.5,−0.1]∧KRT17∈[−3.2,2.8]∧LGALS1∈[−3.4,−0.2]∧LOXL2∈[−6.4,−3.0]∧MCM6∈[−5.3,−2.5]∧MDM2∈[−4.3,−0.7]∧MMP10∈[−9.7,−1.8]∧MMP13∈[−10.4,−2.8]∧MMP2∈[−5.0,−0.5]∧MTOR∈[−3.9,−2.3]∧MYC∈[−3.7,−0.6]∧MYNN∈[−5.1,−3.0]∧NR1D2∈[−4.9,−2.3]∧P4HA2∈[−5.1,−2.4]∧PSMD9∈[−4.3,−2.2]∧RMI2∈[−5.1,−2.9]∧RPA2∈[−4.5,−2.1]∧SFN∈[−1.6,3.1]∧SLC3A2∈[−4.3,−2.3]∧SMDT1∈[−3.9,−2.2]∧SNAI1∈[−7.6,−3.8]∧SPP1∈[−9.0,−3.9]∧TCF3∈[−4.0,−2.1]∧TP53∈[−3.8,−1.0]∧XRCC4∈[−5.2,−3.5]∧XRCC5∈[−2.8,−0.9]∧YAP1∈[−3.6,−0.2]\mathit{ANXA5}\in[-3.0,-0.9]\land\mathit{ATM}\in[-3.8,-1.6]\land\mathit{ATP5G3}\in[-1.1,1.0]\land\mathit{BNIP3}\in[-6.1,-2.6]\land\mathit{BSG}\in[-9.4,-5.4]\land\mathit{CD24}\in[-8.8,-4.5]\land\mathit{CXCL12}\in[-6.2,-1.3]\land\mathit{DCBLD1}\in[-7.5,-4.5]\land\mathit{DKK3}\in[-5.3,-1.4]\land\mathit{FANCA}\in[-6.7,-3.1]\land\mathit{FN1}\in[-4.6,1.8]\land\mathit{HIF1A}\in[-7.1,-4.6]\land\mathit{HK2}\in[-6.7,-2.6]\land\mathit{HSPA4}\in[-3.9,-2.2]\land\mathit{INHBA}\in[-7.9,-1.3]\land\mathit{ITGB1}\in[-2.5,-0.1]\land\mathit{KRT17}\in[-3.2,2.8]\land\mathit{LGALS1}\in[-3.4,-0.2]\land\mathit{LOXL2}\in[-6.4,-3.0]\land\mathit{MCM6}\in[-5.3,-2.5]\land\mathit{MDM2}\in[-4.3,-0.7]\land\mathit{MMP10}\in[-9.7,-1.8]\land\mathit{MMP13}\in[-10.4,-2.8]\land\mathit{MMP2}\in[-5.0,-0.5]\land\mathit{MTOR}\in[-3.9,-2.3]\land\mathit{MYC}\in[-3.7,-0.6]\land\mathit{MYNN}\in[-5.1,-3.0]\land\mathit{NR1D2}\in[-4.9,-2.3]\land\mathit{P4HA2}\in[-5.1,-2.4]\land\mathit{PSMD9}\in[-4.3,-2.2]\land\mathit{RMI2}\in[-5.1,-2.9]\land\mathit{RPA2}\in[-4.5,-2.1]\land\mathit{SFN}\in[-1.6,3.1]\land\mathit{SLC3A2}\in[-4.3,-2.3]\land\mathit{SMDT1}\in[-3.9,-2.2]\land\mathit{SNAI1}\in[-7.6,-3.8]\land\mathit{SPP1}\in[-9.0,-3.9]\land\mathit{TCF3}\in[-4.0,-2.1]\land\mathit{TP53}\in[-3.8,-1.0]\land\mathit{XRCC4}\in[-5.2,-3.5]\land\mathit{XRCC5}\in[-2.8,-0.9]\land\mathit{YAP1}\in[-3.6,-0.2] ”“ ATM∈[−3.8,−1.6]∧HIF1A∈[−7.1,−4.6]∧KRT17∈[−3.2,2.8]∧MMP13∈[−10.4,−2.8]∧MMP2∈[−5.0,−0.5]∧PSMD9∈[−4.3,−2.2]∧RMI2∈[−5.1,−2.9]∧SNAI1∈[−7.6,−3.8]∧XRCC4∈[−5.2,−3.5]∧YAP1∈[−3.6,−0.2]\mathit{ATM}\in[-3.8,-1.6]\land\mathit{HIF1A}\in[-7.1,-4.6]\land\mathit{KRT17}\in[-3.2,2.8]\land\mathit{MMP13}\in[-10.4,-2.8]\land\mathit{MMP2}\in[-5.0,-0.5]\land\mathit{PSMD9}\in[-4.3,-2.2]\land\mathit{RMI2}\in[-5.1,-2.9]\land\mathit{SNAI1}\in[-7.6,-3.8]\land\mathit{XRCC4}\in[-5.2,-3.5]\land\mathit{YAP1}\in[-3.6,-0.2] ”Subgroup size: 47→4547\to 45 Exceptionality: 12.0529→11.926412.0529\to 11.9264
“ ALDH1A1∈[−7.0,−3.4]∧BSG∈[−8.9,−3.9]∧CA9∈[−9.5,−4.5]∧CBX4∈[−5.3,−3.7]∧CLDN4∈[−4.9,−0.4]∧CXCL12∈[−4.3,−1.5]∧ENO1∈[−0.3,1.2]∧ENO2∈[−5.8,−2.4]∧EPHA1∈[−7.2,−4.6]∧EPOR∈[−10.2,−6.6]∧ERCC5∈[−5.2,−3.0]∧HIF1A∈[−5.7,−3.7]∧LOX∈[−2.9,−1.0]∧MMP9∈[−2.8,0.6]∧MYNN∈[−5.2,−2.8]∧PGK1∈[−2.0,−0.2]∧PTEN∈[−6.5,−5.1]∧RAD23B∈[−1.0,1.1]∧RAD50∈[−3.6,−1.6]∧RB1∈[−5.0,−2.4]∧SLC5A1∈[−10.0,−3.3]∧TCF3∈[−4.0,−2.1]∧XRCC1∈[−5.2,−4.1]∧XRCC4∈[−6.2,−4.1]∧XRCC5∈[−2.2,−1.0]\mathit{ALDH1A1}\in[-7.0,-3.4]\land\mathit{BSG}\in[-8.9,-3.9]\land\mathit{CA9}\in[-9.5,-4.5]\land\mathit{CBX4}\in[-5.3,-3.7]\land\mathit{CLDN4}\in[-4.9,-0.4]\land\mathit{CXCL12}\in[-4.3,-1.5]\land\mathit{ENO1}\in[-0.3,1.2]\land\mathit{ENO2}\in[-5.8,-2.4]\land\mathit{EPHA1}\in[-7.2,-4.6]\land\mathit{EPOR}\in[-10.2,-6.6]\land\mathit{ERCC5}\in[-5.2,-3.0]\land\mathit{HIF1A}\in[-5.7,-3.7]\land\mathit{LOX}\in[-2.9,-1.0]\land\mathit{MMP9}\in[-2.8,0.6]\land\mathit{MYNN}\in[-5.2,-2.8]\land\mathit{PGK1}\in[-2.0,-0.2]\land\mathit{PTEN}\in[-6.5,-5.1]\land\mathit{RAD23B}\in[-1.0,1.1]\land\mathit{RAD50}\in[-3.6,-1.6]\land\mathit{RB1}\in[-5.0,-2.4]\land\mathit{SLC5A1}\in[-10.0,-3.3]\land\mathit{TCF3}\in[-4.0,-2.1]\land\mathit{XRCC1}\in[-5.2,-4.1]\land\mathit{XRCC4}\in[-6.2,-4.1]\land\mathit{XRCC5}\in[-2.2,-1.0] ”“ ALDH1A1∈[−7.0,−3.4]∧CA9∈[−9.5,−4.5]∧CBX4∈[−5.3,−3.7]∧ENO1∈[−0.3,1.2]∧EPHA1∈[−7.2,−4.6]∧LOX∈[−2.9,−1.0]∧MMP9∈[−2.8,0.6]∧MYNN∈[−5.2,−2.8]∧SLC5A1∈[−10.0,−3.3]∧TCF3∈[−4.0,−2.1]∧XRCC1∈[−5.2,−4.1]\mathit{ALDH1A1}\in[-7.0,-3.4]\land\mathit{CA9}\in[-9.5,-4.5]\land\mathit{CBX4}\in[-5.3,-3.7]\land\mathit{ENO1}\in[-0.3,1.2]\land\mathit{EPHA1}\in[-7.2,-4.6]\land\mathit{LOX}\in[-2.9,-1.0]\land\mathit{MMP9}\in[-2.8,0.6]\land\mathit{MYNN}\in[-5.2,-2.8]\land\mathit{SLC5A1}\in[-10.0,-3.3]\land\mathit{TCF3}\in[-4.0,-2.1]\land\mathit{XRCC1}\in[-5.2,-4.1] ”Subgroup size: 12→1212\to 12 Exceptionality: 27.4009→26.071827.4009\to 26.0718
“ ATM∈[−4.7,−1.6]∧BIRC5∈[−4.2,−1.0]∧BNIP3L∈[−3.2,−1.2]∧BSG∈[−9.2,−3.9]∧CA9∈[−10.7,−2.9]∧CBX4∈[−5.3,−3.1]∧CD24∈[−8.3,−4.5]∧CDKN3∈[−6.1,−2.7]∧CXCL12∈[−7.5,−2.0]∧DCBLD1∈[−7.5,−4.5]∧ENO2∈[−6.2,−2.4]∧ERCC5∈[−5.7,−3.0]∧HK2∈[−6.7,−2.2]∧HSPA4∈[−4.0,−2.2]∧KDR∈[−6.0,−2.4]∧KRT17∈[−3.0,5.1]∧LOX∈[−7.1,−1.5]∧LOXL2∈[−6.4,−1.8]∧MCM6∈[−4.8,−2.5]∧MME∈[−9.6,−4.0]∧MRE11A∈[−6.1,−4.0]∧MTOR∈[−4.2,−2.3]∧MYC∈[−3.7,−0.5]∧MYNN∈[−5.0,−2.3]∧NBN∈[−3.9,−1.6]∧NOTCH1∈[−6.3,−1.9]∧RAD23B∈[−1.4,1.1]∧RAD50∈[−4.0,−1.6]∧RASSF6∈[−8.1,−1.6]∧RB1∈[−5.0,−2.3]∧RELA∈[−3.5,−1.4]∧SDHA∈[−6.2,−2.9]∧SERPINB2∈[−8.0,−0.6]∧TCF3∈[−3.9,−2.1]∧TP53∈[−5.0,−1.0]∧XPA∈[−5.3,−2.6]∧XPC∈[−5.7,−3.7]∧XRCC1∈[−5.2,−2.5]∧XRCC4∈[−5.8,−3.5]∧XRCC5∈[−2.8,−0.8]\mathit{ATM}\in[-4.7,-1.6]\land\mathit{BIRC5}\in[-4.2,-1.0]\land\mathit{BNIP3L}\in[-3.2,-1.2]\land\mathit{BSG}\in[-9.2,-3.9]\land\mathit{CA9}\in[-10.7,-2.9]\land\mathit{CBX4}\in[-5.3,-3.1]\land\mathit{CD24}\in[-8.3,-4.5]\land\mathit{CDKN3}\in[-6.1,-2.7]\land\mathit{CXCL12}\in[-7.5,-2.0]\land\mathit{DCBLD1}\in[-7.5,-4.5]\land\mathit{ENO2}\in[-6.2,-2.4]\land\mathit{ERCC5}\in[-5.7,-3.0]\land\mathit{HK2}\in[-6.7,-2.2]\land\mathit{HSPA4}\in[-4.0,-2.2]\land\mathit{KDR}\in[-6.0,-2.4]\land\mathit{KRT17}\in[-3.0,5.1]\land\mathit{LOX}\in[-7.1,-1.5]\land\mathit{LOXL2}\in[-6.4,-1.8]\land\mathit{MCM6}\in[-4.8,-2.5]\land\mathit{MME}\in[-9.6,-4.0]\land\mathit{MRE11A}\in[-6.1,-4.0]\land\mathit{MTOR}\in[-4.2,-2.3]\land\mathit{MYC}\in[-3.7,-0.5]\land\mathit{MYNN}\in[-5.0,-2.3]\land\mathit{NBN}\in[-3.9,-1.6]\land\mathit{NOTCH1}\in[-6.3,-1.9]\land\mathit{RAD23B}\in[-1.4,1.1]\land\mathit{RAD50}\in[-4.0,-1.6]\land\mathit{RASSF6}\in[-8.1,-1.6]\land\mathit{RB1}\in[-5.0,-2.3]\land\mathit{RELA}\in[-3.5,-1.4]\land\mathit{SDHA}\in[-6.2,-2.9]\land\mathit{SERPINB2}\in[-8.0,-0.6]\land\mathit{TCF3}\in[-3.9,-2.1]\land\mathit{TP53}\in[-5.0,-1.0]\land\mathit{XPA}\in[-5.3,-2.6]\land\mathit{XPC}\in[-5.7,-3.7]\land\mathit{XRCC1}\in[-5.2,-2.5]\land\mathit{XRCC4}\in[-5.8,-3.5]\land\mathit{XRCC5}\in[-2.8,-0.8] ”“ BIRC5∈[−4.2,−1.0]∧BSG∈[−9.2,−3.9]∧CA9∈[−10.7,−2.9]∧CD24∈[−8.3,−4.5]∧DCBLD1∈[−7.5,−4.5]∧LOX∈[−7.1,−1.5]∧LOXL2∈[−6.4,−1.8]∧MCM6∈[−4.8,−2.5]∧MME∈[−9.6,−4.0]∧MTOR∈[−4.2,−2.3]∧MYC∈[−3.7,−0.5]∧MYNN∈[−5.0,−2.3]∧RB1∈[−5.0,−2.3]∧TCF3∈[−3.9,−2.1]∧XPC∈[−5.7,−3.7]∧XRCC1∈[−5.2,−2.5]\mathit{BIRC5}\in[-4.2,-1.0]\land\mathit{BSG}\in[-9.2,-3.9]\land\mathit{CA9}\in[-10.7,-2.9]\land\mathit{CD24}\in[-8.3,-4.5]\land\mathit{DCBLD1}\in[-7.5,-4.5]\land\mathit{LOX}\in[-7.1,-1.5]\land\mathit{LOXL2}\in[-6.4,-1.8]\land\mathit{MCM6}\in[-4.8,-2.5]\land\mathit{MME}\in[-9.6,-4.0]\land\mathit{MTOR}\in[-4.2,-2.3]\land\mathit{MYC}\in[-3.7,-0.5]\land\mathit{MYNN}\in[-5.0,-2.3]\land\mathit{RB1}\in[-5.0,-2.3]\land\mathit{TCF3}\in[-3.9,-2.1]\land\mathit{XPC}\in[-5.7,-3.7]\land\mathit{XRCC1}\in[-5.2,-2.5] ”Subgroup size: 85→8785\to 87 Exceptionality: 11.2782→11.126011.2782\to 11.1260
Appendix
Table 6: Rules learned on the postoperative RCT cohort before and after pruning using a Jaccard similarity threshold of 0.950.95 for membership indicators along with the respective changes in subgroup sizes and exceptionality.
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