FLEXI-Haz: Semiparametric Deep Survival Inference Beyond Proportional Hazards

Asaf Ben Arie; Malka Gorfine

stat.MLDec 11, 2025

FLEXI-Haz: Semiparametric Deep Survival Inference Beyond Proportional Hazards

Authors: Asaf Ben Arie, Malka Gorfine

Organizations: Department of Statistics and Operations Research, Tel Aviv University, Israel

Abstract

Flexible survival prediction does not by itself ensure valid inference for covariate effects or subject-specific survival probabilities. We develop FLEXI-Haz, a semiparametric deep neural network model for right-censored data that combines an interpretable linear component in the log-hazard with a flexible function of time and adjustment covariates. This formulation permits nonproportional nuisance effects while retaining interpretable regression parameters. Under regularity conditions, we establish nuisance convergence rates that are minimax optimal up to logarithmic factors over composite Holder classes, together with root-n-consistent, asymptotically normal, and semiparametrically efficient estimation of the regression parameter. For adjustment covariates with finite support, we derive the explicit influence function for cumulative hazard and construct a cross-fitted one-step estimator yielding asymptotically valid pointwise confidence intervals for subject-specific survival. Simulations demonstrate reduced bias and improved coverage relative to a proportional hazards comparator under time-varying nuisance effects. An analysis of the Rotterdam breast cancer data illustrates the consequences of flexible adjustment for inference on treatment associations.

Figures & tables

$p$	$n$	$\gamma,\rho$	Parameter/Method	Est.	Emp. SD	Est. SE	90%CI	95% CI
1	4000	$0,0$	FLEXI–Haz	0.404	0.026	0.024	87.5 (82.9, 92.1)	94.0 (90.7, 97.3)
			Zhong et al.	0.412	0.029	0.024	82.0 (76.7, 87.3)	90.5 (86.4, 94.6)
1	4000	$0,0.5$	FLEXI–Haz	0.404	0.035	0.033	88.0 (83.5, 92.5)	94.5 (91.3, 97.7)
			Zhong et al.	0.393	0.034	0.033	85.5 (80.6, 90.4)	93.0 (89.5, 96.5)
1	4000	$0,0.75$	FLEXI–Haz	0.405	0.050	0.046	86.5 (81.8, 91.2)	95.0 (92.0, 98.0)
			Zhong et al.	0.399	0.044	0.046	91.0 (87.0, 95.0)	96.0 (93.3, 98.7)

Table 1: Simulation results for estimation and inference on

\theta_{o}=0.4055

for

p=1

, and

\mbox{\boldmath

{\theta}

}_{o}^{\top}=(1,-1)

for

p=2

. Parentheses contain 95% Wald confidence intervals for the empirical coverage probabilities, based on 200 Monte Carlo replications.

Marginal survival	$t_{s}$	90% CI	SD	95% CI	SD
0.1	6.7126	0.9377	0.0741	0.9676	0.0531
0.2	4.2934	0.9120	0.0861	0.9548	0.0638
0.3	3.0192	0.9167	0.0769	0.9565	0.0595
0.4	2.2003	0.9121	0.0755	0.9595	0.0523
0.5	1.6035	0.8903	0.0889	0.9488	0.0589
0.6	1.2330	0.8997	0.0912	0.9520	0.0611

Table 2: Empirical pointwise coverage of the proposed confidence intervals for the survival function, based on 200 replications with training sample size

n=8000

Treatment	FLEXI–Haz	Zhong et al. (2022)	Cox PH
Hormonal	$\mathbf{-0.364\,(-0.58,\ -0.15)}$	$\mathbf{-0.327\,(-0.54,\ -0.11)}$	$-0.132\,(-0.29,\ 0.02)$
Chemo	$\mathbf{-0.321\,(-0.55,\ -0.10)}$	$-0.205\,(-0.43,\ 0.02)$	$-0.091\,(-0.23,\ 0.05)$

Table 3: Estimated treatment effects and

95\%

confidence intervals for the Rotterdam data.

Treatment	ER	FLEXI–Haz	Zhong et al. (2022)	Cox PH
Hormonal	Low	$-0.197\,(-0.49,\ 0.10)$	$-0.198\,(-0.50,\ 0.10)$	$0.093\,(-0.12,\ 0.30)$
Hormonal	High	$\mathbf{-0.569\,(-0.87,\ -0.27)}$	$\mathbf{-0.493\,(-0.78,\ -0.21)}$	$\mathbf{-0.318\,(-0.53,\ -0.11)}$
Chemo	Low	$\mathbf{-0.322\,(-0.60,\ -0.05)}$	$-0.223\,(-0.48,\ 0.03)$	$-0.094\,(-0.26,\ 0.07)$
Chemo	High	$\mathbf{-0.307\,(-0.59,\ -0.02)}$	${-0.241\,(-0.51,\ 0.03)}$	$-0.093\,(-0.29,\ 0.10)$

Table 4: Estimated ER-specific treatment effects and

95\%

confidence intervals for the Rotterdam breast cancer data.

Figure 1: Estimated survival curves for 16 Rotterdam patients, conditional on their grouped nuisance covariates and primary covariates. Dashed blue curves show the plug-in FLEXI–Haz estimates; orange step curves show the cross-fitted one-step estimates. Shading represents pointwise confidence intervals.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

		$n=4000$			$n=8000$
$\gamma$	$\rho$	FLEXI–Haz	Zhong et al.	LR	FLEXI–Haz	Zhong et al.	LR
0	0	$100\times 2$	$25\times 3$	$0.05$	$150\times 2$	$50\times 2$	$0.10$
0	0.5	$150\times 2$	$500\times 5$	$0.01$	$150\times 2$	$300\times 2$	$0.005$
0	0.75	$150\times 2$	$500\times 6$	$0.005$	$150\times 2$	$25\times 2$	$0.10$
0.5	0	$300\times 2$	$150\times 4$	$0.005$	$300\times 3$	$100\times 2$	$0.01$
0.5	0.5	$100\times 2$	$500\times 2$	$0.0001$	$150\times 2$	$300\times 5$	$0.001$

Appendix

Table 5: Selected main-network architectures for Setting 1. Each entry

w\times L

denotes

L

hidden layers with

w

units per layer. LR denotes the selected learning rate for Zhong et al.

Component	Parameter	Value
Main network	Optimizer	Adam
	Learning rates	$10^{-3}\to 10^{-4}\to 10^{-5}$
	Epoch limit / patience	500 per phase / 30
	Batch size	$\lceil N_{\mathrm{exp,tr}}/213\rceil$
	Initializations per replication	5
Projection network	Folds	5, by subject

Appendix

Table 6: Fixed training settings for FLEXI–Haz in Setting 1.

Depth	Width sequences
1	$(16)$ , $(32)$ , $(64)$ , $(128)$ , $(256)$
2	$(16,16)$ , $(16,32)$ , $(32,32)$ , $(32,64)$ , $(64,64)$ , $(64,128)$ , $(128,128)$
3	$(16,16,16)$ , $(16,16,32)$ , $(32,32,32)$ , $(32,32,64)$ , $(64,64,64)$ , $(64,64,128)$ , $(128,128,128)$
4	$(16,16,16,16)$ , $(16,16,32,32)$ , $(32,32,32,32)$ , $(32,32,64,64)$ , $(64,64,64,64)$
5	$(16,16,16,16,16)$ , $(16,16,32,32,32)$ , $(32,32,32,32,32)$

Appendix

Table 7: The 27 candidate hidden-layer configurations used in Setting 2.

$n$	Method	Hidden widths	Learning rate(s)
2000	FLEXI–Haz	$(16,16,16)$	$10^{-3}\to 10^{-4}\to 10^{-5}$
4000	FLEXI–Haz	$(32,32,32)$	$10^{-3}\to 10^{-4}\to 10^{-5}$
8000	FLEXI–Haz	$(128,128,128)$	$10^{-3}\to 10^{-4}\to 10^{-5}$
2000	Zhong et al.	$(32,32)$	$0.03222255749$
4000	Zhong et al.	$(16,16,16,16)$	$0.02204518664$
8000	Zhong et al.	$(64,64)$	$0.04726183068$

Appendix

Table 8: Architectures and learning rates used in Setting 2. For FLEXI–Haz, the projection networks use the same hidden widths as the main network; their learning rate is fixed at

10^{-3}

. FLEXI–Haz uses the displayed three-phase schedule, whereas the proportional comparator uses a constant selected learning rate.

Analysis	Fold	FLEXI–Haz	Zhong et al.	LR
Overall effects	1	$(200,200)$	$(25,25,25)$	$0.01$
	2	$(150,150)$	$(25,25,25)$	$0.01$
	3	$(100,100)$	$(100,100)$	$0.0005$
	4	$(200,200)$	$(25,25,25)$	$0.0005$
	5	$(100,100)$	$(25,25)$	$0.01$
ER interactions	1	$(200,200)$	$(25,25,25)$	$0.01$

Appendix

Table 9: Selected main-network architectures for the Rotterdam analyses. An architecture lists the widths of successive hidden layers. LR is the reported Zhong main-network learning rate. Fold labels 1–5 correspond to labels 0–4 in the supplied configuration file.

Variable	Group 1	Group 2	Group 3	Group 4
Age (years)	$\leq 45$	$46$ – $54$	$55$ – $65$	$>65$
Positive lymph nodes	$0$	$1$ – $3$	$\geq 4$	—
ER	$\leq 3$	$4$ – $9$	$10$ – $49$	$\geq 50$
PGR	$\leq 9$	$10$ – $99$	$\geq 100$	—

Appendix

Table 10: Covariate grouping for the Rotterdam survival-inference analysis in Figure 1.

Rank	Age	Meno	Size	Grade	Nodes	PGR	ER	$n$	Events	Censored
1	$>65$	Post	$\leq 20$	3	0	$\geq 100$	$\geq 50$	46	26	20
2	$\leq 45$	Pre	20–50	3	0	$\leq 9$	$\leq 3$	38	20	18
3	55–65	Post	$\leq 20$	3	0	$\geq 100$	$\geq 50$	37	12	25
4	55–65	Post	$\leq 20$	3	0	10–99	$\geq 50$	36	9	27
5	$>65$	Post	20–50	3	0	$\geq 100$	$\geq 50$	36	20	16
6	$\leq 45$	Pre	$\leq 20$	3	0	$\geq 100$	$\geq 50$	35	16	19

Appendix

Table 11: Covariate values and counts for the 16 most frequent Rotterdam adjustment profiles.

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