Beyond Consistency: Inference for the Relative risk functional in Deep Nonparametric Cox Models

Sattwik Ghosal; Xuran Meng; Yi Li

stat.MLMar 25, 2026

Beyond Consistency: Inference for the Relative risk functional in Deep Nonparametric Cox Models

Authors: Sattwik Ghosal, Xuran Meng, Yi Li

Organizations: Department of Biostatistics, University of Michigan

Abstract

There remain theoretical gaps in deep neural network estimators for the nonparametric Cox proportional hazards model. In particular, it is unclear how gradient-based optimization error propagates to population risk under partial likelihood, how pointwise bias can be controlled to permit valid inference, and how ensemble-based uncertainty quantification behaves under realistic variance decay regimes. We develop an asymptotic distribution theory for deep Cox estimators that addresses these issues. First, we establish nonasymptotic oracle inequalities for general trained networks that link in-sample optimization error to population risk without requiring the exact empirical risk optimizer. We then construct a structured neural parameterization that achieves infinity-norm approximation rates compatible with the oracle bound, yielding control of the pointwise bias. Under these conditions and using the Hajek--Hoeffding projection, we prove pointwise and multivariate asymptotic normality for subsampled ensemble estimators. We derive a range of subsample sizes that balances bias correction with the requirement that the Hajek--Hoeffding projection remain dominant. This range accommodates decay conditions on the single-overlap covariance, which measures how strongly a single shared observation influences the estimator, and is weaker than those imposed in the subsampling literature. An infinitesimal jackknife representation provides analytic covariance estimation and valid Wald-type inference for relative risk contrasts such as log-hazard ratios. Finally, we illustrate the finite-sample implications of the theory through simulations and a real data application.

Figures & tables

Panel of Figure 1 — Figure 1: A fully connected neural network with $L=3$ and $\mathbf{p}=(5,4,4,3,1)$ .

Case	Setting	Bias	MAE	EmpSD	SE	CP	AIL
1	$n=800,\ r=\lfloor n^{0.85}\rfloor$	-0.02	0.36	0.48	0.50	0.87	1.98
	$n=800,\ r=\lfloor n^{0.90}\rfloor$	-0.02	0.21	0.45	0.48	0.92	1.86
	$n=800,\ r=\lfloor n^{0.95}\rfloor$	-0.03	0.11	0.40	0.44	0.95	1.72
2	$n=800,\ r=\lfloor n^{0.85}\rfloor$	0.11	0.26	0.47	0.49	0.89	1.91
	$n=800,\ r=\lfloor n^{0.90}\rfloor$	0.05	0.14	0.42	0.44	0.93	1.74
	$n=800,\ r=\lfloor n^{0.95}\rfloor$	0.04	0.11	0.36	0.40	0.95	1.59

Table 1: Simulation summary for pointwise inference on

g_{0}(\mathbf{x})

across test points. EmpSD is the empirical standard deviation of

\widehat{g}_{B}(\mathbf{x})

over replications. SE is the mean estimated standard error. CP is the empirical coverage probability of nominal 95% intervals, and AIL is the mean interval length.

Panel of Figure 2 — Figure 2: Estimation and inference summary for Model 3 with $n=800$ , $r=\lfloor n^{0.90}\rfloor$ , $B=1000$ , DNN architecture $(p_{0},128,64,1)$ . Panel (a) plots $\widehat{g}_{B}(\mathbf{x})$ versus $g_{0}(\mathbf{x})$ across test points. Panel (b) compares $\mathrm{EmpSD}$ with the estimated $\mathrm{SE}$ , highlighting corrected and uncorrected versions when available.

Panel of Figure 3 — Figure 3: Empirical performance across the subsampling index $\alpha$ . At smaller values ( $\alpha\leq 0.8$ ), high MAE degrades the CP, validating the theoretical lower bound $\alpha_{\text{lower}}$ . Conversely, as $\alpha\to 1$ , the CP drops significantly despite stable MAE, and variance estimates diverge, empirically validating the upper bound $\alpha_{\text{upper}}$ .

Case	Setting	Bias	MAE	EmpSD	SE	CP	AIL
1	$n=800,\ r=\lfloor n^{0.85}\rfloor$	-0.45	0.65	0.69	0.72	0.94	2.83
	$n=800,\ r=\lfloor n^{0.90}\rfloor$	-0.23	0.52	0.62	0.67	0.95	2.61
	$n=800,\ r=\lfloor n^{0.95}\rfloor$	-0.22	0.50	0.55	0.63	0.97	2.46
2	$n=800,\ r=\lfloor n^{0.85}\rfloor$	-0.17	0.59	0.71	0.69	0.94	2.72
	$n=800,\ r=\lfloor n^{0.90}\rfloor$	-0.02	0.45	0.55	0.64	0.97	2.51
	$n=800,\ r=\lfloor n^{0.95}\rfloor$	0.06	0.43	0.53	0.56	0.96	2.21

Table 2: Simulation summary for inference on the log-hazard ratio contrast

\psi(\mathbf{x}_{\textrm{test},i},\mathbf{x}_{\textrm{test},j})

Case	$n$	Method	MAE	EmpSD	SE	CP	AIL
Case 1	800	CoxPH	0.11	0.03	0.14	0.95	0.57
		RSF	0.69	0.02	0.20	0.53	1.03
		GBR	0.33	0.03	0.35	0.89	1.37
		DeepSurv	0.39	0.06	2.52	0.99	2.63
		ESM	0.11	0.40	0.44	0.95	1.72
	1000	CoxPH	0.10	0.03	0.13	0.95	0.50

Table 3: Estimation Performance and Uncertainty Quantification (Replications

R=200

\textrm{Bootstrap}=200

) for Model 1 and 3. Pointwise metrics are averaged over the test set

x_{test}

Method	Mean C	C-index SE	C-index AIL	Mean AUC	AUC SE	AUC AIL
ESM	0.7316	0.0018	0.0074	0.7987	0.0039	0.0171
DeepSurv	0.7284	0.0019	0.0076	0.7928	0.0043	0.0186
CoxPH	0.7223	0.0020	0.0080	0.7944	0.0050	0.0214
RF	0.7221	0.0020	0.0080	0.7928	0.0058	0.0247
GBR	0.7209	0.0019	0.0076	0.7865	0.0043	0.0184

Table 4: Comparative Performance of Survival Models on the BLCS Inference Set

Panel of Figure 4 — Figure 4: Comparative evaluation of model performance. The ensemble shows strong ranking stability and stable estimation accuracy over time.

Panel of Figure 5 — Figure 5: Stage-specific contrast HR studies. In each panel, curves correspond to Stages 2, 3, and 4, with shaded regions indicating IJ-based 95% confidence bands.

Survival Analysis

Survival analysis models when events such as disease progression, failure, or death occur, accounting for incomplete follow-up and censoring.

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FLEXI-Haz: Semiparametric Deep Survival Inference Beyond Proportional Hazards

Asaf Ben Arie, Malka Gorfine

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

Survival Analysis Semiparametric Inference

Open paper AskPDF arXiv

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.

Abstract

Survival Analysis

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