Neural Frailty Machine: Beyond proportional hazard assumption in neural survival regressions
Authors: Jiawei Qiao
Ruofan Wu
Mingzhe Wu
Wen Yu
Ming Zheng
Tengfei LIU
Tianyi Zhang
Weiqiang Wang
Venue: Submitted to ICLR 2023 (2023)
Keywords: survival analysis
sieve method
theory

Abstract: We present neural frailty machine (NFM), a powerful and flexible neural modeling framework for survival regressions. The NFM framework utilizes the classical idea of multiplicative frailty in survival analysis to capture unobserved heterogeneity among individuals, at the same time being able to leverage the strong approximation power of neural architectures for handling nonlinear covariate dependence. Two concrete models are derived under the framework that extends neural proportional hazard models and nonparametric hazard regression models. Both models allow efficient training under the likelihood objective. Theoretically, for both proposed models, we establish statistical guarantees of neural function approximation with respect to nonparametric components via characterizing their rate of convergence. Empirically, we provide synthetic experiments that verify our theoretical statements. We also conduct experimental evaluations over 6 benchmark datasets of different scales, showing that the proposed NFM models outperform state-of-the-art survival models in terms of predictive performance. 

Under review as a conference paper at ICLR 2023
NEURAL FRAILTY MACHINE: BEYOND PROPOR- TIONAL HAZARD ASSUMPTION IN NEURAL SURVIVAL
REGRESSIONS
Anonymous authors Paper under double-blind review
ABSTRACT
We present neural frailty machine (NFM), a powerful and flexible neural model- ing framework for survival regressions. The NFM framework utilizes the classical idea of multiplicative frailty in survival analysis to capture unobserved hetero- geneity among individuals, at the same time being able to leverage the strong approximation power of neural architectures for handling nonlinear covariate de- pendence. Two concrete models are derived under the framework that extends neural proportional hazard models and nonparametric hazard regression models. Both models allow efficient training under the likelihood objective. Theoretically, for both proposed models, establish statistical guarantees of neural function approximation with respect to nonparametric components via characterizing their rate of convergence. Empirically, we provide synthetic experiments that verify our theoretical statements. We also conduct experimental evaluations over 6 bench- mark datasets of different scales, showing that the proposed NFM models outper- form state-of-the-an survival models in terms of predictive performance.
INTRODUCTION
Regression analysis of time-to-event data (Kalbfleisch & Prentice 2002) has been among the most important modeling tools for clinical studies and has witnessed growing interest in areas like corporate finance (Duffie et al. 2009). recommendation systems (Jing Smola 2017). and com- putational advertising (Wu et al. 2015). The key feature that differentiates time-to-even, data from other types of data is that they are often incompletely observed, with the most prevailing form of incompleteness being the right censoring mechanism (Kalbfleisch & Prentice, 2002). In the right censoring mechanism, the duration time than the observation time instead of being recorded precisely. It is well known in the community of survival analysis that even in the case of linear regression, naively discarding the censored observa- tions produces estimation results that statistically biased (Buckley & James 1979), at the same time losses sample efficiency if the censoring proportion is high.
sampled subject is (sometimes) only known to be larger
Cox's proportional hazard (CoxPH) model (Cox 1972) using the convex objective of negative partial likelihood (Cox, 1975) is the de facto choice in modeling right censored time-to-event data (hereafter abbreviated as censored data without misunderstandings). The model is semiparametric (Bickel et al. 1993) in the sense that the baseline hazard function needs no parametric assumptions. The original formulation of CoxPH model assumes a linear form and therefore has limited flexibility since the truth is not necessarily lincar. Subsequent studies extended CoxPH model to nonlinear variants using ideas from nonparametric regression (Huang) 1999 Cai al. 2007 2008), ensemble learning (Ishwaran et al. 2008), and neural networks (Faraggi & Simon 1995 Katzman et al., 2018)) While such extensions allowed a more flexible nonlinear dependence structure with the covariates, the learning objectives were still derived under the proportional hazards (PH) assumption, which was shown to be inadequate in many real-world scenarios (Gray) 20000). The most notable case was the failure of modeling the phenomenon of crossing hazards (Stablein & Koutrouvelis, 1985). It is thus of significant interest to explore extensions of CoxPH that both allow nonlinear dependence over covariates and relaxations of the PH assumption.

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among the most important research topics in modern survival analysis, in that they provide a principled way of extending CoxPH model via incorporating multiplicative random effect to capture unobserved heterogeneity. The resulting parameterization contains many useful variants of CoxPH like the proportional odds model (Bennett 1983 under specific choices of frailty families. While the theory of frailty models has been well- established (Murphy) 1994 1995,
Frailty models (Wienke 2010
Duchateau & Janssen 2007)
Parner 1998 Kosorok et al. 2004), most of them focused on the linear case. Recent developments on applying neural approaches to survival analysis (Katzman et al., 2018) Kaamme et al. 2019 Tang et al. 2022, Rindt et al. 2022) have shown promising results terms of empirical predictive performance, with most of them lacking theoretical discussions.
Therefore, it is of significant interest to build more powerful frailty models via adopting techniques
in modern deep learning (Goodfellow et al. 2016) with provable statistical guarantees.
In this paper, we present general framework for neural extensions of frailty models called the neural frailty machine (NFM). Two concrete neural architectures are derived under the frame- work: The first one adopts the proportional frailty assumption, allowing an intuitive interpretation of the neural CoxPH model with a multiplicative random effect. The second one further relaxes the proportional frailty assumption and could be viewed gression (NHR) (Cox & O' Sullivan 1990] Kooperberg et al. 1995), sometimes referred to as "fully neural" models under the context of neural survival analysis (Omi et al. 2019). We summarize our contributions as follows.
extension of nonparametric hazard
We propose the neural frailty machine (NFM) framework as a principled way of incorporating unobserved heterogeneity into neural survival regression models. The framework includes many commonly used survival regression models as special cases.
We derive two model architectures based on the NFM framework that extend neural CoxPH mod- els and neural NHR models. Both models allow stochastic training and scale to large datasets.
We show thcoretical guarantees for the two proposed models via characterizing the rates con- vergence of the proposed nonparametric function estimators. The proof technique is different from previous theoretical studies on neural survival analysis and is applicable to many other types of neural survival models.
We conduct extensive studies on various benchmark datasets at different scales. Under standard performance metries, both models are empirically shown to perform competitively, matching or outperforming state-of-the-art neural survival models.
2
RELATED WORKS
2.1
NONLINEAR EXTENSIONS OF CoxPH
Most nonlinear extensions of CoxPH model stem from the equivalence of partial likelihood and semiparametric profile likelihood (Murphy & Van der Vaart 2000) of CoxPH model, resulting in nonlinear variants that essentially replaces the linear term in partial likelihood with nonlinear vari- ants: Huang (1999) used smoothing splines. et al. (2007 2008) used local polynomial regression (Fan & Gijbels (996)). The empirical success of tree- based models inspired subsequent develop- ments like Ishwaran al. (2008) that equip tree-based models such as gradient boosting trees and random forests with losses in the form of negative log partial likelihood. Early developments of neural survival analysis Faraggi & Simon (1995) adopted similar extension strategies and obtained neural versions of partial likelihood. Later attempts Katzman et al. (2018) suggest using the suc- cessful practice of stochastic training which is believed to be the heart of the empirical success of modern neural methods (Hardt et al., 2016). However, stochastic training under the partial likeli- hood objective highly non- privialvial, as mini-batch versions of log partial likelihood Katzman et al. 2018) are no longer valid stochastic gradients of the full-sample log partial likelihood (Tang et al. 2022
2.2
BEYOND COXPH IN SURVIVAL ANALYSIS
In linear survival modeling, there are standard alternatives to CoxPH such as the accelerated failure time (AFT) model (Buckley & James 1979 Ying 1993), the extended hazard regression model
Amoli & Ciampi 1987), and the family of linear transformation models (Zeng & Lin
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2006
While these models allow certain types of nonlinear extensions, the resulting form of (condi-
tional) hazard function is still restricted to be of a specific form. The idea of nonparametric hazard regression (NHR) (Cox & O' S'Sullivan 1990] Kooperberg et al. [995; Strawderman & Tsiatis 1996)
further improves the flexibility of nonparametric survival analysis via directly modeling the condi- tional hazard function by nonparametric regression techniques such spline approximation. Neural
versions of NHR have been developed lately such as the CoxTime model Kvamme et al. (2019)
Rindt al. (2022) used a neural network to approximate the conditional survival function and could be thus viewed as another trivial extension of NHR.
Aside from developments in NHR, et al. (2018) proposed a discrete-time model with its objec- tive being a mix of the discrete likelihood and rank-basee score; Zhong et al. (2021a) proposed neural version of the extended hazard model, unifying both neural CoxPH and neural AFT model; Tang et al. (0022) used an ODE approach to model the hazard and cumulative hazard functions.
2.3
THFORETICAL JUSTIFICATION OF NEURAL SURVIVAL MODELS
Despite the abundance of neural survival models, assessment of their theoretical properties remains nascent. In Zhong et al. (2021b). the authors developed minimax theories of partially linear cox model using neural networks as the functional approximator. [Zhong et al. (2021a) provided con- vergence guarantees of neural estimates under the extended hazard model. The theoretical devel- opments therein rely on specific forms of objective function (partial likelihood and kernel pseudo- likelihood) and are not directly applicable the standard likelihood-based objective which is fre- quently used in survival analysis.
3
METHODOLOGY
3.1 THE NEURAL FRAILTY MACHINE FRAMEWORK
Let T > be the interested event time with survival function denoted by S(1) - P(T > 1) associ- ated with a feature(covariiat) vector Rd. Suppose that T' a continuous random variable and let f(t) be its density function. Then X(t)= = (t)) is the hazard function and A(t) = fo A(s)ds is the cumulative hazard function. Aside from the covariate Z, use positive scalar random variable w E R+ to express the unobserved heterogeneity corresponding to individuals, or frailty.
In this paper we will assume the following generating scheme of T via specifying its conditional hazard function:
Z.) wii(t Z).
Here 7 is an unspecified non-negative function, and we let the distribution of w be parameterized by one-dimensional parameter IR. P The formulation (T) is quite general and contains several important models in both traditional and neural survival analysis:
1. When follows parametric distributional assumptions, and btt, Z) A(t)e 2 reduces to the standard proportional frailty model (Kosorok et al. 2004). A special case is when w is degenerate, i.e., has randomness, then the model corresponds to the classic CoxPH model.
2. When w is degenerate and D is arbitrary, the model becomes equivalent to nonparametric hazard regression (NHR) (Cox & O' S'ulllivan 1990, Kooperberg et al.) 1995). In NHR, the function parameter of interest is usually the logarithm of the (conditional) hazard function.
In this paper we construct neural approximations to the logarithm of i, i.e., r(t, 2) = log (t, Z). The resulting models called Neural Frailty Machines (NFM). Depending on the prior knowl- edge of the function v/ we propose two function approximation schemes:
For example in medical biology, it was observed that genetically identical animals kept in as similar an environment as possible will typically not behave the same upon exposure to environmental carcinogens (Bren-
2002)
The choice of one mne-dimeniional frailty family is mostly for simplicity and clearness of theoretical deriva- tions. Note that there exist mullti-dimessonnal frailty families like the PVF family (Wienke 2010)) Generalizing families would require additional sets of regularity conditions, and will
our theoretical results such kinds of he left future explorations.
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The proportional frailty (PF) scheme assumes the dependence of on event time and covariates to be completely decoupled, i.e.,
ll, 2) = h(()) + (Z).
Proportional-style assumption over hazard functions has been shown to be useful inductive bias in survival analysis. We will treat both h and m in (2) as function parameters, and device two multi-layer perceptrons (MLP) to approximate them separately.
The fully neural (FN) scheme imposes no a priori assumptions over v and is the most general version of NFM. It is straightforward to see that the most commonly used survival models, such as CoxPH, AFT, EH, or PF models are included in the proposed model space special cases. We treat v(t, Z) as the function parameter with input dimension + 1 and use a multi-layer perceptron (MIP) as the function approximator to v. Similar approximation schemes with respect to the hazard function have been proposed in some recent works (Omi et al. 2019) Rindt et al. 2022). referred to as "fully neural approaches" without theoretical characterizations.
The choice of frailty family There are many commonly used families of frailty distributions Kosorok et al., 2004; Duchateau & Janssen 2007; [Winnee 2010)) among which the most popu- lar one is the gamma frailty, where w follows a gamma distribution with mean 1 and variance 0. We briefly introduce some other types of frailty families in appendix A
3.2 PARAMETER LEARNING UNDER CENSORED OBSERVATIONS
In time-to-event modeling scenarios, the event times are typically observed under right censoring. Let C' be the right censoring time which is assumed be conditionally independent of the event time T' given Z, i.e., T IL Z. In data collection, one can observe the minimum of the survival time and the censoring time, that is, observe T T A C well as the censoring indicator where a A b min(a, b) for constants a and b and I(-) stands for the indicator function. We assume n independent and identically distributed (i.i.d.) copies of (T, 6, Z) are used as the training sample (T;, 6,, Z E [n]]. where we use [n] to denote the set (1,2...., n... Additionally, we assume the iid.
I(T < CC),
unobserved frailties are independent and identically distributed, i.e., Wi ~ fe(w), i E [n]. Next, we derive the learning procedure based on the observed log-likelihood (OLL) objective under both PF and FN scheme. To obtain the observed likelihood, we first integrate the conditional survival function given the frailty:
S(t = Bu [e- (e-wSh (g =: e d)))
Here the frailty transform Go(x) - log (T [e= is defined as the negative of the logarithm of the Laplace transform of the frailty distribution. The conditional cumulative hazard function is thus Att Go (((). For the PF scheme of NFM, we use two MLPs h h(t: W b²) and m = m Z: W b"d) as function approximators to v and m, parameterized by (Wh b²) and (Wm)) m)) respectively. According to standard results censored data likelihood (Kalbfleisch & Prentice 2002), we write the learning objective under the PF scheme as:
L(W", bb, W nTh 0)
1
8, loggee go iEin]
e mIZ,)
(" T.
(iodds + +,,T((I))+ &.m(Z))
(4)
("r
Ti
Ji(s) ass ds )
-Ga
(iZi)
Here we define (x)))
8g Go(z). Let (Wh 6A mm 6. Om) be the maximizer of and fur-
ther denote (1) = Ti(1; Wh b and (Z) m( z; Wm bgr. The resulting estimators for conditional cumulative hazard and survival functions are:
APF(t = Gā. ( ds
(PF = -5 (1)2)
(5)
Here we adopt the conventional notation that W is the collection of the weight matrices of the MI. in all layers, and b corresponds to the collection of the bias vectors in all layers.
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For the FN scheme, use D ptt, Z; W", b²) to approximate (tt, Z) parameterized by (W b.)) The OLL objective is written as:
og log go (f% (l 5, "", W".b") W",
C(W". b", 6)
Σ
72
(6)
Go (na
el 8") ds
Let (W n² 6% 0n) be the maximizer of (6). and further denote i (t, 2) it,, Z; W bT). The conditional cumulative hazard and survival functions are therefore estimated as:
AFN(1 = Gā. (T. PP
S = -
(7)
The evaluation of objectives like ( and its gradient requires computing a definite integral of an exponentially transformed MLP function. Instead of using exact computations that are available for only restricted type of activation functions and network structures, we use numerical integration for such kinds of evaluations, using the method of Clenshaw- -arttis quadrature (Boyd) (200)). which has shown competitive performance and efficiency in recent applications to monotonic neural networks WWeennkel & Louppe 2019).
The interpretation of frailty terms differs in the two schemes. In the PF scheme, intro- ducing the frailty effect strictly increases the modeling capability (i.e., the capability of modeling crossing hazard) in comparison to CoxPH or neural variants of CoxPH (Kosorok et al.) (2004). In the FN scheme, is arguable that in the i.i.d. case, the marginal hazard function is a reparameterization of the hazard function in the context of NHR. Therefore, we view the incorporation of frailty effect as injecting a domain- -speciffe inductive bias that has proven to be useful in survival analysis and time-to-event regression modeling and verify this claim empirically in section [3.2] Moreover, frailty becomes especially helpful when handling correlated or clustered data where the frailty term as- sumed to be shared among certain groups of individuals (Parner 1998). Extending NFM to such scenarios is valuable and we left it to future explorations.
Remark 1.
THEORETICAL RESULTS
In this section, we present theoretical properties of both NFM estimates by characterizing their rates of convergence when the underlying event data follows corresponding model assumptions. The proof technique is based on the method of sieves (Shen Wong 1994, Shen 1997; Chen 2007) that views neural networks approximation properties (Yarotsky) 2017). Since both models produce estimates of function pa- rameters, we need to specify a suitable function space work with. Here choose the following Hölder ball as was also used in previous works on nonparametric estimation using neural networks Schmedt-Hiebber 2020: Farrell et al. 2021 Zhong al. 2021b)
special kind of nonlinear sieve (Chen) 2007) that satisfies desirable
W (2)
f:
max esssup D (f(x)) asja|g =EX
(8)
where the domain X' is assumed to be subset of d-dimensional euclidean space. = (a1,... ........ a.) d-dimensional tuple of nonnegative integers satisfying |a| = a1 + ar and D f ginls B.!
is the weak derivative of f. Now assume that M is reasonably large constant, and let θ be a closed interval over the real line. We make the following assumptions for the true parameters under both schemes:
Condition 1 (True parameter, PF scheme). The euclidean parameter 0g E + R, and the two function parameters my E ( 1J²), by e Will (10, ([0,T]). and 7 > is the ending time of the study duration, which is usually adopted in the theoretical studies in survival analysis (Van der Vaari 2000).
Condition 2 (True parameter, FN scheme). The euclidean parameter 0g E O C R. and the function parameter My E W (00, 7] x [-1,
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Next, we construct sieve spaces for function parameter approximation via restricting the complexity of the MLPs to "scale" with the sample size n.
Condition 3 (Sieve space, PF scheme). The sieve space Hn is constructed as a set of MLPs satisfying hi E W ([0,T)) with depth of order O(logn) and total number of parameters of order O(nzia log The sieve space Mn is constructed as a set of MLPs satisfying m. Wi Mm. ([-1, with depth of order Ollogn) and total number of parameters of order O(n log n). Here Mr and Mn are sufficiently large constants such that every function in ([-1, ([-1,11]6 1]d) and W ((). could be accurately approximated by functions inside Hr and Mns according to (Yarotsky 2017] Theorem 1).
of MLPs sat- isfying E W ([0, with depth of order ((logn) n) and total number of parameters of order O(n sea+1 logn). Here My is a sufficiently large constant such that Vn satisfies approximation properties, analogous to condition 3
Condition 4 (Sieve space, FN scheme). The sieve space Vn2 is constructed
For technical reasons, will assume the nonparametric function estimators constrained to fall inside the corresponding sieve spaces, i.e., hn Hn, m, M, and Vn, This will not affect the implementation of optimization routines as was discussed in Farrell et al. (201). Furthermore,
restrict the estimate 0n θ in both PF and FN schemes.
Additionally, we need the following regularity condition on the function Go(z):
Condition 5. Gz(z) is viewed as a bivariate function G θ B + R, where B is compact set on 8 R. The functions Go(z), 00 Go (z)r) 8 Ge (z). (2)) 50 θ x B.
8 log ge (z). du
8 log (x are bounded on
dz
We define two metries that measures convergence of parameter estimates: For the PF scheme, let Φ0 = (ho, mp. fb) be the true parameters and の. = (h,,, m., 0n) be the estimates. We abbreviate F.o.7== the conditional probability distribution of (T),)) given Z under the true parameter,
and IP == as the conditional probability distribution of (T, given Z under the estimates.
Define the following metric
dpF (pn. 100 (0) BENPZ
2= 2=; HT (P
Poo |Poo.z=a)] )
where (P Q2) (ViT= dTP Vd
is the squared Hellinger distance between probability distributions I and Q. The case for the FN scheme is similar: Let eho (No, 00) be the parameters and 2, (D,, (x) be the estimates. Analogous the definitions above, define Pvo.2= the true conditional distribution given Z = 2, and P Z= be the estimated conditional distribution, we will use the following metric in the FN scheme:
dFN
(ens wo) = -- EanP2 (P 7 P Poo.Z==)
(10)
Now we state our main theorems. We denote IP as the data generating distribution and use to hide loga-toga factors in the big-O notation.
Theorem (Rate of convergence, PF scheme). In the PF scheme, under condition I
3
we have
that dpF (Zn: do) Öp (n 2P+2d
Theorem (Rate of convergence, FN scheme). In the FN scheme, under condition 2 that dFN (,., v/) Öp 28+2242
we have
Remark The idea of using Hellinger distance to measure the convergence rate of sieve MLEs was proposed in Wong & Shen ( (1995). Obtaining rates under a stronger topology such as Lz is possible the likelihood function satisfies certain conditions such as the curvature condition (Farrell et al.) 2021). However, such kind of conditions are in general too stringent for likelihood- baaied objectives, instead, we use Hellinger convergence that has minimal requirements. Consequently. our proof strategy is applicable to many other survival models that rely neural function approximation such as Rindt et al. (2022), with some modification to the regularity conditions. For proper choices of metrics in sieve theory, see also the discussion in Chen (007 Chapter 2).

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1000 (PF)
(PF)
10000 (PF)
ground ostimared
ground
ground
vit.
w
60 index
100
50 Index
100
50
100
index
N 1000 (FN)
N 5000 (FN)
N 10000 (FN)
ground truch
ground tuth
ground truth
ostmmred
estimnnted
estimated
()
vit, 2)
w
100
50
100
Index
index
index
Figure 1: Visualizations of synthetic data results under the NFM framework. The plots in the first row compare the empirical estimates of the nonparametrie component v(t, Z) against its true value evaluated 100 hold-out points, under the PF scheme. The plots in the second row obtained using the FN scheme, with analogous semantics to the first row.
EXPERIMENTS
In this section, assess the empirical performance of NFM. We first conduct synthetic experiments for verifying the theoretical convergence guarantees developed section 4 To further illustrate the empirical efficacy of NFM, evaluate the predictive performance of NFM over benchmark datasets ranging from small scale to large scale, against state-of- of-theart baselines.
5.1 SYNTHETIC EXPERIMENTS
We conduct synthetic experiments to validate proposed theory. The underlying data generating scheme is as follows: First, we generate a 5-dimensional feature Z that is independently sampled from the uniform distribution the event time takes the form of the proportional frailty model (2) with h(t) = t and m m22) 0.2, 0.3, 0.4, 0.5). The frailty generated according
the interval [0, 1]. The (true) conditional hazard function of
sin((Z,3) + ( where to gamma distribution with mean and variance equal to 1. We use this generating model to assess the recovery guarantce of both NFM modeling schemes via inspecting the empirical recovery of rtt, Z). For the PF scheme, we have more underlying information about the generating model, and we present an additional assessment regarding the recovery of m(Z) in appendix D.1 We generate three training datasets of different scales, with E {1000, 5000, 10000). A censoring mechanism applied such that the censoring ratio is around 40% for each dataset. The assessment will be made on fixed test sample of 100 hold-out points that are independently drawn from the generating scheme the event time. We report more detailed description of the implementation of the data generating scheme and model architectures in appendix C.2 We present the results of our synthetic data experiments in figure The evaluation results suggest that both NFM schemes are capable of approximating complicated nonlinear functions using moderate amount of data, i.e., 1000.

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Table 1: Survival prediction results measured in IBS and INBLL metric (%) on four small-scale survival datasets. In each column, the boldfaced score denotes the best result and the underlined score represents the second- result.
Model
METABRIC
RoGGBS
FLCHAIN
SUPPORT
IBS
INBLL
IBS
INBLL
IBS
INBLL
IBS
INBLL
CoxPH
26.462030
49.57 22.00
18.25...44
58.7621.11
10.05
33.18=1.6
20.54.0.38
59.58 ss
GBM
16.61=0.2
49.87
17.8320.44
52.78.1.11
9.98 40.37
32.85 =2.8
19.18
56.46.0
RSF
16.62 16.6220.1
49.61
17.8920.42
52.77
9. 9.96
32.92-100
19.11
56.28
Surr
16.55=o.as
19.85xs5.02
17.80
12.22.22
33. 18
19.20
56,48
CoxTime
16.54_o.ss
49.67
17.80
52.56611.47
20.28
34.18=1.63
19.17
56.45
DeepHit
17.50
52.10
19.61
56.67
11.83
37. 37.72-1.00
20.6610033
60.06
DeepEH
16.56=0.05
19.42-x1.53
17.6220052
52 1.27
20.37
33. 33.30 10
19.30
50.67-00.44
SuMo-net
16.49=0.8s
49.74+22.1
17.77=0.47
52.62 1.111
10.47
33.20 13.20:1.10
19.40
56.8T=0.se
SODEN
16.52.0.63
49.39
17.05
50.45
10.13
30.37
19.07
56.15
NEM-PF
16.33
49.07
17.60
52. 12.12
9.96
32 02.84
19.14
56.35
NEM-FN
16. 16.11g0.81
48.2112.0
17.66
12.21
10.05
33.11=1:10
18.97
55.87 +1.50
5.2 REAL-WORLD DATA EXPERIMENTS
Datasets We use five survival datasets and one non-survival dataset for evaluation. The survival datasets include the Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) (Curtis et al., 2 the Rotterdam tumor bank and German Breast Cancer Study Group (RotG- BSGu et al. 1995). the Assay Of Serum Free Light Chain (FLCCAIN) (Dispenzieri et al.) 2012). the Study to Understand Prognoses Preferences Outcomes and Risks of Treatment (SUP- PORT) (Knaus al., 1995). and the Medical Information Mart for Intensive Care (MIMIC-III) (Johnson et al.] 2016). For all the survival datasets, the event of interest is defined as the mortality after admission. In our experiments, as small-scale datasets and MIMIC-III as a moderate scale dataset. We additionally use the KKBOX dataset vavamee al. 2019) large-scale evaluation. In this dataset, an event time is observed if a customer churns from the KKBOX platform. We summarize the basic statistics of all the datasets table
view METABRIC, RotGBSG, FLCHAIN, and SUPPORT
Baselines We compare NFM with 9 baselines. The first one is the linear CoxPH model (Cox 1972,
Gradient Boosting Machine (GBM) (Friedman 2001 Chen & Guestrin 2016) and Random Survival Forests (RSF) (Ishwaran et al. 2008 are two bree- nonparametric survival regression methods. DeepSurv (Katzman et al.J (018)) and CoxTime (Kvamme al.) 2019) are two models that adopt neural variants of partial likelihood as objectives. SuMo-net (Rindt et al. (022) is a neural variant of NHR. We additionally chose three latest state-of-the-ar neural survival models: DeepHit (Lee) et al. 2018). DeepEH (Zhong et al. (021a), and SODEN (Tang et al., (022). Among the chosen baselines, DeepSurv and SuMo-net are viewed as implementations of neural CoxPH and neural NHR and are therefore of particular interest for the empirical verification of the efficacy of frailty. A more thorough performance comparison with a larger set of baselines is provided in appendix D.3
Evaluation strategy We two standard metrics in survival predictions for evaluating model per- formance: integrated Brier score (IBS) and integrated negative binomial log-likelihood (INBLL). Both metries are derived from the following:
1 n
e(O. (0,S((4))) ((4Z)) St,, =1) E(1, S(1 (1Z))(T > >()
S(E,t1,t2)
dt. (1I)
(T.)
Sc(t)
Where Sc is an estimate of the survival function Sc- of the censoring variable, obtained by the Kaplan-Meier estimate (Kaplan & Meier, 1958) of the censored observations on the test data. e: (0,1 1} [0, 1] + is some proper loss function for binary classification (Gneiting & Raftery 2007]. The IBS metric corresponds to being the square loss, and the INBLL metric corresponds to l being the negative binomial (Bemoulli) log- likelihoohd (Graf et al., 1999). Both IBS and INBLL are proper scoring rules the censoring times and survival times are independent. We additionally
Otherwise, one may pose covariate-dependent model the censoring time and use (4) instead of (t))). We adopt the Kaplan-Meier approach since it's still the prevailing practice in evaluations of survival predictions.
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Table Survival prediction results measured in IBS and INBLL metric (%) on two larger datasets. In each column, the boldfaced score denotes the best result and the underlined score represents the second- best result. Two models are not reported, namely SODEN and DeepEH, as we found empirically that their computational//mmmoon cost is significantly worse than the rest, and we fail to obtain reasonable performances over the two datasets for these two models.
Model
MIMIC-III
KKBOX
IBS
INBLL
IBS
INBLL
CoxPH
20.40
60.02.0.00
12.00
39.40 29.40000.00
GBM
17.70
52.30
11.8110.00
38. 38.151.000
RSF
17.7910.19
53.34
14.4622.000
44.3910.00
DeepSurv
18.5840.92
55.98
11.31 +0.06
35. 25.28
Cox
17.68+1.a6
52.08+3306
10.70
33.10
DecpHit
19.80
59.03-g4.20
16.00+0.54
48.644
SuMo-net
18.62+1.23
54.51 42.97
11.58
36.61
NEM-PF
16.2820.28
49.18
11.02
35.102o022
NFM-FN 17.47
51.48 1.23
10.63
32.81 10.14
report the result of another widely used metric, the concordance index (C-index), in appendix D Since all the survival datasets do not have standard train/test splits, we follow previous practice (Zhong et al. (0211) that uses 5-fold cross-validation (CV): 1 fold is for testing, and 20% of the rest is held out for validation. our experiments, we observed that single random split into 5 folds does not produce stable results for most survival datasets. Therefore we perform 10 different CV runs for each survival dataset and report average metrics as well as their standard deviations. For the KKBOX dataset, we use the standard train/valid/ten splits that are available via the pyсox package Kvamme et al. 2019) and report results based on 10 trial runs.
Experimental setup We follow standard preprocessing strategies (Katzman al. 2018 et al.) 2019; Zhong et al. (2011)) that standardize continuous features into zero mean and unit vari- ance, and do one-hot encodings for all categorical features. We adopt MLP with ReLU activation for all function approximators, including ht f in PF scheme, and 7 in FN scheme, across all datasets, with the number of layers (depth) and the number of hidden units (width) within each layer being tunable. We tune the frailty transform over several standard choices detailed in appendix C.3 We find that the gamma frailty configuration performs reasonably well across all tasks and is recom- mended to be the default choice. A more detailed description of the tuning procedure, as well as training configurations for baseline models, reported in appendix C.3
K Kvamee
Results we report experimental results of small-scale datasets table ! and results of two larger datasets in table 2 The proposed NFM framework achieves the best performance 5 of the 6 datasets. The improvement over baselines is particularly evident in METABRIC, SUPPORT, and MIMIC-III datasets.
Benefits of frailty to better understand the additional benefits introducing the frailty formulation, compute the (relative) performance gain of NFM-PF and NFM-FN, against their non-frailty counterparts, namely DeepSurv (Katzman et al. 2018) and SuMo-net (Rindt et al. 2022). The evaluation is conducted for all three metrics mentioned in this paper. The results are shown in table 7
The results suggest a solid improvement in incorporating frailty, as the relative increase in performance could be over 10% for both NFM models. more detailed discussion presented in section D.5
CONCLUSION
In this paper, we make principled explorations on applying the idea of frailty models in modern survival analysis to neural survival regressions. A flexible and scalable framework called NFM is proposed that includes many useful survival models as special cases. Under the framework, we study two derived model architectures both theoretically and empirically. Theoretically, we obtain the rates of convergences of the nonparametric function estimators based on neural function approx- imation. Empirically, we demonstrate the superior predictive performance of the proposed models by evaluating several benchmark datasets.

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REFERENCES
Laura Antolini, Patrizia Boracchi, and Elia Biganzoli. A time-dependent discrimination index for survival data. Statistics in medicine, 24(24)3927-3944, 2005.
Peter L Bartlett, Nick Harvey, Christopher Liaw, and Abbas Mchrabian. Nearly-tight ve-dimension and pseudodimension bounds for piecewise linear neural networks. The Journal of Machine Learning Research, 20(1):2285 2019.
Steve Bennett. Analysis of survival data by the proportional odds model. Statistics in medicine, 2 (2):273-277, 1983.
Peter Bickel, Chris AJ Klaassen, Ya' acov Ritov, J Klaassen, Jon A Wellner, and YA Acov Ritov. Efficient and adaptive estimation for semiparametric models, volume Springer, 1993.
John P Boyd. Chebyshev and Fourier spectral methods. Courier Corporation, 2001.
Paul Brennan. can we measure it? Carcinogenesis, 23(3):381-387, 2002.
interaction and aetiology of cancer: what does it mean and how
Jonathan Buckley and lan James. Linear regression with censored data. Biometrika, 66(3):429-436, 1979.
Jianwen Cai, Jianqing Fan, Jiancheng Jiang, and Haibo Zhou. Partially linear hazard regression for multivariate survival data. Journal of the American Statistical Association, 102(478):538-551, 2007.
Jianwen Cai, Jianqing Fan, Jiancheng Jiang, and Haibo Zhou. Partially linear hazard regression with varying coefficients for multivariate survival data. Journal of the Royal Statistical Society: Series B (Statistical Methodology). 70(1):141-553 2008.
Tianqi Chen and Carlos Guestrin. Xgboost: A scalable tree boosting system. In Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining, 785-794, 2016.
Xiaohong Chen. Large sample sieve estimation of semi-nonparametric models. Handbook of econo- metrics, 6:5549-5632, 2007.
Xiaohong Chen and Xiaotong Shen. Sieve extremum estimates for weakly dependent data. Econo- metrica, pp. 289-314, 1998.
David R Cox. Regression models and life-tables. Journal of the Royal Statistical Society: Series (Methodological)) 34(2):187-202 1972.
David R Cox. Partial likelihood. Biometrika, 62(2):269-27) 276, 1975.
Dennis D Cox and Finbarr O' Sullivan. Asympiotic analysis of penalized likelihood and related estimators. The Annals of Statistics, pp. 1676-1695, 1990.
Christina Curtis, Sohrab P Shah, Suet-Feung Chin, Gulisa Turashvili, Oscar M Rueda, Mark J Dun- ning, Doug Speed, Andy Lynch, Shamith Samarajiwa, Yinyin Yuan, et al. The genomic and transcriptomic architecture of 2,000 breast tumours reveals novel subgroups. Nature, 486(7403): 346-352, 2012.
Jack Cuzick. Rank regression. The Annals of Statistics, pp. 1369- 1389, 1988.
Dorota M Dabrowska and Kjell A Doksum. Partial likelihood in transformation models with cen- sored data. Scandinavian journal of statistics, pp. 1-23, 1988.
Dominie Danks and Christopher Yau. Derivative-based neural modelling of cumulative distribu- tion functions for survival analysis. In International Conference on Artificial Intelligence and Statisties, pp. 7240-7256. PMLR, 2022.
10

---

Under review as a conference paper at ICLR 2023
Angela Dispenzieri, Jerry A Katzmann, Robert A Kyle, Dirk R Larson, Terry M Themeau, Colin L Colby, Raynell Clark, Graham Mead, Shaji Kumar, L Joseph Melton II et al. Use of non- clonal serum immunoglobulin free light chains to predict overall survival in the general popula- tion. In Mayo Clinic Proceedings, volume 87, pp. 517-523. Elsevier, 2012.
Luc Duchateau and Paul Janssen. The frailty model. Springer Science & Business Media, 2007.
Darrell Duffie, Andreas Eckner, Guillaume Horel, and Leandro Saita. Frailty correlated default. The Journal of Finance, 64(5):2089 2223, 2009.
Jamshid Etezadi-Amoli and Antonio Ciampi. Extended hazard regression for censored survival data with covariates: spline approximation for the baseline hazard function. Biometrics, pp. 181-192, 1987.
Jaanqing Fan and Irene Gijbels. Local Polynomial Modelling and Its Applications: Monographs on Statistics and Applied Probability 66, volume 66. CRC Press, 1996.
David Faraggi and Richard Simon. A neural network model for survival data. Statistics in medicine, 14(1):73-22, 1995.
Max H Farrell, Tengyuan Liang, and Sanjog Misra. Deep neural networks for estimation and infer- ence. Econometrica, 89(1)111-2113, 2021.
Jerome H Friedman. Greedy function approximation: gradient boosting machine. Annals of
statistics, pp. 1189-1232, 2001.
Tilmann Gneiting and Adrian E Raftery. Strictly proper scoring rules, prediction, and estimation. Journal of the American statistical Association, 102(477):359-378 2007.
Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep learning. 2016.
Erika Graf, Claudia Schmoor, Willi Sauerbrei, and Martin Schumacher. Assessment and comparison of prognostic classification schemes for survival data. Statistics in medicine, 18(17-18):2529 2545, 1999.
Robert J Gray. Estimation of regression parameters and the hazard function in transformed linear survival models. Biometrics, 56(2):571-576, 2000.
Stefan Groha, Sebastian M Schmon, and Alexander Gusev. A general framework for survival anal- ysis and multi-state modelling. arXiv preprint arXiv:2006.04893, 2020.
Moritz Hardt, Ben Recht, and Yoram Singer. Train faster, generalize better: Stability of stochastic gradient descent. International conference on machine learning, pp. 1225-1234. PMLR, 2016.
Philip Hougaard. Life table methods for heterogeneous populations: distributions describing the heterogeneity, Biometrika, 71(1):75-83, 1984.
Jian Huang. Efficient estimation of the partly linear additive cox model. The annals of Statistics, 27 (5):1536-1563, 1999.
Hemant Ishwaran, Udaya B Kogalur, Eugene H Blackstone, and Michael S Lauer. Random survival forests. The annals of applied statistics, 2(3):841-86)) 2008
How Jing and Alexander J Smola. Neural survival recommender. In Proceedings of the Tenth ACM International Conference on Web Search and Data Mining. pp. 515-524, 2017.
Alistair EW Johnson, Tom J Pollard, Lu Shen, Li-wei H Lehman, Mengling Feng, Mohammad Ghassemi, Benjamin Moody, Peter Szolovits, Leo Anthony Celi, and Roger G Mark. Mimic-iii, freely accessible critical care database. Scientific data, 3(1):1-9, 2016.
John D Kalbfleisch and Ross L Prentice. The Statistical Analysis of Failure Time Data, volume 360.
John Wiley & Sons, 2002.
Edward L Kaplan and Paul Meier. Nonparametric estimation from incomplete observations, Journal of the American statistical association, 53(282))457-481, 1958.

---

Under review as a conference paper at ICLR 2023
Jared L Katzman, Uri Shaham, Alexander Cloninger, Jonathan Bates, Tingting Jiang, and Yuval Kluger. Deepsurv: personalized treatment recommender system using a cox proportional hazards deep neural network. BMC medical research methodology, 18(1):1-12, 2018.
William A Knaus, Frank E Harrell, Joanne Lynn, Lee Goldman, Russell S Phillips, Alfred F Con- nors, Neal Dawson, William J Fulkerson, Robert M Califf, Norman Desbiens, et al. The support prognostic model: Objective estimates of survival for seriously ill hospitalized adults. Annals of internal medicine, 122(3):191-203: 1995.
Charles Kooperberg, Charles Stone, and Young K Truong. Hazard regression. Journal of the American Statistical Association, 90(429):78 94, 1995.
Michael R Kosorok, Bee Leng Lee, and Jason P Fine. Robust inference for univariate proportional hazards frailty regression models. The Annals of Statistics, 32((): 1448-1491, 2004.
Hävard Kvamme, Ømnll Borgan, and Ida Scheel. Time-to-event prediction with neural networks and regression. arXiv preprint arXiv: 1907. 1907000897. 2019.
Changhee Lee, William Zame, Jinsung Yoon, and Mihaela Van Der Schaar. Deephit: A deep learn- ing approach to survival analysis with competing risks. In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018.
Susan A Murphy. Consistency in proportional hazards model incorporating a random effect. The Annals of Statistics, 22(2):712-731, 1994.
Susan A Murphy. Asymptotic theory for the frailty model. The annals of statistics, pp. 182-198, 1995.
Susan A Murphy and Aad W Van der Vaart. On profile likelihood. Journal of the American Statis- tical Association, 95(450):449 465, 2000.
Chirag Nagpal, Steve Yadlowsky, Negar Rostamzadch, and Katherine Heller. Deep cox mixtures for survival regression. In Machine Learning for Healthcare Conference, pp. 674-708. PMLR, 2021.
Takahiro Omi, naonori ueda, and Kazuyuki Aihara. Fully neural network based model for general temporal point processes. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-But E. Fox, and R. Garnett (eds.). Advances in Neural Information Processing Systems, volume 32. Cur- http:/ttt:////o/// ran Associates, Inc., 2019. (1
neurips httpppsccccpaape///111..
File/39e497 33211442331333333283333333333333
Erik Pameer Asymptotic theory for the correlated gamma-frailty model. The Annals of Statistics, 26 (1):183-214, 1998.
Sebastian Pölsterl. seikit-survival: library for time-to-event analysis built on top of scikit- learn. Journal of Machine Learning Research, 21(212): 2020. WRRLhttt::///
sapers/v21/253 22122220222222222222 mm
Sanjay Purushotham, Chuizheng Meng, Zhengping Che, and Yan Liu. Benchmarking deep learning models large healtheare datasets. Journal of biomedical informatics, 83:1 13:114, 2018.
D Rindt, R Hu, D Steinsaltz, and D Sejdinovic. Survival regression with proper scoring rules and monotonic neural networks. Proceedings of Machine Learning Research. Journal of Machine Learning Research, 2022.
Johannes Schmidt-Hieber. Nonparametric regression using deep neural networks with relu activation function. The Annals of Statistics, 48(4): 1875-1897, 2020.
Xiaotong Shen. On methods sieves and penalization. The Annals of Statistics, 25(6):2555-2555 1997.
Xiaotong Shen and Wing Hung Wong. Convergence rate of sieve estimates. The Annals of Statistics, pp. 580-615, 1994.
Donald M Stablein and IA Koutrouvelis. two-sample test sensitive to crossing hazards in uncen- sored and singly censored data. Biometrics, pp. 643-652, 1985.
12

---

Under review as a conference paper at ICLR 2023
Harald Steck, Balaji Krishnapuram, Cary
Philippe Lambin, and Vikas C Raykar. On ranking in survival analysis: Bounds on the concordance index. In J. Platt, D. Koller, Y. Singer, and S. Roweis
in Neural Information Processing Systems, volume 20. Curran As- httpp///ss.ccc/aaee////////1/
sociates, Inc., 2007.
33e8075e 335000177777eeecraaaaaaaa4444223
pdf
Robert L Strawderman and Anastasios Tsiatis. On the asymptotic properties of flexible hazard estimator. The Annals of Statistics, 24(1):41-66 1996.
Weijing Tang. Jiaqi Ma, Qiaozhu Mei, and Ji Zhu. Soden: A scalable continuous-time survival model through ordinary differential equation networks. Journal of Muchine Learning Research, 23(34):1-29, 2022. JRLbtt:://mmml org/ http:/ppppss///3200pp0e/e 900. html
Aad W Van der Vaart. Asymptotic statistics, volume 3. Cambridge university press, 2000.
AW van der Vaart, A.W. van der Vaart, A. van der Vaart, and J. Wellner. Weak Conver- gence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. Springer, 1996. ISBN 9780387946405.. httthttts///ooo gowgle.oooo...oooooooooooooo.o.. com/books?id (GCenCW9qmp4C)
Antoine Wehenkel and Gilles Louppe. Unconstrained monotonic neural networks. Advances in neural information processing systems, 32, 2019.
Andreas Wienke. Frailty models in survival analysis. Chapman and Hall/CRC, 2010.
Wing Hung Wong and Xiaotong Shen. Probability inequalities for likelihood ratios and convergence rates of sieve mles. The Annals of Statisties, pp. 339-362, 1995.
Wush Chi-Hsuan Wu, Mi- Yen Yeh, and Ming-Syan Chen. Predicting winning price in real time bidding with censored data. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1305-1314, 2015.
Dmitry Yarotsky. Error bounds for approximations with deep relu networks. Neural Networks, 94: 103-114, 2017.
Zhiliang Ying. large sample study of rank estimation for censored regression data. The Annals of Statistics, pp. 76-99, 1993.
Donglin Zeng and DY Lin. Efficient estimation of semiparametric transformation models for count- ing processes. Biometrika, 93(3):627-640, 2006.
Qixian Zhong, Jonas W Mueller, and Jane-Ling Wang. Deep extended hazard models for survival analysis. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan (eds.), Advances in Neural Information P Procesiig Systems, volume 34, pp. 15111-15124. Curran Asso- ciates, Inc., 2021a. URL htt./httpsss//procceenne neuripe cctppaapr//22///file 666cfff0bbaa 3o6 Papeee--apr..... pdf
Qixian Zhong, Jonas W Mueller, and Jane-Ling Wang. Deep learning for the partially linear cox model. The Annals of Statistics, 2021b.
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A EXAMPLES OF FRAILTY SPECIHICATIONS
We list several commonly used frailty models, and specify their corresponding characteristics via their frailty transform Ga:
Gamma frailty: Arguably the gamma frailty is the most widely used frailty model Murphy 1994 1995
Parner 1998) : Wienke (2010); Duchateau & Janssen (2007), with
1 log(1 + 0z)), 0 2 0. 0
(12)
When 0 = 0, Go(z) = limg-so Go(z) is defined as the (pointwise) limit. A notable fact of the gamma frailty specification is that when the proportional frailty (PF) assumption (2) is met, f = 0, the model degenerates to CoxPH Otherwise if 0 1, the model corresponds to the proportional odds (PO) model (Bennett) 1983).
Box-Co Cox transformation frailty: Under this specification, we have
Ga( (1 + (1+ 020.
(13)
0
The case of = 0 is defined analogously to that of gamma frailty, which corresponds to the PO model under the PF assumption. When 0 1, the model reduces CoxPH under the PF assump- tion.
(G)(a) frailty: This is an extension of gamma frailty (Kosorok al., 2004) and includes other types of frailty specifications like the inverse gaussian frailty Hougaard (1984), with
1-a =0 a0
fx (
Ge(z)
0 0, 1),
(14)
In the one-dimensional parameter paradigm, the parameter a is assumed known instead of being learnable. When = 1/2, we obtain the gamma frailty model. When a - 0, the limit corresponds the inverse Gaussian frailty.
Satistiability of regularity condition 5 In Kosorok et al. 22004 Proposition 1), the authors verified the regularity condition of gamma and IGG(a) frailties. Using similar argument, it is straightfor- ward to verify the regularity of Box-Cox transformation frailty.
B PROOFS OF THEOREMS
B. PRELIMINARY
Additional definitions The theory of empirical processes (van der Vaart et al.) 1996) will be in- volved heavily in the proof. Therefore we briefly introduce some common notations: For a function class F. define N (e, F, || to be the covering number of JF with respect norm || under radius e, and define Nn (e, 11.. to be the bracketing number of F with respect to norm || under radius e. We use VC (F) denote the VC-dimension of F. Moreover, we use the notation S to denote S Cb for some positive constant C.
Before proving theorem and we introduce some additional notations that will useful through- out the proof process.
In the PF scheme, define
J" chields
((T, 6, Z; m. 0) =ō logge
mm())
+ +Th(TT) + dm(()
on
(" chlo)dss
- Go
m(z
where we denote ge = GO)).. Under the definition of the sieve space stated in condition
wc
restate the parameter estimates
(or:. mn, 0m)
1 Σ ((T,, 6,,, Zo h m, 0).
arginax M,,
T vE(x)
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Similarly, in the FN scheme, we define
I(T, 6, 2; 5lg (fo (lss))dss)))
+ bullT, Go (eeess
Co
Under the definition of the sieve space stated in condition 4 we restate the parameter estimates as
Pn (2,(t.))8..)
argmax DEV.. 7
[(T,, 6, 2,;00),
iE|n]
We denote the conditional density function and survival function of the event time î given Z by fi (t) and S (t), respectively. Similarly, we denote the conditional density function and survival function of the censoring time given by fciz(t) and Saiz(t) Under the assumption that T IL C Z, the joint conditional density of the observed time T and the censoring indicator ó given can expressed as the following:
PTT, Z) =
(T)Z(T) (TI))
where xr)z((T) (T) is the conditional hazard function of the survival time T given Under the model assumption of PF scheme, p(T, 5 2) can be expressed by
p(T, 8 Z; h, m, 0) exp (((T, 6, Z; h, 0)) feiz(T) (
For @o (ho, mo, 0a) and an estimator क (h, m, 0), the defined distance dpF (o. Φ0) can be explicitly expresses by
Votc.. 2;h,m,0 V ho, mo, 00)
2
deN (3. vi)) vo)))) = Ez
u(dT d8)
Here the dominating measure is defined such that for any (measurable) function r(T, 6)
/ ()Tdd do) '-" r(T, 1)dT + [ r(T, 0)dT
Under the model assumption of FN scheme, 5 Z) can be expressed by PT, 8 Z;u, 0) exp (I(T, 5, M; 0)) faiz(T)) Sc(z)
For too = (vo, 6o) and an estimator = (i, 0), the defined distance dFN ( (5. can be explicitly
expresses by
Ez / //WTT,01 2;0,0) - VP(,8 Z;vo, 0o)
dFN (0.th)=
\
pldT dō)
B.2 TECHNICAL LEMMAS
The following lemmas are needed for the proof of Theorem and Hereafter for notational conve- nience, will use h, m for arbitrary elements in the corresponding sieve space listed in condition T for arbitrary element in the sieve space listed in condition 4 and Ø for arbitrary element in ④
Lemma 1. Under condition bounded:
3 for (T,5, Z) E [0, x {0.1) x [-,,1] the following terms
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1. I(T, 6, Z; mo, Bo) with true parameter (ho, ma: Bu)
2. (T,, 6, Z; m. with parameter estimates (hr, m, 0) in any sieve space listed in condition
Lemma 2. Under condition 2 4 for (T, 2) E [0,-] x (0. x 1-,, the following terms
are bounded:
1. ((T, 6, Z;M,, 8o) with true parameter (Pos 8g)
2. I(T, 6, 2;0,0) () with parameter estimates (D, 0) in any sieve space listed in condition 4
Lemma 3. Under condition 3 5 let (h, m, 0). (fy mi, 0.), ) and (h2 m2g 02) be arbitrary three
parameter triples inside the sieve space defined in condition the following two inequalities hold.
[I(T, 6, ho, mo. 0o) 1TT, 2; h m, 0) I 10o 0 llho - A1100 + 1| - milloe
[[[,T, 6, Z; h,, m.. 01) I(T, 6, 2, m2 82) 2 101 02l
11A1 hall + ll m2
Lemma 4. Under condition 2 4
let (1) 0). (01, 01). and (Dz, 02) be arbitrary three parameter
tuples inside the sieve space defined in condition 4. the following inequalities hold.
[I(TT, 6, Z 8b) - L(T, Z; 1oc S 100 - 0 + 1110 01100
||(T,, 6, 0,) - I(T, 22 02)1 - 10 021 + | D211
Lemma 5 (Approximating error of PF scheme). In the PF scheme, for any n. there exists an ele- ment in the corresponding sieve space Tn = ho, T, R,,0),, satisfying dpp (Tn oo)
O 72
Lemma (Approximating error scheme). In the FN scheme, for any n, there exists an element in the corresponding sieve space n. (N, vo) satisfying dFN (naton, v/o) O गामा.
class of functions satisfying that N(E,F, |]) < co for Ve > We define (e, II) to be the minimal number of e-balls =(f = {g : 11 - fll < E} needed to
Lemma 7. Suppose
cover F with radius and further constrain that F. Then we have
F111,,11-1 s (e, 11- N
Lemma 8. Suppose F is class of functions satisfying that Nn(E, F, l1 < co for Ve 0. We define F, lloo) to be the minimal number of brackets [t, u] needed to cover F with ||| - ulloo < E and further constrain that f E F, = and u = f 2. Then we have
Ni(e, F, (lo0) S FF, 1loo) S No G F,
Furthermore, we have (E, F, (1x0) = Ñ(5 F, F, 1100).
Lemma 9 (Model capacity of PF scheme). Let Fr = [([, 8, Z; h m. 0) h E Hz, m E Mn, e {). Under condition 5 with 20+1 that
29
and sm
29
there exist constants Cy and cm > 0 such
20
(n (l,0) S - N(cne 1/a Hns (121))2))))) x N (cme 1/em Mameel/eem... -12).
(Cmell/em
Lemma 10 (Model capacity of FN scheme). Let Gn = {[T Z; 0) 1 E Vn. Ø E 0). Under 23 condition 5 with 28+d+1. there exists constant cv > such that
_N(cus//e Vnn, 112))
Nn In (11o))
B.3 PROOFS OF THEOREM AND
Proof of theorem]
7
The proof is divided into four steps.
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We denote du (ho. me, Bb) and Ø (h, 0). where h Hn.m M, and 0 e. For arbitrary small E > 0, we have that
Step 1
inf dor (0r000)))
E [(T, Z; ho, mo, 0o) -I(T, (,
m,
Ex [ET,11 [log (0T, 8 Z; 22hhomo, ma, 0o) logp(T,0 2:7, m, ]]
inf doF (0.00)))
Ez [DKL (5 || PPo.22]
inf dor (0.00)))
(Păz ll (Poo.)) 2 2H2 Poo.2). Thus, we further obtain that
Using the fact that DKL
inf dor(0.0))2
E ((T, 6, ho, mo. 0o) ((T, 8, 2: 2: m.
2
inf
2E2
H2 (P |l Poo.2)
dor (0.00)22
dp (0.00)
= 2 inf
dps (0.00)22
2 25..
Step 2
Consider the following derivation.
Var [[T, 6, 2; ho, mo, 0o) 1(T, 2;,, m, 0)
sup )o=(
E (ICT, Z; ho. ma, 0o) I(T, Z: n m, 0)
sup doF (3.00))
Ez Er, logg(T, 6, Z; ho, ma, 0o) log Dogp(T, (,, 7, m, R)]²
= sup def (ō.0))
|/ ((T,0;22ho, mo, 0o) log V
") uldr dō)
p(T, 5, Z; h; mo, 0o) p(T, Z; h, m. 0)
=4 sup dop ( (0)0)s Se
Ez
By Taylor's expansion on logæ, x, there exists E(T, 8, 2)
between på (T, 8, 2; bo, mo, 0o) and
p² (T, Z;h, m. 8) pointwisely such that
P(T, 8, Z; bo, ma, 00) (1og V
p(T, 5, Z: ho, mo, On)
p(T, 8, 2; h m,
2
log Vp(T, 6, 2; ho, ma, 0o) - llgg V p(T, 6, Z; m 0)
p(T, 6, Z; ho, mo. 0o)
p(T, 8, Z; ho, mo, 00)
<Vp(T, 2; (ohhommy ma, 0o) / p(T, Z; m,
2
E(T, 2,²
Since
P(T, 5, Z: ho, mo, mp,00)
p(T, 6, Z; h;h, m,
by lemma 1(T, 6, Z; ho mo. 0b) and [(T, 6, Z: h. 2) are bounded among(0, 7] x {0, x m. 1] uniformly on all I = (h, m. 0). Thus, there exist constants C₁ and C₂ such that 0 < Gi < p(T, 6, Z: ho, mp, (T) 7: 7; m, C2. This leads to the fact that
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p(T, 8, Z; ho, mo. 86) EyT, is bounded. We further obtained that
p(T, 6, Z; ho, mo. Bu) log V //pp/T,,, Z: hu, mu, 8b) log / p(T Z:h, m. m.
mo. x p(T,,, mo, 00) V p(I, 0, 2:
Thus, have that
Var(I ho mo. 90) I(T, 6, 2:h, m. 0))
sup do= 10:0550
/ / 6, Z: mo.6o) VP(T, 2, m. (, 0)
sup AoF (5
Ez
MlT x d6)
dpF (7)
sup (3.00) <E7
Step 3 We define that Fn = (1TT, 2;h, m. I(T, ho mo To) To E Hn. m E Mn, 0 @}. Here (T,, T00, 00) have been defined in lemma Obviously, we have that log Nin (E, Fn. || (loo) log (E, || lloo). where FF is defined in lemma By lemma 9 we further have that
log (n, Il (1,o) log - log N 1/s, Hn, || 112) + log ((cms Ma, -122..
According to Bartlett et al. 2019 Theorem 7), under condition 3 we have that the VC-dimension of H,, and Mn satisfy that VC (H..) S कनन log² and (CM.) S n log n. Thus, obtain that
VC(Ha) log S nota log n log
1 E
log N(cnet/on Hn. 11 112) x
8h
and
log (Cme MR |l (12) (CM) log 1 S roa log" log 1 Sy we obtain that log (E, Fr, || (lxe) S n of log" log 1
Thus,
Step 4 By the Cauchy-Schwards inequality, we have that
v E [TT 5, Z; n, my म. 0p) 1(T, 5, Z; ho, my, 0u)l
< [[ 8, Z; T ho, T, v,) - I(T, 5, Z; ho, mo. 21)) à
Similar to the second part and by lemma 5 we further have that
[[T,, 8, Z; TT bo, Mo, Bo) I(T, Z: bo, mo, 00)] 2 v dpF (N Φ0) S n 28422
Now let
8
log log
2
2,9 + 2d
log n
By Step 1,2,3 and Shen & Wong 1994 Theorem 1), we have
dPF
fnn 60)
max (n - .dpF (o Φo),
R. [[(T, 6, 7: T hoo ma 0o) - I(T, 8. Z; ho, mo, 0o)
By lemma dpF @n, Φ0) = Onn zta)
By Step 4, / E [( 7; nr, nn ma, n, 0o) I(T, 8, 7; ho, mg. 0o) = - |パ -(- Thus, we
have dpF Proof of theorem?
@mn Gu) = O(n अमंं log" n) = Ö(n -==
2
The proof is divided into four steps.
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Step
We denote tu (2o, 06) and & -),0), where D Vn and @ G. For arbitrary
0 < E < 1, we have that
inf dru (o,
E ((T, ó, 2; 0o) - 1(T,8,
22
Br,a|z logp(T,8) 8 Z; - ( 2; 2, ]]]
inf dnu(W.No))z
Ez
inf (vu(V. )2c
Ez DKI. (Fo.z IIll P.00.2)]
Using the fact that K L(P
322
(.)..) 2 5.2 || 2. Thus, further obtain that [[((I, 8, Zi 0o) 1(T, 8, 2;
inf dev (ov(voo))2
2
inf deu (o.())))2
2Ez HP 3, PPooi))
dbN (o)
= 2 inf daw des (v.v0)) 22
2 2
Step 2
We consider the following derivation.
I(T, 6, Z; 00) 1(T,6,2;,0) 0]
sup
Var
S sup
E
dev dov(
ET,Zi (l0gB/T: (10gP((,6,2,,00 2;4,
log 5, 2; 0) '1
sup deu ( Se
Ez
[/ [ 2: Ao)
|TT,5,
=4 4
sup dru( (e))) ) Se
Ez
uldT dō)
(,T,6,2;2,0)
By Taylor's expansion on logæ, there exists nTT, 0, 2)
between 8p(T,8,2 2:000
and
Vo(T, 8, 2:0,0) pointwisely such that
P(T, 8, Z;v0, Bo0,3 v0,
p(T, 2;0,00000) V
p 8, 2;0,0)
[log / p(T, Z:vo, 00) - log / p(T, 8, 2:0, 8)
2
=P(T, =,(T,6,2;00- (6,2;00000)
PTT, 6, 2; 00)
(VP(T, 2,u;0 80) P((, 5, 2;0, 0)
Since p(T, 6, Z; No, 0n)/p(T 8, D; 0) e (00
by lemma L(T, Z; vo, 0o)
and ((T, 6, 7; 0) are bounded [0, r] x {0, 1] x 1_ uniformly for all v (2, 0). Thus
there exist constants C3 and C₁ such that < C3 S PT 5. 1/. pp)/ Z: D. 9) S C.. This
leads to the fact that p(T, 8, Z; 60) nTT is bounded. We further have that
p(T, 6, Z:vo. 0o) log v p(T, ó, Z; 00) log pTT, 2:0,0
$
/ VTT S, 5, 7:20, 00) (TT 5, 7;0,0
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Thus, have that
Var [(TT, 6, 2;00, -1TT,6,2;;;00
sup
doul (o.oooo
Ex / p(TT, Z;v,
2 MldT x d6)
v 6,
sup Con(vvo)ss
dFN wo)
sup de( (u.(vvv)se 6
Step
We define that Gn = (U(T,6 6, Z: T; 0
I(T, 5, Z; no 00) Vn, E O). Here
(Tal0, 00) have been defined in lemma Obviously, we have that log Ni Gn. || lloo)
log Nij (E,9u, In. (loo], where is defined in lemma 10 By lemma 10 we further obtain that
log (E, |l (1o0) X log log Noce //>, Vn. 112).
According to Barrlett et al. (2019) Theorem 7), under condition 4 we have that the VC-dimension of Vn satisfies that VC VC()n) S m #+ log n. Thus, we obtain that log N (ene Vn |l (122) VC(Vz) log n log" log
8y Furthermore, we get that log (I, G.. || (1so) S n F+ log nlog :.
Step 4 By the
incquality, we have that
VE[(TT 8, Z; To. 00) - L(T, 6, Z; 6o)1 - E ((TT, 8, Z: To Tn - [[T, M; B0))
Similar to the second part and by lemma 6 further obtain that
VE[I(T, 2; Vu, 0u) I(T. 6, Z; vo, 0o)l S vdFN (Tn v/o) S 16+2,क7
Now let
3
log log n
log By step 1,2.3 and Step 1,2,3 and Shen & Wong (9994 Theorem 1), Fnn vi
233 2d
dFN
max (n TT, dFN (Tnvoo wh),
/ /E[((T 8, Z; Mo, T 0o) 1(T, 6, 2; 80)]
By 6 dFN (Tn No. vh) = O(n 34241
By Step 4, E[i(T, 5, Z2 vo, 0o) - I(T, Z; 0b) = O(n 2242242). Thus, we have
deN Vny 35n = O(n Iorlarz log? n) = Ö(n 37F2245
B.4 PROOFS OF TECHNICAL LEMMAS
Proof of lemma 7 Since hT) E W (10, (10, r)) and mZ) W ([-1, 1]²), we have that ho( z
M, mo () < M and ( So ho(s)ds S 2M Let B = [0, JeM we have that
[[T, 6, Z: ho. mo. 0o)l
T molz) ("
[hoo(T) Imo(Z)) + Ge (mo()) s" co(e) ds
ho(l) ds
log goo
2. + sup llog ga, (x)) + sup zEB
Go, (x)
#e8
By condition 5, we have that 1(T, 6, Z; ho, ma, 0o) is bounded for (T, 2) E [0. x {0. x
[-1, 1]d. The proof of the boundness of I(T, 6, Z: h m, 0) is similar.
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Proof of lemma lo e²(x.2) ds S TeM Let B = [0, TeM], we have that
2 Since (T( E W ( 1-1,11%) we have vo(T, 2) 1
M and
[[T, 5, 2o Bb)l
(f cmo(s.2) ds
+ |Po + Goo (%/- crole22)ds
< log Go
$ MM sup xEB
log 0o (x) sup Go, (z)]. =EB
By condition 5 we have that I(T, 6, Z; vo; 0o) is bounded among (T, [0, r] (0. 1/x111111111 1j
The proof of the boundness of (T,, 6, Z; D, 0) is similar.
Proof of lemma 3 By definition, we have that
[T, 6, Z; ho, mo. Bu) I(T, 6, Z: m. 0)
mol /-² chols) ds
Logga Em(Z) l-" Mioldss + hho(T) - -((T))
< log gau
!'" chols) do
Gg (2(2)) ("aairdde
+ +/mo (2) - -fa(z) + Goo mm)(Z)
Let = [0, r max(c22 2M RaM,eMMa CM Mm )1 By Taylor's expansion, we can further show that [(T, 0, Z; h m, 0)
[(T, 8, Zi ho, mo, 0o)
0 log 9 (
0o - 0
< sup $c0,8e6
0g
log 9a (7)
mo) lo holt) ml 1" N(s) ds
+ sup
8EB.8EB
(G |00
+ (T) T((T)) + (ma - m(
sup OE0,EE
0g
(G
mmall) ("² I (ols) do - 62(2) 1" Jioo
+ sup BEB
Again, by Taylor's expansion, we have that
ema(Z) [-" ho(s) mm(2) f-" ((s)
emal2) ( -cholo)-etiss)dd - + (emo(2) - 1- chio) ds
-
< M Te (MX,
Ino -
+ No. ma M ml
Finally, we obtain that
[[T, 8, Z; ho, mo, 00)
(TT, 6, Z; h m, 0)1
i log 9 (7)
MMM Te max (M. - -ollooo + Ma Maax(M.Mm)) -mlloo
< sup dE0.EEB
a
log (ā)
0o - 0| + +||ho(T) - -i((T) + +1mo(2)- mm(2)
+ sup 8ee,aen
0g
@G
0o ō
+ sup
0g
8eB,
@Ga(ā)
M Te max MM - Tolloo + Mr. maxx(M,Ma)]lon - m01100
+ sup 8EB
0z
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Taking supremum on both sides, we conclude that
[1 6, Z: ho, mo, 0b) I(T, 6, 2,h, m, l S 100 - 0 11/10 - -11l20 + 11mo - milac.
The proof the second inequality is similar.
Proof of lemma
4
By definition, have that
[[T, 8, 2o Bb) 1(T, 220,0)
(%" c
log goo
ds
llog 96
-
(oo(2;
-(T,
Go
cooluz)dss Gō
Let B = (0, r max(e [MM)) By Taylor's expansion, we can further show that
[T, Z: vo, 0)) I(T, Z;0,))
i Olog (2)
Ölog Ologgg(2) (g)
f' cuo(a.2) (' c(2,2)
0b 0
$ sup 0E0.EEB
+
sup
0g
Oz
0E0
DG
+luo 2) ()
(TT, 2) + sup
100 0
Og
(Gg(T)
fr T
cuo(,.2) ds
F(a, ds
+ sup 8EB
Again, by Taylor's expansion,
L'" cola ds /' ((s,2) ds max M,) 1ll00 000000
Finally, we obtain that
I(T, 6, Z: Bb) I(T, 0
I g6 (ō)
log g8 (ō)
sup 0E0.EEB
0o 0
+
sup
ma. 1ll0 2
0g
Oz
dE
(G²
+|ooTT, Z) (T Z)
+
sup
6o
0g
d
(G₂(I)
Ma.(M
sup dEe.FEB
a₂
Taking supremum on both sides, we conclude that
[T, 6, Po 6o) I(T. 8, 2;0, 10 S 10o - -0 + 1111 00
The proof the second inequality is similar.
Proof of lemma 5 According to Yarotsky 2017
Theorem 1), there exist approximating functions
hor and i such that [7 -holpo O FT and fm*=molloo = O (n (-p Let T,,/10 = 7
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Ma f and TT 86- We have that dpF (Tn do)
Ez S IVP(T, 5 2; nnlo, Talli, To) - Vp(T, 6 h; mo. uu) x dō)
Ez
x do)
<
Ez [ faz((T) ((T)))) x d6)
By lemma T and
we have that
|| ho. _ 11000
< |,, - Bull + || - bolloc + || -mmll
= =0 (n- nn= s+d
Since (c]2(T)²-6 S (T) + and (T)2(T)6 I 1, we also have that
Ezz V
( x dô)
Ez
[
+1+ dō)
/2 + 2T.
Thus, we obtain that dpF @n Φ0) = 0 00. (n= 3T2
Proof of lemma 6 According to Yarotsky 2017
Theorem 1), there exists an approximating func-
tion D" such that 1-- volloo = n W
Let 7n20 "* and o, fy. We have that
deN (nn sho)
/ /VVDCT,6 Z; TnVo, Tn.o) VP(T,6 8 Z;vo, Bu) pldT do)
Ez
Ez /
/eddrrszmmmmm -eap:m000000000.............. e310T,6.22.mm00))
(czz(7)1=S0(((T) x dō)
J (TTAZ.00000)
I
< 2
Rz
x d6)
By lemma 2 and
have that | =.((T,6.2:000666
|1
S 1m 0ol
10 Volla 00
O n 345+1
Since (C]z(T)2-6 S (T)Z(T) + and Sciz (T) < 1, we also have that
| (o227)-3S01(()ym xdo) do)
[
Ez
Ez
(11+fcz(()))t dō)
/2 2T.
Thus, we obtain that dFN (TT 2/ = 0 (n उसककT
Proof of lemma 7 The left inequality is trivial according to the definition of covering number. We need show that the correctness of the right inequality.
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Suppose that we have (g. 2) 1..., .N, where N = - F. 11-11) are the minimal number of
2. 5-ball that covers F. Then there exists at least one fi E F such that fi E E).. Consider the following&
bolls ( = I. For arbitrary f F F Bgg. f), we have that Il - fill s Il - 9ill + [ - gell S 8. Thus {), 1..,, N forms a e-covering of F. By definition, we have that FF, II- S F.I1 II).
Proof of lemma g The proof of the first two inequalities follows exactly the same steps of lemma 7
Here we just need to mention the rest of the statement that (E, || |1a)) = [5. F,
lcc) We
first choose set of 5-covering balls [B(,i, 5)}. i 1,.... NN. where N₁ (5 F. || [e ). Now
2 we construct set of brackets {[4, wi]), = 1..., NN, where 4 = - & and u; Si + Noting
that the bracket {14, ui]} is exactly the same as B(G,, 2). The set {1u, u.] = 1,...... Ni covers
F, which leads to (e, F, (bb)) S ( F, I| Iloo). Likewise, we have that Nn, F, || 1b 2
5 al Consequently, we have that (e, F, [1l0) = (F, (1o0).
LI
Proof of lemma 9
By lemma 8 first we have that Fe, 11-11)0))) S Me, Fr. ll (1.)) By lemma
3 there exists constant C1 0 such that for arbitrary h1 fu2 E Hn. m2 E Mn and 01, 02 e θ,
we have that
(I, 6, 2;hi, 01) I(T, 6, Z; T,,,, m2, 02) S ca 021 1lm Rallo + | m2
= we which indicates that as long as |0: 021 S 301" lh1 - hrelloo S e and ll - mello S 3e1
se1
have that [|((T, 8, Z; hu, mi, 01) (T,, 8, Zi h2 m2, 02) S.. Consequently, we have that (E, l1bo) S No 3c1
.. || )1 Mi1, 3c1
Hm. |l 11o0) x Nil 3e1 ,Mn ll lloe).
Since e is a compact set on R. by lemma and traditional volume argument, we have that (E 0, 11o0) S (EE Get 1 (l00) S
g
For GE Hn:l ll,o). by lemma 8. we have that N6 Ha, ll (1oo) = Ñ =
(a
3eg Hn. |l 11o0).
Sej By Chen & Shen 1998 Lemma 2), there exists a constant C2 > such that l1h, T22l S
02 [h
h2 which leads to N (3e. ( Hn.ll 11o0) N (M (3/10)) Hn, l12). By lemma 7
gd/on
further have that N(
el/on (3010g) Hor |l (12) S N 2(30112) Hns Ml l12)) Let on = 2(301ez)
gl/un
We have that (1e Hn: ll (lxo) & N(chet/on, Hn, -12).
Similarly, there exists constant Cm. 0 such that N Mn. 1l. 11o0) S N(cm= "an 11.111)))).
Thus, finally we can obtain that
Nn,E,Fmm 11. (100) S N(che
1/0n Hor -l. (1112)) N (Cme! /t/e//mmmm,e ,r 112)-
Proof of lemma 70 By lemma 8 first we have Nn(s, Gn, || (1o0) S Nn(e, Gn, 1lo)). By lemma
there exists a constant C3 > such that for arbitrary 21, T2 Vn and 01, B2 E O, we have that
||([T, 6, 7; &, 01) I(T, 6, 7;0,, 02)llcc 1 cal ca(00 10 0z1 + |0 - 0z 1211oc],
which indicates that as long as 10, 02 < 2ca and 1| 22 < e. we have that
[I 6, 2; 0.) I(T, 6, 7: D2 0z) 1l S Thus, we have:
Nn(e, Gn. ||. 1lx) s No E_ 0. 10.11-11 11oo) x N E Vn. ll lloo)
203
203
Since θ is compact set on R. by lemma 8 and traditional volume argument, we have that Nu (25 O, |loo) (G, O, lxx) _
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Table 3: Descriptive statistics of benchmark datasets
METABRIC
RotGBSG
FLCHAIN
SUPPORT
MIMIC-III
KKBOX
Size
1904
2232
6524 0.699
8873 0.320
35953
2646746
Censoring rate Features
0.423
0.432
0.901
0.280
14
26
15
For N1 5
Brs Vn, || 11,0), ...),,r by lemma 8 we have that Nip 2ng Vny (1x)) ) - Ñ 2cg Vn,
1lo)) By
Chen & Shen 19988 Lemma 2), there exists a constant >> 0 such that |0 82 ll < call
D2 1h
which leads to N c 2cg Vo: 11x)) S N( [20ace] Va, 112). By lemma
el/ow
we further
have ( (2eges)"/aa Vn, (12) S ( gh/ov Vn 112). Let Cp 2(2cgee)"/00 we have that
gl/ov
( 5 Va |1 (Nces Vn. 112)).
Thus, finally obtain that
Ni Insll -16)) - N(costles Vnn -12).
C EXPERIMENTAL DETAILS
C.I DATASET SUMMARY
We report summaries of descriptive statistics of the 6 benchmark datasets used in section 5.2 in table
C.2 DETAILS OF SYNTHETIC EXPERIMENTS
Since the true model assumed to be of PF form, we generate event time according to the following transformed regression model (Dabrowska & Doksum
1988):
log H (T) m m2) +
(15)
where (I)) -
- So cls) with d defined in The error term is generated such that has cumulative hazard function Gg. The formulation
15) is the equivalent to ( (Dabrowska & Dok- sum 1988; Cuzick 1988) Kosorok et al., 2004). In our experiments, the covariates are of dimension 5, sampled independently from the uniform distribution over [0, 1]. We set h(t)
and hence (t) = e:_ The function form m(Z) ) is set to be m(Z) = sin( (3)) + (sin(Z).,3) where (0.1, 0.2, 0.3, 0.4, 0.5). Then censoring time C is generated according to
log HCC) - m( + +
(16)
which reuses covariate Z, and draws independently noise vector ec such that the censoring ratio is controlled at around 40%. We generate three datasets with E {1000, 5000, 10000} respectively.
Hyperparameter configurations We specify below the network architectures and optimization con- figurations used in all the tasks:
PF scheme: For both in and h. we use 64 hidden units for n = 1000, 128 hidden units for n 5000 and 256 hidden units for n 10000. We train each model for 100 epochs with batch size 128, optimized using Adam with learning rate 0.0001, and no weight decay.
64 hidden units for 1000, 128 hidden units for 5000 and 256 hidden units for n = 10000. We train each model for 100 epochs with batch size 128, optimized using Adam with learning rate 0.0001, and weight decay.
FN scheme: For both 2,
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C.3 DETAILS OF PUBLIC DATA EXPERIMENTS
Dataset preprocessing For METABRIC, RoGGBSG, FLCHAIN, SUPPORT and KKBOX dataset, take the version provided in the pyсox package (Kvamme al. 2019). We standardize contin- uous features into zero mean and unit variance and do one-hot encodings for all categorical features. For the MIMIC-III dataset, we follow the preprocessing routines in Purushotham al. (2018) which extracts 26 features. The event of interest is defined as the mortality after admission, and the cen- sored time is defined as the last time of being discharged from the hospital. The definition similar to that in Tang et al. (2022). But since the dataset is not open sourced, according to our implemen- tation the resulting dataset exhibits a much higher censoring rate (90.2% as compared to 61.0% as reported in the SODEN paper (Tang al. 2022)) Since the major purpose of this paper is for the proposal of the NFM framework, We use our own version the processed dataset to further verify the predictive performance of NFM.
Hyperparameter configurations We follow the general training template that uses MLP as all nonparametric function approximators (i.e., fi and in the PF scheme, and in the FN scheme), and train for 100 epochs across all datasets using Adam as the optimizer. The tunable parameters and their respective tuning ranges are reported as follows:
Number of layers (network depth) We tune the network depth L E (2,3,4) 4).. Typically, the per- formance of two-layer MLPs is sufficiently satisfactory.
Number of hidden units in each layer (network width) We tune the network width W (2*,5 S S 10].
Optional dropout We optionally apply dropout with probability p (0.1,0.2,0. 0.3, 0.7J.
Batch size We tune batch size within the range {128, 256, 512), in the KKBOX dataset, also tested with larger batch sizes [1024}.
Learning rate and weight decay We tune both the learning rate and weight decay coefficient of Adam within range (0.01, 0.001, 0.0001}.
Frailty specification We tested gamma frailty, Box-Cox transformation frailty, and IGG(() frailty with a E {0, 0.25, 0.75). Here note that (GG(0 is equivalent gamma frailty. We also empiri- cally tried to set to be a learnable parameter and found that this additional flexibility provides little performance improvement regarding the datasets used for evaluation.
C.4 IMPLEMENTATIONS
We use pytorch to implement NFM. The source code is provided in the supplementary mate- rial. For the baseline models:
We use the implementations of CoxPH, GBM, and RSF from the sksurv package Pölsterl (2020). for the KKBOX dataset, we use the XGBoost library (Chen & Guestrin, 2016 to imple- ment GBM and RSF, which might yield some performance degradation.
We the pyсox package to implement DeepSurv, CoxTime, and DeepHit models.
We the official code provided in the SODEN paper (Tang et al.) 2022) to implement SODEN.
We obtain results of SuMo and DeepEH based on our re-implemenntaioos..
D ADDITIONAL EXPERIMENTS
D.1 RECOVERY ASSESSMENT OF m(Z) IN PF SCHEME
We plot empirical recovery results targeting the m function in 2 figure The result demonstrates
satisfactory recovery with moderate amount of data, i.e., n 1000.
D.2 RECOVERY ASSESSMENT OF SURVIVAL FUNCTIONS
To assess the recovery performance of NFM with respect to survival functions, we consider the following setup: under the same data generation framework in section C.2 compute the test
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1000
5000
10000
ground truth estimated
ground truth
ground truth estimated
2
estimnated
Wppt we MM My Myiceee
Wir Aud WV Myy Mathayl p
"Mppr M MDW Wh
1
8 1
50
100
50
100
50 index
100
index
index
Figure 2: Visualizations of synthetic data results under the PF scheme of NFM framework, regarding empirical recovery of the m function in
N = 1000
N = 5000
10000
ground truh
ground truth estimated
ground tuth
estimated
estimated
0.75
0.75
0.75
S(1)2)
0.25
0.25
0.25
= 1000
5000
10000
ground truh
ground Inuh
ground mith
estimaned
estimated
estimated
0.70
0.75
D.75
5(1)2)
0.25
0.25
D.25
Figure 3: Visualizations of synthetic data results under the NFM framework. The plots in the first row compare the empirical estimates of the survival function St) against its true value with Z being the average of the features of the 100 hold-out points, under the PF scheme. The plots in the second row are obtained using the FN scheme, with analogous semantics to the first row.
feature 2 as the sample mean of all the 100 hold-out test points. And plot S(t) against the ground truth S()] regarding both PF and FN schemes. The results are shown in figure 3 The results suggest that both scheme provides accurate estimation of survival functions when the sample size is sufficiently large.
D.3 PERFORMANCE EVALUATION UNDER IBS AND INBLL
In this subsection we augment the experimental results in section 5 with three recent staee-of-the-art baseline models: SurvNode Groha et al. (2020), DCM (Nagpal et al. 202I) and DeSurv (Danks & Yau 2022). The results demonstrated in table for four small scale datasets, and table 5 two larger datasets. The proposed NFM framework achieves the best performance on of the 6 datasets, which is consistent with the findings in table
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Table 4: Survival prediction results measured in IBS and INBLL metric (%) four small-scale survival datasets. In each column, the boldfaced score denotes the best result and the underlined score represents the second-best result.
Model
MITTABIIC
ROGGBGG
FLCHAIN
SUPPORT
IBS
INBLL
IBS
INBLL
IBS
INBLL
IBS
INBLL
CoxPH
16.46gu.00
49.57
18.25
53.76
10.05
33.18
20.54
59.58
GBM
16.61
49.87=2.4
17.83 17.83
52.78
9.98
9.98
32.88
19.18 19.18
56. 36.46200.
16.62.20.4
49.61 54
17.89
52.77
9.96 20.57
32.92-1.05
19.11
56. 28.28:14
DeepSurv
16.55.a0.98
49.85=a.02
17.80=0.49
52.62+1.25
10.09
33.28=11115
19.20
36,48
CoxTime
16.54
49.67
17.80
52.566
10.28
34.18=1.3
19.17
56.45
DeepHit
17.50gu.8z
52.10 22.10 16
19.61 20.38
56.67
19.83
37.72=1.02
20.66.00.3
60.062. 72
DeepEH
16.56:00.65
49.42221.5
17 17.6222052
52.08
10.11
33.30
19.30
56.67
SuMo-net
16.4940.8a
49.74=2.21
17.77
52.62
10.07
33.20=1.19
19.40.2..8
56. 50.80
SODEN
16.5220.63
49.39
17.05.0.61
50.45
10.13
33.37
19.07
56.15
SurvNode
16.67
49.73
17.42:0055
51.70g1.16
10.40:00.2
3/137
19.58
57.49
DCM
16.58
49.488+222
17.66
52.26:1.28
10.13
$3.40=1..88
19.29
56, 56.68
DeSurv
16.71
49.61 22.61=2.33
17.98
53.28
10.06
33. 33.18
19.50)
57.28
NEM-PF
16.33
49.07
17.60 20.55
52.12
9.96 20.39
32 84
19.14,0030
56.35
NEM-FN
16.1120.44
48.21 +2.04
17.66:0.52
52.41
10.05
33.11 11.10
18.97
55.87
Table 5: Survival prediction results measured in IBS and INBLL metric (%) on two larger datasets. In each column, the boldfaced score denotes the best result and the underlined score represents the second- best result. Two models are reported, namely SODEN and DeepEH, we found empirically that their computational/memmon cost is significantly worse than the rest, and we fail to obtain reasonable performances over the two datasets for these two models.
Model
MIMIC-I11
KKBOX
IBS
INBLL
IBS
INBLL
CoxPH
20.40
60.0210.00
12.60=0.0m
39.40...m
GBM
17.7040.00
52.30.10.00
11.81
38.15 15.11
RSF
17.7940.19
53.31
14.46=20.00
44.39
DeepSurv
18.58+4.02
55.98+2.43
11.31
35.28
CoxTime
17.68+1.36
52.08
10.70
33.10 13.10
DeepHit
19.80+1.31
59.03:+4.20
16.00=0.34
18.6141..4
SuMo-net
18.6241.23
54.51
11.58_0.11
36.61
DCM
18.02+0.49
52.83
10.71
33. 33.2444..06
DeSurv
19.
54.69442.88
10.77
33.22 00.12
NFM-PF
16.2840.36
49.18 0..02
11.022o.11
35. 10.0.22
NFM-FN
17.47 40..7
51.48 11.23
10. 10.63
32.81 10.10.
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Under review as a conference paper at ICLR 2023
Table 6: Survival prediction results measured in C-index (%) on all the 6 benchmark datasets. In each column, the boldfaced score denotes the best result and the underlined score represents the second- best result. The average rank of each model reported in the rightmost column. We did not manage obtain reasonable results for DeepEH and SODEN two larger datasets MIMIC-III and KKBOX, and we set corresponding ranks to the worst on those datasets.
Model
METABRIC
RoIGBSG
FLCHAIN
SUPPORT
MIMIC-II
KKBOX
Ave. Rank
CoxPH
1..4
66.14_1146
19.09
56.89 40.91
56.89
74.91 10.00
83.01
11.33
GBM
64.02+1.79
66.35-1.16
79.47 1.08
61.46440.80
a5.20
85.84
7.17
RSF
64.47
67.33_1.34
78.75=1.07
61.63+0.84
75.47
=0.00
8.00
DeepSurv
63.95+2.12
67.20
79.04
60.91 +0.80
80.08+0.44
20.09
8.50
CoxTime
1..22+1.00
-1.35
18.95
61.54+0.87
78.78+0.62
87.3 +0.24
5.00
DeepHit
16.3
1.007
78.48:1109
63.20 2.....85
79.16.0.59
86.12
6.50
DeepEH
66.59+2.0
67.98
78.71#1.44
61.51+1.04
SuMo-net
64.82+1.80
07.20-1.11
79.28 +.002
10.77
76.23+1.06
84.77 00002
7.00
SODEN
64.82+1.05
66..97=0.75
79.00=00.96
61.10+0.50
10.17
SurvNode
44.64
67.30_1.65
76. 20.98
55.37 55.37
+0.77
11.50
DCM
152
60.75=1.a5
20,79
62.19 49.9
76.45+0.34
83. 83.48
8.33
DeSurv
65.88
67 11.40
78.97
61.47
80.97 80.97#0.20
86.1 00000
5.67
NEM-PF
64.9811.87
67.77 1:35
79.45
61.3310.s
19.5610.15
86 16.231100
4.67
NEM-EN
66. 63 11.82
67.73_.. 1.29
79.29 10.93
62.21 10.41
80.18 10.20
86.61
2.16
Table 7: Relative improvement of NFM models in comparison to their non-frailty counterparts, measured in IBS, INBLL, and C-index.
Dataset
NFM-PF DeepSurv
NFM-FN SuMo-net
IBS
INBLL
C-index
IBS
INBLL
C-index
METABRIC
+1.33%
+1.56%
+1.61%
+2.30%
+3.08%
+2.79%
RotGBSG
+1.11%
+0.95%
+0.84%
+0.62%
+0.40%
+0.79%
FLCHAIN
+1.29%
+1.32%
+0.52%
+0.20%
+0.27%
+0.01%
SUPPORT
+0.31%
+0.23%
+0.69%
+2.22%
+1.76%
+0.05%
MIMIC-III
+12.38%
+12.15%
-0.64%
+6.18%
+5.56%
+5.18%
KKBOX
+2.56%
+0.51%
+0.75%
+8.20%
+10.38%
+2.17%
D.4 PERFORMANCE EVALUATIONS UNDER THE CONCORDANCE INDEX (C-INDEX)
The concordance index (C-index) Antolini al. 2005) is yet another evaluation metric that is commonly used in survival analysis. The C-index estimates the probability that, for a random pair of individuals, the predicted survival times of the individuals have the same ordering as their true survival times. Formally, C-index is defined as
3(T) 2.) S(T, 1 T, -1
(17)
We report performance evaluations based on C-index over all the benchmark datasets in table 6 From table 6 it appears that there's clear winner regarding the C-index metric across the 6 selected datasets. We conjecture this phenomenon to be closely related to the loose correlation between the C-index and the likelihood-based learning objective, as was observed in Rindt et al. (002)). Therefore we compute the average rank of each model as an overall assessment of perfor- mance, as illustrated in the last column table 6 The results suggest that the two NFM models perform better on average.
D.5 BENEFITS OF FRAILTY
We compute the (relative) performance gain of NFM-PF and NEM-FN, against their non-frailty counterparts, namely DeepSurv (Katzman et al., 2018 and SuMo-net (Rindt et al. 2022) based on results in table [ table and 6 The results are shown in table 7 The results suggest solid improvement incorporating frailty, especially for IBS and INBLL metrics, the relative increase in performance could be over 10% for both NFM models. For the IBS and INBLL metrics, the per- formance improvement consistent across all datasets. The only performance degradation appears
29

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Under review as a conference paper at ICLR 2023
on the MIMIC-III dataset evaluated under C-index. This phenomenon is also understandable: Since the DeepSury model utilized variant of partial likelihood (PL) for model training, as previous works (Steck al. 2007) pointed out that PL type objective is closely related to the ranking prob- lem. As C-index could be considered a certain type of ranking measure, it is possible that DeepSury obtains better ranking performance than NFM-type models which are trained using scale-sensitive likelihood objective.
30


Reviews:
[{"review_id": "qMM_V6GznMF", "reviewer": "Reviewer_xG5j", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This work proposes a hazard estimator which models the unobserved heterogeneity in individuals using frailty models. They propose two function approximation schemes where in one they assume the covariate effect and time effect to be multiplicative and in the other no assumptions are made. They also provide theoretical results on the convergence rate of their estimators.", "summary_of_the_review": "I find this work novel and interesting. Nonetheless, it only handles time-static covariates which is a major drawback given the recent advances in the time-varying survival analysis literature.", "strength_and_weaknesses": "I found this paper interesting. It is well-written and the theoretical results seem interesting. However, section 4 is quite disconnected. The authors seem to get tangled up into explaining their theory and in my eyes lose sight of the paper flow. I would have appreciated a better narrative of what is the significance of their results. In other words, what do we gain from knowing the rate of convergence? Does it show that the proposed method is efficient? Or on the contrary it shows that it is slow?\n\nGiven the survival analysis literature has paid considerable attention to time-varying covariates recently, a drawback of this work is failing to consider time-varying covariates. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "f8QhG4D2lip", "reviewer": "Reviewer_nTQp", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors propose a neural network parameterized proportional frailty model. The proportional frailty model uses a random variable to express the unobserved heterogeneity corresponding to individuals. The main contribution is to introduce the neural networks into proportional frailty model. ", "summary_of_the_review": "I am not very convinced that why we need to introduce such a frailty model both makes the representation power smaller and computation time larger in a neural hazard model. More metrics and baselines should also be included. But I still think it's above the borderline since it's an interesting direction that includes some prior knowledge in deep models. ", "strength_and_weaknesses": "Strength:\n1. The authors introduce deep networks in to frailty models.\n2. The authors provide theoretical analysis and synthetic evaluations to verify the correctness of their approach.\n\nWeakness:\n1. I am still not very convinced that why we want to introduce a random variable to represent frailty. It reduces the representation power of the neural network based hazard model. Typically, the author uses a gamma family, why not use a more flexible family? In the real-world experiments, why do we need such a frailty model?I\n2. The authors use a numerical integration to compute the integral in likelihood. What's the speed of this method compared to other approaches? Will it be very slow?\n3. The authors should also include concordance as an evaluation metric in real-world study. It's a commonly used metric in survival analysis.\n4. The authors may consider other recent SOTA methods, for example, survival mixture density networks (Han et al, 2022), DeSurv (Danks et al, 2022).\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "MYdTrUOzzeR", "reviewer": "Reviewer_oRcF", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper introduces a new neural network technique for survival analysis termed the neural frailty model, which in turn is inspired by the literature on frailty models. The authors provide a theoretical analysis for the convergence rate of the conditional distributions of the event data and demonstrate state-of-the-art performance on common survival datasets.", "summary_of_the_review": "The paper is strong in theory, but somewhat weak in the experiments section. In particular, claims are stronger than they should be and the current literature should be correctly presented. With these issues I don't think the paper is ready to be published.", "strength_and_weaknesses": "__Strengths__:\n- An interesting investigation into the area of frailty models from a neural network perspective\n- The approach comes with interesting theoretical results concerning convergence rates. \n- The simulation study covers the common survival literature, using standard datasets as well as comparing to many common approaches.\n\n__Weaknesses__:\n- Soden (Tang et. al. 2022) is a special case of earlier work of Groha et. al. (2020) which needs to be cited. Those authors also report better results on METABRIC than Soden or NFM here. \n- The results in Table 1 have strongly overlapping confidence sets, casting doubts on the performance difference. Those might as well be explained by seed choice or hyperparameter tuning. Any claims on \"superior predictive performance\" should therefore be adjusted. (The term state-of-the-art should be fine, but explicitly superior seems too much.)\n\n[1] https://arxiv.org/abs/2006.04893\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "1aX0NdAACo", "reviewer": "Reviewer_yX4d", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes gamma frailty for modulating proportional hazards (NFM-PF) or nonparametric hazards  (NFM-FN) functions. The assumed hazard functions  are modeled with neural networks where model parameters are learned via maximum likelihood.  Additionally, the paper provides  theoretical convergence analysis for the NFM-PF and NFM-FN optimising schemes. Experimental results on real-world datasets show improvements per metrics integral brier score  (IBS) and integrated negative binomial log-likelihood (INBLL).", "summary_of_the_review": "Frailty survival models have been proposed before. Its unclear how we should use the derived theoretical convergence rates for NFM-PF and NFM-FN schemes. Additionally, the experimental results are unconvincing. ", "strength_and_weaknesses": "**Strengths**\n- The paper studies survival analysis, an important problem in many domains\n- Modulating hazard functions with frailty models is interesting \n- Theoretical convergence analysis of NFM-PF and NFM-FN optimising schemes may be of interest to some researchers\n- The proposed approach seems easy to implement\n\n**Weakness**\nSee references below. \n\n*Technical Limitations*\n- Its unclear why the gamma frailty model is advantageous compared to other frailty models apart from its popularity.\n- The paper leverages Clenshaw-Curtis quadrature for numerical integration. It's unclear why other solvers are not considered, *e.g.*, adjoint [5].\n- Its unclear how we should use the derived convergence rates for NFM-PF and NFM-FN schemes\n\n*Underwhelming Experiments*\n- Its unclear why the standard C-Index [7] is not used for model evaluation\n- For completeness, the paper should also compare to models that directly estimate event times $p(t|x)$ parametrically [e.g., AFT, 1, 4] or non parametrically [2, 3].\n-  Paper should provide qualitative results of estimated hazards $\\lambda(t|Z)$, survival $S(t|Z)$ and  $p(t|Z)$ compared against ground truth times, particularly for the synthetic datasets\n-  The synthetic experiment is not representative of real-world datasets:\n(i) What happens when the frailty model is misspecified? Currently, the assumed frailty model matches the data generation mechanism;\n(ii) The assumed very noisy hazard function seems unrealistic. Alternative data generation distributions described in [6] should be considered;\n(iii) The performance gains in terms of IBS and INBLL seem marginal\n\n**References**\n- [1] Ranganath et al., \"Deep survival analysis\", In Machine Learning for Healthcare Conference, 2016\n- [2] Chapfuwa et al., \"Adversarial time-to-event modeling\", ICML, 2018\n- [3 Miscouridou et al., “Deep survival analysis: Nonparametrics and missingness,” Machine Learning for Healthcare Conference, 2018\n- [4] Nagpal et al., \"Deep Cox mixtures for survival regression\", Machine Learning for Healthcare Conference, 2021\n- [5] Chen et al.,  \"Neural ordinary differential equation\",  NeurIPS2018\n- [6] Bender et al., \"Generating survival times to simulate Cox proportional hazards models\". Statistics in medicine (2005)\n- [7] Harrell et al., “Regression modelling strategies for improved prognostic prediction,” Statistics in medicine, 1984", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject