Deep Extended Hazard Models for Survival Analysis

1 Introduction . Across areas such as biomedical science and reliability engineering , survival data analysis is critically used to study the time until certain events occur ( e.g . patient death in clinical applications , component failure in industrial applications ) . A key issue stems from the fact that many occurrence-times fail to be recorded due to natural data censoring . For example , a patient may relocate and drop out of a longitudinal clinical study , leading to the loss of follow-up observations for the remaining period of the study . For such right censored data , we only know that the true event time is greater than the last available observation time for this individual ( but not by how much ) , and thus standard regression approaches are not applicable to most survival data . A naive approach that removes all censored individuals from the data would induce bias , as these tend to be individuals with longer event times . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Learning the distribution of the event time in the presence of censorship is a fundamental task in survival analysis . A wide array of statistical methodologies have been developed to study this distribution . Among them , Kaplan and Meier [ 37 ] considered a product of successive conditional probabilities to estimate the survival function , which encodes tail probabilities of the event time distribution . As a nonparametric approach , the Kaplan-Meier estimator assumes no particular form for either the underlying distribution of event time or the relationship between individuals ’ observed feature-values and the outcome event time . Alternatively , the Nelson-Aalen estimator considers the cumulative hazard function from a counting process perspective [ 1 , 48 ] . The classical Nelson-Aalen estimator was improved by leveraging more powerful supervised learning methods , in particular random forests , which lead to the random survival forest model [ 33 ] . While the Kaplan-Meier estimator has been extended to assess the effect of multiple subject features ( covariates ) on the associated event time [ 5 ] , it struggles to work with even moderately highdimensional features . A semi-parametric approach , the Cox proportional hazards ( CoxPH ) model [ 14 ] , has thus been popularly adopted as a dimension reduction approach . However , the CoxPH model is restrictive as it assumes that the conditional log hazard functions for individuals are parallel to each other . When this assumption fails to be realistic , the accelerated failure time ( AFT ) model [ 35 , 62 ] often emerges as an alternative , which simply assumes a linear regression model for the log event time against the individuals ’ feature-values . Like the CoxPH model , the AFT model is also semi-parametric because the error distribution in the regression model is unspecified . An even simpler approach is to adopt a parametric approach for survival data by assuming that the distribution of event time obeys a certain known law , such as the Weibull or Gamma distribution [ 44 ] . While non-parametric methods are more flexible , a semi-parametric or parametric approach can be more effective when there are limited data , if the distribution of event times can be closely approximated by a certain semi-parametric or parametric form . Rapid advances in data collection and analysis have encouraged researchers to explore the potential of deep learning in the field of survival analysis . Faraggi and Simon [ 20 ] proposed to replace the linear predictor of the CoxPH model with a single hidden-layer neural network , but this extension failed to provide reliable improvements in terms of concordance index [ 63 ] . Under the same CoxPH framework , Katzman et al . [ 38 ] and Kvamme et al . [ 41 ] applied neural networks of greater complexity to obtain more flexible nonlinear models . Zhu et al . [ 67 ] and Tarkhan et al . [ 57 ] used convolutional architectures to explore the relationship between event time and unstructured features like images , and domain-specific variants of these models have been developed for particular applications , such as genomic data [ 10 , 26 ] , clinical research [ 49 ] , pedestrian waiting time in urban areas [ 34 ] and heart failure rehospitalization [ 36 ] . Beyond the CoxPH framework , deep learning has been used to model survival data in various alternative forms , including : neural network versions of the Beta-Logistic model [ 32 ] , recurrent neural networks applied with time-discretization [ 22 , 51 ] , generative adversarial modeling of timeto-event distributions [ 8 ] , generalized forms of regression [ 40 , 43 ] , and more flexible extensions of the Kaplan-Meier estimator [ 9 , 56 ] and the Weibull distribution [ 50 ] . Among existing deep learning methods for survival analysis , the distribution free approaches [ 9 , 12 , 28 , 40 , 43 ] make almost no assumptions on the underlying survival function structure , but these very general models may exhibit poor sample complexity , requiring very large data to accurately estimate the survival function . In this paper , we adapt neural networks to generalize the extended hazard ( EH ) model [ 11 ] , which includes both the CoxPH and AFT models as special cases . The proportional hazards assumption are often violated in practice and can lead to misinterpretation of the results , especially when there is heterogeneity among individuals [ 2 ] . In contrast , the AFT framework directly considers the relationship between feature values and the event time , which results in a time-dependent hazard ratio that is more realistic . Keiding et al . [ 39 ] also show that , relative to CoxPH , AFT models are more stable when accounting for unobserved features . Other studies of CoxPH and AFT models have however reported conflicting results [ 46 ] , and thus practitioners may find it difficult to choose which model to use in their applications . The EH model [ 19 , 58 , 59 ] , under which feature values can affect both the baseline hazard rate and hazard ratio , flexibly combines the CoxPH and AFT models into a more general framework . EH models also subsume a wider range of survival models such as Weibull and Gamma distributions . Compared with the current practice of deep learning under the CoxPH or AFT framework , our deep extensions of EH model retain the advantages of standard EH model over CoxPH and AFT , and achieve better empirical performance on real survival data . While deep learning has enjoyed a number of breakthroughs in survival data analysis , mathematical understanding of its success is still lagging far behind . Leveraging neural networks ’ capacity for nonparametric function approximation [ 4 , 17 , 29 , 53 , 65 ] , this paper provides theoretical support for our Deep Extended Hazard ( DeepEH ) model , which replaces the two linear risk predictors in the standard EH model with two nonparametric functions ( cf . h1 and h2 in ( 4 ) ) . Using neural networks to model these nonparametric functions , we show that the resulting estimators of h1 , h2 and survival functions are asymptotically consistent and enjoy fast convergence rates . Specially , the convergence rates are determined by the intrinsic dimension of the underlying functions rather than the original high-dimension input features . Our results formally illustrate how neural networks are able to identify low-dimensional structure of the data , hence circumventing the curse of dimensionality . 2 Background . In survival analysis , we are interested in modeling the time T until some event of interest ( i.e . survival time ) , whose probability distribution is typically characterized in one of four equivalent ways : probability density function , survival function , hazard function , or cumulative hazard function . The survival function S ( t ) is the probability of a individual surviving beyond time t : S ( t ) = P ( T > t ) = 1− ∫ t 0 f ( s ) ds , where f is the probability density function of survival times . The hazard function , denoted by λ ( t ) , characterizes the chance that an event occurs in an infinitesimal interval after time t , given it has not yet occurred at time t : λ ( t ) = lim δ→0 P ( t ≤ T < t+ δ|T ≥ t ) δ . This results in the relationship : S ( t ) = exp { −Λ ( t ) } , where Λ ( t ) = ∫ t 0 λ ( s ) ds is the cumulative hazard function . In practice , a survival study of n subjects will contain less than n completely observed event times due to right censoring . For instance , the event of some individuals may not have occurred yet at the time when the study is concluded or the subject may have dropped out of the study . For the i-th subject in the study , Ti and Ci denote , respectively , the event time and the potential censoring time ; and Xi ∈ Rp denotes the observed features ( covariates ) . Thus , the observed data from a typical survival study contain independent observations D = { Xi , Oi , ∆i } ni=1 , where the observed time Oi = min ( Ti , Ci ) and the event indicator ∆i = 1 , if the observed time is Ti , i.e . Ti ≤ Ci , otherwise ∆i = 0 , the subject is censored at Ci . Given the feature values Xi , the event time Ti and censoring time Ci are typically assumed to be independent . We aim to estimate the survival function and hazard function using censored data in which the true event times may be unavailable . Conditional on the features X , the overall log-likelihood of the censored survival data D [ 15 ] is : Ln = 1 n n∑ i=1 [ ∆i log f ( Oi|Xi ) + ( 1−∆i ) logS ( Oi|Xi ) ] = 1 n n∑ i=1 [ ∆i log λ ( Oi|Xi ) − Λ ( Oi|Xi ) ] , ( 1 ) where f ( t|X ) , S ( t|X ) , λ ( t|X ) and Λ ( t|X ) denote the conditional density , survival , hazard , and cumulative hazard function of T |X , respectively , and the last equality is derived from the aforementioned relationship between the ( cumulative ) hazard function and the density/survival function . To obtain maximum likelihood estimates , one can assume that the conditional distribution of T follows a particular parametric distribution . However to avoid restrictive assumptions , it is advantageous to use more flexible alternatives such as the CoxPH and AFT models . 2.1 Cox Proportional Hazard Model . Given features X , the Cox proportional hazard ( CoxPH ) model [ 13 ] for event time T , is a semiparametric model whose hazard function has a multiplicative form in terms of a baseline hazard function λ0 ( t ) and an exponential risk function of X : λ ( t|X ) = λ0 ( t ) exp ( β > X ) , where the baseline hazard function λ0 ( t ) and parameter β are both unknown . The function λ0 ( t ) represents the baseline hazard rate when all features take the baseline value of zero . It is difficult to estimate λ0 and β directly from the likelihood function ( 1 ) because the optimization with respect to λ0 , a nonparametric function , would be challenging . In fact , this likelihood is unbounded due to the unconstrained nature of the nonparametric function λ0 . Luckily the partial likelihood method [ 13 , 14 ] has been developed to handle the parametric component of the model without involving the nonparametric component λ0 ( t ) . Specifically , the partial likelihood technique first estimates β by maximizing the log-partial likelihood : Ln ( β ) = 1 n n∑ i=1 ∆i [ β > Xi − log ( n∑ j=1 Yj ( Oi ) e β > Xj ) ] , where Yj ( t ) = 1 ( Oj ≥ t ) . Intriguingly , the log-partial likelihood does not involve the nonparametric baseline hazard function λ0 , which is a major advantage in accomplishing dimension reduction . Once the parameter β has been estimated through the log partial likelihood , the cumulative baseline hazard function Λ0 ( t ) = ∫ t 0 λ0 ( s ) ds can be estimated through the Breslow estimator [ 6 ] . Here the simple parameter β0 can be estimated directly through the partial likelihood without requiring nonparametric estimation of λ0 . Thanks to the appeal of this straightforward estimation , the Cox proportional hazards model has dominated the field of survival analysis for nearly fifty years .

Summary: This work proposes DeepEH, a survival analysis framework that formulates the hazard function as \lambda(t|X) = \lambda_0(t.exp(h_1(x))).exp(h_2(x)). This formulation has the Cox PH and AFT as special cases. The authors estimate h_1 and h_2 by neural networks and provide theoretical support for the consistency and convergence rate of the resulting survival function estimator.