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 paper introduces the Deep Extended Hazard Model as a new approach to learning a survival model, based on learning 2 neural nets, to estimate the 2 terms of the extended hazard function.  It explains how this model extends the EH model, which extends both the Cox proportional hazard model and Accelerated Failure Time.  It then provides first a theoretical analysis, following by an empirical exploration of these models