Towards simple time-to-event modeling: optimizing neural networks via rank regression

1 INTRODUCTION . Time-to-event analysis , also known as survival or failure time analysis , is a major statistical approach in various fields such as biostatistics , medicine , and economics to estimate either risk scores or the distribution of event time , given a set of features of subjects ( Viganò et al. , 2000 ; Cheng et al. , 2016 ; Dirick et al. , 2017 ; Li et al. , 2021 ) . There are benefits of assessing risk or quantifying survival probabilities but , for all that , time-to-event analysis itself is challenging because of the existence of censoring . In real-world studies , a subject ( e.g . a patient in medical research ) can drop out before events of interest ( e.g . death ) happen , so that one can not follow them up ( Leung et al. , 1997 ) . The presence of censoring in survival data can create a serious challenge in applying standard statistical modeling strategies . Usually , the censoring process is assumed to be non-informative in that it is irrelevant of the underlying failure process given features , but should be properly accounted for , otherwise leading to biased results . The most popular and standard approach for modeling time-to-event data is to use Cox ’ s proportional hazards ( CoxPH ) model . CoxPH relates a conditional hazard to given features in a multiplicative form between the baseline hazard function and exponentiated regression component , and consequently learns relative risks . It often works on the assumptions of proportional hazards and time-invariant covariate-effects , which are difficult to follow in the real world ( Aalen , 1994 ) . Statistical testing procedures , such as Schoenfeld ’ s test , are usually conducted to examine these underlying assumptions since many Cox-based analyses are vulnerable to violation of model assumptions . ( Aalen & Gjessing , 2001 ; Kleinbaum & Klein , 2010 ) . The accelerated failure time model ( AFT ) or accelerated life model relates the logarithm of the failure time linearly to the features . As a result of its direct physical interpretation and the connection with linear models , this model provides an attractive alternative to the CoxPH for the regression analysis of censored failure time data . Unlike CoxPH , standard AFT model parametrizes the underlying time-to-event distribution up to a set of finite-dimensional parameters such as Weibull and log-normal ( Lee & Wang , 2003 ) . However , imposing distributional assumption is too strict in real data analysis and can reduce the attractiveness inherent in the AFT model , mostly underperforming Cox-based analysis ( Cox , 2008 ; Kleinbaum & Klein , 2010 ) . Recently , based on statistical theories and the advent of deep learning techniques , various time-to-event models have been explored to circumvent the necessity of assumptions such as linearity , single risk , discrete time , and fixed-time effect ( Katzman et al. , 2018 ; Lee et al. , 2018 ; Ren et al. , 2019 ; Kvamme & Borgan , 2019 ; Avati et al. , 2020 ; Tarkhan et al. , 2021 ; Rahman et al. , 2021 ) . For example , Cox-Time ( Kvamme et al. , 2019 ) and DATE ( Chapfuwa et al. , 2018 ) alleviate the most fundamental but strict assumptions of the CoxPH and parametric AFT models by achieving nonproportional hazards and non-parametric event-time distribution , respectively . Cox-Time exploits the neural network as a relative risk function to model interactions between time and covariates . The authors also show that the proposed loss function is a good approximation for the Cox partial loglikelihood . DATE is a conditional generative adversarial network for implicitly specifying a timeto-event distribution of ATF model . It does not require the pre-specified distribution in parametric form , instead , the generator can learn it from the data with the adversarial loss function . Incidentally , various deep learning-based approaches have been spotlighted to improve performance by resolving issues such as temporal dynamics and calibration ( Lee et al. , 2019 ; Nagpal et al. , 2021 ; Gao & Cui , 2021 ; Kamran & Wiens , 2021 ; Hu et al. , 2021 ) . Therefore , it became important to utilize welldesigned objective functions that fit not only statistical backgrounds but also optimization of neural networks . In this paper , we introduce a Deep AFT Rank-regression for Time-to-event prediction model ( DART ) , a deep learning-based semiparametric AFT model trained with an objective function originated from Gehan ’ s rank statistic . The model does not require specifying event time distribution while keeping the advantage of the standard AFT model that directly predicts event time . With a simple form of the loss function , by constructing comparable rank pairs , the optimization of DART is efficient compared to other deep learning-based time-to-event models . Experimental results show that DART is not only well-calibrated but also competitive in event order prediction performance even compared to hazard-based models . Furthermore , we believe that this work can be widely applied in the community while giving prominence to advantages of the AFT model that is relatively unexplored . 2 RELATED WORKS . We first overview time-to-event modeling focusing on the loss functions of Cox-Time and DATE models to highlight the difference in concepts before introducing our method . The primary interest of time-to-event analysis is to estimate survival quantities like survival function S ( t ) = P ( T ≥ t ) or hazard function h ( t ) = limδ→0 P ( t ≤ T ≤ t + δ|T ≥ t ) /δ , where T ∈ R+ denotes time-to-event random variable . In most cases , due to censored observations , those quantities can not be directly estimated with standard statistical inference procedure . In the presence of right censoring , Kaplan & Meier ( 1958 ) and Aalen ( 1978 ) provided consistent nonparametric survival function estimators , exploiting right-censoring time random variable C ∈ R+ . Researchers then can get stable estimates for survival quantities with data tuples { yi , δi , Xi } Ni=1 , where yi = min ( Ti , Ci ) is the observed event time with censoring , δi = I ( Ti ≤ Ci ) is the censoring indicator , and a vector of features Xi ∈ RP . Here , N and P denote the number of instances and the number of features , respectively . While those nonparametric methods are useful , one can improve predictive power by incorporating feature information in a way of regression modeling . Cox proportional-hazards ( CoxPH ) and acceleratedfailure-time ( AFT ) frameworks are the most common approaches in modeling survival quantities utilizing both censoring and features . 2.1 HAZARD-BASED MODELS . A standard CoxPH regression model ( Cox , 1972 ) formulates the conditional hazard function as : h ( t|Xi ) = h0 ( t ) exp ( βTXi ) , ( i = 1 , . . . , N ) , ( 1 ) where h0 ( · ) is an unknown baseline hazard function which has to be estimated nonparametrically , and β ∈ RP is the regression coefficient vector . It is one of the most celebrated models in statistics in that β can be estimated at full statistical efficiency while achieving nonparametric flexibility on h0 under the proportionality assumption . Note the model is semiparametric due to the unspecified underlying baseline hazard function h0 . Letting Ri be the set of all individuals “ at risk ” , meaning that are not censored and have not experienced the event before Ti , statistically efficient estimator for regression coefficients can be obtained minimizing the loss function with respect to β : LCoxPH ( β ) = ∑ i δi log ∑ j∈Ri exp [ βTXj − βTXi ] , ( 2 ) which is equivalent to the negative partial log-likelihood function of CoxPH model . Based on this loss function , Kvamme et al . ( 2019 ) proposed a deep-learning algorithm for the hazard-based predictive model , namely Cox-Time , replacing βTXj and βTXi with g ( yj , Xj ; θ ) and g ( yi , Xi ; θ ) , respectively . Here , g ( · ) denotes the neural networks parameterized by θ , and Ri would be replaced by R̃i , representing the sampled subset ofRi . With a simple modification of the standard loss function in Eq . ( 2 ) , Cox-Time can alleviate the proportionality for relative risk , showing empirically remarkable performance against other hazard-based models in both event ordering and survival calibration . 2.2 ACCELERATED-FAILURE-TIME MODELS . The conventional AFT model relates the log-transformed survival time to a set of features in a linear form : log Ti = β TXi + i , ( i = 1 , . . . , N ) , ( 3 ) where i is an independent and identically distributed error term with a common distribution function F0 ( · ) that is often assumed to be Weibull , exponential , log-normal , etc . As implied in Eq . ( 3 ) , AFT model takes a form of linear modeling and provides an intuitive and physical interpretation on event time without detouring via the vague concept of hazard function , making it a powerful alternative to hazard-based analysis . However , imposing a parametric distributional assumption for i is a critical drawback of the model , for which model in Eq . ( 3 ) could be a subclass of the hazard-based models . To alleviate linearity and parametric distributional assumptions , several works brought the concept of generative process and approximated the error distribution via neural networks like generative adversarial network ( GAN ) ( Miscouridou et al. , 2018 ; Chapfuwa et al. , 2018 ) . Especially , Chapfuwa et al . ( 2018 ) proposed a deep adversarial time-to-event ( DATE ) model , which specifies the loss function as : LDATE ( θ , φ ) = E ( X , y ) ∼Fnc [ Dφ ( X , y ) ] + EX∼Fnc , ξ∼Fξ [ 1−Dφ ( X , Gθ ( X , ξ ; δ = 1 ) ) ] + λ2E ( X , y ) ∼Fc , ξ∼Fξ [ max ( 0 , y −Gθ ( X , ξ ; δ = 0 ) ) ] ( 4 ) + λ3E ( X , y ) ∼Fnc [ ‖t−Gθ ( X , ξ ; δ = 1 ) ‖1 ] where θ , φ denotes the parameter set associated with a generator Gθ and a discriminator Dφ , respectively , ( λ2 , λ3 ) are hyperparameters to tune censoring trade-off , Fnc ( X , y ) and Fc ( X , y ) are empirical joint distributions for non-censored cases and censored cases , respectively , and Fξ is the simple distribution , such as uniform distribution . The generatorGθ implicitly defines event time distribution . Despite DATE achieves prominent survival calibration via the sample-generating process , the objective function is quite complicated and the GAN framework is inherently prone to mode collapse , i.e. , the generator learns only a few modes of the true distribution while missing other modes ( Srivastava et al. , 2017 ) . Also , when optimizing neural networks with multiple loss functions , it is difficult to balance and there might be conflicts ( i.e . trade-off ) with each term ( Dosovitskiy & Djolonga , 2020 ) . Therefore , their loss function might be difficult to be optimized as intended and requires a burdening training time , and consequently not be suitable for large-scale time-to-event analysis . In the statistical literature , there have been many attempts to directly estimate regression coefficients in the semiparametric AFT model , where the error distribution F0 is left unknown , rather than imposing specific parametric distribution or exploiting generative models . In this work , we bridge non-linear representation learning and an objective function for estimation of semiparametric AFT model , which is originated from Gehan ’ s rank statistic . By extensive quantitative analysis , we have shown the beauty of simplicity and compatibility of rank-based estimation , along with outstanding experimental performance .

Summary: The paper extends the previously proposed linear semiparametric AFT model based on Gehan’s rank statistic to a nonlinear setup. The nonlinear model termed Deep AFT Rank-regression for Time-to-event prediction model (DART) is parameterized by a neural network. Experimental results across four datasets show a competitive advantage over baselines per concordance index (C-Index), integrated Brier score (IBS), and training time.