A Deep Learning Approach for Dynamic Survival Analysis with Competing Risks

1 INTRODUCTION . Survival analysis informs our understanding of the relationships between the ( distribution of ) first hitting times of events ( such as death , onset of a certain disease , etc . ) and the covariates , and enables us to issue corresponding risk assessments for such events . Clinicians use survival analysis to make screening decisions or to prescribe treatments , while patients use the information about their clinical risks to adjust their lifestyles in order to mitigate such risks . However , designing the best clinical intervention for a patient is a daunting task , as the appropriate level of interventions or the corresponding outcome often depend on whether this patient is susceptible to or suffers from competing risks . For example , studies in ( Koene et al . ( 2016 ) ) have shown that various treatments , such as chemotherapy , for breast cancer increase the risk of a cardiovascular events . To refer to the same example in cystic fibrosis ( CF ) , the decision on lung transplantation , which is particularly recommended for patients with end-stage respiratory failure , must jointly account for deaths from other CF-associated failures ( e.g. , CF-associated liver failure ) since they share a number of risk factors ( Kobelska-Dubiel et al . ( 2014 ) ) . Meanwhile , as a growing number of electronic health records ( EHRs ) have been deployed in hospitals1 , modern clinical data in EHR often comprises longitudinal measurements ; especially with chronic diseases , patients are followed up over the span of years , usually as part of regular physical examinations . Information contained in these longitudinal ( follow-up ) measurements is of significant importance such that interpreting these measurements in their historical context can offer an explanation for how the underlying process of clinical events progresses ( Rizopoulos et al . ( 2017 ) ; Suresh et al . ( 2017 ) ) . For example , forced expiratory volume ( FEV1 ) , and its development , is a crucial biomarker in assessing the severity of CF as it allows clinicians to describe the progression of the disease and to anticipate the occurrence of respiratory failures ( Nkam et al . ( 2017 ) ; Li et al . ( 2017 ) ) . Therefore , to provide a better understanding of disease progression , it is essential to incorporate longitudinal measurements of biomarkers and risk factors into a model . Rather than 1EHRs are deployed in more than 75 % of hospitals in the United States , according to the recent data brief by the Offcie of National Coordinator ( ONC ) ( Henry et al . ( 2016 ) ) . discarding valuable information recorded over time , this allows us to make better risk assessments on the clinical events of interest . In light of the discussion above , we design a deep neural network for dynamic survival analysis with competing risks , which we call Dynamic-DeepHit , that learns , on the basis of the available longitudinal measurements , a flexible data-driven distribution of first hitting times of various events of interest . An important aspect of our method is that it naturally handles situations in which there are multiple competing risks where more than one type of event plays a role in the survival setting . To enable survival analysis with competing risks , Dynamic-DeepHit employs a network architecture that consists of a single shared subnetwork and a family of cause-specific subnetworks . We incorporate historical information of each patient using a recurrent neural network ( RNN ) structure in the shared subnetwork , which allows our method to update personalized and cause-specific risk predictions as additional measurements are collected in a fully dynamic fashion . We conduct a set of experiments on real-world datasets showing that our model outperforms state-of-the-art survival models in discriminating individual risks for different causes . In addition , we provide variable importance and dynamic risk predictions of our method to yield clinical usefulness in interpreting the associations between covariates and the survival . Related works : Since the Cox proportional hazard model ( Cox ( 1972 ) ) was first introduced , a variety of methods have been developed for survival analysis , ranging from statistical models to deep learning techniques ( Fine & Gray ( 1999 ) ; Ishwaran et al . ( 2008 ) ; Lee & Whitmore ( 2010 ) ; Yu et al . ( 2011 ) ; Lee et al . ( 2018 ) ; Luck et al . ( 2017 ) ; Katzman et al . ( 2016 ) ; Alaa & van der Schaar ( 2017 ) ; Bellot & van der Schaar ( 2018 ) ) . Especially , deep networks have been shown to achieve significantly improved performance in survival analysis ( Katzman et al . ( 2016 ) ; Luck et al . ( 2017 ) ; Alaa & van der Schaar ( 2017 ) ; Bellot & van der Schaar ( 2018 ) ; Lee et al . ( 2018 ) ) owing to the ability to represent complicated associations between features and outcomes . However , all of these methods provide static survival analysis : they use only the current information to perform the survival predictions and most of the works focus on a single risk rather than multiple risks . In this paper , we extend the idea of ( Lee et al . ( 2018 ) ) , which finds the estimated joint distribution of the first hitting time and competing events utilizing a deep network , into the longitudinal setting where repeated measurements are available . Few methods have been developed to use the longitudinal time-to-event data in order to enable dynamic survival analysis under competing risks . The first strand of literatures includes landmarking ( Heagerty & Zheng ( 2005 ) ; Zheng & Heagerty ( 2005 ) ; van Houwlingen ( 2007 ) ; van Houwelingen & Putter ( 2008 ) ) and joint models ( Henderson et al . ( 2000 ) ; Ibrahim et al . ( 2010 ) ; Tsiatis & Davidian ( 2004 ) ; Brown et al . ( 2005 ) ; Barrett & Su ( 2017 ) ) . Landmarking refrains from modeling the time-dependent aspect of longitudinal variables and , instead , obtains survival probabilities from the survival model fitted to subjects who are still at risk at the time points of interest ( i.e. , landmarking times ) . On the other hand , joint models explicitly model the longitudinal process and leverage their predictions as inputs in a separate survival process used for predicting survival probabilities . However , both approaches , and their variations ( van Houwelingen & Putter ( 2008 ) ; Tsiatis & Davidian ( 2004 ) ; Brown et al . ( 2005 ) ; Barrett & Su ( 2017 ) ) , make strong assumptions about the underlying stochastic models for the survival process ( in landmarking ) or for both the longitudinal and survival processes ( in joint models ) . Hence , model mis-specifications ( e.g. , typically , a linear mixed model and a Cox proportional hazard model ) limit their ability to learn and infer complex interactions between covariates and survival times , which are common in many diseases with heterogeneity . The second strand models the longitudinal data using variants of RNNs ( Choi et al . ( 2016a ; b ) ; Razavian et al . ( 2016 ) ; Lipton et al . ( 2016 ) ) and avoids the need for explicit model specifications , which results in performance gain in terms of predictive accuracy . However , these models can not properly cope with time-to-event data where the goal is to find the probability of the first hitting event occurring at different times of our interest . Instead , given longitudinal measurements , they view making risk predictions of one or multiple event ( s ) as solving a single or multiple label ( s ) classification problem at each time stamp , e.g. , whether an event occurs or not at the measurement time ( Choi et al . ( 2016a ; b ) ) , within a predefined time-window ( Razavian et al . ( 2016 ) ) , or at the end of the available longitudinal measurements ( Lipton et al . ( 2016 ) ) . To our best knowledge , this paper is the first to investigate a deep learning approach for dynamic time-to-event analysis with competing risks on the basis of repeated measurements ( longitudinal time-to-event data ) . 2 BACKGROUND . 2.1 LONGITUDINAL TIME-TO-EVENT DATA . We consider a dataset D = { ( X i , τ i , ki ) } Ni=1 comprising time-to-event ( survival ) data for N subjects who have been followed up for a certain amount of time . For each subject i , X i is a set of longitudinal observations including both static and time-varying covariates , τ i = min ( T i , Ci ) is the time information with T i ∈ T indicating the time-to-event and Ci ∈ T indicating the time-tocensoring , and ki ∈ K being the event or censoring that occurred at τ i . Define X i ( t ) = { xi ( tij ) : 0 ≤ tij ≤ t for j = 1 , · · · , M i } a set of longitudinal observations until time t where xi ( tj ) is covariates recorded at time tj . Here , we distinguish notations between time stamps j = 1 , · · · , M i and the corresponding actual times tij = ti1 , · · · , tiMi , since measurements are not necessarily observed at regular intervals . Then , we use X i = X i ( tiMi ) to denote a whole set of longitudinal observations available for subject i until the last measurement time tiMi of that subject for notational simplicity . A set of possible survival times is denoted as T = { 0 , 1 , · · · , Tmax } with Tmax being a predefined maximum time horizon , where we treat survival time as discrete2 ( e.g. , a resolution of one month ) and the time horizon as finite ( e.g. , no patients lived longer than 100 years ) . A set of possible events is K = { ∅ , 1 , 2 , · · · , K } , with ∅ denoting right-censoring as survival data is frequently right-censored due to subjects being lost to follow-up . We assume that every subject experiences exactly one event among K ≥ 1 possible events of interest within T . This includes cause-specific deaths due to CF , where deaths from other causes are competing risks for death due to respiratory failure ( Gooley et al . ( 1999 ) ) . Figure 1 depicts a time-to-event dataset comprising histories of longitudinal measurements with competing risks , where subjects are aligned based on the synchronization event . The aforementioned characteristics are highlighted with annotations . 2.2 CUMULATIVE INCIDENCE FUNCTION . The cause-specific cumulative incidence function ( CIF ) is key to survival analysis under the presence of competing risks . As defined in ( Fine & Gray ( 1999 ) ) , the CIF expresses the probability that a particular event k∗ ∈ K occurs on or before time τ∗ conditioned on the history of longitudinal measurements X ∗ . The fact that longitudinal measurements have been recorded up to t∗M∗ implies survival of the subject up to this time point . Thus , the CIF is defined as follows : Fk∗ ( τ ∗|X ∗ ) , P ( T ≤ τ∗ , k = k∗|X ∗ , T > t∗M∗ ) = ∑ m≤τ∗ P ( T = m , k = k∗|X ∗ , T > t∗M∗ ) . ( 1 ) Whenever a new measurement is recorded for this subject at time t > t∗M∗ , we can update ( 1 ) accounting for that information in a dynamic fashion . 2Discretization is performed by transforming continuous-valued times into a set of contiguous time intervals , i.e. , T = τ implies T ∈ [ τ , τ + δt ) where δt implies the resolution . Similarly , the survival probability of a subject at time τ∗ given X ∗ ( i.e. , the probability that a subject does not experience any event before or at time τ∗ ) can be derived by S ( τ∗|X ∗ ) , P ( T > τ∗|X ∗ , T > t∗M∗ ) = 1− ∑ k 6=∅ Fk ( τ ∗|X ∗ ) . ( 2 ) Our goal is to estimate the CIF , F̂k∗ ( τ∗|X ∗ ) , from the dataset D , in order to analyze the causespecific risk given the history of observations and to issue dynamic risk predictions .

Summary: The authors present a novel deep learning representation for jointly modelling longitudinal measurements and dynamic time-to-event analysis where there are competing risks for a given event. The authors incorporate patient-level historical data using an RNN which allows updating of individual-level (i.e. personalized) risk predictions as additional data points are collected. This method (Dynamic-DeepHit) makes no assumptions about the underlying stochastic processes. The authors further evaluate the clinical utility of these methods in terms of interpretability of variable importance and dynamic risk predictions.