Deep Neural Networks for Semiparametric Frailty Models via H-likelihood
For prediction of clustered time-to-event data, we propose a new deep neural
network based gamma frailty model (DNN-FM). An advantage of the proposed model
is that the joint maximization of the new h-likelihood provides maximum
likelihood estimators for fixed parameters and best unbiased predictors for
random frailties. Thus, the proposed DNN-FM is trained by using a negative
profiled h-likelihood as a loss function, constructed by profiling out the
non-parametric baseline hazard. Experimental studies show that the proposed
method enhances the prediction performance of the existing methods. A real data
analysis shows that the inclusion of subject-specific frailties helps to
improve prediction of the DNN based Cox model (DNN-Cox).
Hangbin Lee, IL DO HA, Youngjo Lee
# Deep Neural Networks for Semiparametric Fraility Models via H-likelihood

###### Abstract

For prediction of clustered time-to-event data, we propose a new deep neural network based gamma frailty model (DNN-FM). An advantage of the proposed model is that the joint maximization of the new h-likelihood provides maximum likelihood estimators for fixed parameters and best unbiased predictors for random frailties. Thus, the proposed DNN-FM is trained by using a negative profiled h-likelihood as a loss function, constructed by profiling out the non-parametric baseline hazard. Experimental studies show that the proposed method enhances the prediction performance of the existing methods. A real data analysis shows that the inclusion of subject-specific frailties helps to improve prediction of the DNN based Cox model (DNN-Cox).

**Keywords:** Deep neural network, Frailty model, H-likelihood, Prediction, Random effect.

## 1 Introduction

Recently, deep neural network (DNN) has provided a major breakthrough to enhance prediction in various areas (LeCun et al., 2015; Goodfellow, 2016). The DNN models allow extensions of Cox proportional hazards (PH) models (Kvamme et al., 2019; Sun et al.,2020). Recently, subject-specific prediction of the DNN models has been studied by including random effects in neural network (NN) predictor (Tran et al., 2020; Mandel et al., 2022). However, these DNN random-effect models have been studied for only complete data. In this paper we propose a new DNN-FM. To the best of our knowledge, there is no literature on the DNN-FM for censored survival data. Lee and Nelder (1996) introduced the h-likelihood for the inference of general models with random effects and Ha, Lee and Song (2001) extended it to the semi-parametric frailty models. We reformulate the h-likelihood to obtain maximum likelihood estimators (MLEs) for fixed unknown parameters and best unbiased predictors (BUPs; Searle et al., 1992; Lee et al., 2017) for random frailties by a simple joint maximization of the profiled h-likelihood, which is constructed by profiling out