Two Methods to Fit Longitudinal Sub-Model of Joint Model

The joint model is a new shared parameter model consisting of longitudinal and survival data. In general, we often use a linear mixed-effects model to estimate the longitudinal sub-model and a Cox proportional hazard model to estimate the surviving sub-model. However, the linear model requires a hypothesis of normal distribution. Consequently, when the assumption is not valid, the estimation result will produce large deviations. In this paper, we use machine learning to estimate the longitudinal model, and the survival model still uses the Cox model. We compare the effects of the linear mixed-effects model and the machine learning method. These two estimation methods are compared by simulated data. The results of the comparison show that the residuals diagnosis outcomes of the machine learning survival sub-model are more in line with the theoretical results, while the residuals of the longitudinal sub-model are more dispersed than the linear model. Additionally, the running time of the machine learning is shorter than that of the linear mixed effect model. Introduction In recent years, more and more researchers are interested in joint models consisting of longitudinal data and time-to-event data, especially in the field of statistics and biological diseases. Longitudinal data refers to the repeated measurement of data by a group of individuals in time or space order. Each individual is observed multiple times at different times or under different test conditions, and the obtained data has the characteristics of time series and cross-section data. In many practical studies, participants were followed up repeatedly to obtain longitudinal data. The event time data refers to the time elapsed from a certain starting point to the occurrence of the event. The event time data we usually study is the time of death, also known as Survival data. There are many approaches on the joint model with longitudinal data and survival data. The most typical one is two-stage method, unfortunately substantial research on this topic indicates that this approach results in biased estimates [12,3,11,15]. In addition, maximum likelihood estimation [2,5] is a common method to estimate parameters in joint model. In terms of the joint model, time cost is a difficult problem to solve. Therefore, in order to reduce the computational cost and improve the efficiency of the model, many researchers have introduced the R software packages. Rizopoulos (2010, 2014) introduced two software packages, JM and JMbayes, respectively based on the expected maximum and Bayesian methods, which plays an active role in promoting the R packages used to fit the joint model. The machine learning methods have the unique advantage when fitting longitudinal sub-model, namely it does not need to consider any assumptions, compared with linear mixing effect. In this paper, we estimate the joint model by means of the machine learning method [21]. And the simulation data including the longitudinal and survival data is used. Through the simulation data, we compare the maximum likelihood with the aid of the JM packages and the machine learning methods, such as the classification tree, nearest neighbor, random forest and support vector regression. What’ s more, the running time is also be used to compare. Joint Model Framework The joint model of longitudinal data and survival data is divided into two parts: longitudinal 2019 9th International Conference on Education and Social Science (ICESS 2019) Copyright © (2019) Francis Academic Press, UK DOI: 10.25236/icess.2019. 174 915 sub-model and survival sub-model. Linear mixed effect model is usually used to fit the longitudinal sub-model, while survival sub-model is almost the Cox proportional risk model [12]. For each subject 1,2, , i n  ,   1 2 , , , T T T i i i iK y y y y  is the K-variate continuous outcome vector with possible right-censoring, where each ik y denotes an   1 ik n  -vector of observed longitudinal measurements for the k-th outcome type:   1 2 , , , ik T ik i k i k in k y y y y  . Each outcome is measured at observed (possibly pre-specified) time ijk t for 1,2, , ik j n    1,2, , k K  , which can differ between subjects and outcomes. Let i T denote the observed failure time for the i -th subject ( 1,2, , i n  ), which is taken as the minimum of the true event time * i T and the censoring time i C , i.e.,   * min , i i i T T C  . Furthermore, we define the failure indicator   * i i i I T C    , where   I is the indicator function that takes the value 1 if the condition * i i T C  is satisfied , and 0 otherwise. Thus, the observed data for the time-to-event outcome consist of the pairs     , , 1,2, , i i T i n   . In terms of the longitudinal sub-model,               2 , 0, ik ik ik ik k T T ik ik k ik ik y t m t t N