Survival Model With Doubly Interval-Censored Data and Time-Dependent Covariates Kaveh Kiani and Jayanthi Arasan Abstract—In this paper a survival model with doubly interval censored (DIC) data and time dependent covariate is discussed. DIC data usually arise in follow-up studies where the lifetime, T = W - V is the elapsed time between two related events, the first event, V and the second event, W where both events are interval censored (IC). The work starts by describing an algorithm that can be used to simulate doubly interval censored data. Following that the parameter estimates of the model are studied via a comprehensive simulation study. Finally the Wald and jackknife confidence interval estimation procedures are explored for the parameters of this model thorough coverage probability study. Index Terms—doubly censored, time dependent covariate, Wald, jackknife. I. I NTRODUCTION T He analysis of doubly interval censored data begins when De Gruttola and Lagakos [9] proposed a non- parametric estimation procedure based on the Turnbulls self-consistency algorithm. Following that, the analysis of DIC data has been studied extensively using nonparametric and semiparametric regression approaches. Reich et al. [13] proposed the likelihood contribution for a doubly interval censored lifetime. In this research we adapted Reich et al.s idea and proposed a parametric model by assuming both initial event time and lifetime follow exponential distribution. It is rather common in any analysis to find time dependent covariates, for example, blood pressure, cholesterol level and age. These covariates that do not remain at a fixed value over time. A time dependent covariate, x(t) may take values that follow a step function thus remaining constant within an interval but changes from one interval to another. Most literature on time varying covariates involve the extension of the semi parametric Cox proportional hazards model because it easily accommodates time varying covariates. This is due to the partial likelihood function, which is determined by the ordered survival times and not by the actual survival times. Authors who have made a contribution include Crowley and Hu [7],Wulfsohn and Tsiatis [20], Murphy and Sen [16], Marzec and Marzec [15], Cai and Sun [5], Zucker and Karr [21], Martinussen et al.[14], Goggins[8] and Hastie and Tibshirani [11]. Apart from the Cox model, there has also been work on time varying covariates with discrete-time using the logistic regression model by authors such as Brown [4], Hankey and Mantel [10] and Pons [18]. Other works involve the acceler- ated failure time model with time varying covariates which was discussed by Cox and Oakes [6], Nelson [17], Robins K. Kiani is with Data processing and Dissemination Department, Statistical Research and Training Center (SRTC), Tehran, Iran, (email: kiani@srtc.ac.ir). J. Arasan is with the Department of Mathematics, Faculty of Science, University Putra Malaysia, Serdang, 43400, Selangor, Malaysia (e-mail: jayanthi@.upm.edu.my). and Tsiatis [19] and Bagdonavicius and Nikulin [3]. Arasan and Lunn ([1],[2]) has discussed the bivariate exponential model with time varying covariate. Kiani and Arasan [12] discussed the Gompertz model with time dependent covariate for mixed case interval censored data. II. THE MODEL DIC data often arise in the follow-up studies where the survival time of interest is time between two events where both events are IC. For instance, infection by a virus as the first event and onset of the disease as the second event. DIC data include right censored(RC) and IC survival time data as special cases. In order to formulate the censoring scheme let V and W be two non-negative continuous random variables representing the times of two related consecutive events where both of them are IC and V ≤ W . Then, the survival time of interest could be defined as, T = W - V . Also, T is a non-negative continuous random variable. Let survivor functions of V , T and W be S(v), S(t) and S(w). Here it is assumed that V and T follow the exponential distribution. Any value that V takes is IC when its exact value is unknown and only an interval (V L ,V R ] is observed where V ∈ (V L ,V R ] and V L ≤ V R with probability 1. Similarly, any value that W takes is IC when the exact value of W is unknown and only an interval (W L ,W R ] is observed where W ∈ (W L ,W R ] and W L ≤ W R with probability 1. Finally, an observation on T is DIC when the exact value of T is unknown and only one interval (W L - V R ,W R - V L ] is observed where T ∈ (W L - V R ,W R - V L ] and W L - V R ≤ W R - V L with probability 1. Let f V (v) and f T (t) be the probability density functions of V and T and f W (w) be the undefined probability density function of W . Following Reich et al. [13], if f T (t) is known and v is given and t = w - v then the joint p.d.f. of V and W would be f (v,w)= f V (v)f T (w - v). So, the likelihood function for a DIC data is L(λ, γ ) =  v R v L  w R w L f (v,w)dwdv =  v R v L  w R w L f V (v)f T (w - v)dwdv. Distributional assumptions on V and T will allow us to obtain the above likelihood function of the observations. Here it is assumed time to first event, V , and survival time, T , follow the exponential distribution. III. TECHNIQUE FOR SIMULATING DOUBLY I NTERVAL-CENSORED DATA This section looks at the simulation of DIC data when the survivor functions of the T and V are known and the Proceedings of the World Congress on Engineering 2018 Vol I WCE 2018, July 4-6, 2018, London, U.K. ISBN: 978-988-14047-9-4 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCE 2018