Summary. The **Weibull** multi-parameter regression (**MPR**) **model** with frailty is developed for **interval** **censored** **survival** **data**. The basic **MPR** **model** which is wholly parametric with non-proportional hazards was developed by Burke and MacKenzie in their 2016 Biometrics paper. We describe the basic **model**, develop the **interval**-**censored** likelihood and extend the **model** to include Gamma frailty. We present a simulation study and re-analyse **data** from the Signal Tandmo- biel study. The **MPR** **model** is shown to be superior to a proportional hazards competitor.

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simplifies the calculation process especially when dealing with larger sets of **data**. Applying several imputation techniques to our **data** was relatively easy to implement. Using R software, we were able to develop a programming code to apply different imputation methods to the **data** sets and proceed with the parametric analysis. In this study, several imputation techniques were used to estimate **survival** function and compared with the one that was obtained by Turnbull based on **interval** **censored** and PIC failure time **data**. In the next two sections, parametric **model** and imputation techniques will be discussed.

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A special case of **interval**-**censored** **data** is current-status **data**, where individuals are seen only once after enrollment. Current-status **data** often arise in cross-sectional surveys, where the purpose is calculation of the distribution of age of onset for a disease or life event. Thus, the observations are either of the form (0, C] or (C, ∞) (i.e., left- or right- **censored**). These **data** are also commonly referred to as case 1 **interval**-**censored** **data** [14]. Current status **data** are common in demography [15, 16], economics, and epidemiology [17, 18]. In the medical sciences, animal tumorigenicity and HIV studies often result in such **data** because the inves- tigator cannot measure the outcome directly or accurately [19]. The proportional hazards models and tests referenced above for analyzing **interval**-**censored** **data** can be used for the analysis of current status **data**. Murphy and van der Vaart [20] considered semiparametric likelihood ratio inference and proposed a test for significance of the regression coefficient in Cox’s regression **model** for current status **data**. Banerjee [21] examined the power of the test under contiguous alternatives.

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For correlated **survival** times, there are two basic modeling approaches; i.e., marginal or frailty modeling. The marginal approach specifies a marginal **model** for each failure time, adopts a working independence assumption in the likelihood construction, obtains point estimates of the regression parameters under this assumption, and then uses the so-called sandwich estimator to obtain standard error estimates (Wei et al., 1989). Various marginal models have been proposed along the lines of this general approach for multivariate **interval**- **censored** **data**; e.g., the proportional hazards (PH) **model** (Goggins and Finkelstein, 2000; Kim and Xue, 2002), the proportional odds (PO) **model** (Chen et al., 2007), the additive hazards **model** (Tong et al., 2008), the linear transformation **model** (Chen et al., 2013), and the additive transformation **model** (Shen, 2015). Moreover, a goodness-of-fit test for assessing the appropriateness of the marginal Cox **model** for multivariate **interval**-**censored** **data** was proposed by Wang et al. (2006). Even though the marginal approach provides robust inference, it does not adequately account for the correlation that naturally exists between the multiple failure times.

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Following Cox and Oakes (1984), Klein and Moeschberger (2003), Nardi and Schemper (2003), parametric models often remain a useful tool as they are fitted much faster and offers more efficient estimates under conditions such as dependency of the **survival** times on the covariates (either fixed or time-varying) and when parameter values are far from zero. Subsequently, simulation proce- dures enables a researcher to asses the performance of a proposed parametric estimator concurrently in determining suitable inferential procedures for the parameters in a specified **model**. This methodology is crucial in order to draw reliable, precise and important information from the sample **data** in hand.

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calculate the conditional expectations of the local log likelihoods, given the observed **data** and the current estimate of the hazard function, and the M-step, in which these expected log likelihoods are maximized with respect to their parameters. On the other hand, this method requires manual entry of a bandwidth parameter that determines the amount of smoothing for the hazard function estimate (Betensky et al., 2002). Further, the analytic standard errors are not derived, necessitating the use of the bootstrap, which is quite computationally intensive in this setting (Cai & Betensky, 2003). Lastly, there are numerical stability problems with local likelihood in regions of sparse **data**, such as the right-hand tail of the hazard function. For the same problem, Cai and Betensky (2003) proposed a penalized spline-based approach. Basically, they weakly parameterized the log-hazard function with a piecewise-linear spline and provided a smoothed estimate of the hazard function by maximizing the penalized likelihood through a mixed **model**- based approach. One disadvantage of this approach is that the variability due to the estimation of the smoothing parameter for small samples seems out of reach in the frequentist framework from the **data**.

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A few methods have been suggested for left truncated and **interval**-**censored** (LTIC) **data**. Turnbull’s character- ization was corrected to accommodate both truncation and **interval**-censoring time points [13]. It was extended to the regression **model** under the proportional assumption [14]. Pan and Chappell noted that NPMLE is inconsis- tent for the early times with LTIC **data**, while conditional NPMLE is consistent [15]. The estimation of the param- eters in the Cox **model** with LTIC **data** and a rank-based test of **survival** function in LTIC were studied [16, 17]. However, the length-biased problem was not considered in those methods.

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In this paper, we investigate the association and joint distribution of two event times with case 2 **interval** **censored** **data** by using spline-based sieve estimation method. Very recently, the spline-based sieve maximum likelihood estimation method has often been used in **survival** analysis. See, for example, Lu et al. [13] and Zhang et al. [28]. Under a copula **model**, we adopt a two-stage approach and apply the spline-based sieve method to estimate the marginal distributions first, and then the association parameter. We make three contributions in this paper: The proposed two-stage estimators are asymptotically consistent; thanks to the spline procedure, the computation of the two-stage estimator for the association parameter is much more efficient than the two-stage semiparametric method in Sun et al. [19]; thanks to the spline procedure, the estimation for the joint distribution of two failure times is smooth and explicit.

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of the disease [ 1 ]. In the management of at risk population (i.e. elderly), it is therefore im- portant to study the time to aMCI conversion, and to identify risk factors associated with it. Several studies were performed within this respect [ 2 , 3 , 4 ]. In particular, we consider here a study [ 5 ] conducted from 1988 to 2008 which included 241 healthy elderly people (average age of 72 years old) and presents several interesting features. Since participants were followed at regular interviews, the endpoint of interest in this study, the time to aMCI conversion, is only known to occur between two successive visits. That is, all the observed **data** are **interval**-**censored**. Participants who do not experience conversion at their last follow-up date are right-**censored**. Also, it is known that even in this at risk population, some individuals will never experience conversion [ 6 ], therefore, a fraction of the population is “immune” to the event, or “cured”, as opposed to “susceptible” or “uncured”. It is interesting to identify which covariate impacts the probability of being susceptible or not, the time until the con- version, or both. We thus need a method that allows such variable selection and analysis. Up to now, these **data** have been analyzed without variable selection and without accounting for a possible cure fraction, but dealing with the **interval**-**censored** nature of the **data**. Most statistical softwares propose methods for right-**censored** **data**, but few of them allow **data** to be **interval**-**censored** [ 7 ]. In a non-parametric setting, the Kaplan-Meier estimator is no longer available as, in most of the cases, the events can no longer be ordered. To overcome this, the Turnbull non-parametric **survival** estimator was elaborated [ 8 ], and only recently a generalization to allow for continuous covariates was proposed [ 9 ]. Regression models have also been studied under that type of censoring [ 10 , 11 , 12 , 13 , 14 , 15 ]. However, all these methods usually make use of complex algorithms or methods, such as Expectation- Maximization (EM) algorithm [ 16 ] , self-consistency algorithm [ 8 ], Iterative Convex Minorant algorithm[ 12 ], or B-spline smoothing techniques [ 13 ]. On the contrary, assuming a specific distribution for the event times makes the analysis much simpler in the presence of **interval**- censoring.

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Bellamy et al. [8] extend parametric event time models to clustered and **interval** **censored** settings by introducing additive frailties to the linear predictor. Frailty comes into play when multiple events are considered for a given unit. Frailty can also count for unobserved covariates. Bellamy et al. implement their algorithm in existing commercial statistical computing software Bellamy et al. [8]. The authors **model** dependency between multiple events of a patient by frailty: doctor visits during the subject's time in study. Bellamy's idea is easily implemented in a Bayesian framework. WinBUGS [9] allows to implement analyses for **interval** **censored** **data** with frailty in the same **model**. A parametric approach via **Weibull** **model** or Accelerated Failure Time (AFT) **model** is easily realized. One can accommodate a frailty to the linear predictor part. But, the implementation of semiparametric proportional hazards models in WinBUGS is cumbersome (see Example Leuk in Example Volume 1 of the WinBUGS software).

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2. By constructing the confidence **interval**, we have condition the uncertainty based on the observations. As a result, the stages involved in constructing a confidence **interval** are the same as those of the Bayesian estimation. This will lead to find a Bayesian based approach to construct a CI. First, a prior distribution should be identified to **model** the uncertainty, and then information from a given **data** is used to design the CI. The unknown parameter is treated as a random variable and the observed **data** are utilized to obtain the posterior distribution. To conduct the above two-step process, let f X ( ) x ; θ be the joint pdf

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This article deals with the estimation of the parameters and reliability characteristics in inverse **Weibull** (IW) distribution based on the random censoring **model**. The censoring distribution is also taken as an IW distribution. Maximum likelihood estimators of the parameters, **survival** and failure rate functions are derived. Asymptotic confidence intervals of the parameters based on the Fisher information matrix are constructed. Bayes estimators of the parameters, **survival** and failure rate functions under squared error loss function using non-informative and gamma infor- mative priors are developed. Furthermore, Bayes estimates are obtained using Tierney-Kadane’s approximation method and Markov chain Monte Carlo (MCMC) techniques. Also, highest pos- terior density (HPD) credible intervals of the parameters based on MCMC techniques are con- structed. A simulation study is conducted to compare the performance of various estimates. Fi- nally, a randomly **censored** real **data** set supports the estimation procedures developed in this article.

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The third part of this dissertation deals with the regression analysis of multivari- ate **interval**-**censored** **data** with informative censoring. Multivariate **interval**-**censored** failure time **data** often occur in the clinical trial that involves several related event times of interest and all the event times suffer **interval** censoring. Different types of models have been proposed for the regression analysis ( Zhang et al.(2008); Tong et al.(2008); Chen et al.(2009); Sun (2006)). However, most of these methods only deal with the situation where observation time is independent of the underlying **survival** time completely or given covariates. In this chapter, we discuss regression analysis of multivariate **interval**-**censored** **data** when the observation time may be related to the underlying **survival** time. An estimating equation based approach is proposed for re- gression coefficient estimate with the additive hazards frailty **model** and the asymptotic properties of the proposed estimates are established by using counting processes. A major advantage of the proposed method is that it does not involve estimation of any baseline hazard function. Simulation results suggest that the proposed method works well for practical situations.

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For parametric methods, it is straightforward to form the likelihood for **interval**-**censored** **data** under the accelerated failure time **model** and standard likelihood based methods may be applied (see Equation 1 ). These methods are provided in the **survival** package using the survreg function ( Therneau and Lumley 2009 ). For right-**censored** **data** a more common regression method is the semi-parametric Cox proportional hazards regression. In this **model** the baseline hazard function is completely nonparametric, but does not need to be estimated. The score test from this **model** is the logrank test. The generalization of the **model** to **interval**- **censored** **data** typically uses the marginal likelihood of the ranks (see Satten 1996 ; Goggins, Finkelstein, and Zaslavsky 1999 ). The only available software for doing these models of which we are aware is an S function ( Goggins 2007 ) – which calls a compiled C program requiring access to a SPARC based workstation – to perform a Monte-Carlo EM algorithm for proportional hazards models described in Goggins et al. ( 1999 ). Another approach to semi-parametric modeling is to specifically estimate the nonparametric part of the **model** with a piecewise constant intensity **model** (see Farrington 1996 ; Carstensen 1996 ). This is the approach taken with the Icens function in the Epi package ( Carstensen, Plummer, Laara, Hills, and et al. 2010 ).

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In the literature, several estimators of **survival** func- tion are available. Currently, the Kaplan-Meier estimate is the simplest method for computing **survival** over time. Although, it is only adequate for right-**censored** **data** (i.e., the event occurs after the last follow-up). Another impor- tant estimator of **survival** is Turnbull’s algorithm [13] which takes into account **interval**-**censored** **survival** **data**. The **survival** curves generated with the Kaplan-Meier esti- mate and Turnbull’s algorithm are both easily interpreted. Various approaches for analyzing **interval**-**censored** **data** have been proposed in the literature. For example, Peto [14] provided a method to estimate a cumulative distribu- tion function from **interval**-**censored** **data**. This method is similar to the life-table technique and to the presented algorithm for estimating **survival** [15]. Semiparametric approaches based on the proportional hazards **model** have been developed for **interval**-**censored** **data** [16–21]. More- over, a wide variety of parametric models can also be used to estimate the distribution of time to an event of interest in the presence of **interval**-censoring **data** [22–24]. In a comprehensive review, Gómez et al. [25] present the most frequently applied non-parametric, parametric, and semiparametric estimating approaches that have been used to analyze **interval**-**censored** **data**. Rodrigues et al. [26] presents an adequate **interval** cen- sored methodology application in the boys’ first use of marijuana **data** set.

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Table 3 presents relative bias, coverage probabilities and inflation of standard errors when either using single midpoint imputation or multiple imputation for **interval** **censored** censoring events. When **interval** censoring is induced by intervals with lengths up to six months, both analytic strategies perform well with negligible bias, cov- erage probabilities of confidence intervals close to the nominal value, and standard errors that are only mar- ginally increased relative to the ordinary situation, i.e. when only ordinary right censoring is present. As a gen- eral tendency, however, the multiple imputation has lower relative bias and better coverage, in particular when censoring becomes more dominant in terms of higher censoring rates and wider censoring intervals. Only in extreme cases of 6% and 9% annual censoring proportions, censoring intervals of one year length, and larger sample sizes of 10,000, does coverage probabilities of the multiple imputation strategy decrease unaccepta- bly to levels around 75% to 80%, albeit not as low as when single midpoint imputation is used. This poor per- formance in extreme cases is to be expected, as the con- ditional censoring distribution is taken to be uniform in the multiple imputation analysis, while censoring times are actually generated from an exponential **model** – it is Table 2 Parameterization of distributions

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Where i is an indicator variable which takes value 1 if observation is **censored** and 0 otherwise. Section III includes the hypothetical **survival** **data** of fifteen patients with censoring suffering from a disease and they were treated by two different treatments. Section IV and Section V consist of Bayesian regression analysis of treatment 1 and treatment 2 using LaplacesDemon Hall [1] package which is available in R R Development Core Team[2]. The goal of LaplacesDemon is to provide a complete and self-contained Bayesian environment within R. The main function of this package which is used in the paper is LaplaceApproximation. This function gives the approximated posterior estimates of the parameters in Bayesian framework. In order to deal with censoring mechanism we have developed a function which works well for the analysis of **survival** **data**. Comparison of **survival** curves for two treatments by using **Weibull** **model** is reported in section VI. Finally in the last a brief discussion and conclusion is given in section VII.

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In Table 2, when we compared the mean squared error (MSE) and absolute bias of the hazard function of **Weibull** distribution with **interval** **censored** **data** by maximum likelihood (MLE), Bayesian using Lindley’s approximation (BL) and Bayesian using Gaussian Quadrature (BG), we found that Bayesian using Gaussian Quadrature is better compare to the others for all cases except when the n=100 with 3 . Moreover, maximum likelihood is better than Bayesian using Lindley’s approximation for all cases except when the n= 25 with 0.5 and 1.5 .