Interval Censoring

Survival analysis statistics have been used to estimate shelf life of foods based on consumers' acceptance or rejection of samples with different storage times. The interval censored nature of the data does not allow regular methods to estimate the number of necessary consumers (N) for these studies. The objective of the present work was to use simulation studies to estimate N considering the error in the estimation, statistical significance and statistical power. Based on previous studies a Weibull model was assumed for the lifetime distribution with a shape parameter r between 0.17 and 0.71. Simulations were calculated supposing shelf-life values in the middle range of an adimensional time scale. Graphs to estimate N under different scenarios are presented. Considering the average r, an alpha value (Type I error) of 5%, a beta value (Type II error) of 20%, the shelf life to be in the middle of the studied time range, and a difference between the true shelf life and the estimated shelf life of 0.5 on a 0-6 time scale, gave an N value of 120 consumers for this particular set of parameters.

Survival analysis statistics have been used to estimate shelf life of foods based on consumers’ acceptance or rejection of samples with different storage times. The interval censored nature of the data does not allow regular methods to estimate the number of necessary consumers (N) for these studies. The objective of the present work was to use simulation studies to estimate

We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion. We then specialize our results to independent, nonidentically distributed observations, Markov processes, stationary Gaussian time series and the white noise model. We apply our general results to several examples of infinite-dimensional statistical models including nonparametric regression with normal errors, binary regression, Poisson regression, an interval censoring model, Whittle estimation of the spectral density of a time series and a nonlinear autoregressive model. POSTERIOR CONVERGENCE RATES 193 dressed by Amewou-Atisso, Ghosal, Ghosh and Ramamoorthi [1] and Choudhuri, Ghosal and Roy . The main purpose of the present paper is to obtain a theorem on rates of convergence of posterior distributions in a general framework not restricted to the setup of i.i.d. observations. We specialize this theorem to several classes of non-i.i.d. models including i.n.i.d. observations, Gaussian time series, Markov processes and the white noise model. The theorem applies in every situation where it is possible to test the true parameter versus balls of alternatives with exponential error probabilities and it is not restricted to any particular structure on the joint distribution. The existence of such tests has been proven in many special cases by Le Cam and Birgé , who used them to construct estimators with optimal rates of convergence, determined by the (local) metric entropy or "Le Cam dimension" of the model. Our main theorem uses the same metric entropy measure of the complexity of the model and combines this with a measure of prior concentration around the true parameter to obtain a bound on the posterior rate of convergence, generalizing the corresponding result of Ghosal, Ghosh and van der Vaart . We apply these results to obtain posterior convergence rates for linear regression, nonparametric regression, binary regression, Poisson regression, interval censoring, spectral density estimation and nonlinear autoregression. van der Meulen, van der Vaart and van Zanten have extended the approach of this paper to several types of diffusion models.

Data analysis sometimes requires the relaxation of parametric assumptions in order to gain modeling flexibility and robustness against mis-specification of the probability model. In the Bayesian context, this is accomplished by placing a prior distribution on a function space, such as the space of all probability distributions or the space of all regression functions. Unfortunately, posterior distributions ranging over function spaces are highly complex and hence sampling methods play a key role. This paper provides an introduction to a simple, yet comprehensive, set of programs for the implementation of some Bayesian nonparametric and semiparametric models in R, DPpackage. Currently, DPpackage includes models for marginal and conditional density estimation, receiver operating characteristic curve analysis, interval-censored data, binary regression data, item response data, longitudinal and clustered data using generalized linear mixed models, and regression data using generalized additive models. The package also contains functions to compute pseudo-Bayes factors for model comparison and for eliciting the precision parameter of the Dirichlet process prior, and a general purpose Metropolis sampling algorithm. To maximize computational efficiency, the actual sampling for each model is carried out using compiled C, C++ or Fortran code.

Estimation for the Birnbaum-Saunders (BS) regression model has been discussed by various authors when data are either complete or subject to Type-I or random censoring. But, this problem has not been considered for the case of interval censoring. In this article, we discuss the estimation of a regression model with current status data when the failure times follow the BS distribution. We estimate the parameters by the method of maximum likelihood, and derive the asymptotic distribution of these estimators. The performance of these estimators is then assessed through Monte Carlo simulations for different sample sizes under two types of monitoring. Finally, an analysis of real data is used to illustrate the proposed method.

In this paper, we consider the problem of estimating unknown parameters of an inverse Weibull distribution when it is known that samples are progressive type-I interval censored. We propose an EM algorithm to obtain maximum likelihood estimates and mid point estimates. For comparison purpose Bayes estimates are also obtained under the square error loss function using non-informative and informative prior situations. A simulation study is conducted to access the performance of the proposed estimators and recommendations are made on the basis of simulation results. A real data set is also analyzed in detail for an illustration purpose. Finally, by making use of expected Fisher information matrix various inspection times and optimal censoring schemes are obtained.

This chapter is about the accelerated failure time model with an interval-censored covariate. Two cases concerning the response variable Y are distinguished: On one hand, Y is possibly right-or left-censored, on the other hand, Y is a doubly-censored variable.

We apply Gaussian process (GP) regression, which provides a powerful non-parametric probabilistic method of relating inputs to outputs, to survival data consisting of time-to-event and covariate measurements. In this context, the covariates are regarded as the `inputs' and the event times are the `outputs'. This allows for highly flexible inference of non-linear relationships between covariates and event times. Many existing methods, such as the ubiquitous Cox proportional hazards model, focus primarily on the hazard rate which is typically assumed to take some parametric or semi-parametric form. Our proposed model belongs to the class of accelerated failure time models where we focus on directly characterising the relationship between covariates and event times without any explicit assumptions on what form the hazard rates take. It is straightforward to include various types and combinations of censored and truncated observations. We apply our approach to both simulated and experimental data. We then apply multiple output GP regression, which can handle multiple potentially correlated outputs for each input, to competing risks survival data where multiple event types can occur. By tuning one of the model parameters we can control the extent to which the multiple outputs (the time-to-event for each risk) are dependent thus allowing the specification of correlated risks. Simulation studies suggest that in some cases assuming dependence can lead to more accurate predictions.

For a version of the interval censoring model, case 2, in which the observation intervals are allowed to be arbitrarily small, we consider estimation of functionals that are differentiable along Hellinger differentiable paths. The asymptotic information lower bound for such functionals can be represented as the squared L 2 -norm of the canonical gradient in the observation space. This canonical gradient has an implicit expression as a solution of an integral equation that does not belong to one of the standard types. We study an extended version of the integral equation that can also be used for discrete distribution functions like the nonparametric maximum likelihood estimator (NPMLE), and derive the asymptotic normality and efficiency of the NPMLE from properties of the solutions of the integral equations.

In some biomedical clinical trials, especially cancer clinical trials, time-to-event data are generated from cancer scan within a specific interval (such as 1 or more months) resulting in interval-censored data. The common practice in analyzing data from this type of trial is to approximate the interval-censored data using the midpoint or right endpoint (i.e. the first observed time) of the interval so that well-known statistical methods developed for rightcensored data may be used for the requisite analyses. This could introduce bias and lead to erroneous conclusions. In this paper we conduct a simulation study to investigate the bias in parameter estimation and inference. Appropriate statistical models for interval-censored data are proposed with application to breast cancer clinical trial data.

Based on a data set obtained in a dental longitudinal study, conducted in Flanders (Belgium), the joint time to caries distribution of permanent first molars was modeled as a function of covariates. This involves an analysis of multivariate continuous doubly-interval-censored data since: (i) the emergence time of a tooth and the time it experiences caries were recorded yearly, and (ii) events on teeth of the same child are dependent. To model the joint distribution of the emergence times and the times to caries, we propose a dependent Bayesian semiparametric model. A major feature of the proposed approach is that survival curves can be estimated without imposing assumptions such as proportional hazards, additive hazards, proportional odds or accelerated failure time.

Based on a data set obtained in a dental longitudinal study, conducted in Flanders (Belgium), the joint time to caries distribution of permanent first molars was modeled as a function of covariates. This involves an analysis of multivariate continuous doubly-intervalcensored data since: i) the emergence time of a tooth and the time it experiences caries were recorded yearly, and ii) events on teeth of the same child are dependent. To model the joint distribution of the emergence times and the times to caries, we propose a dependent Bayesian semiparametric model. A major feature of the proposed approach is that survival curves can be estimated without imposing assumptions such as proportional hazards, additive hazards, proportional odds, or accelerated failure time.

Estimating causal inferences in observational studies with time varying covariates require methods that can address complexities such as non-random allocation of patients' to treatment groups, possible censoring of, exposure variables e.g., time to antiretroviral therapy (ART) initiation, and outcome variables e.g., mortality. Marginal Structural Cox Proportional Hazard Model (MSCM) adopts inverse probability of treatment and censoring weights to correct for non-random allocation of treatment groups, patient attrition or administrative censoring and confounding. We set out to evaluate the causal effects of 'time to ART initiation' on the survival of human immunodeficiency (HIV) infected patients enrolled in Jaramogi Oginga Odinga Teaching and Referral Hospital (JOOTRH), a high HIV-burden hospital in Western Kenya.A retrospective review of observational data was conducted at JOOTRH for patients aged ≥15 years enrolled between May 2011 and December 2013. Patients were categorized as 'early initiators' if they were initiated on ART within 12 weeks of determination of ART eligibility and 'late initiators' if they were initiated on ART after.We compared the results from MSCM to those from unadjusted standard Cox Proportional Hazard (Cox-PH) model and adjusted standard Cox-PH model (adjusting for age, gender, CD4 category, WHO stage, education level, marital status, and weight) to assess the extent to which standard methods can bias estimates for causal effects.Of 786 patients enrolled; 606 (77%) were ART-eligible of whom 444 (73%) had documenteddates of ART initiation. Among patients who were initiated on ART, 41 (9%) were late-initiators. There were3 deaths and 9.2 person-years of follow-up and 31 deaths and 115.8 person-years of follow-up among 'late-initiators' and 'early-initiators' respectively. Compared to 'early-initiators', the hazard of dying among 'late-initiators' computed using unadjusted standard Cox-PH model, adjusted standard Cox-PH model and MSCM was1.6 (95% CI 0.5-5.4) and 1.9 (95% CI 0.5-6.7) and 3.2 (95% CI 1.7-6.4) respectively.Delays in ART initiation was associated with an increased risk of mortality only after employing MSCM. When randomized trials are not feasible due to costs or ethical concerns, causal inference could still be deduced from longitudinal observational data by applying appropriate causal models.

In some biomedical clinical trials, especially cancer clinical trials, time-to-event data are generated from cancer scan within a specific interval (such as 1 or more months) resulting in interval-censored data. The common practice in analyzing data from this type of trial is to approximate the interval-censored data using the midpoint or right endpoint (i.e. the first observed time) of the interval so that well-known statistical methods developed for rightcensored data may be used for the requisite analyses. This could introduce bias and lead to erroneous conclusions. In this paper we conduct a simulation study to investigate the bias in parameter estimation and inference. Appropriate statistical models for interval-censored data are proposed with application to breast cancer clinical trial data.

Interval censored life history data arise when the events of interest are only detectable at periodic assessments. When interest lies in the occurrence of two such events, bivariate interval censored event time data are obtained. We describe how to fit a four state Markov model useful for characterizing the association between two interval censored event times. The approach treats the two events symmetrically and enables one to fit multiplicative intensity models which give estimates of covariate effects as well as relative risks characterizing the association between the two events. An EM algorithm is described for estimation for which the maximization step can be carried out with standard software. The method is illustrated by application to data from a trial of HIV patients where the events are the onset of viral shedding in the blood and urine among individuals infected with cytomegalovirus.

It is widely believed that setting monetary policy through a majority voting commitee has major bene…ts. It is argued that a monetary policy committee can take personalities out of monetary policy decisions, while at the same time o¤ering the opportunity to consider and debate a range of possible policy responses. Critical to understanding these claims would be an assessment of how such a committee functions. In this paper we consider interval censored responses of individual committee members when there is both heterogeneity and interaction among members. In addition to heterogeneity in policy responses arising from several di¤erent sources, we identify signi…cant interactions between committee members. JEL Classi…cation: E42, E43, E50, E58.

We consider the problem of estimating the intensity functions for a continuous time 'illness-death' model with intermittently observed data. In such a case, it may happen that a subject becomes diseased between two visits and dies without being observed. Consequently, there is an uncertainty about the precise number of transitions. Estimating the intensity of transition from health to illness by survival analysis (treating death as censoring) is biased downwards. Furthermore, the dates of transitions between states are not known exactly. We propose to estimate the intensity functions by maximizing a penalized likelihood. The method yields smooth estimates without parametric assumptions. This is illustrated using data from a large cohort study on cerebral ageing. The age-specific incidence of dementia is estimated using an illness-death approach and a survival approach.

Many authors in recent years have proposed extensions of familiar survival analysis methodologies to apply in dependent data settings, for example, when data are clustered or subject to repeated measures. However, these extensions have been considered largely in the context of right censored data. In this paper, we discuss a parametric frailty model for the analysis of clustered and interval censored failure time data. Details are presented for the speciÿc case where the underlying time to event data follow a Weibull distribution. Maximum likelihood estimates will be obtained using commercially available software and the empirical e ciency of these estimators will be explored via a simulation study. We also discuss a score test to make inferences about the magnitude and signiÿcance of over-dispersion in clustered data settings. These methods will be illustrated using data from the East Boston Asthma Study.

This paper discusses the analysis of interval-censored failure time data, which has recently attracted a great amount of attention (Li and Pu, Lifetime Data Anal 9:57-70, 2003; Sun, The statistical analysis of interval-censored data, 2006; Tian and Cai, Biometrika 93(2):329-342, 2006; Zhang et al., Can J Stat 33:61-70, 2005). Interval-censored data mean that the survival time of interest is observed only to belong to an interval and they occur in many fields including clinical trials, demographical studies, medical follow-up studies, public health studies and tumorgenicity experiments. A major difficulty with the analysis of interval-censored data is that one has to deal with a censoring mechanism that involves two related variables. For the inference, we present a transformation approach that transforms general interval-censored data into current status data, for which one only needs to deal with one censoring variable and the inference is thus much easy. We apply this general idea to regression analysis of interval-censored data using the additive hazards model and numerical studies indicate that the method performs well for practical situations. An illustrative example is provided.

In biostatistics applications interest often focuses on the estimation of the distribution of a timevariable T. If one only observes whether or not T exceeds an observed monitoring time C, then the data structure is called current status data, also known as interval censored data, case I. We consider this data structure extended to allow the presence of both time-independent covariates and time-dependent covariate processes that are observed until the monitoring time. We assume that the monitoring process satisfies coarsening at random.

PDF version ISBN 952-9528-68-X http://ethesis.helsinki.fi HELSINGIN YLIOPISTON VERKKOJULKAISUT Helsinki 2001 I wish to present my gratitude to my supervisor Elja Arjas for his support and patience during all these years of my postgraduate studies. My co-operation with experts in dentistry, Jorma Virtanen, Markku Larmas and Hannu Hausen, has also been fruitful, starting from when I was working at the University of Oulu. The working environment was good there, most of all due to the staff, especially Andrei Andreev, Dario Gasbarra (who were my colleagues also in Helsinki) and Liping Liu, in the department of statistics to whom I am grateful. Over the past years I have worked at the Rolf Nevanlinna Institute, and would like to thank all of the staff for a good working atmosphere, especially members of the biometry division: Kari Auranen, Mervi Eerola, Bob O'Hara, Jukka Ranta, Samuli Ripatti and Mikko Sillanpää. A good computing environment has been vital in my work, so Pekka Kangas and Matti Taskinen have my special gratitude. In the rat tumorigenicity study the help of Timo Hakulinen and Marja Mutanen was important. The constructive feedback of the pre-examiners, Jochen Mau and Erik Parner is gratefully acknowledged. The support of my family and my friends has been important. Thank Scandinavian Journal of Statistics for the permission to reproduce Article I.

Adequate knowledge of timing and pattern of permanent tooth emergence is useful for diagnosis and treatment planning in paediatric dentistry and orthodontics and is essential in forensic dentistry. Based on a data set obtained in a large dental longitudinal study, conducted in Flanders (Belgium), the joint emergence distribution of seven teeth was modeled as a function of gender and caries experience on primary teeth. Besides establishing the marginal dependence of emergence on the covariates, there was also interest in examining the impact of the covariates on the association among emergence times. This allows to establish the preferred rankings of emergence and their dependence on covariates, but it necessitates to model the covariance matrix (of a multivariate normal distribution) as a function of covariates. Modeling the covariance matrix has received recently quite some attention in the literature. Indeed, in a variety of statistical models allowing the covariance matrix to depend on covariates can improve the fit of the model to the data considerably and can imply a dramatic change in the conclusion, especially for non-linear models. However, modeling must to be done with care since the positive definiteness of the covariance matrix needs to be assured. Further, it is preferable that all regression parameters of the model are interpretable. The modified Cholesky representation of the covariance matrix, as suggested by , splits up the covariance matrix into two parts where the parameters can be interpreted given a natural ranking of the responses. This approach was used to model 2 tooth emergence data. However, the emergence times were recorded in an interval-censored manner. Hence, we opted for a Bayesian implementation of the Data Augmentation algorithm.

The transparency of the monetary policymaking process at the Bank of England has provided very detailed data on both the votes of individual members of the Monetary Policy Committee and the information on which they are based. In this paper we consider interval censored responses of individual committee members in the context of a model in which inflation forecast targeting is used but there is both heterogeneity and interaction among the members of the committee. We find substantial heterogeneity in the policy reaction function across members. Further, we identify significant interactions between individual decisions of the committee members. The nature of these interdependencies inform about information sharing and strategic interactions within the Bank of England's Monetary Policy Committee.

When an examination is needed to determine if an event has occurred there will be a loss of efficiency in using the resulting interval-censored data instead of the exact event times. In designing follow-up intervals this loss for longer intervals needs to be weighed against extra visits required by shorter intervals. We obtain results to quantify this for the estimation of the median and mean survival and for covariates in parametric regression models with equally spaced examination times. Asymptotic information loss for the log-normal and Weibull distributions are similar when comparisons are made between corresponding members of the two families. For distributions with coefficients of variation of 50% or more, a choice of interval from 0.25 to 0.7 times the median is recommended.

In some biomedical clinical trials, especially cancer clinical trials, time-to-event data are generated from cancer scan within a specific interval (such as 1 or more months) resulting in interval-censored data. The common practice in analyzing data from this type of trial is to approximate the interval-censored data using the midpoint or right endpoint (i.e. the first observed time) of the interval so that well-known statistical methods developed for rightcensored data may be used for the requisite analyses. This could introduce bias and lead to erroneous conclusions. In this paper we conduct a simulation study to investigate the bias in parameter estimation and inference. Appropriate statistical models for interval-censored data are proposed with application to breast cancer clinical trial data.

The non-parametric Bayes estimator with Dirichlet process prior of a survival function based on right censored data was considered by and many others. We obtain the non-parametric Bayes estimator of a survival function when data are right, left or interval censored. The resulting Bayes estimator with Dirichlet process prior has an explicit formula. In contrast, there is no explicit formula known for the non-parametric maximum likelihood estimator (NPMLE) with such data. In fact, we show that the NPMLE with doubly/interval censored data cannot, in general, equal to the limit of Bayes estimator for any sequence of priors. Several examples are given, showing that the NPMLE and the non-parametric Bayes estimator may or may not be the same, even when the prior is 'non-informative'.

The non-parametric Bayes estimator with Dirichlet process prior of a survival function based on right censored data was considered by and many others. We obtain the non-parametric Bayes estimator of a survival function when data are right, left or interval censored. The resulting Bayes estimator with Dirichlet process prior has an explicit formula. In contrast, there is no explicit formula known for the non-parametric maximum likelihood estimator (NPMLE) with such data. In fact, we show that the NPMLE with doubly/interval censored data cannot, in general, equal to the limit of Bayes estimator for any sequence of priors. Several examples are given, showing that the NPMLE and the non-parametric Bayes estimator may or may not be the same, even when the prior is 'non-informative'.

We consider the problem of estimating the joint distribution function of the event time and a continuous mark variable when the event time is subject to interval censoring case 1 and the continuous mark variable is only observed in case the event occurred before the time of inspection. The nonparametric maximum likelihood estimator in this model is known to be

The relatively short durations of the initial pivotal randomized placebo-controlled trials involving atomoxetine HCl for the treatment of attention-deficit/hyperactivity disorder (ADHD) provided limited insight into the time courses of ADHD core symptom responses to this nonstimulant, selective norepinephrine reuptake inhibitor. The aim of this analysis was to evaluate time courses of treatment responses or remission, as assessed by attainment of prespecified scores on the ADHD Rating Scale-IV-Parent Version: Investigator Administered and Scored (ADHDRS-IV-PI) and the Clinical Global Impressions-ADHD-Severity (CGI-ADHD-S) scales, during up to 1 year of atomoxetine treatment in children with ADHD.

Consider the model in which the data consist of possibly censored lifetimes, and one puts a mixture of Dirichlet process priors on the common survival distribution. The exact computation of the posterior distribution of the survival function is in general impossible to obtain. This paper develops and compares the performance of several simulation techniques, based on Markov chain Monte Carlo and sequential importance sampling, for approximating this posterior distribution. One scheme, whose derivation is based on sequential importance sampling, gives an exactly iid sample from the posterior for the case of right censored data. A second contribution of this paper is a battery of programs that implement the various schemes discussed in this paper. The programs and methods are illustrated on a data set of interval-censored times arising from two treatments for breast cancer.

The marginal Cox model approach is perhaps the most commonly used method in the analysis of correlated failure time data . It assumes that the marginal distributions for the correlated failure times can be described by the Cox model and leaves the dependence structure completely unspecified. This paper discusses the assessment of the marginal Cox model for correlated interval-censored data and a goodness-of-fit test is presented for the problem. The method is applied to a set of correlated interval-censored data arising from an AIDS clinical trial.

Based on a data set obtained in a large dental longitudinal study, conducted in Flanders (Belgium), the joint emergence distribution of seven teeth was modeled as a function of gender and caries experience on primary teeth. Besides establishing the marginal dependence of emergence on the covariates, there was also interest in examining the impact of the covariates on the association among emergence times. This allows the establishment of the preferred rankings of emergence and their dependence on covariates. To this end, the covariance matrix was modeled as a function of covariates. Modeling the covariance matrix in this way needs to ensure the positive definiteness of the covariance matrix and it is preferable that the regression parameters of the model are interpretable. The modified Cholesky decomposition of the covariance matrix, as suggested by Pourahmadi, splits up the covariance matrix into two parts where the parameters can be interpreted, given a natural ranking of the resp...

The semi-Markov assumption emphasizes the importance of time spent in a state. In order to compute this type of multistate model, most transition times are always considered to be exactly identified or right censored. However, in the longitudinal analysis of chronic diseases, investigators are often confronted with interval-censored data (transition times are known to have occurred in some interval). Thus, the two key issues are the modeling of the duration dependence and the interval censoring. In this article, we define a semi-Markov model, allowing for interval censoring, for parametric hazard functions with a ∪or ∩-shape and for determination of initial states according to covariates. Our modeling approach is specific to each transition, so as to obtain a more coherent model. Parallel to competing risks models, the multistate model takes into account several final events. We consider an example of kidney transplant recipient follow-up to illustrate the utility of the method.

Adequate knowledge of timing and sequence of tooth emergence is of clinical significance for diagnosis and treatment planning in pediatric dentistry and orthodontics and is very useful in forensic dentistry. On the basis of mean ⁄ median emergence times, 'classical' or 'preferred' sequences of emergence have been reported (1-8). However, in the clinical-pedodontic and orthodontic-setting, one is much more interested in individual sequences of tooth emergence. Mean ⁄ median emergence orders can be poor predictors of individual emergence sequences in view of individual variability (9-11). Furthermore, estimating individual sequences of tooth emergence naturally leads to an estimate of the prevalence of the different sequences in a straightforward manner. Statistical inference on emergence times (of permanent teeth) might be quite complicated since the exact time of emergence of a tooth into the oral cavity is never known exactly but rather it takes

Data from sensory shelf-life studies can be analyzed using survival statistical methods. The objective of this research was to introduce Bayesian methodology to sensory shelf-life studies and discuss its advantages in relation to classical (frequentist) methods. A specific algorithm which incorporates the interval censored data from shelf-life studies is presented. Calculations were applied to whole-fat and fat-free yogurt, each tasted by 80 consumers who answered ''yes'' or ''no'' to whether they would consume samples with different storage times. The Bayesian approach was used to obtain the failure functions and the estimated shelf lives of the yogurts. Two approaches were considered: first non-informative priors were used, and then results from the fat-free yogurt were used to construct a more informative prior for the whole-fat product. For both approaches a Weibull distribution was assumed. For the non-informative priors the results were analogous to those obtained by the classical methods. When an informative prior was used to estimate the whole-fat shelf-life parameters, there was a tendency to closer credible limits in the percentile distributions. Advantages of the Bayesian approach are discussed: no need for asymptotical assumptions and the possibility of improving the precision of the results by introducing external information.

Based on a data set obtained in a dental longitudinal study, conducted in Flanders (Belgium), the joint time to caries distribution of permanent first molars was modeled as a function of covariates. This involves an analysis of multivariate continuous doubly-intervalcensored data since: i) the emergence time of a tooth and the time it experiences caries were recorded yearly, and ii) events on teeth of the same child are dependent. To model the joint distribution of the emergence times and the times to caries, we propose a dependent Bayesian semiparametric model. A major feature of the proposed approach is that survival curves can be estimated without imposing assumptions such as proportional hazards, additive hazards, proportional odds, or accelerated failure time.

Consider the model in which the data consist of possibly censored lifetimes, and one puts a mixture of Dirichlet process priors on the common survival distribution. The exact computation of the posterior distribution of the survival function is in general impossible to obtain. This paper develops and compares the performance of several simulation techniques, based on Markov chain Monte Carlo and sequential importance sampling, for approximating this posterior distribution. One scheme, whose derivation is based on sequential importance sampling, gives an exactly iid sample from the posterior for the case of right censored data. A second contribution of this paper is a battery of programs that implement the various schemes discussed in this paper. The programs and methods are illustrated on a data set of interval-censored times arising from two treatments for breast cancer.

Age at which cattle become faecal culture positive for Mycobacterium avium subsp. paratuberculosis (Map) can be used as a proxy parameter for age at onset of faecal shedding, which is an important parameter in the control of Map in cattle herds. To investigate the age at becoming faecal culture positive, survival analysis methods were applied. The analyses were carried out on asynchronous interval censored data of faecal culture results of samples collected from 18,979 female Holstein-Frisian cattle in 353 Dutch herds between 1996 and 2002. The data were analysed with a Weibull proportional hazards model.