Joint models for longitudinal and time to event data are frequently used in many observational studies such as clinical trials with the aim of investigating how biomarkers which are recorded repeatedly in time are associated with time to an event of interest. In most cases, these joint models only consider a univariate time to event process. However, many clinical trials of patients with cancer, involve multiple recurrences of a single event together with a single terminal event experienced by patients over time. Therefore, this article proposes joint modelling approachs for longitudinal and multi-state data. The approach considers two sub-models that are linked by a common latent random variable. The first sub-model is linear mixed effect model that defines the longitudinal process and the second sub-model is a proportional intensity function for the multi-state process. Furthermore, on the proportional intensity model, two different formulations are used to define dependence structure between longitudinal and multi-state processes. In this article, a semi-Markov process that consider the time spent in the current state is defined for the transitions between states. Moreover, the time spent in each transient state is assumed to have Gompertz distribution. A Bayesian method using Markov Chain Monte Carlo (MCMC) is developed for parameter estimation and inferences. The deviance information criterion (DIC) is also derived for Bayesian model selection and comparison. Finally, our proposed joint modeling approach is evaluated through a simulation study and is applied to real datasets (colorectal and colorectal.Longi) which present a random selection of 150 patients from a multi-center randomized phase III clinical trial FFCD 2000-05 of patients diagnosed with metastatic colorectal cancer.

Longitudinal and time to event data are frequently encountered in many medical studies. Clinicians are more interested in how longitudinal outcomes influences the time to an event of i nterest. To study the association between longitudinal and time to event data, joint modeling approaches were found to be the most appropriate techniques for such data. The approaches involves the choice of the distribution of the survival times which in most cases authors prefer either exponential or Weibull distribution. However, these distributions have some shortcomings. In this paper, we propose an alternative joint model approach under Bayesian prospective. We assumed that survival times follow a Gompertz distribution. One of the advantages of Gompertz distribution is that its cumulative distribution function has a closed form solution and it accommodates time varying covariates. A Bayesian approach through Gibbs sampling procedure was developed for parameter estimation and inferences. We evaluate the finite samples performance of the joint model through an extensive simulation study and apply the model to a real dataset to determine the association between markers(tumor sizes) and time to death among cancer patients without recurrence. Our analysis suggested that the proposed joint modeling approach perform well in terms of parameter estimations when correlation between random intercepts and slopes is considered.

Background: Sojourn time refers to the amount of time a HIV patient spends in each clinical state in a single stay before he/she makes a transition to another state.  HIV can be broken down into a number of intermediate states, based on CD4 counts. The four states of the Markov process of HIV are commonly defined as: S1: CD4 count > 500 cells/microlitre of blood; S2: 350 < CD4 count ≤ 500 cells/microlitre of blood; S3: 200 < CD4 count ≤ 350 cells/microlitre of blood; S4: CD4 count ≤ 200 cells/microliter of blood. Aims: The aim of the study was to estimate sojourn and transition between clinical states of patients under ART in Namibia using homogenous semi-Markov processes, on data obtained from MoHSS. Methods: A retrospective study design was used to obtain data on 2422 patients who were observed 11028 times, during 2008 to 2017 follow up period. The four staged semi-Markov model was employed to estimate sojourn times and transition between clinical states. Results: Results indicates that 1637 (67.6%) were female and 785 (32.41%) were male .657(27.13%) patients started ART in state 1, 683(28.19%) patients  started ART in state 2, 677(27.95%) patients  started ART in state 3 and 405(16.72%) patients started ART in state 4, at treatment commencement (t = 0). As expected, the probabilities of transiting from good to worse states increased with time. After 6 months, the probabilities of transiting from state 1 to 3, and from state 1 to 4 are 0.023 and 0.004 respectively. Whereas after 12 months, the probabilities of transiting from state 1 to 3, and from state 1 to 4 are 0.059 and 0.010 respectively. As time increased the probabilities to remain in the same state is decreasing (probabilities of remaining in state 1 after 6, 12 and 18 months is 0.804, 0.698 and 0.633). Sojourn times for states 1, 2, 3 and 4 were 22, 8, 10 and 15 months respectively. Conclusions: Sojourn time is of interest in HIV modeling, as it gives a signal of how HIV is progressing. Longer sojourn times indicates slow HIV progression and shorter sojourn times indicates rapid HIV progression. As time increases, transition probabilities from good states to worse states increases.

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