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Bayesian Joint Models for Longitudinal and Survival Data

This paper takes a quick look at Bayesian joint models (BJM) for longitudinal and survival data. A general formulation for BJM is examined in terms of the sampling distribution of the longitudinal and survival processes, the conditional distribution of the random effects and the prior distribution. Next a basic BJM defined in terms of a mixed linear model and a Cox survival regression models is discussed and some extensions and other Bayesian topics are briefly outlined.


Introduction
Longitudinal data are observations of one or more variables measured over time of each of the individuals in the study. They include observations between and within individuals that allow the assessment of general patterns of the target population as well as specific individual characteristics. These data are multivariate, clustered, and repeated measures. Longitudinal data of each individual could be understood as a time series. Panel data are longitudinal data in economic scenarios.
Survival times measure follow-up times from a defined starting point to the occurrence of a given event or endpoint of interest. Standard statistical techniques cannot be applied to survival data because they are subject to censoring and/or truncation schemes that usually do not provide complete experimental information.
Joint modeling of longitudinal and survival data is an increasing and productive area of statistical research that examines the association between longitudinal and survival processes. It enhances survival interests with the inclusion of internal time-dependent covariates assessed through longitudinal models as well as longitudinal objectives by allowing for the inclusion of non-ignorable dropout mechanisms through survival tools (Verbeke and Davidian, 2009;Ye and Yu, 2014). Joint models were introduced during the 1990s as a consequence of the research about the human immunodeficiency virus infection and acquired immune deficiency syndrome (HIV/AIDS), and cancer studies. Since then, they have been applied to a great variety of studies, mostly in epidemiological and biomedical areas.
The two key elements of the Bayesian reasoning (with regard to the frequentist one) are the conception of probability, that allows to measure uncertainty associated to parameters, models, hypotheses, missing data, etc. in probabilistic terms, and the use of Bayes' theorem to sequentially update probabilities as more relevant information is obtained. As a result, Bayes inference offers a wide and attractive framework to joint models of longitudinal and survival analysis: posterior inferences for any outcome of interest depending on the parameters that makes unnecessary asymptotic approximations, a simple framework to easily incorporate historical data into the inferential process, or prediction of observable quantities directly assessed in probabilistic terms among others (Ibrahim et al, 2001).
The literature on joint models of longitudinal and survival data is enormous (Rizopoulos, 2012) is an excellent text within the frequentist methodology). Bayesian literature is not as extensive but too numerous to be cited here. References in this paper are only a very small part of all those that should appear in any more extensive work on the area. In any case, a reminder of the early works on the topic (Faucett and Thomas, 1996;Ibrahim et al., 2001;Wang and Taylor, 2001) is always more than convenient to have a historical perspective of the beginning research in the subject.

Bayesian joint models 2.1 General formulation
A BJM for longitudinal and survival data is a joint probability distribution for the observable longitudinal (y) and survival variables (s) as well as for the corresponding subject-specific random effects vector (b) and the parameters and hyperparameters (θ) of the model. This density could be generally expressed as where p(y, s | b, θ) is the sampling distribution of (y, s) given (b, θ), p(b | θ) the conditional distribution of the random effects given θ, and p(θ) the prior distribution for θ. Note that the subsequent formulation in the frequentist framework would not include the vector θ and therefore its definition would be only written in terms of p(y, s, b), p(y, s | b) and p(b).
There are different proposals for the specification of p(y, s | b, θ). All them provide a wide framework of the relationship between the longitudinal and the survival processes which facilitates the modeling into longitudinal and survival submodels with various types of connectors between them (Verbeke and Davidian 2009). The most popular structures for that relation are the so-called conditional models, share-parameter models, random-effects models, and the joint latent class models. Copula based models have also addressed that issue recently (Li et al., 2019).
Conditional models were the first to be used in the initial studies on the subject. They express the joint sampling distribution of the longitudinal and survival process as follows: Selection models are used when the interest of the study lies in the survival process while pattern-mixture models are appropriate for longitudinal purposes.
Shared-parameter models connect the longitudinal and the time-to-event processes by means of a common set of subject-specific random effects. This approach postulates conditional independence between the longitudinal and survival processes given the random effects and the parameters: A clear disadvantage of these models is the stiffness of the correlation structure between the longitudinal and the survival processes. The random-effects approach (Henderson et al., 2000) has the same design that the shared-parameter models but allows for more flexibility (and thus more complexity) for the connection between the survival and the longitudinal processes, enabling that a part of the random effects associated to both processes are not common.
The joint latent class model (Proust-Lima et al., 2014) is based on finite mixtures: heterogeneity among the individuals is classified into a finite number of homogeneous latent clusters, which share the same longitudinal trajectory and risk function. Both processes are conditionally independent within the subsequent latent group as follows: where G is the random variable that measures the uncertainty on the membership of each individual to each group, usually modelled by means of a multinomial logistic model.
The complete specification of a BJM includes the conditional distribution of the random effects, which can be both time-dependent and time-independent, and the choice of an a prior distribution for θ. After their specification and the obtention of data, Bayes' theorem updates the current information in terms of the posterior distribution p(b, θ | data) which allows posterior inferences for any outcome of interest depending on b and/or θ and it is the main element for assessing posterior predictive distributions for new observable longitudinal and survival elements from or separately via the marginal posterior predictive distributions p(y | data) and p(s | data).

A basic Bayesian joint model
The basic BJM is composed of the two most paradigmatic longitudinal and survival models such as the linear mixed model and the Cox survival regression model. The natural territory of the linear mixed effects model is the normal distribution, whose conditional mean accounts for common population terms and individual-specific elements, the random effects, which are unique to each individual separately. Let y ij denote the longitudinal response variable for the ith, i = 1, . . . , N individual registered at the time t ij , j = 1, . . . , n i . We assume the following linear mixed model for y i = (y i1 , . . . , y ini ) ′ : where X i is a n i × q matrix of covariates associated to the parametric vector β, Z i is a matrix of covariates associated to random effects b i , and I is the n i × n i identity matrix. Random effects The Cox proportional hazards model expresses the hazard function h i (t) of the survival time T i of individual i as the product of a common baseline hazard function, h 0 (t), which determines the shape of h i (t), and an exponential term with the relevant covariates as follows: where M i (t) = {m i (l), 0 ≤ l ≤ t} represents the true longitudinal trajectory of individual i up to time t, d i is a vector of baseline covariates associated to coefficients γ, and α is the coefficient of association between the longitudinal and the survival process. Obviously, if α were zero the survival and the longitudinal process would be independent. The last element of the BJM is the prior distribution for θ which includes all the parameters and hyperparameters of the longitudinal model, β, Σ b and σ 2 , as well as the subsequent of the survival process, γ, α and the ones in h 0 (t). Prior independence is the most simple assumption for the joint prior distribution for θ. Priors for β, γ and α are commonly elicited as normal distributed centered at zero with wide variances, inverse Wishart for Σ b and inverse gamma for σ 2 (Guo and Carlin, 2004), although options such as uniform or half-Cauchy distributions for variances seem more appropriate (Gelman, 2006).
Simple extensions of the basic BJM introduce complexity into the longitudinal and the survival modeling such as the distribution of the random effects, the baseline hazard function, and the connectors between the longitudinal and the survival processes. More flexible longitudinal trajectories in terms of splines are in Tang

Some extensions and other topics
Bayesian inference offers a natural environment to address complex models with sophisticated longitudinal and/or survival structures. In the case of the survival submodels, administrative right censoring is the most usual pattern but also interval-censoring and left truncation (Armero et al., 2018) can be considered. More complex structures include cure rate models (Yu et al.), 2004 or even spatial terms in the hazard function (Martins et al., 2016). Survival submodels with more that one event of interest include, among others, recurrent and competing risks models (Hu et