Stable Non-Linear Generalized Bayesian Joint Models for Survival-Longitudinal Data

Abstract

Joint models have received increasing attention during recent years with extensions into various directions; numerous hazard functions, different association structures, linear and non-linear longitudinal trajectories amongst others. Here, we present a partially linear joint model with a Bayesian smoothing spline component to capture non-linear longitudinal trajectories where the longitudinal data can assume non-Gaussian distributions. Our approach is stable with regards to the knot set as opposed to most well-known spline models. We implement this method using the R-INLA package and show that most joint models with shared Gaussian random effects are part of the class of latent Gaussian models (LGMs). We present an illustrative example to show the use of the stable partially linear joint model and an application to real data using a skew-normal partially linear joint model. This paves the way for efficient implementation of joint models with various complex model components using the same methodology and computational platform.

Appendix

1.1 Computational considerations for joint models in using INLA

The likelihood of a joint model b asically consists of two types of likelihoods and this can be facilitated in the INLA framework. It is essential to construct the response matrix and the covariate matrices correctly for the estimation procedure. For the purpose of this paper, we will present only the case where the joint model consists of longitudinal and survival submodels. This can be extended to include more marginal submodels in the case of multiple endpoint modeling.

Within the context of this paper, consider the following structured predictors of the longitudinal and survival submodels, respectively:

$$ \begin{array}{@{}rcl@{}} \eta^{L}_{ijk}&=&\alpha(t_{ijk})+\boldsymbol{\beta}^{T}\boldsymbol{X}_{ijk}+w_{ij}+v_{ij}t_{ijk}\\ {\eta^{S}_{l}}(s)&=&\boldsymbol{\gamma}^{T}\boldsymbol{Z}_{l}+f(w_{l},v_{l}s). \end{array} $$

(A.1)

Consider the case where the data consists of N_i,i = 1,...,N observations for each of the N individuals, so that in total there are N_L longitudinal observations and correspondingly N_S = N event times and censoring indicators (s_i,c_i),i = 1,...,N. The data is then composed as a list in which each variable consists of N_L + N_S elements. To achieve this, we include zeros for fixed effects if the covariate is not included in that specific submodel and NA’s for the random effects. In the case of Eq. A.1, the new response is defined as a list of the y_ijk and (s_i,c_i). The fixed effect covariates are constructed as $(\boldsymbol {X},\boldsymbol {0}_{1,...,N_{S}})$ and $(\boldsymbol {0}_{1,...,N_{L}},\boldsymbol {Z})$ while the random effects are constructed as $(\boldsymbol {\alpha },\boldsymbol {NA}_{1,...,N_{S}})$.

The main contribution in this area is the estimation of α. Most of the commonly used approaches to estimate the non-linear trend invloves the use of knots. This method was also used in Kim et al. (2017). In this paper we propose the use of a time-continuous spline model manifested as a second-order random walk presented in Section 4.

1.2 Example: Simulated joint model

In this example we simulated data from the following scenario:

$$ \begin{array}{@{}rcl@{}} \eta^{L}(t)=f(t)+v_{i}\\ \eta^{S}=0.01Age+0.5v_{i} \end{array} $$

where $v_{i}\sim N(0,{\sigma ^{2}_{v}})$ are the subject-specific random effects that are shared in this joint model, Age is a linear covariate and f(t) is a non-linear mean longitudinal trajectory. The aim of this example is to illustrate the practical method to fit a joint model in R-INLA and compare the estimated trajectories from R-INLA and jplm with the true underlying mean trajectory f(t).

We investigated two scenario’s with different forms of f(t). For each scenario we present a graphical presentation of the estimated trajectories based on different simulated samples from the same underlying process.

• In Fig. 6 four samples and their results from f(t) = t² are illustrated.

• In Fig. 7 two samples and their results from f(t) = t^− 0.25 are illustrated.

It is clear from Figs. 6 and 7 that the random walk two model captures the true underlying longitudinal trajectory in the joint model much better than the spline approach proposed and embedded in the jplm function.

Figure 6

Estimated and true trajectories with the simulated data for f(t) = t²

Figure 7

Estimated and true trajectories with the simulated data for f(t) = t^− 0.25

The R code is available at http://www.r-inla.org/examples/case-studies/van-niekerk-bakka-and-rue-2019.

1.3 Computational framework information

The R code used to obtain the results as presented in Sections 6 and 7 is available at http://www.r-inla.org/examples/case-studies/van-niekerk-bakka-and-rue-2019http://www.r-inla.org/examples/case-studies/van-niekerk-bakka-and-rue-2019.

The computational time needed was for an Apple Macbook Pro i5 3.1GHz with 16GB 2133 MHz LPDDR3.