Joint Modeling of Multivariate Longitudinal Data and Competing Risks Using Multiphase Sub-models

Abstract

In many clinical studies that involve follow-up, it is common to observe one or more sequences of longitudinal measurements, as well as one or more time to event outcomes. A competing risks situation arises when the probability of occurrence of one event is altered/hindered by another time to event. Recently, there has been much attention paid to the joint analysis of a single longitudinal response and a single time to event outcome, when the missing data mechanism in the longitudinal process is non-ignorable. We, in this paper, propose an extension where multiple longitudinal responses are jointly modeled with competing risks (multiple time to events). Our shared parameter joint model consists of a system of multiphase non-linear mixed effects sub-models for the multiple longitudinal responses, and a system of cause-specific non-proportional hazards frailty sub-models for competing risks, with associations among multiple longitudinal responses and competing risks modeled using latent parameters. The joint model is applied to a data set of patients who are on mechanical circulatory support and are awaiting heart transplant, using readily available software. While on the mechanical circulatory support, patient liver and renal functions may worsen and these in turn may influence one of the two possible competing outcomes: (i) death before transplant; (ii) transplant. In one application, we propose a system of multiphase cause-specific non-proportional hazard sub-model where frailty can be time varying. Performance under different scenarios was assessed using simulation studies. By using the proposed joint modeling of the multiphase sub-models, one can identify: (i) non-linear trends in multiple longitudinal outcomes; (ii) time-varying hazards and cumulative incidence functions of the competing risks; (iii) identify risk factors for the both types of outcomes, where the effect may or may not change with time; and (iv) assess the association between multiple longitudinal and competing risks outcomes, where the association may or may not change with time.

Appendices

Appendix A: Non-Linear Time Function

We now briefly describe the following non-linear equation to model the non-linear (or linear) temporal trend of the longitudinal process (Sect. 2—Eq. 2). Development of the generic equation (12) started with differential equation [17, 45] that was formed from different dynamics models such as biochemical reaction rate, allometric growth, and population growth. They also found that this generic equation has relationship with some statistical distributions. By setting $\nu$ to $\pm \,1$ or limiting case, $0/ \infty$, special and simpler cases of models are found. All these linear and various non-linear models are nested by dint of the common generic differential equation. This equation is originally used by Hazelrig et al. [17] to model cumulative mortality and then later used by Blackstone et al. [7] in multiphase hazard modeling (see “Data Analysis”—Sect. 4.2). In their modeling, both had used a 3-parameter time function. Since our data analysis experience [2, 5, 6, 16, 25] showed that a reduced 2-parameter equation adequately fits the longitudinal process, we use the following family of 2-parameter equation:

$$\begin{aligned} G(t, \varvec{{\Theta }})=\frac{ |\nu | - \nu }{2 |\nu |} + \frac{\nu }{|\nu |} \exp \left[ - \left( \frac{ t}{\kappa \log ^{\nu }(2)} \right) ^{-1/\nu } \right] , \end{aligned}$$

(12)

where $\nu \ne 0$ is a shaping parameter, and $\kappa >0$ is a scaling parameter. In survival terminology, Blackstone et al. [7] called $\kappa$ as the half life time. Natural constraints on G are that $G(0, \varvec{{\Theta }})=0$ and $G(t, \varvec{{\Theta }}) \rightarrow 1$ as $t \rightarrow \infty$. Note that, when $\nu >0, \; G(t, \varvec{{\Theta }})$ simplifies as

$$\begin{aligned} G(t, \varvec{{\Theta }}) = \exp \left[ - \left( \frac{ t}{\kappa \log ^{\nu }(2)} \right) ^{-1/\nu } \right] , \end{aligned}$$

and when $\nu <0$,

$$\begin{aligned} G(t, \varvec{{\Theta }}) = 1 - \exp \left[ - \left( \frac{ t}{\kappa \log ^{\nu }(2)} \right) ^{-1/\nu } \right] . \end{aligned}$$

Fig. 6

Three scenarios of shapes of $g(t,\varvec{{\Theta }})$ for different values of $\nu$ and $\kappa$. Note that, by changing $\kappa$ one can change the peaking time

Blackstone et al. [7] in their multiphase hazard modeling approach used $g(t, \varvec{{\Theta }})=\frac{\partial G(t, \varvec{{\Theta }})}{\partial t}$ to model early phase of hazard and $h(t, \varvec{{\Theta }})=\frac{g(t, \varvec{{\Theta }})}{1-G(t, \varvec{{\Theta }})}$ to fit late phase of the hazard. The motivation to use these equations, in survival terminology, is that if $G(t, \varvec{{\Theta }})$ is cumulative hazard $g(t, \varvec{{\Theta }})$ is the hazard. Note that, since $G(t, \varvec{{\Theta }})$ is bounded above by 1, if one uses $G(t, \varvec{{\Theta }})$ as cumulative hazard, it forms an incomplete time distribution. Suppose $G(t, \varvec{{\Theta }})$ is cumulative distribution, and $h(t, \varvec{{\Theta }})$ is a proper hazard function. We can use $G(t, \varvec{{\Theta }})$ or any of its transformation or combination of transformations to fit a longitudinal process. In our data analysis experience in fitting longitudinal process, we found that, in most cases, $g(t, \varvec{{\Theta }})$ can be used as early time function $T(t, \varvec{{\Theta }})$ and $h(t, \varvec{{\Theta }})$ can be used as late time function $T(t, \varvec{{\Theta }})$. Examples of different shapes of $T(t, \varvec{{\Theta }})=g(t, \varvec{{\Theta }})$ and $T(t, \varvec{{\Theta }})=h(t, \varvec{{\Theta }})$ are given in Figs. 6 and 7.

Fig. 7

Three scenarios of shapes of $h(t,\varvec{{\Theta }})$ for different values of $\nu$ and $\kappa$. Note that, when $\nu =-\,1$, $\; h(t,\varvec{{\Gamma }})$ reduces to a constant function

Appendix B: Temporal Decomposition

We illustrate the temporal decompositions of the bivariate longitudinal process that resulted in the overall temporal trends depicted in Fig. 3.

Fig. 8

Estimated decomposition of the overall temporal of GFR and Log(Bilirubin)

For GFR, there is an early peaking phase followed by a late increasing phase (Top—Fig. 8), and for Log(Bilirubin) the analysis yielded an early decreasing phase followed by an increasing phase that plateaued after 4 months (Bottom—Fig. 8).