I have been struggling with grasping the intuition behind joint models, and I hope someone can elucidate a particular aspect of theirs. Joint models, first of all, are essentially combinations of linear model + survival model, in order to be able to estimate "the time left" to an "event", at any arbitrary point within a time series. A good intuitive description is found here.

The purpose of joint models is clear, as the above link demonstrates. A doctor might want to know, at any moment, how much longer (based on a patient's treatment and his/her biomarker history/time series data) they can be expected to survive.

However, following this article, I would extremely appreciate help understanding the trajectory function mentioned on page 2.

To make sure I am on the right track, I created some dummy data of 3 patients: the first 2 receive a treatment and survive longer (later Time-of-event), while the last does not, and, unfortunately, dies earlier. There also exists an arbitrary Bio-marker that is being measured (like white blood cell count).

On the same page, the linear model aspect of joint models is presented,

$$Y_{ij} = X_{ij} + \epsilon_{ij}\, ,$$

where $Y_{ij}$ is an "observed outcome" (so I assume it is some Bio-marker), and $X_{ij}$ is some "trajectory function" that stands for the following:

$$X_{ij} = \theta_{0i} + \theta_{1i} * t_{ij} + \gamma Z_i\, .$$

$Z_i$ is the presence (or not) of the treatment (1 or 0). However, the rest of the equation makes no sense to me. I agree that the Bio-marker should be affected by the presence of the treatment, but why would one assume that the Bio-marker grows with time so naively, in the first place? Wouldn't an ARIMA model (or any more robust time-series approach) be more believable for $X_{ij}$?

If anyone could elaborate on the trajectory function (why it is assumed to be this way, what other structures could be used instead, etc), I would extremely appreciate it.


A couple of points:

  • Joint models combine a linear mixed model (not a simple linear model) with a survival (typically a relative risk) model. Namely, in the specification of the model for $Y_i$, the coefficients $\theta_{i0}$ and $\theta_{i1}$ are random effects that are different from subject to subject. These specification translates to accounting for the correlations in the repeated measurements of the biomarker values of each subject.
  • The specification presented in this paper is just one/simple formulation of the linear mixed model part of the joint model. Depending on the features of your data (i.e., how the shapes of the subject-specific biomarker profiles look like, and how strong the correlations are in the repeated measurements of each subject), you could fit more elaborate models that postulate that the biomarker has a nonlinear evolution over time. This can be done by including polynomials or splines in the specification of the fixed- and random-effects parts of the model.
  • You can find more detailed info on joint models in my course notes: http://www.drizopoulos.com/ -> Teaching.
  • 1
    $\begingroup$ Thank you for taking an interest in my question, and for linking your own notes on the subject (I have been going over them quite fervently now). Going through them, a few questions have arisen that, I think express more specific aspects of my general confusion regarding joint models, and which I hope you can answer. They are all based on your notes here: drizopoulos.com/courses/EMC/ESP72.pdf $\endgroup$ – Coolio2654 Apr 29 '19 at 19:57
  • $\begingroup$ 1) On pg. 85, after having stated that $y_i$ stands for longitudinal markers (like white blood cell count, or sex), and $\beta_i$ and $b_i$ are the fixed and random effects, respectively, what then do $x_i$ and$z_i$ stand for? They seem to represent some real data, but $y_i$ is already stated to represent the longitudinal data. I grasp the linear mixed-effects aspect of the model, but not this part. $\endgroup$ – Coolio2654 Apr 29 '19 at 20:05
  • $\begingroup$ 2) On pg. 87, you state that "random effects explain all [the] interdependencies [for joint models]." I am having a severely difficult time conceptualizing this. Could you give more elaboration on what this means, or describe a possible alternate interdependency arrangement within Joint Models, with different strength and weaknesses? $\endgroup$ – Coolio2654 Apr 29 '19 at 20:05
  • $\begingroup$ Regarding (1), $y_i$ is indeed a longitudinally measured biomarker (most often sex cannot be consider such a biomarker because it does not change over time). For this marker you postulate a mixed model, which has a fixed-effects part described by $x_i$ and $\beta$ and random-effects part by $z_i$ and $b_i$. The former describes the average of $y_i$ over time and the latter the correlations in the repeated measurements of the biomarker of each subject over time. Check Chapter 2 in the course notes and/or my course notes for my Repeated Measurements course. $\endgroup$ – Dimitris Rizopoulos Apr 30 '19 at 4:38
  • $\begingroup$ Regarding (2) the random effects are unobserved variables that explain why the observed outcomes, namely the longitudinal outcome and the time-to-event are associated. One way to think of it is that the random effects represent a summary of all the environmental, genetic and other factors that you have not observed in your study that explain why a particular subject has, say high levels in the biomarker that increase over time, which in turn leads to a higher risk (hazard) of having the event. $\endgroup$ – Dimitris Rizopoulos Apr 30 '19 at 4:42

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