1
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.
$\endgroup$
  • 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.