Simple elaboration on Joint Models' linear equations/trajectory functions 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: 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.

