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.

• 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.
• 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. – Coolio2654 Apr 29 '19 at 20:05
• 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. – Dimitris Rizopoulos Apr 30 '19 at 4:38