# Joint modelling of longitudinal and survival data: question on shared random effect

I have a question on the interpretation of joint models of longitudinal and survival data. According to my literature review,

(1st) There is model of the type $$$$\begin{cases} y_i(t) = m_i(t) + \epsilon_i(t) \\ m_i(t) = \textbf{x}_i(t)^\top \boldsymbol{\beta} + \textbf{z}_i(t)^\top\textbf{b}_i \\ \textbf{u}_i \sim N(0, \textbf{D}) , \epsilon_i(t) \sim N(0, \sigma^2_{\epsilon}) \\ h_i(\mathcal{M}_i(t), \textbf{w}_i) = h_0(t)\exp(\textbf{w}_i^\top \boldsymbol{\gamma} + \alpha\{m_i(t)\}) \end{cases}$$$$ where $$y_i(t)$$ is usually a linear mixed-effects model and $$h_i(\mathcal{M}_i(t), \textbf{w}_i)$$ is a survival submodel.

So the longitudinal process and the survival process are linked by a function of random effect.

(2nd) I have also seen a model whereby : $$$$\begin{cases} y_i(t) = m_i(t) + \epsilon_i(t) \\ m_i(t) = \textbf{x}_i(t)^\top \boldsymbol{\beta} + \textbf{z}_i(t)^\top\textbf{b}_i \\ \textbf{u}_i \sim N(0, \textbf{D}) , \epsilon_i(t) \sim N(0, \sigma^2_{\epsilon}) \\ h_i(t, \textbf{w}_i) = h_0(t)\exp(\textbf{w}_i^\top \boldsymbol{\gamma} + \boldsymbol{\alpha}\textbf{b}_i) \end{cases}$$$$

So what is common to both longitudinal and the survival model is the random effect.

My question is: What are the differences in terms of interpretation in both models? When do we use the 1st model and the 2nd model?

The first model says that the hazard of an event at time point $$t$$ is associated with the underlying level of the longitudinal outcome at $$t$$. By underlying we mean without the measurement error (biological variation in some contexts). The coefficient $$\alpha$$ is the corresponding log hazard ratio for one unit increase of the longitudinal outcome at time $$t$$.
Presumably, in the second model, the term should be $$\boldsymbol{\alpha}^\top \mathbf{b}_i$$, not $$\mathbf{b}_i$$. This model says that the hazard of an event depends on the random effects from the model for the longitudinal outcome. The formulation makes more sense when you have only random intercepts or random intercepts and random slopes in the linear mixed model. In this case, you would have $$\alpha_1 b_{i0} + \alpha_2 b_{i1}$$. The coefficient $$\alpha_1$$ would be the log hazard ratio for a unit increase in the random intercept $$b_{i0}$$, and $$\alpha_2$$ the log hazard ratio for a unit increase in the random slope $$b_{i1}$$ (given each time that all other covariates and the other random effect are fixed). This formulation loses its nice interpretability when you include nonlinear terms in the design matrix for the random effects $$\mathbf{z}_i(t)$$ (because then you cannot interpret the $$\boldsymbol{\alpha}$$'s in isolation).