Table 1.

List of random effects model considered for the independent and joint analyses and frailty terms

Random effects models fitted
Linear random effects model (Model 1)	${\text{Prose}}_{it}={\alpha }_{0i}+{\alpha }_{1i}{T}_{it}+{\epsilon }_{it}$ (1) ${\alpha }_{0i}={\alpha }_{00}+{\gamma }_{01}B.{\mathit{age}}_i+{\gamma }_{02}{\mathit{Educ}}_i+{\gamma }_{03}{\mathit{Gender}}_i+{\gamma }_{04}\mathit{Inc}.{\mathit{Dem}}_i+{\nu }_{0i}$ (1.1) ${\alpha }_{1i}={\alpha }_{11}+{\gamma }_{11}B.{\mathit{age}}_i+{\gamma }_{12}{\mathit{Educ}}_i+{\gamma }_{13}{\mathit{Gender}}_i+{\gamma }_{14}\mathit{Inc}.{\mathit{Dem}}_i+{\nu }_{1i}$ (1.2)
Quadratic random effects model (Model 2):	${\text{Prose}}_{it}={\alpha }_{0i}+{\alpha }_{1i}{T}_{it}+{\alpha }_{2i}{T}_{it}^2+{\epsilon }_{it}$ (2) ${\alpha }_{0i}={\alpha }_{00}+{\gamma }_{01}B.{\mathit{age}}_i+{\gamma }_{02}{\mathit{Educ}}_i+{\gamma }_{03}{\mathit{Gender}}_i+{\gamma }_{04}\mathit{Inc}.{\mathit{Dem}}_i+{\nu }_{0i}$ (2.1) ${\alpha }_{1i}={\alpha }_{11}+{\gamma }_{11}B.{\mathit{age}}_i+{\gamma }_{12}{\mathit{Educ}}_i+{\gamma }_{13}{\mathit{Gender}}_i+{\gamma }_{14}\mathit{Inc}.{\mathit{Dem}}_i+{\nu }_{0i}$ (2.2) ${\alpha }_{2i}={\alpha }_{22}+{\gamma }_{21}B.{\mathit{age}}_i+{\gamma }_{22}{\mathit{Educ}}_i+{\gamma }_{23}{\mathit{Gender}}_i+{\gamma }_{24}\mathit{Inc}.{\mathit{Dem}}_i+{\nu }_{0i}$ (2.3)
Fixed change point model (Model 3)	${\text{Prose}}_{it}={\alpha }_{0i}+{\alpha }_{1i}{T}_{it}\text{I}(T_{it}\le \tau )+[{\alpha }_{0i}+{\alpha }_{1i}\tau +{\alpha }_{2i}(T_{it}-\tau )]\text{I}(T_{it}>\tau )+{\epsilon }_{it}$ (3) ${\alpha }_{0i}={\alpha }_{00}+{\gamma }_{01}B.{\mathit{age}}_i+{\gamma }_{02}{\mathit{Educ}}_i+{\gamma }_{03}{\mathit{Gender}}_i+{\gamma }_{04}\mathit{Inc}.{\mathit{Dem}}_i+{\nu }_{0i}$ (3.1) ${\alpha }_{1i}={\alpha }_{11}+{\gamma }_{11}B.{\mathit{age}}_i+{\gamma }_{12}{\mathit{Educ}}_i+{\gamma }_{13}{\mathit{Gender}}_i+{\gamma }_{14}\mathit{Inc}.{\mathit{Dem}}_i+{\nu }_{0i}$ (3.2) ${\alpha }_{2i}={\alpha }_{22}+{\gamma }_{21}B.{\mathit{age}}_i+{\gamma }_{22}{\mathit{Educ}}_i+{\gamma }_{23}{\mathit{Gender}}_i+{\gamma }_{24}\mathit{Inc}.{\mathit{Dem}}_i+{\nu }_{0i}$ (3.3)
Random change point model (Model 4)	${\text{Prose}}_{it}={\alpha }_{0i}+{\alpha }_{1i}{T}_{it}\text{I}(T_{it}\le {\tau }_i)+[{\alpha }_{0i}+{\alpha }_{1i}{\tau }_i+{\alpha }_{2i}(T_{it}-{\tau }_i)]\text{I}(T_{it}>{\tau }_i)+{\epsilon }_{it}$ (4) ${\alpha }_{0i}={\alpha }_{00}+{\gamma }_{01}B.{\mathit{age}}_i+{\gamma }_{02}{\mathit{Educ}}_i+{\gamma }_{03}{\mathit{Gender}}_i+{\gamma }_{04}\mathit{Inc}.{\mathit{Dem}}_i+{\nu }_{0i}$ (4.1) ${\alpha }_{1i}={\alpha }_{11}+{\gamma }_{11}B.{\mathit{age}}_i+{\gamma }_{12}{\mathit{Educ}}_i+{\gamma }_{13}{\mathit{Gender}}_i+{\gamma }_{14}\mathit{Inc}.{\mathit{Dem}}_i+{\nu }_{0i}$ (4.2) ${\alpha }_{2i}={\alpha }_{22}+{\gamma }_{21}B.{\mathit{age}}_i+{\gamma }_{22}{\mathit{Educ}}_i+{\gamma }_{23}{\mathit{Gender}}_i+{\gamma }_{24}\mathit{Inc}.{\mathit{Dem}}_i+{\nu }_{0i}$ (4.3)
Frailty terms	$\begin{array}{l}{W}_i(t)=R_0{\nu }_{0i};W_i(\text{t})=R_1{\nu }_{1i};W_i(t)=R_2{\nu }_{2i}\\ W_i(t)=R_0{\nu }_{0i}+R_1{\nu }_{1i}\\ W_i(t)=R_0{\nu }_{0i}+R_2{\nu }_{2i}\\ W_i(t)=R_2{\nu }_{2i}+R3{\nu }_{3i}\\ W_i(t)=R_0{\nu }_{0i}+R_1{\nu }_{1i}+R_2{\nu }_{2i}\end{array}$