Bayesian semiparametric joint model of multivariate longitudinal and survival data with dependent censoring

Abstract

We consider a novel class of semiparametric joint models for multivariate longitudinal and survival data with dependent censoring. In these models, unknown-fashion cumulative baseline hazard functions are fitted by a novel class of penalized-splines (P-splines) with linear constraints. The dependence between the failure time of interest and censoring time is accommodated by a normal transformation model, where both nonparametric marginal survival function and censoring function are transformed to standard normal random variables with bivariate normal joint distribution. Based on a hybrid algorithm together with the Metropolis–Hastings algorithm within the Gibbs sampler, we propose a feasible Bayesian method to simultaneously estimate unknown parameters of interest, and to fit baseline survival and censoring functions. Intensive simulation studies are conducted to assess the performance of the proposed method. The use of the proposed method is also illustrated in the analysis of a data set from the International Breast Cancer Study Group.

Appendices

Appendix A: The conditional likelihood of survival submodel with dependent censoring

If neither true survival time nor censoring time can be observed for i-th individual before the follow-up termination time $\tau$, i.e $\varrho _i=0$, based on the monotonic semiparametric normal transformation (2.11), it is easy to obtain the following conditional likelihood for the i-th individual with $\varrho _i=0$

$$\begin{aligned} \begin{aligned} \textrm{Pr}(T_i, \varrho _i=0| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho )&=\textrm{Pr}(T_{i1}>\tau , T_{i2}>\tau | {\varvec{b}}_i, {\varvec{\theta }}_s,\rho )\\&={\widetilde{\varPhi }}_2({\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }})\\&=1-\varPhi (Z_{i1})-\varPhi (Z_{i2})+{\varPhi }_2({\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }}), \end{aligned} \end{aligned}$$

(A.1)

where ${\varvec{Z}}_{i}=(Z_{i1},Z_{i2})^{\!\top \!}\sim N_2({\varvec{0}},{\varvec{\varUpsilon }})$ with ${\varvec{\varUpsilon }}=\left( \begin{array}{cc} 1 &{} \rho \\ \rho &{} 1 \\ \end{array} \right) ,$ and $z_{im}=\varPhi ^{-1}\{1-S_{im}(\tau )\}$ for $m=1$ and 2.

If the survival time is observable, i.e $\varrho _i=1$ and $\delta _i=1$, it is easy to obtain the following joint probability density function for the i-th individual with $(\varrho _i,\delta _i)=(1,1)$

$$\begin{aligned} \begin{aligned} \textrm{Pr}(T_i,\varrho _i=1, \delta _i=1| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho )&=\textrm{Pr}(T_{i}^*=T_i, C_{i}>T_i)\\&=\textrm{f}_{i1}(T_i)\textrm{Pr}(C_{i}>T_i|T_{i}^*=T_i)\\&={\widetilde{\varPhi }}(Z_{i2}|Z_{i1})\textrm{f}_{i1}(T_{i}), \end{aligned} \end{aligned}$$

(A.2)

where ${\widetilde{\varPhi }}(Z_{i2}|Z_{i1})=1-\varPhi (Z_{i2}|Z_{i1})$ with $\varPhi (Z_{i2}|Z_{i1})$ being the conditional cumulative distribution function of $Z_{i2}$ given $Z_{i11}$, and $\textrm{f}_{i1}(T_{i})=\lambda _{i1}(T_{i})S_{i1}(T_{i})$. Similarly, If the censoring time is observable, i.e $\varrho _i=1$ and $\delta _i=0$, it follows that

$$\begin{aligned} \begin{aligned} \textrm{Pr}(T_i, \varrho _i=1, \delta _i=0| {\varvec{b}}_i,\rho , {\varvec{\theta }}_s)={\widetilde{\varPhi }}(Z_{i1}|Z_{i2})\textrm{f}_{i2}(T_i), \end{aligned} \end{aligned}$$

(A.3)

where $\textrm{f}_{i2}(T_{i})=\lambda _{i2}(T_{i})S_{i2}(T_{i})$. By combining Eqs. (A.1)–(A.3), one has Eq. (2.12).

Appendix B: Second order partial derivatives about the conditional likelihood function of survival submodel with dependent censoring

In order to implement MH algorithm from posterior distributions, it is necessary to compute

$$\begin{aligned} \begin{aligned} \frac{\ln \{\textrm{Pr}(T_i,\varrho _i,\delta _i| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho )\}}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}=&\mathop {\sum }\limits _{\varrho _i=0}\frac{\partial ^2 \ln \{{\widetilde{\varPhi }}_2({\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }})\}}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}\\&+\mathop {\sum }\limits _{\varrho _i=1,\delta _i=1}\left[ \frac{\partial ^2 \ln \{{\widetilde{\varPhi }}(Z_{i2}|Z_{i1})\}}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}+ \frac{\partial ^2 \ln \{\textrm{f}_{i1}(T_{i})\}}{\partial \widetilde{{\varvec{\theta }}} \partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}\right] \\&+\mathop {\sum }\limits _{\varrho _i=1,\delta _i=0}\left[ \frac{\partial ^2 \ln \{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})\}}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}+ \frac{\partial ^2 \ln \{\textrm{f}_{i2}(T_{i})\}}{\partial \widetilde{{\varvec{\theta }}} \partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}\right] , \end{aligned} \end{aligned}$$

(A.4)

where $\widetilde{{\varvec{\theta }}}$ stands for ${\varvec{\varphi }}_m$, ${\varvec{\alpha }}_m$ or ${\varvec{\gamma }}_m$ in survival process ($m=1$) or censoring process ($m=2$), or ${\varvec{\beta }}_k$ or ${\varvec{b}}_i$ shared in longitudinal and survival (or censoring) process. Equation (A.4) involves the following second order partial derivatives (A.5–A.7):

$$\begin{aligned} \begin{aligned} \frac{\partial ^2 \ln \{{\widetilde{\varPhi }}_2({\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }})\}}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}=&\mathop {\sum }\limits _{m=1}^2\frac{\partial \ln \{{\widetilde{\varPhi }}_2({\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }})\}}{\partial Z_{im}}\left( \frac{\textrm{d} Z_{im}}{\textrm{d} S_{im}}\frac{\partial ^2 S_{im}}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}+\frac{\textrm{d}^2 Z_{im}}{\textrm{d} S_{im}^2}\frac{\partial S_{im}}{\partial \widetilde{{\varvec{\theta }}}}\frac{\partial S_{im}}{\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}\right) \\&+\quad \mathop {\sum }\limits _{m=1}^2\mathop {\sum }\limits _{j=1}^2\frac{\partial ^2 \ln \{{\widetilde{\varPhi }}_2({\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }})\}}{\partial Z_{im}\partial Z_{ij}^{\!\top \!}}\frac{\textrm{d} Z_{im}}{\textrm{d} S_{im}}\frac{\textrm{d} Z_{ij}}{\textrm{d} S_{ij}}\frac{\partial S_{im}}{\partial \widetilde{{\varvec{\theta }}}}\frac{\partial S_{ij}}{\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}, \end{aligned} \end{aligned}$$

(A.5)

where

$$\begin{aligned} \frac{\partial \ln \left\{ {\widetilde{\varPhi }}_2\left( {\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }}\right) \right\} }{\partial Z_{im}}=-\frac{\phi (Z_{im}){\tilde{\varPhi }}\left( Z_{i[-m]}|Z_{im}\right) }{{\widetilde{\varPhi }}_2({\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }})}, \end{aligned}$$

$Z_{i[-m]}$ denotes the vector obtained by deleting the m-th element from ${\varvec{Z}}_{i}$,

$$\begin{aligned} \begin{aligned} \frac{\partial ^2 \ln \left\{ {\widetilde{\varPhi }}_2\left( {\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }}\right) \right\} }{\partial Z_{im}\partial Z_{ij}}=&\frac{\phi (Z_{im})\left\{ Z_{im}{\tilde{\varPhi }}\left( Z_{i[-m]}|Z_{im}\right) - \rho \phi \left( Z_{i[-m]}|Z_{im}\right) \right\} }{{\widetilde{\varPhi }}_2({\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }})}\\&-\left\{ \frac{\phi (Z_{im}){\tilde{\varPhi }}\left( Z_{i[-m]}|Z_{im}\right) }{{\widetilde{\varPhi }}_2({\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }})}\right\} ^2 \end{aligned} \end{aligned}$$

for $m=j=1~\text {or}~2$, and

$$\begin{aligned} \frac{\partial ^2 \ln \left\{ {\widetilde{\varPhi }}_2\left( {\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }}\right) \right\} }{\partial Z_{im}\partial Z_{ij}}=\frac{\phi _2({\varvec{Z}}_{i}|{\varvec{0}},{\varvec{\varUpsilon }})}{{\widetilde{\varPhi }}_2({\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }})}-\frac{\phi (Z_{im})\phi (Z_{ij}){\tilde{\varPhi }}\left( Z_{ij}|Z_{im}\right) {\tilde{\varPhi }}\left( Z_{im}|Z_{ij}\right) }{\left\{ {\widetilde{\varPhi }}_2({\varvec{Z}}_i|{\varvec{0}},{\varvec{\varUpsilon }})\right\} ^2} \end{aligned}$$

for $m\ne j$, $m,j\in \{1,2\}$, ${\textrm{d} Z_{im}}/{\textrm{d} S_{im}}=-{1}/{\phi (Z_{im})}$ and ${d^2 Z_{im}}/{d S_{im}^2}={Z_{im}}/{[\phi (Z_{im})]^2}$. In addition,

$$\begin{aligned} \begin{aligned} \frac{\partial ^2 \ln \{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})\}}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}=&\mathop {\sum }\limits _{m=1}^2\frac{\partial \ln \{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})\}}{\partial Z_{im}}\left( \frac{\textrm{d} Z_{im}}{\textrm{d} S_{im}}\frac{\partial ^2 S_{im}}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}+\frac{\textrm{d}^2 Z_{im}}{\textrm{d} S_{im}^2}\frac{\partial S_{im}}{\partial \widetilde{{\varvec{\theta }}}}\frac{\partial S_{im}}{\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}\right) \\&+\mathop {\sum }\limits _{m=1}^2\mathop {\sum }\limits _{j=1}^2\frac{\partial ^2 \ln \{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})\}}{\partial Z_{im}\partial Z_{ij}}\frac{\textrm{d} Z_{im}}{\textrm{d} S_{im}}\frac{\textrm{d} Z_{ij}}{\textrm{d} S_{ij}}\frac{\partial S_{im}}{\partial \widetilde{{\varvec{\theta }}}}\frac{\partial S_{ij}}{\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}, \end{aligned} \end{aligned}$$

(A.6)

where

$$\begin{aligned} \frac{\partial \ln ({\widetilde{\varPhi }}(Z_{i1}|Z_{i2}))}{\partial Z_{i1}}= & {} - \frac{{\phi }(Z_{i1}|Z_{i2})}{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})},\\ \frac{\partial \ln ({\widetilde{\varPhi }}(Z_{i1}|Z_{i2}))}{\partial Z_{i2}}= & {} \frac{\rho {\phi }(Z_{i1}|Z_{i2})}{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})},\\ \frac{\partial ^2 \ln \{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})\}}{\partial Z_{i1}^2}= & {} -\left\{ \frac{\phi (Z_{i1}|Z_{i2})}{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})}\right\} ^2+\frac{(Z_{i1}-\rho Z_{i2}){\phi }(Z_{i1}|Z_{i2})}{(1-\rho ^2){\widetilde{\varPhi }}(Z_{i1}|Z_{i2})},\\ \frac{\partial ^2 \ln \{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})\}}{\partial Z_{i2}^2}= & {} \rho ^2\frac{\partial ^2 \ln \{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})\}}{\partial Z_{i1}^2}, \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2 \ln \{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})\}}{\partial Z_{i1}\partial Z_{i2}}= \frac{\partial ^2 \ln \{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})\}}{\partial Z_{i2}\partial Z_{i1}}= -\rho \frac{\partial ^2 \ln \{{\widetilde{\varPhi }}(Z_{i1}|Z_{i2})\}}{\partial Z_{i1}^2}. \end{aligned}$$

One can evaluate $\frac{\partial ^2 \ln \{{\widetilde{\varPhi }}(Z_{i2}|Z_{i1})\}}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}$ in the similar framework given above. Moreover,

$$\begin{aligned} \frac{\partial ^2 \ln \{\textrm{f}_{im}(T_{i})\}}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}= & {} \frac{\partial ^2 \ln (\lambda _{im})}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}+\frac{\partial ^2 \ln (S_{im})}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}\nonumber \\= & {} \frac{\partial ^2 \ln (\lambda _{im})}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}+\frac{1}{S_{im}}\frac{\partial ^2 S_{im}}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}-\frac{1}{S_{im}^2}\frac{\partial S_{im}}{\partial \widetilde{{\varvec{\theta }}}}\frac{\partial S_{im}}{\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}, \end{aligned}$$

(A.7)

where $S_{im}=S_{im}(T_i)$ and $\lambda _{im}=\lambda _{im}(T_i)$, which are specified respectively in (2.8) and (2.9),

$$\begin{aligned} \frac{\partial ^2 \ln (\lambda _{im})}{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}=\left\{ \begin{aligned}&-\frac{{\widetilde{\mathbb {B}}}(T_i) {\widetilde{\mathbb {B}}}^{\!\top \!}(T_i)}{\{{\varvec{\varphi }}_m^{\!\top \!}{\widetilde{\mathbb {B}}}(T_i)\}^2},&\widetilde{{\varvec{\theta }}}={\varvec{\varphi }}_m\\&0,&\widetilde{{\varvec{\theta }}}\ne {\varvec{\varphi }}_m \end{aligned} \right. , \end{aligned}$$

and when $\widetilde{{\varvec{\theta }}}$ respectively takes various parameter vector, ${\partial S_{im}}/{\partial \widetilde{{\varvec{\theta }}}}$ and ${\partial ^2 S_{im}}/{\partial \widetilde{{\varvec{\theta }}}\partial \widetilde{{\varvec{\theta }}}^{\!\top \!}}$ can be computed as follows:

$$\begin{aligned} \begin{aligned} \frac{\partial S_{im}}{\partial {\varvec{\varphi }}_m}=&-S_{im}\int _0^{T_i}\textrm{exp}\big \{{\varvec{\alpha }}_m^{\!\top \!}{\varvec{\eta }}_i(t,{\varvec{b}}_i)+{\varvec{\gamma }}_m^{\!\top \!}{\varvec{\xi }}_{i}\big \}\textrm{d}\mathbb {{B}}(t), \\ \frac{\partial ^2 S_{im}}{\partial {\varvec{\varphi }}_m \partial {\varvec{\varphi }}_m^{\!\top \!}}=&S_{im}\left[ \int _0^{T_i}\textrm{exp}\big \{{\varvec{\alpha }}_m^{\!\top \!}{\varvec{\eta }}_i(t,{\varvec{b}}_i)+{\varvec{\gamma }}_m^{\!\top \!}{\varvec{\xi }}_{i}\big \}\textrm{d}\mathbb {{B}}(t)\right] ^{\otimes 2}, \\ \frac{\partial S_{im}}{\partial {\varvec{\alpha }}_m}=&-S_{im}\int _0^{T_i}\eta _i(t,{\varvec{b}}_i)\textrm{exp}\big \{{\varvec{\alpha }}_m^{\!\top \!}{\varvec{\eta }}_i(t,{\varvec{b}}_i)+{\varvec{\gamma }}_m^{\!\top \!}{\varvec{\xi }}_{i}\big \}{\varvec{\varphi }}_m^{\!\top \!}\textrm{d}\mathbb {{B}}(t),\\ \frac{\partial ^2 S_{im}}{\partial {\varvec{\alpha }}_m \partial {\varvec{\alpha }}_m^{\!\top \!}}=&S_{im}\left[ \int _0^{T_i}{\varvec{\eta }}_i(t,{\varvec{b}}_i)\textrm{exp}\big \{{\varvec{\alpha }}_m^{\!\top \!}{\varvec{\eta }}_i(t,{\varvec{b}}_i)+{\varvec{\gamma }}_m^{\!\top \!}{\varvec{\xi }}_{i}\big \}{\varvec{\varphi }}_m^{\!\top \!}\textrm{d}\mathbb {{B}}(t)\right] ^{\otimes 2}\\&-S_{im}\int _0^{T_i}{\varvec{\eta }}_i(t,{\varvec{b}}_i)^{\otimes 2}\textrm{exp}\big \{{\varvec{\alpha }}_m^{\!\top \!}{\varvec{\eta }}_i(t,{\varvec{b}}_i)+{\varvec{\gamma }}_m^{\!\top \!}{\varvec{\xi }}_{i}\big \}{\varvec{\varphi }}_m^{\!\top \!}\textrm{d}\mathbb {{B}}(t),\\ \frac{\partial S_{im}}{\partial {\varvec{\gamma }}_m}=&-S_{im} \varLambda _{im}(T_i)\xi _i,~~~~ \frac{\partial ^2 S_{im}}{\partial {\varvec{\gamma }}_m \partial {\varvec{\gamma }}_m^{\!\top \!}}=S_{im}\varLambda _{im}^2(T_i)\xi _i^{\otimes 2}-S_{im}\varLambda _{im}(T_i)\xi _i^{\otimes 2}, \\ \frac{\partial S_{im}}{\partial {\varvec{\beta }}_k}=&-\alpha _{mk} S_{im}\int _0^{T_i}{\varvec{R}}_i(t)\textrm{exp}\big \{{\varvec{\alpha }}_m^{\!\top \!}{\varvec{\eta }}_i(t,{\varvec{b}}_i)+{\varvec{\gamma }}_m^{\!\top \!}{\varvec{\xi }}_{i}\big \}{\varvec{\varphi }}_m^{\!\top \!}\textrm{d}\mathbb {{B}}(t),\\ \frac{\partial ^2 S_{im}}{\partial {\varvec{\beta }}_k \partial {\varvec{\beta }}_k^{\!\top \!}}=&\alpha _{mk}^2 S_{im}\left[ \int _0^{T_i}{\varvec{R}}_i(t)\textrm{exp}\big \{{\varvec{\alpha }}_m^{\!\top \!}{\varvec{\eta }}_i(t,{\varvec{b}}_i)+{\varvec{\gamma }}_m^{\!\top \!}{\varvec{\xi }}_{i}\big \}{\varvec{\varphi }}_m^{\!\top \!}\textrm{d}\mathbb {{B}}(t)\right] ^{\otimes 2}\\&-\alpha _{mk}^2 S_{im}\int _0^{T_i}{\varvec{R}}_i(t)^{\otimes 2}\textrm{exp}\big \{{\varvec{\alpha }}_m^{\!\top \!}{\varvec{\eta }}_i(t,{\varvec{b}}_i)+{\varvec{\gamma }}_m^{\!\top \!}{\varvec{\xi }}_{i}\big \}{\varvec{\varphi }}_m^{\!\top \!}\textrm{d}\mathbb {{B}}(t),\\ \frac{\partial S_{im}}{\partial {\varvec{b}}_i}=&-S_{im}\int _0^{T_i}{\varvec{W}}_i^{\!\top \!}(t){\varvec{\alpha }}_{m} \textrm{exp}\big \{{\varvec{\alpha }}_m^{\!\top \!}{\varvec{\eta }}_i(t,{\varvec{b}}_i)+{\varvec{\gamma }}_m^{\!\top \!}{\varvec{\xi }}_{i}\big \}{\varvec{\varphi }}_m^{\!\top \!}\textrm{d}\mathbb {{B}}(t),\\ \frac{\partial ^2 S_{im}}{\partial {\varvec{b}}_i \partial {\varvec{b}}_i^{\!\top \!}}=&S_{im}\left[ \int _0^{T_i}{\varvec{W}}_i^{\!\top \!}(t){\varvec{\alpha }}_{m}\textrm{exp}\big \{{\varvec{\alpha }}_m^{\!\top \!}{\varvec{\eta }}_i(t,{\varvec{b}}_i)+{\varvec{\gamma }}_m^{\!\top \!}{\varvec{\xi }}_{i}\big \}{\varvec{\varphi }}_m^{\!\top \!}\textrm{d}\mathbb {{B}}(t)\right] ^{\otimes 2}\\&-\alpha _{mk}^2 S_{im}\int _0^{T_i}\{{\varvec{W}}_i^{\!\top \!}(t){\varvec{\alpha }}_{m}\}^{\otimes 2}\textrm{exp}\big \{{\varvec{\alpha }}_m^{\!\top \!}{\varvec{\eta }}_i(t,{\varvec{b}}_i)+{\varvec{\gamma }}_m^{\!\top \!}{\varvec{\xi }}_{i}\big \}{\varvec{\varphi }}_m^{\!\top \!}\textrm{d}\mathbb {{B}}(t), \end{aligned} \end{aligned}$$

where $a^{\otimes 2}=aa^T$ for any vector a. Based on Eqs. (A.5)–(A.7), it is easy to calculate

$$\begin{aligned}{} & {} \frac{\partial ^2 \ln \{\textrm{Pr}( T_i,\varrho _i, \delta _i| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho )\}}{\partial {{\varvec{\varphi }}_m}\partial {\varvec{\varphi }}_m^{\!\top \!}},\quad \frac{\partial ^2 \ln \{\textrm{Pr}( T_i, \varrho _i, \delta _i| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho )\}}{\partial {{\varvec{\alpha }}_m}\partial {\varvec{\alpha }}_m^{\!\top \!}},\\{} & {} \frac{\partial ^2 \ln \{\textrm{Pr}( T_i,\varrho _i, \delta _i| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho )\}}{\partial {{\varvec{\gamma }}_m}\partial {\varvec{\gamma }}_m^{\!\top \!}},\quad \frac{\partial ^2 \ln \{\textrm{Pr}( T_i,\varrho _i, \delta _i| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho )\}}{\partial {{\varvec{\beta }}_k}\partial {\varvec{\beta }}_k^{\!\top \!}} \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2 \ln \{\textrm{Pr}(T_i,\varrho _i, \delta _i| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho )\}}{\partial {{\varvec{b}}_i}\partial {\varvec{b}}_i^{\!\top \!}}. \end{aligned}$$

Appendix C: Bayesian inference on JMLS with dependent censoring

To obtain Bayesian estimates of unknown parameters, baseline hazard functions and random effects in our considered JMLS with dependent censoring, the Gibbs sampler is employed to draw a sequence of random observations from the joint posterior distribution $\pi ({\varvec{\theta }}, {\varvec{B}}|{\varvec{D}}, \rho )$ presented in Eq. (2.16). The block Gibbs sampler is conducted by iteratively sampling observations from the following conditional distributions: $\pi ({\varvec{\theta }}_l|{\varvec{\theta }}_s, {\varvec{\theta }}_{\varepsilon }, {\varvec{B}}, {\varvec{D}}, \rho )$, $\pi ({\varvec{\theta }}_s|{\varvec{\theta }}_l, {\varvec{B}}, {\varvec{D}}, \rho )$, $\pi ({\varvec{\varOmega }}|{\varvec{B}})$ and $\pi ({\varvec{B}}|{\varvec{\theta }}_l,{\varvec{\theta }}_s,{\varvec{D}}, \rho )$. The conditional distributions required in implementing the Gibbs sampler are presented as follows.

Block Gibbs Sampler (A): Conditional distribution related to ${\varvec{\theta }}_l$

Let ${\varvec{\theta }}_l=\{{\varvec{\beta }}, {\varvec{\varSigma }}\}$, where ${\varvec{\beta }}=({\varvec{\beta }}_1,\ldots ,{\varvec{\beta }}_K)$ in which ${\varvec{\beta }}_k=(\beta _{k0},\beta _{k1},\ldots ,\beta _{kr})^{\!\top \!}$ for $k=1,\ldots ,K$. From Eq. (2.16), the conditional distribution

$$\begin{aligned} \pi ({\varvec{\beta }}_{k}|{\varvec{\theta }}_s, {\varvec{\beta }}_{[-k]}, {\varvec{\varSigma }},{\varvec{B}},{\varvec{H}}_{\beta _k},{\varvec{D}},\rho ) \end{aligned}$$

is proportional to

$$\begin{aligned} \begin{aligned} \textrm{etr}\left[ -\frac{1}{2}\mathop {\sum }\limits _{i=1}^{n}\mathop {\sum }\limits _{j=1}^{n_{i}}{\varvec{\varSigma }}^{-1}{\left\{ {\varvec{y}}_{ij} -{\varvec{\beta }}^{\!\top \!}{\varvec{R}}_i(t_{ij})-{\varvec{W}}_{i}^{\!\top \!}(t_{ij}){\varvec{b}}_i\right\} ^{\otimes 2}}\right] \prod _{i=1}^{n}\textrm{Pr}(T_i, \varrho _i, \delta _i| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho ), \end{aligned} \end{aligned}$$

where ${\varvec{\beta }}_{[-k]}$ is parameters’ matrix ${\varvec{\beta }}$ with the k-th column deleted. Because the above equation is not a familiar distribution, it is rather difficult to directly sample from the conditional distribution $\pi ({\varvec{\beta }}_{k}|{\varvec{\theta }}_s, {\varvec{\beta }}_{[-k]}, {\varvec{\varSigma }},{\varvec{B}},{\varvec{H}}_{\beta _k},{\varvec{D}})$. Therefore, the well-known MH algorithm is adopted to simulate observations from the above given conditional distribution, which is implemented as follows. Given the current value ${\varvec{\beta }}_k^{(\ell )}$ at the $\ell$-th step, a new candidate ${\varvec{\beta }}_k$ is generated from the proposal distribution $N_{p}({\varvec{\beta }}_k^{(\ell )},\sigma _{\beta _k}^2{\varvec{\varXi }}_{\beta _k})$ with $\sigma _{\beta _k}^2$ set to control the acceptance rate, and then the drawn candidate ${\varvec{\beta }}_k$ is accepted with probability

$$\begin{aligned} \text{ min }\left\{ 1,\frac{\pi ({\varvec{\beta }}_{k}|{\varvec{\theta }}_s, {\varvec{\beta }}_{[-k]}, {\varvec{\varSigma }},{\varvec{B}},{\varvec{H}}_{\beta _k},{\varvec{D}},\rho )}{\pi ({\varvec{\beta }}_{k}^{(\ell )}|{\varvec{\theta }}_s, {\varvec{\beta }}_{[-k]}, {\varvec{\varSigma }},{\varvec{B}},{\varvec{H}}_{\beta _k},{\varvec{D}},\rho )}\right\} , \end{aligned}$$

where

$$\begin{aligned} {\varvec{\varXi }}_{\beta _k}=\left[ \varSigma _{i=1}^{n}\varSigma _{j=1}^{n_i}\sigma ^{kk}{\varvec{R}}_{i}(t_{ij}){\varvec{R}}_{i}(t_{ij})^{\!\top \!}-\frac{\partial ^2 \ln \{\textrm{Pr}(T_i, \varrho _i, \delta _i| {\varvec{b}}_i,\rho , {\varvec{\theta }}_s)\}}{\partial {{\varvec{\beta }}_k}\partial {\varvec{\beta }}_k^{\!\top \!}}|_{\beta _k=\beta _k^{(\iota )}}\right] ^{-1} \end{aligned}$$

with $\sigma ^{kk}$ being the (k, k)-th entry of ${\varvec{\varSigma }}^{-1}$, and ${\partial ^2 \ln \{\textrm{Pr}(T_i, \varrho _i, \delta _i| {\varvec{b}}_i,\rho , {\varvec{\theta }}_s)\}}/{\partial {{\varvec{\beta }}_k}\partial {\varvec{\beta }}_k^{\!\top \!}}$ given in Appendix B.

From the prior distribution of ${\varvec{\varSigma }}$ and ${\varvec{\varepsilon }}_{ij}\sim N_K({\varvec{0}},{\varvec{\varSigma }})$, it is easily shown that

$$\begin{aligned} {\varvec{\varSigma }}|{\varvec{\beta }}, {\varvec{B}}, {\varvec{Y}} \sim \textrm{IW}_K\left[ a_1+N, {\varvec{\varSigma }}^0+ \mathop {\sum }\limits _{i=1}^{n}\mathop {\sum }\limits _{j=1}^{n_i}\left\{ {\varvec{y}}_{ij} -{\varvec{\beta }}^{\!\top \!}{\varvec{R}}_i(t_{ij})-{\varvec{W}}_{i}^{\!\top \!}(t_{ij}){\varvec{b}}_i\right\} ^{\otimes 2}\right] . \end{aligned}$$

where $N=\mathop {\sum }\limits _{i=1}^n n_i$.

Block Gibbs Sampler (B): Conditional distribution related to ${\varvec{\theta }}_s$

Let ${\varvec{\theta }}_s=\{({\varvec{\alpha }}_{m},{\varvec{\gamma }}_{m},{\varvec{\varphi }}_{m}): m=1,2\}$. ${\varvec{\alpha }}_{m},{\varvec{\gamma }}_{m}$ and ${\varvec{\varphi }}_{m}$ can be iteratively sampled from their corresponding conditional distributions, which are given as follows. It follows from Eq. (2.16) that the conditional distributions $\pi ({\varvec{\alpha }}_{m}|{\varvec{\beta }},{\varvec{\gamma }}_{m},{\varvec{\varphi }}_{m},\rho ,{\varvec{B}},{\varvec{D}},\rho )$ and $\pi ({\varvec{\gamma }}_{m}|{\varvec{\beta }},{\varvec{\alpha }}_{m},{\varvec{\varphi }}_{m},\rho , {\varvec{B}},{\varvec{D}},\rho )$ are proportional to $\prod _{i=1}^{n}\textrm{Pr}(T_i,\varrho _i, \delta _i| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho )$. Similarly to drawing ${\varvec{\beta }}_k$, it is easy to sample from the above posterior distributions via MH algorithm, so the details are omitted.

Conditional distribution $\pi ({\varvec{\varphi }}_{m}|{\varvec{\beta }},{\varvec{\alpha }}_{m},{\varvec{\gamma }}_{m},{\varvec{B}},{\varvec{D}},\rho )$ is proportional to

$$\begin{aligned} \textrm{exp}\left( -\frac{1}{2\varsigma _m^2}{\varvec{\varphi }}_m^{\!\top \!}{\varvec{H}}_{\varphi _m}^{-1}{\varvec{\varphi }}_m\right) \textrm{I}({\varvec{A}}_m{\varvec{\varphi }}_m>0)\prod _{i=1}^{n}\textrm{Pr}(T_i,\varrho _i, \delta _i| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho ), \end{aligned}$$

where ${\varvec{H}}_{\varphi _m}$ is a $(L+s) \times (L+s)$ second difference penalized matrix with rank $L+s-2$ (Lang and Brezger 2004). Given the current value ${\varvec{\varphi }}_m^{(\ell )}$, it is usual to sample from the above posterior with MH algorithm, in which the truncated multivariate normal distribution $\textrm{TMN}_{L+s}({\varvec{\varphi }}_m^{(\ell )},\sigma _{\varphi _m}^2{\varvec{\varXi }}_{\varphi _m}) \textrm{I}({\varvec{A}}_m{\varvec{\varphi }}_m>0)$ is adopted as the proposal distribution due to the linear constraints specified by Eq. (2.7), where $\sigma _{\varphi _m}^2$ is set to control the acceptance rate and Fishery information matrix

$$\begin{aligned} {\varvec{\varXi }}_{\varphi _m}=\left[ {\varvec{H}}_{\varphi _m}^{-1}-\varSigma _{i=1}^{n}\frac{\partial ^2 \ln \{\textrm{Pr}(T_i, \varrho _i, \delta _i| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho )\}}{\partial {{\varvec{\varphi }}_m}\partial {\varvec{\varphi }}_m^{\!\top \!}}|_{\varphi _m=\varphi _m^{(\iota )}}\right] ^{-1} \end{aligned}$$

with ${\partial ^2 \ln (\textrm{Pr}(T_i, \varrho _i, \delta _i| {\varvec{b}}_i,\rho , {\varvec{\theta }}_s))}/{\partial {{\varvec{\varphi }}_m}\partial {\varvec{\varphi }}_m^{\!\top \!}}$ given in Appendix B. However, to compute probability of acceptance, it is inevitable to involve high-dimensional integral, which may cause low-efficient sampling and inaccurate result, especially for large $L+s$. Based on Gibbs sampler (Geman and Geman 1984), instead of sampling the whole ${\varvec{\varphi }}_m$ directly, we propose to sample each component of ${\varvec{\varphi }}_m$ one by one, which may achieve good sampling effect at cost of moderate computational time. Given the current j-th component ${\varphi }_{mj}^{(\ell )}$ in the $\ell$-th iterative, a new candidate ${\varphi }_{mj}$ is generated from the truncated normal $\textrm{TN}({\varphi }_{mj}^{(\ell )}, \sigma _{m}^2{\varvec{\varXi }}_{\varphi _m}^{jj})\textrm{I}\{\varphi _{mj}\in (a_{mj}, +\infty )\}$ with ${\varXi }_{\varphi _m}^{jj}$ being the (j, j)-the entry of ${\varvec{\varXi }}_{\varphi _m}$ and $\sigma _{m}^2$ set to control the acceptance rate, $a_{mj}$ can be obtained via solving the inequalities ${\varvec{A}}_m{\varvec{\varphi }}_m>0$ in (2.7) with respect to ${\varphi }_{mj}$, and then the drawn candidate ${\varphi }_{mj}$ is accepted with probability

$$\begin{aligned} \text{ min }\left[ 1,\frac{\pi ({\varvec{\varphi }}_{mj}|{\varvec{\varphi }}_{m[-j]},{\varvec{\beta }},{\varvec{\alpha }}_{m},{\varvec{\gamma }}_{m}, {\varvec{B}},{\varvec{D}},\rho )\{1-\varPhi (a_{mj}|{\varphi }_{mj}^{(\ell )}, \sigma _{m}^2{\varvec{\varXi }}_{\varphi _m}^{jj})\}}{\pi ({\varvec{\varphi }}_{mj}^{(\ell )}|{\varvec{\varphi }}_{m[-j]}^{(\ell )},{\varvec{\beta }},{\varvec{\alpha }}_{m},{\varvec{\gamma }}_{m}, {\varvec{B}},{\varvec{D}},\rho )\{1-\varPhi (a_{mj}|{\varphi }_{mj}, \sigma _{m}^2{\varvec{\varXi }}_{\varphi _m}^{jj})\}}\right] , \end{aligned}$$

where ${\varvec{\varphi }}_{m[-j]}$ denotes ${\varvec{\varphi }}_{m}$ with the j-th component deleted.

The conditional distribution of the smoothing parameter $\varsigma _m^2$ is given by

$$\begin{aligned} \pi (\varsigma _m^{-2}|{\varvec{\varphi }}_m) \sim \textrm{Gamma}\{a_{\varsigma }^m+0.5(L+s-2),b_{\varsigma }^m+0.5{\varvec{\varphi }}_m^{\!\top \!}{\varvec{H}}_{\varphi _m}^{-1}{\varvec{\varphi }}_m\}. \end{aligned}$$

Block Gibbs Sampler (C): Conditional distribution related to random effects set ${\varvec{B}}$ and random effects covariance matrix ${\varvec{\varOmega }}$

Conditional distribution $\pi ({\varvec{b}}_{i}|{\varvec{\theta }}, {\varvec{\varOmega }}, {\varvec{D}}_i)$ is in proportion to

$$\begin{aligned} \textrm{etr}\left[ -\frac{1}{2}{\varvec{\varOmega }}^{-1}{\varvec{b}}_i^{\otimes 2}-\frac{1}{2}\mathop {\sum }\limits _{j=1}^{n_{i}} {\varvec{\varSigma }}^{-1}{\left\{ {\varvec{y}}_{ij} -{\varvec{\beta }}^{\!\top \!}{\varvec{R}}_i(t_{ij})-{\varvec{W}}_{i}^{\!\top \!}(t_{ij}){\varvec{b}}_i\right\} ^{\otimes 2}}\right] \prod _{i=1}^{n}\textrm{Pr}(T_i, \varrho _i, \delta _i| {\varvec{b}}_i, {\varvec{\theta }}_s,\rho ). \end{aligned}$$

Similarly, the MH algorithm is used to sample ${\varvec{b}}_i$ from the above conditional distribution for $i=1,\ldots ,n$. Based on that ${\varvec{b}}_i{\mathop {\sim }\limits ^{\mathrm{i.i.d.}}} N_{q}({\varvec{0}}, {\varvec{\varOmega }})$ and the prior specified in (2.13), it is readily seen that

$$\begin{aligned} {\varvec{\varOmega }}|{\varvec{B}} \sim \textrm{IW}_q\left( a_2+n, {\varvec{\varOmega }}^0+\mathop {\sum }\limits _{i=1}^{n}\ b_i^{\otimes 2}\right) . \end{aligned}$$

Appendix D: Summaries about variables in the IBCSG data (Tang et al. 2017)

• 1. Four untransformed longitudinal QOL indicators

• $y_{1}$: physical well-being on a scale of zero (lousy) to hundred (good);

• $y_{2}$: mood on a scale of zero (miserable) to hundred (happy);

• $y_{3}$: appetite on a scale of zero (none) to hundred (good);

• $y_{4}$: perceived coping (How much effort does it cost you to cope with your illness?) on a scale of zero (a great deal) to hundred (none).

• 2. Observed event time $T_{im}$ in survival submodel

• $T_{i}$: the monitored overall survival time, abbreviated as ‘OS’.

• 3. Covariates in JMLS

• $R_{i1}$: the number of positive nodes of the tumor, abbreviated as ‘#Positive nodes’;

• $R_{i2}$: three versus six initial cycles of oral cyclophosphamide, methotrexate and fluorouracil, abbreviated as ‘#Initital cycle’;

• $R_{i3}$: the reintroduction of three single courses of delayed chemotherapy, abbreviated as ‘Reintroduction’;

• $R_{i4}$: the interaction of the number of initial cycles and reintroduction, abbreviated as ‘#INIR’;

• $R_{i5}$: whether the residency is Switzerland, abbreviated as ‘Residency: Switzerland’;

• $R_{i6}$: whether the residency is Sweden, abbreviated as ‘Residency: Sweden’;

• $R_{i7}$: the age of premenopausal woman, abbreviated as ‘#Age’;

• $R_{i8}$: the estrogen receptor (ER) status (negative/positive), abbreviated as ‘ER’.