Multivariate Survival Model

Kinds

Copula Model

Definition

Let $T_1,T_2,\dots ,T_n$ be i.i.d. survival time with CDF $F(t_1,t_2,\dots ,t_k)$, PDF $f(t_1,t_2,\dots ,t_k)$, and Survival Function $S(t_1,t_2,\dots ,t_k)$, $S_j(t_j)=P(T_j>t_j)$ be marginal survival function, $F_j(t_j)=P(T_j\le t_j)$ be marginal CDF, and $C_1,C_2,\dots ,C_n$ be i.i.d. censoring time.

Copula model define the joint survival function as the copula function whose arguments are marginal survival functions.

When $k=2$, $S(t_1,t_2)=K(S_1(t_1),S_2(t_2))$ where $K$ is the copula function with uniform marginals.

Copula Functions

Clayton’s copula function $S(t_1,t_2)=(S_1(t_1{)}^{-\varphi }+S_2(t_2)-1{)}^{-1/\varphi }$ where $\varphi \in {\mathbb{R}}^{+}$

Crowder’s copula function $S(t_1,t_2)=(1+{\lambda }_1{t}_1^{{\gamma }_1}+{\lambda }_2{t}_2^{{\gamma }_2}{)}^{-\nu }$

Hougaard’s copula function $S(t_1,t_2)=(1+\varphi {\Lambda }_1(t_1)+\varphi {\Lambda }_2(t_2){)}^{-1/\varphi }$

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Competing Risks Model

Definition

Competing risks (multiple modes of failures) model is designed to accommodate the multiple causes to the same event.

Let $(T_i,K_i),i=1,2,\dots ,n$ be the survival time and failure mode (competing risk type) for each individual, where $K_i\in \{1,2,\dots ,k\}$

The mode-specific hazard function is defined as ${\lambda }_j(t)={\lim }_{\Delta t\to 0}\frac{P(T<t+\Delta t,K=j|T>t)}{\Delta t}$ where $j=1,2,\dots ,k$

The marginal hazard function and marginal cumulative hazard function are defined as $\lambda (t)=\sum _{j=1}^k{\lambda }_j(t)$ $\Lambda (t)=\sum _{j=1}^k{\Lambda }_j(t)$

Likelihood

We can observe $(Y_i,{\delta }_i,K_i),i=1,2,\dots ,n$, where $Y_i=\min (T_i,C_i)$, ${\delta }_i=I(T_i\le C_i)$ is censoring indicator, and $K_i$ is failure mode.

The likelihood function is defined as $L={\prod }_{i=1}^n{\prod }_{j=1}^k{f}_j(y_i{)}^{{\delta }_{ij}}S(y_i{)}^{1-{\delta }_i}$ where ${\delta }_{ij}=I(T_i\le C_i,K_i=j)$

Estimators

The Nelson-Aalen Estimator for competing risk model is defined as ${\hat{\Lambda }}_j(t)=\sum _{i:y_i\le t}\frac{{\delta }_{ij}}{n_i}$ where $j=1,2,\dots ,k$

The survival function is estimated by $\hat{S}(t)=\exp \left[-\sum _{j=1}^k{\hat{\Lambda }}_j(t)\right]$

The sub-distribution function is estimated by ${\hat{F}}_j(t)=\sum _{i:y_i\le t}\hat{S}(t_i)\frac{{\delta }_{ij}}{n_i}$ where $j=1,2,\dots ,k$

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Examples

When $k=2$, ${\lambda }_j(t)={\lim }_{\Delta t\to 0}\frac{P(T_j<t+\Delta t|T_1>t,T_2>t)}{\Delta t}$ where $j=1,2$

${\lambda }_{1|2}(t_1|t_2)={\lim }_{\Delta t\to 0}\frac{P(T_1<t_1+\Delta t|T_1>t_1,T_2=t_2)}{\Delta t}$ where $t_1>t_2$

${\lambda }_{2|1}(t_2|t_1)={\lim }_{\Delta t\to 0}\frac{P(T_2<t_2+\Delta t|T_1=t_1,T_2>t_2)}{\Delta t}$ where $t_1<t_2$