Survival Models with Surviving Fractions

Definition

Consider an event such as death from a specific cause on the incidence of a particular disease.
Define a binary variable $z$ where $z=1$ indicates that an individual will experience the event eventually, and $z=0$ indicates that the individual will never experience the event.
Let $T$ denote the time of occurrence of the event with PDF $f_{\theta }$ and CDF $F_{\theta }$, $T$ is defined only when $z=1$, and $T$ and $z$ are independent. We observe $(Y_i,{\delta }_i),i=1,\dots ,n$ where $Y_i=\min (T_i,C_i)$ and ${\delta }_i=\{\begin{array}{ll}1& T_i\le C_i\text{: uncensored}\\ 0& T_i{C}_i\text{: censored}\end{array}$.

Suppose that the probability of $z$ is $p$, $p=P(z=1)$.

No Covariate Case

$\begin{array}{rl}P({\delta }_i=1,T_i=Y_i,z_{i=1})& =p{f}_{\theta }(y_i)\\ P({\delta }_i=0,T_i{C}_i,z_{i=1})& =(1-p)+p{S}_{\theta }(y_i)\end{array}$

The likelihood is defined as $L(\theta )=\prod _{i=1}^n[p{f}_{\theta }(y_i){]}^{{\delta }_i}[(1-p)+p{S}_{\theta }(y_i){]}^{1-{\delta }_i}$

With Covariates Case

The likelihood is defined as $L(\theta )=\prod _{i=1}^n[p({\mathbf{x}}_i)f_{\theta }(y_i|{\mathbf{x}}_i){]}^{{\delta }_i}[(1-p({\mathbf{x}}_i))+p({\mathbf{x}}_i)S_{\theta }(y_i|{\mathbf{x}}_i){]}^{1-{\delta }_i}$