Type 1 Censoring

Definition

Type 1 Censoring

when $c_1=\cdots =c_n=t_c$ ($c_i$’s are constant)

Suppose that $T_1,T_2,\dots ,T_n$ are i.i.d. random variables represent survival times with CDF $F$, and $C_1,C_2,\dots ,C_n$ are the censoring times. And let ${\delta }_i=\{\begin{array}{ll}1& T_i\le C_i\text{: uncensored}\\ 0& T_i>C_i\text{: censored}\end{array}$ be the censoring indicator. We observe $(Y_i,{\delta }_i),i=1,\dots ,n$ where $Y_i=\min (T_i,C_i)$.

In a type 1 censoring setting, the censored times $C_i$ are fixed constants, not random variables.

The PDF of the observation $Y_i$ is derived as $\begin{array}{rl}P(Y_i=C_i,{\delta }_i=0)& =P(T_i>C_i)=S(y_i)\\ P(Y_i=T_i,{\delta }_i=1)& =f(y_i)\end{array}$

Likelihood of Type 1 Censoring Data

The Likelihood Function of type 1 censoring data is defined as $L(\theta )=\prod _{i=1}^{n}f(y_i{)}^{{\delta }_i}S(y_i{)}^{1-{\delta }_i}=\prod _{i=1}^n\lambda (y_i{)}^{{\delta }_i}S(y_i)=\prod _{i=1}^n\lambda (y_i{)}^{{\delta }_i}\exp [-\Lambda (y_i)]$