Right Censoring

Kinds

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 $L(\theta )=\prod _{i=1}^{n}f(y_i{)}^{{\delta }_i}S(y_i{)}^{1-{\delta }_i}=\prod _{i=1}^n\lambda (y_i{)}^{{\delta }_i}S(y_i)=\prod _{i=1}^n\lambda (y_i{)}^{{\delta }_i}\exp [-\Lambda (y_i)]$

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Type 2 Censoring

Definition

Type 2 Censoring

when $r=3$

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 2 censoring setting, we observe first $r$ out of $n$ experiment $T_1,\dots ,T_r$. In other words, for the order statistics $T_{(i)}$ of $T_i$, we only observe $T_{(1)}\le T_{(2)}\le \cdots \le T_{(r)}$. Where $C_1=C_2=\cdots =C_n$ are not constants, but random variables.

Likelihood of Type 2 Censoring Data

The likelihood function of type 2 censored data can be computed using the same equation used for type 1 censored data but computing the joint-PDF of the Order Statistic $T_{(1)}\le \cdots \le T_{(r)}$ is easier.

The joint PDF of $T_{(1)}\le \cdots \le T_{(r)}$ is derived as $L(\theta )=\frac{n!}{(n-r)!}\left[\prod _{i=1}^{r}f(t_{(i)})\right]S(t_{(r)}{)}^{n-r}$

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Random Censoring

Definition

Random Censoring

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, which can be both random variable or constant. 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.

In a random censoring setting, we only observe $(Y_i,{\delta }_i),i=1,\dots ,n$ where $Y_i=\min (T_i,C_i)$, and the censoring times are $C_1,\dots ,C_n$ are i.i.d. Random Variable follows PDF $g_{\gamma }$ and CDF $G_{\gamma }$.

The PDF of the observation $Y_i$ is derived as $\begin{array}{rl}P(Y_i=t,{\delta }_i=0)& =P(C_i=t,T_i>C_i)=g_{\gamma }(t)S_{\theta }(t)\\ P(Y_i=t,{\delta }_i=1)& =P(T_i=t,T_i\le C_i)=f_{\theta }(t)(1-G_{\gamma }(t))\end{array}$ where $\theta$ is the parameter of interest, and $\gamma$ is the: nuisance parameter

Likelihood of Random Censoring Data

The Likelihood Function of random censored data is defined as $L(\theta )=\prod _{i=1}^n[f_{\theta }(t)(1-G_{\gamma }(t)){]}^{{\delta }_i}[S_{\theta }(t)g_{\gamma }(t){]}^{1-{\delta }_i}=c\cdot \prod _{i=1}^n{f}_{\theta }(t{)}^{{\delta }_i}{S}_{\theta }(y_i{)}^{1-{\delta }_i}$ where $c$ is a constant.

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