Event observation, censoring, truncation — Examples

In Model A, the outcome $T=1$ has probability $1\mathrm{/}2$ and gives $(U,\delta )=(1,1)$ because $C=4$. The outcome $T=3$ has probability $1\mathrm{/}2$ and gives $(U,\delta )=(1\mathrm{/}2,0)$ because $C=1\mathrm{/}2<T$.

In Model B, $T=1$ again gives $(U,\delta )=(1,1)$ with probability $1\mathrm{/}2$, while $T=2$ gives $(U,\delta )=(1\mathrm{/}2,0)$ with probability $1\mathrm{/}2$.

For the truncation example, $P(T>1)={\int }_1^3(1\mathrm{/}3)dt=2\mathrm{/}3$.

Both observed laws assign probability $1\mathrm{/}2$ to each of $(1,1)$ and $(1\mathrm{/}2,0)$, although the censoring time is a deterministic function of $T$ in each model.

Conditional on inclusion, the event-time density is $(1\mathrm{/}3)\mathrm{/}(2\mathrm{/}3)=1\mathrm{/}2$ on $(1,3)$ and zero outside, whereas the unselected density is $1\mathrm{/}3$ on $(0,3)$. Thus left truncation changes the sampled event-time law.

The event-time laws differ: $P_A(T\le 2)=1\mathrm{/}2$ and $P_B(T\le 2)=1$. Equal observed laws with different event-time laws contradict the refuted statement.

The finite censoring construction refutes the claim about equal observed laws, and the uniform calculation shows that truncation changes the sampling distribution by selection. Informative censoring requires its joint observation law to be modeled.

Integrating the density gives $F_{\lambda }(t)=1-e^{-\lambda t}$ and $S_{\lambda }(t)=e^{-\lambda t}$ for $t\ge 0$; in particular $F_{\lambda }(0)=0$ and $S_{\lambda }(0)=1$.

With fixed censoring time $C=2$, an event observed at $U=1$ has event factor $f_{\lambda }(1)=\lambda e^{-\lambda }$ and censor factor $1$. A censored observation at $U=2$ has event factor $S_{\lambda }(2)=e^{-2\lambda }$ and censor factor $1$ relative to the point-mass censoring law.

An interval observation $1<T\le 2$ has probability $F_{\lambda }(2)-F_{\lambda }(1)=(1-e^{-2\lambda })-(1-e^{-\lambda })=e^{-\lambda }-e^{-2\lambda }$.

At fixed inspection time $C=1$, current status $\delta =1$ has event factor $F_{\lambda }(1)=1-e^{-\lambda }$, and $\delta =0$ has event factor $S_{\lambda }(1)=e^{-\lambda }$.

After delayed entry $L=1$, with fixed censoring time $C=3$, an event observed at $U=2$ has conditional event factor $f_{\lambda }(2)\mathrm{/}{S}_{\lambda }(1)=\lambda e^{-2\lambda }\mathrm{/}{e}^{-\lambda }=\lambda e^{-\lambda }$. A censored observation at $U=3$ has conditional event factor $S_{\lambda }(3)\mathrm{/}{S}_{\lambda }(1)=e^{-2\lambda }$.

If instead the sample is right-truncated at $R=2$, the selected-sample density is $\lambda e^{-\lambda t}\mathrm{/}(1-e^{-2\lambda })$ for $0\le t<2$, since $P(T<2)=F_{\lambda }(2)=1-e^{-2\lambda }$. This is a conditional event-time density, not a right-censoring contribution.

The censoring and truncation examples use different event factors: ordinary right censoring uses $f_{\lambda }(u)$ or $S_{\lambda }(u)$, delayed entry divides by $S_{\lambda }(1)$, and right truncation divides by $P(T<2)$.