Event observation, censoring, truncation

An event time is a measurable real-valued map $T$ with values in $[0,\mathrm{\infty })$. A covariate is a measurable map $Z$ into a specified standard-Borel space $(\mathcal{Z},\mathcal{B})$, where standard Borel means measurably isomorphic to a Polish space (Standard Borel spaces). A censoring or inspection time is a measurable real-valued map $C$ with values in $[0,\mathrm{\infty })$. When conditional laws are used, they are probability kernels $K_T(z,\cdot )$ and $K_C(z,\cdot )$ in the sense of Conditional law given a random element and Measure kernel and probability kernel. They are part of the specified model; their existence for real-valued targets and factorization through standard-Borel $Z$ follow under The Axiom of Choice from Existence of regular conditional distributions for standard borel targets and Regular conditional kernels factor through a standard borel conditioning variable.

The notation $T\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp C\mid \sigma (Z)$ has the meaning in Conditional independence given a sigma-algebra: for bounded measurable $a,b$, $\mathbb{E}[a(T)b(C)\mid \sigma (Z)]=\mathbb{E}[a(T)\mid \sigma (Z)]\mathbb{E}[b(C)\mid \sigma (Z)]$ almost surely. It is a model assumption about the conditional observation law, not a consequence of adjusting for $Z$ and not an empirical conclusion.

Under right censoring, the recorded pair is $U=\min (T,C)$ and $\delta =1\{T\le C\}$. Thus $\delta =1$ records an event at $U=T$, including a tie, while $\delta =0$ records censoring at $U=C$ and implies $T>C$. Left censoring at a known threshold $L$ records only $T\le L$. Interval censoring at known endpoints $L<R$ records $L<T\le R$. Current-status observation at an inspection time $C$ records only ${\delta }_C=1\{T\le C\}$.

Left truncation, also called delayed entry at $L$, selects a unit into the sample only when $T>L$; observation begins after that selection. Right truncation at $R$ selects a unit only when $T<R$. These are selection rules before the observed sample is formed, whereas censoring coarsens an event time after sampling. The strict and weak inequalities in these definitions fix the endpoint conventions. A random observation endpoint is called noninformative here only when it is fixed by design or is generated independently of $T$ conditional on the stated covariates under a parameter-distinct law; otherwise its joint observation law must be modeled.

Assume the Axiom of Choice and the Axiom of Countable Choice. Let $Z,T,C$ be random elements, with $Z$ taking values in a standard-Borel space (Standard Borel spaces) and $T,C$ taking values in $[0,\mathrm{\infty })$ with its Borel sigma-algebra. Consider the conditional experiment given the observed covariate $Z=z$. For each $(\theta ,\eta )$ in a product parameter space $\mathrm{\Theta }\times H$, let $K_{\theta }^T(z,\cdot )$ and $K_{\eta }^C(z,\cdot )$ be the specified conditional event-time and censor-time laws. Assume $T\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp C\mid \sigma (Z)$ under each parameter pair. Assume $K_{\theta }^T(z,dt)=f_{\theta }(t\mid z)dt$ with a jointly measurable density in $(z,t)$, and that every $K_{\eta }^C(z,\cdot )$ is dominated by one fixed sigma-finite measure $\nu$ with a jointly measurable density $g_{\eta }(z,c)$. The event law depends only on $\theta$ and the censor law only on $\eta$. All displayed conditional identities hold for $P_{Z;\theta ,\eta }$-almost every $z$ under each fixed parameter pair $(\theta ,\eta )$, where $P_{Z;\theta ,\eta }$ is the marginal law of $Z$ under that pair as in Law or distribution of a random element. All density equalities are in their almost-everywhere versions for that pair; no common exceptional set over an uncountable parameter set is asserted.

For right-censored data $(U,\delta )$, define the reference measure $\rho$ on $[0,\mathrm{\infty })\times \{0,1\}$ by $\rho (B\times \{1\})=\mathrm{Leb}(B)$ and $\rho (B\times \{0\})=\nu (B)$. Put $S_{\theta }(u\mid z)=K_{\theta }^T(z,(u,\mathrm{\infty }))$ and $G_{\eta }(u\mid z)=K_{\eta }^C(z,[u,\mathrm{\infty }))$. In the scientific model contract of Scientific statistical model contracts, one analysis unit is one subject; the full-data random quantities are $T,C$, and the observation map records $(Z,U,\delta )=(Z,\min (T,C),1\{T\le C\})$, so the record contains the minimum and status rather than both full times. This theorem conditions on observed $Z$ and asserts no target-population estimand or empirical adequacy. The conditional density of $(U,\delta )$ with respect to $\rho$ is

$h_{\theta ,\eta }(u,1\mid z)=f_{\theta }(u\mid z)G_{\eta }(u\mid z),h_{\theta ,\eta }(u,0\mid z)=S_{\theta }(u\mid z)g_{\eta }(z,u)\mathrm{.}$

Consequently, up to the censor-only factor, the event-model likelihood contribution is $f_{\theta }(U\mid Z{)}^{\delta }{S}_{\theta }(U\mid Z{)}^{1-\delta }$, where the two cases mean $f_{\theta }(U\mid Z)$ when $\delta =1$ and $S_{\theta }(U\mid Z)$ when $\delta =0$. This is a conditional likelihood given $Z$; a parameter-dependent marginal law of $Z$ would supply a separate factor.

For an arbitrary conditional event-time law ${\mu }_{\theta ,z}$ on $[0,\mathrm{\infty })$ with distribution function $F_{\theta }(t\mid z)={\mu }_{\theta ,z}((-\mathrm{\infty },t])$, for $0\le L\le R<\mathrm{\infty }$,

$P(L<T\le R\mid Z=z)=F_{\theta }(R\mid z)-F_{\theta }(L\mid z),P(T\le L\mid Z=z)=F_{\theta }(L\mid z),$

and $P(T=t\mid Z=z)=F_{\theta }(t\mid z)-F_{\theta }(t^{-}\mid z)$. For fixed inspection endpoints, or endpoints conditionally independent of $T$ given $Z$ under a law with no event parameter $\theta$, these are the event factors for interval and left-censored observations. In current-status observation at a fixed inspection time $c$, the event contribution is $F_{\theta }(c\mid z{)}^{\delta }{S}_{\theta }(c\mid z{)}^{1-\delta }$, with $S_{\theta }(c\mid z)=P(T>c\mid Z=z)$ and $\delta =1\{T\le c\}$. An informative inspection process requires its own joint model.

For delayed entry at $L$, let $W$ denote all pre-entry covariates; take $Z=W$ when $L$ is fixed by design and $Z=(W,L)$ when random $L$ is observed. In the random-entry case, assume that before selection the conditional joint law of $(T,L)$ given $W=w$ factors as $K_{\theta }^T(w,dt)Q_{\eta }(w,d\mathrm{\ell })$, where the entry kernel $Q_{\eta }$ does not depend on the event parameter $\theta$. Thus $L\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp T\mid W$ under this parameter-distinct entry model, and the event law given $Z=(W,L)$ remains $K_{\theta }^T(W,\cdot )$. For each parameter pair, require $S_{\theta }(L\mid W)>0$ for almost every $(W,L)$ under its pre-selection law, select only units with $T>L$, and assume post-entry right censoring satisfies $T\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp C\mid \sigma (Z)$ with $C\ge L$. Conditional on inclusion $T>L$, the event density and survivor terms are $f_{\theta }(u\mid W)\mathrm{/}{S}_{\theta }(L\mid W)$ for an event and $S_{\theta }(u\mid W)\mathrm{/}{S}_{\theta }(L\mid W)$ for censoring, for $u>L$. Thus the right-censored event contribution is divided by $S_{\theta }(L\mid W)$. An event branch has $u>L$, while a censoring branch may have $u=L$ when the censoring law has an atom at entry; the survivor ratio $S_{\theta }(u\mid W)\mathrm{/}{S}_{\theta }(L\mid W)$ applies for every $u\ge L$.

For right truncation at fixed $R$, or at an observed threshold conditionally independent of $T$ given $Z$ under a law with no event parameter $\theta$, inclusion is conditional on $T<R$ with $P(T<R\mid Z=z)>0$. Its conditional event-time law is ${\mu }_{\theta ,z}(A\cap (-\mathrm{\infty },R))\mathrm{/}P(T<R\mid Z=z)$; in the continuous-density case the selected-sample density is $f_{\theta }(t\mid z)\mathrm{/}{F}_{\theta }(R^{-}\mid z)$ for $t<R$. This selection likelihood is distinct from the right-censoring likelihood. Independence of censoring, inspection, or entry is a model assumption, not a theorem about an empirical study.