Filtrations for Event History Data

Why Information Needs a Time Index

An event-history record is revealed over time. Before a transition occurs, the analyst does not know that it will occur; once it is observed, the event and its type become part of the record. A filtration formalizes this expanding information. It is the mathematical structure that lets us state precisely what is known at each time, and what was already known immediately before that time.

This distinction is essential in continuous-time event-history analysis. A quantity calculated using an event that occurs at time $t$ is available at $t$, but it was not available just before $t$. In contrast, quantities based only on recorded history before $t$ belong to the information just before $t$. Filtrations organize this distinction without making assumptions about the probability model for the events.

Definition: A filtration $(\mathcal F_t)_{t\geq 0}$ is a family of sigma-algebras on a probability space such that $\mathcal F_s\subseteq\mathcal F_t$ whenever $s\leq t$. The sigma-algebra $\mathcal F_t$ represents the information available by time $t$. The event-history filtration is the filtration generated by the recorded event histories and other information specified as available over time.

A sigma-algebra records which events can be distinguished using the available information. For example, if $\mathcal F_t$ contains the observed transition counts through $t$, then knowing $\mathcal F_t$ includes knowing which transitions have been recorded by then. It does not mean that future transitions are known. A stochastic process $V(t)$ is adapted to $(\mathcal F_t)$ if $V(t)$ is measurable with respect to $\mathcal F_t$ at every $t$; in practical terms, its value at time $t$ can be determined from information available by $t$.

Building a Filtration From Recorded Histories

Suppose there are $n$ subjects in a study with a finite set of states. For subjects $i=1,\ldots,n$, let $N_{i,rs}(u)$ denote the recorded number of transitions from state $r$ to state $s$ by time $u$. Let $Y_i(u)$ denote the at-risk indicator defined for the study’s observation window. Include baseline information, such as a recorded baseline covariate $Z_i$, and any observation or censoring information that has been recorded by time $u$. These quantities describe the observed histories; they do not reveal events after a subject’s observation has ended.

One raw event-history filtration is

\mathcal F_t^0 = \sigma\!\left( Z_i,\, N_{i,rs}(u),\, Y_i(u),\, \text{other recorded observation information through }u :\ i=1,\ldots,n,\ r\ne s,\ 0\leq u\leq t \right).

Here $\sigma(\cdot)$ means the sigma-algebra generated by the listed random quantities. The exact list depends on the data collection process. For example, a time-varying measurement belongs in $\mathcal F_t^0$ only once it has been observed by $t$. A future scheduled measurement is not included merely because its schedule is known, although the schedule itself may be included if it is known at baseline. The at-risk indicator can be included when it is determined by recorded entry, event, and observation-window information; it is often redundant with that information but can be convenient notation.

The filtration must be chosen to match the scientific observation scheme. It is not the same as the complete, unobserved history that would be available if every event were observed forever. If a participant is censored, the observed history includes that observation ending, not the participant’s unobserved event time after censoring. The filtration is a description of information, not an assumption that censoring is independent of event times.

Definition: The information available immediately before time $t>0$ is represented by the left-limit sigma-algebra $$ \mathcal F_{t-}=\sigma\!\left(\bigcup_{0\leq s<t}\mathcal F_s\right). $$ The information available at time $t$ is $\mathcal F_t$. Thus $\mathcal F_{t-}\subseteq\mathcal F_t$; information first recorded exactly at $t$ belongs to $\mathcal F_t$, not to $\mathcal F_{t-}$.

For an event count $N_{i,rs}(t)$, its value at $t$ counts recorded transitions by time $t$, including any transition at exactly $t$. Its left limit $N_{i,rs}(t-)$ counts transitions strictly before $t$. This is the event-history version of the distinction between “by $t$” and “before $t$.” When multiple types of information can first arrive at a time point, the same distinction applies to each recorded component.

Worked Examples of Information Before and At a Time

Worked Example: A Two-Transition History

Consider one hypothetical subject who starts in state 0, makes an observed transition $0\to1$ at time 4 days, then makes an observed transition $1\to2$ at time 9 days, and is administratively followed through day 12. Let $N_{01}(t)$ and $N_{12}(t)$ be the corresponding transition counts.

Immediately before day 4, neither transition has occurred. Therefore $N_{01}(4-)=0$ and $N_{12}(4-)=0$, and the state recorded just before day 4 is state 0. At day 4, the new transition is observed, so $N_{01}(4)=1$, $N_{12}(4)=0$, and the recorded state becomes state 1. In terms of information, the occurrence and type of the day-4 transition are in $\mathcal F_4$ but not in $\mathcal F_{4-}$.

Immediately before day 9, the recorded history includes the first transition but not the second: $N_{01}(9-)=1$ and $N_{12}(9-)=0$. At day 9, $N_{12}(9)=1$. The fact that follow-up continues to day 12 is part of the known observation plan or recorded observation history; the day-9 event does not reveal any additional event after day 9. This example assumes that the stated transitions and times are observed exactly and that the administrative end of follow-up is known.

Worked Example: Competing Events and Censoring

Suppose three hypothetical subjects are followed from time 0. Subject A has an observed event of type 1 at day 5; subject B is censored at day 5 with no event recorded; and subject C remains under observation beyond day 5. Assume the event type and censoring time are recorded as they occur.

Just before day 5, the history contains no day-5 event or censoring record for A or B. At day 5, the filtration gains two different records: an event of type 1 for A and the end of observation for B. The two records are not interchangeable. For A, the observed history identifies an event and its type. For B, it identifies that observation ended without an event having been observed by that time. It does not identify B’s event time after censoring.

The filtration for the full cohort contains both records at day 5, along with the earlier histories of all three subjects. This construction does not require censoring to be independent of event risk; such an assumption, if needed for a particular estimator, is a separate modeling condition. Here the only assumption is that the records described are observed and time-stamped as stated.

Worked Example: Delayed Entry and an Observation Window

Consider a hypothetical subject who enters observation at day 3, is observed in state 0 through day 7, and has a recorded transition $0\to1$ at day 7. Let $Y(t)$ indicate whether the subject is under observation and at risk under the study’s definition.

Before day 3, this subject contributes no observed at-risk history; the filtration can still contain the known entry time if it is recorded at baseline or when entry occurs. From day 3 up to but not including day 7, the recorded history indicates observation in state 0 and $Y(t)=1$, under the stated risk-set convention. Immediately before day 7, the transition at day 7 is not yet part of the history. At day 7, the transition record enters $\mathcal F_7$, and the state and at-risk status are updated according to the study’s definitions.

This example assumes that entry and transition times are observed and that the risk indicator is constructed from the actual observation window. If entry were known only retrospectively, the filtration would need to reflect when that information became available rather than treating it as known earlier.

Why the Left Limit Has This Form

The definition of $\mathcal F_{t-}$ is not just shorthand. It makes the “strictly before” convention precise. For an event-history count with right-continuous paths, take any sequence $s_m<t$ increasing to $t$. The count $N_{i,rs}(s_m)$ is known at time $s_m$, so it is measurable with respect to $\mathcal F_{s_m}$, and hence with respect to $\mathcal F_{t-}$. The values along this sequence converge to the left limit $N_{i,rs}(t-)$. Therefore the left-limit count is measurable with respect to the information immediately before $t$.

N_{i,rs}(t-)=\lim_{m\to\infty}N_{i,rs}(s_m), \qquad s_m\uparrow t,\quad s_m<t.

By contrast, $N_{i,rs}(t)$ is included among the generators of $\mathcal F_t$, because the recorded history through $t$ includes any event at exactly $t$. If a transition occurs at $t$, then $N_{i,rs}(t)-N_{i,rs}(t-)=1$ for that transition count, while the left-limit count does not include it. If there is no such transition, the two values agree for that component. This argument justifies the information distinction without imposing a hazard or intensity model.

In applications, data may be recorded only at discrete visits or rounded to a time unit. Then “at $t$” means the information that the protocol assigns to that recorded time point. If several events share a recorded time because of rounding, their order may not be identifiable. A filtration cannot recover ordering information that the recording system did not capture.

Raw Filtrations and the Usual Conditions

The raw filtration $(\mathcal F_t^0)$ is generated directly by the stated observed history. In continuous-time probability, it is common to work instead with a completed, right-continuous filtration. Completion adds subsets of probability-zero events, and right-continuity imposes $\mathcal F_t=\bigcap_{u>t}\mathcal F_u$. These conventions are often called the usual conditions when both are imposed.

The choice should be explicit. The raw filtration is often the clearest description of what the data have recorded by each time. A completed, right-continuous version is useful for many continuous-time probability results, but mathematical augmentation does not mean that a study has actually observed future outcomes. In standard event-history settings without unusual information appearing at deterministic time boundaries, the distinction is often technically harmless for analysis; it should not be silently used to alter the operational meaning of the data.

Specify what is observed.
List baseline information, transition records, observation-window information, and any measurements, together with when each becomes available.

Generate information through time.
At $t$, include only listed quantities recorded by $t$. Do not include an unobserved future event merely because it later appears in the completed data set.

Separate before from at the time.
Use $\mathcal F_{t-}$ for history strictly before $t$ and $\mathcal F_t$ for history through $t$, including records arriving exactly at $t$.

State any mathematical augmentation.
If a result requires completion or right-continuity, say which version of the filtration is being used and preserve the study’s observed-data interpretation.

Diagnostics, Alternatives, and Limitations

A practical diagnostic is to ask, for every quantity used at time $t$: could an analyst determine this from records available by $t$? If not, it does not belong in $\mathcal F_t$ as an observed quantity. For $\mathcal F_{t-}$, ask the stricter question of whether the information was available strictly before $t$. This check catches common timing errors, including using the transition at $t$ to describe the risk set just before $t$, or treating a post-censoring outcome as observed.

The event-history filtration is a framework for organizing information, not a statistical model or estimator. It does not specify transition probabilities, conditional event rates, or whether censoring is independent. Those questions require additional assumptions and methods. Nor does the filtration itself correct inaccurate dates, missing event types, or informative observation schedules.

The main alternative is to work with the complete-data history, which includes latent event times or outcomes that were not observed. That may be useful when defining a probability model, but it is not the filtration available to an analyst using the recorded data. Another alternative is to enlarge the filtration with additional known information, such as a treatment assignment or a baseline variable. This is appropriate only when that information is genuinely available at the stated time. Enlarging the filtration changes what counts as “known” and can change the conditions of later stochastic-process results.

Common pitfall: It is tempting to say that an event at time $t$ is part of the history “up to $t$” and therefore also part of the history immediately before $t$. That confuses $\mathcal F_t$ with $\mathcal F_{t-}$. The correction is to use the left-limit history for strictly earlier information; a record first observed at $t$ belongs to $\mathcal F_t$, not $\mathcal F_{t-}$.

Check Your Understanding

Answer each question using the distinction between observed history through a time and history strictly before it.

A subject has one observed transition at day 6. Which count includes that transition: $N(6-)$ or $N(6)$?
A subject is censored at day 8. Does the event-history filtration reveal whether that subject has an event at day 10?
What must be true before a time-varying measurement can be included in $\mathcal F_5$?
In one sentence, distinguish the raw event-history filtration from a completed, right-continuous version.

Solutions. (1) $N(6)$ includes the transition at day 6; $N(6-)$ counts only transitions strictly before day 6. (2) No. The observed history records the end of observation at day 8, not the unobserved event status at day 10. (3) The measurement must have been observed by day 5 under the stated recording convention; a future measurement cannot be included just because it is eventually collected. (4) The raw filtration is generated directly by the specified recorded histories, whereas the completed, right-continuous version additionally incorporates probability-zero subsets and right-continuity as mathematical conventions.