Censoring Indicators and At-Risk Histories

Why Follow-Up Status Needs Its Own Notation

In longitudinal data, a measurement can be absent because the participant is no longer observed, not because the underlying outcome ceased to exist. These are different facts: the outcome belongs to the full-data process, while follow-up status says which parts of that process are available to the analyst. A precise record of follow-up status is therefore needed before describing observed histories or analyzing incomplete longitudinal records.

We use a discrete schedule of visits $t=0,1,\ldots,K$. Visit 0 is baseline. At each visit, a vector $Z_t$ collects the variables scheduled to be recorded, such as covariates, treatment, and outcome. This tutorial concerns loss of observation that, once it occurs, prevents later scheduled measurements from being observed. It does not assume that the missingness mechanism is ignorable, and it does not define weights for correcting it.

Definition: Let $C_t$ be the indicator that a participant remains under observation at visit $t$: $C_t=1$ if the scheduled record through visit $t$ is observed, and $C_t=0$ otherwise. Assume baseline observation, so $C_0=1$. For $t=0,\ldots,K-1$, let $D_t=1$ indicate loss of observation after visit $t$ and before visit $t+1$, among participants observed at $t$. Set $D_t=0$ for a participant already lost before $t$. The at-risk indicator for the next observation transition is $R_t=C_t$; thus, only those with $R_t=1$ can have $D_t=1$.

This convention makes $D_t$ an interval indicator and $C_t$ a visit-level status. In particular, $D_t$ is not the indicator that a participant has ever been censored by visit $t$. Loss after visit $t$ leaves the visit-$t$ record available, but prevents the visit-$t+1$ record from being observed.

Encoding Monotone Loss of Observation

The defining feature of this censoring process is that observation status is absorbing: after a participant is lost, the participant does not reappear at a later scheduled visit. The transition rule is

C_{t+1}=C_t(1-D_t), \qquad t=0,\ldots,K-1.

If $C_t=0$, the convention $D_t=0$ gives $C_{t+1}=0$. If $C_t=1$, then $D_t=0$ means observation continues and $D_t=1$ means the next visit is not observed. Iterating the transition gives the product representation.

Result: Under $C_0=1$ and the transition rule above, $$ C_t=\prod_{s=0}^{t-1}(1-D_s), \qquad t=1,\ldots,K. $$ Consequently, $C_{t+1}\leq C_t$ at every transition. The observed visits form a prefix of the scheduled visits: if visit $t$ is unobserved, then every later visit is unobserved.

Proof. For $t=1$, the transition rule gives $C_1=C_0(1-D_0)=1-D_0$, which is the stated product. Suppose the product formula holds at $t$. Then

C_{t+1}=C_t(1-D_t) =\left\{\prod_{s=0}^{t-1}(1-D_s)\right\}(1-D_t) =\prod_{s=0}^{t}(1-D_s).

Induction proves the formula for all $t=1,\ldots,K$. Since $C_t$ and $1-D_t$ are each either 0 or 1, $C_{t+1}=C_t(1-D_t)\leq C_t$. Once $C_t=0$, the same recurrence gives $C_{t+1}=0$, and applying it repeatedly gives zero at all later visits. $\square$

For a participant with at least one unobserved visit, define the first unobserved visit as $T_C=\min\{t\in\{1,\ldots,K\}:C_t=0\}$. If every scheduled visit is observed, set $T_C=K+1$. Then $C_t=1$ exactly when $t<T_C$. This gives a second way to describe the record: $T_C=t$ means loss occurred between visits $t-1$ and $t$, while $T_C=K+1$ means no loss occurred during scheduled follow-up.

Observed Histories and At-Risk Histories

Write the full-data history through visit $t$ as $\bar Z_t=(Z_0,\ldots,Z_t)$. It is the history that would be defined if every scheduled measurement were available. The observed history is the portion actually recorded. Under the monotone process defined above, a complete history through visit $t$ is available precisely for participants with $C_t=1$.

Definition: The at-risk history at visit $t$ is $\bar Z_t$ restricted to participants with $R_t=C_t=1$. For these participants, $Z_0,\ldots,Z_t$ have been observed under the scheduled-measurement convention. If $C_t=0$, the observed history ends at visit $T_C-1$; later components of the full-data history are not recorded.

The phrase “at risk” here refers only to being under observation for the next censoring transition. It does not, by itself, mean that a participant is free of an outcome event or belongs to a survival-outcome risk set. That distinction matters whenever the data also include an event process.

The observed history at $t$ is nested within the history at $t+1$: someone observed through $t+1$ must also have been observed through $t$. The reverse need not hold. Thus, the set of participants contributing an observed $Z_t$ can only stay the same or shrink across visits. This describes the data structure; it does not say why participants leave observation or what statistical correction is appropriate.

Worked Example: Locating the Censoring Interval

A participant has visit-status indicators $C_0=1,C_1=1,C_2=1,C_3=0,C_4=0$. Find the interval indicator sequence and identify the observed history.

From $C_{t+1}=C_t(1-D_t)$, when $C_t=1$ we have $D_t=1-C_{t+1}$. Thus $D_0=1-1=0$, $D_1=1-1=0$, and $D_2=1-0=1$. Since the participant is already lost at visit 3, the convention gives $D_3=0$. The sequence is therefore $(D_0,D_1,D_2,D_3)=(0,0,1,0)$.

The first unobserved visit is $T_C=3$, so loss occurred after visit 2 and before visit 3. The available history is $\bar Z_2=(Z_0,Z_1,Z_2)$; $Z_3$ and $Z_4$ are not observed. The value $D_3=0$ does not mean the participant returned to follow-up: it records the convention that no new loss transition is counted after the participant has already left observation.

Worked Example: Counting At-Risk Participants by Visit

Four participants have status patterns through visits 0 to 3 as follows: participant A has $(1,1,1,1)$; B has $(1,1,1,0)$; C has $(1,1,0,0)$; and D has $(1,0,0,0)$. Count how many are at risk at each visit.

At visit 0, all four have $C_0=1$, so $\sum_i R_{i0}=4$. At visit 1, A, B, and C remain observed, giving $\sum_i R_{i1}=3$. At visit 2, A and B remain observed, giving $\sum_i R_{i2}=2$. At visit 3, only A remains observed, giving $\sum_i R_{i3}=1$.

The counts are $4,3,2,1$, which are nonincreasing, as required. They count participants whose visit-specific records are observed; they are not counts of outcome events or treatment groups. If the question concerns a particular variable measured only on some participants who otherwise remain in follow-up, these counts alone would not describe that variable’s missingness.

Worked Example: Detecting a Nonmonotone Pattern

A data file shows $C_0=1,C_1=1,C_2=0,C_3=1$. Can this pattern be generated by the absorbing loss process?

No. At the transition from visit 2 to visit 3, the rule requires $C_3=C_2(1-D_2)$. Since $C_2=0$, this gives $C_3=0$ for either allowed value of $D_2$, contradicting the recorded $C_3=1$. The pattern therefore cannot represent monotone loss of observation as defined here.

The discrepancy could reflect a coding error, a return to observation after a missed visit, or a variable-specific missing measurement rather than loss of all follow-up. Those possibilities have different data structures. They should be investigated rather than forced into an absorbing censoring indicator.

What the Encoding Guarantees—and What It Does Not

The product formula provides a simple audit of a longitudinal record. For each participant, check that $C_0=1$, that all $C_t$ are binary, and that $C_{t+1}\leq C_t$. If these conditions hold, define $D_t=C_t-C_{t+1}$ for each transition. When $C_t=1$, this difference is either 0 or 1 and equals the interval loss indicator. When $C_t=0$, monotonicity forces $C_{t+1}=0$, so the difference is 0, agreeing with the convention for an already-lost participant.

This check establishes internal consistency of the status indicators, not correctness of the underlying data or validity of an analysis. It cannot establish whether loss depends on measured or unmeasured variables, whether a later outcome can be recovered, or whether participants remaining under follow-up represent those no longer observed. Those are substantive and statistical questions beyond the bookkeeping identities proved here.

A common mistake is to code $D_t=1$ at every visit after the first missed visit. That changes the meaning of $D_t$: it turns an interval loss event into a cumulative “already lost” indicator. The correction is to let $D_t$ mark the transition into loss once, and use $C_t$ to record continued observation. For example, the status pattern $(1,1,0,0)$ corresponds to $D=(0,1,0)$, not $D=(0,1,1)$.

Another mistake is to treat a single missed measurement as automatic loss of all subsequent observation. That is appropriate only under the monotone follow-up structure assumed here. If later visits are observed after an earlier missed visit, the recorded pattern is not monotone and needs a representation that permits intermittent missingness. Finally, the retention indicator is not an outcome-event indicator: an observed event can occur while $C_t=1$, and a participant can leave observation without having experienced the event under study.

Compared with recording only the last observed visit, the sequence $C_0,\ldots,C_K$ makes the visit-level status explicit and supports direct checks of the absorbing structure. Compared with separate missingness flags for each variable and visit, it represents a stronger, whole-follow-up loss process: one $C_t$ applies to the scheduled record, not just one measurement. The simpler representation is useful when its assumptions fit, but it cannot describe intermittent or variable-specific missingness.

Set the time convention.
Specify which scheduled visit is baseline and whether a loss between visits $t$ and $t+1$ leaves the visit-$t$ record available.

Record visit status.
Use $C_t=1$ for observed through visit $t$, and define $R_t=C_t$ for eligibility to contribute the next transition.

Check the absorbing pattern.
Verify $C_0=1$ and $C_{t+1}\leq C_t$. A later return to $C=1$ violates this encoding.

Recover interval loss.
For participants still observed at $t$, $D_t=1-C_{t+1}$; after loss, use the stated $D_t=0$ convention.

Check Your Understanding

Answer each question using the visit-level convention in this tutorial. Short solutions follow.

For $C=(1,1,0,0)$ across visits 0 to 3, what are the interval indicators $D_0,D_1,D_2$, and when did loss occur?
Why does $C_t=0$ imply $C_{t+1}=0$ under the stated transition rule?
What is the at-risk history at visit 2 for a participant with $C=(1,1,1,0)$ across visits 0 to 3?
Can the pattern $(1,0,1)$ be represented by an absorbing loss-of-observation indicator? Explain.

Solutions. (1) $D_0=0$, $D_1=1$, and $D_2=0$: loss occurs after visit 1 and before visit 2. (2) The transition equation gives $C_{t+1}=0(1-D_t)=0$. (3) Since $C_2=1$, the participant is at risk at visit 2 and the observed history is $(Z_0,Z_1,Z_2)$. (4) No. It returns to 1 after being 0, contradicting $C_{t+1}\leq C_t$; the data require a different missingness representation or correction.

Censoring Indicators and At-Risk Histories

What You'll Learn