Administrative Censoring At Interim Cutoffs

Why an Interim Cutoff Changes the Logrank Data

A group sequential logrank analysis uses the event and censoring information available at a specified interim analysis. When that analysis is defined by a fixed calendar date, participants enrolled at different times have different opportunities for follow-up. A participant who has not had an event by the cutoff is not known to remain event-free indefinitely: follow-up is simply administratively stopped at the date of the data cutoff.

The cutoff determines which outcomes are observable and how long each participant can contribute to risk sets. An event known to have occurred by the cutoff is counted. A participant who has not had an observed event is censored at the end of observed follow-up, whether that endpoint is the administrative cutoff or an earlier loss to follow-up. Censoring does not count as an event, but it removes the participant from later risk sets.

The time origin must match the trial's analysis plan. In the common setup where time is measured since randomization, \(H_i\) is the participant-specific follow-up duration from randomization to the calendar cutoff. If the analysis instead uses calendar time, risk sets must also account for when participants entered observation. Whatever the time scale, a risk set must contain only people who are under observation and event-free immediately before the event time being analyzed.

From Cutoff Records to Score and Information

For event time \(t_j\), let \(n_{1j}\) and \(n_{0j}\) be the numbers at risk in the treatment and control groups immediately before that time. Write \(n_j=n_{1j}+n_{0j}\). Let \(d_j\) be the number of events at that time, and \(D_{1j}\) the number of those events in the treatment group. The logrank score uses the observed treatment-group events minus their null expected number. Let \(\mathcal J(A)\) index the event-time contributions formed from events observed by calendar cutoff \(A\); \(t_j\) remains measured on the chosen analysis time scale, and each risk set is formed on that scale.

Here the sum is over event times observable by the cutoff. With tied events, the usual risk-set allocation variance contribution is

This is the standard tied-event variance under the null allocation of the \(d_j\) events among the \(n_j\) people at risk. For a single event at a time with \(n_j>1\), \(d_j=1\), it reduces to \(V_j=n_{1j}n_{0j}/n_j^2\); if \(n_j=d_j=1\), then \(V_j=0\). The event-based information at the cutoff is \(I(A)=V(A)\). A censoring time contributes no event term to \(U(A)\) and no separate event-time variance term. Its effect is indirect: after censoring, that participant is absent from subsequent risk sets and can no longer contribute to their score or information.

The familiar logrank standardization, when \(V(A)>0\), is \(Z(A)=U(A)/\sqrt{V(A)}\). It is the accumulated score and variance from the records available at that cutoff that enter this calculation, not the number enrolled by that date alone.

Why the Variance Formula Applies at the Cutoff

At a given event time, under the null hypothesis and the standard logrank risk-set model, the \(d_j\) events are allocated among the \(n_j\) people at risk with treatment-group count \(D_{1j}\). Conditional on the risk set and the number of tied events, this count has a hypergeometric distribution. Its mean gives the expected treatment events, and its variance gives the tied-event contribution.

Consequently, the event-time score is \(D_{1j}-E(D_{1j})\), and its variance is the expression above. The cutoff does not change these event-time formulas. It determines which event times are included and which observed records determine each risk set.

This calculation relies on appropriate event-time measurement, correct risk-set construction, and censoring that is non-informative for the event process under the analysis model, conditional on the factors being accounted for. A fixed administrative date does not automatically guarantee non-informative censoring. For example, if enrollment date is associated with prognosis because of calendar-time changes in care, or enrollment timing selects different kinds of participants, the resulting follow-up pattern may be related to event risk. The trial's analysis assumptions and any needed stratification or adjustment must address such features.

Implementation Checks and Common Mistakes

Before calculating a cutoff-specific score, check the date and time conventions, the analysis time scale, and the status of each participant. Confirm that no event after the cutoff has been labeled as observed, that a participant has not been censored before an event they experienced, and that each risk set includes exactly those under observation and event-free immediately before its event time. With staggered enrollment, verify each participant's available follow-up from the actual enrollment date; do not assign everyone the same maximum duration unless the dates justify it.

A common mistake is to treat every participant without an event at the interim cutoff as if they had a permanent event-free outcome. The incorrect reasoning is: “No event is recorded, so this participant counts as a non-event through the end of the trial.” The correction is that the participant is observed event-free only through their censoring time. They are omitted from later risk sets, and their status may be updated at a later cutoff when additional follow-up becomes available.

Another error is to count administrative censoring as an event or to add a separate score contribution for each censored participant. Censoring has no observed-versus-expected event term. It changes later risk-set sizes, which can change the score and variance contributed by later events. A related diagnostic is information equal to zero when there are no event times with both treatment and control participants at risk: in that situation, the logrank standardization \(U/\sqrt{V}\) is not defined.

Administrative censoring should also be distinguished from other analysis choices. Excluding participants who have short follow-up, rather than retaining their observed time and censoring them, discards usable information and can select a non-representative subset. Conversely, treating an interim censor as a permanent dropout at all later looks discards follow-up that may subsequently become available. An event-driven analysis uses an event target rather than a fixed calendar date to schedule a look; it still requires accurate event and censoring records. Neither scheduling choice repairs an incorrect risk set or informative censoring.