Information Accrual Versus Calendar Time

Why the Calendar Is Not the Information Scale

In a time-to-event trial, participants enter at different dates, remain under observation for different lengths of time, and may experience the event at very different times after entry. A month on the calendar therefore does not generally supply a fixed amount of evidence about a treatment contrast. A month with few observed events may add little information; a later month with many events may add much more.

This distinction matters for group sequential monitoring. A monitoring plan may specify analyses after particular event targets, but the calendar date at which a target is reached depends on when events occur. Even then, the event count is not information itself: the amount of information contributed by an event depends on the risk set and on the statistic being used. The introductory material on event targets and calendar dates distinguishes the target from its expected date; here the focus is why evidence accrues unevenly between those dates.

For an event-driven analysis, the event count is often a convenient operational scale: as events accumulate, a survival comparison usually becomes more precise. But the calendar does not tick forward in equal increments of information. Enrollment, time since each participant entered, underlying event risk, censoring, treatment allocation within risk sets, and the analysis statistic can all affect the rate of information accrual.

Where Logrank Information Comes From

Consider a two-arm randomized trial analyzed with the ordinary logrank score. At an event time, let \(Y_1\) and \(Y_0\) be the numbers at risk in the treatment and control arms immediately before the event, and let \(Y=Y_1+Y_0\). Assume for this explanation that there are no tied event times. Under the null hypothesis of equal event hazards, conditional on one event occurring at that time, the probability that it occurs in the treatment arm is \(Y_1/Y\). The event's treatment-arm score contribution is the observed treatment event indicator minus that expected proportion.

The conditional variance of this contribution is the variance of a Bernoulli variable with success probability \(Y_1/Y\). This gives the event's contribution to the logrank information under the null:

Summing these conditional variances over event times gives the usual logrank variance, which is the information for the score test under the null. With approximately balanced risk sets, \(Y_1\) and \(Y_0\) are close, and each event contributes close to \(1/4\). Thus, in a balanced trial, the information is often roughly proportional to the number of events. If risk sets become unbalanced, an event contributes less. For instance, the fraction \(Y_1Y_0/Y^2\) is largest at balance and falls as one arm has fewer participants at risk.

This argument describes the ordinary logrank statistic under its null variance calculation. For a proportional-hazards model, that information also supports approximate precision calculations for the log hazard ratio. Under non-proportional hazards, the logrank variance remains a measure of score variability, but the statistic's treatment-effect interpretation and expected signal require care. A weighted logrank statistic also changes the contribution of events according to its prespecified time weights. In those settings, the raw event count can be a less faithful measure of information relevant to the chosen analysis.

A Short Derivation: Why Balanced Events Give About One Quarter

Write \(q=Y_1/Y\), the treatment fraction of the risk set. For one event, the logrank score contribution has conditional variance \(q(1-q)\). Completing the square shows why balance maximizes it:

Equality holds only when \(q=1/2\). If each of \(D\) events occurs with a risk set that is approximately balanced, summing the contributions gives information near \(D/4\). This is an approximation, not a universal identity. It assumes an ordinary logrank analysis, and it can be inaccurate when treatment proportions in risk sets differ materially from one half, or when weighted event contributions are used.

Event Counts Are a Proxy, Not a Definition

The risk-set calculation makes the limitation of event counting explicit. The same number of events can contribute different total information if the treatment composition of the risk sets differs. In addition, changing weights or a different analysis statistic can change how events contribute. Event targets are useful because events usually drive the precision of survival comparisons, but a planned event fraction does not automatically equal the same information fraction.

The calculation also suggests a useful interpretation: an event occurring when both arms have substantial representation at risk is generally more informative for a between-arm comparison than an event occurring when one arm has very few participants at risk. Event totals conceal that distinction.

Slow and Fast Accrual of Information

Information can accrue slowly during a calendar period with few events and rapidly during a period with many events. That pattern may arise because participants have only recently entered and have had limited time at risk, because event risk varies over time since entry, or because the mix of participants still under observation changes. These are reasons for variation, not a recipe for predicting event counts: projecting those counts requires additional assumptions about enrollment, survival, and follow-up.

Assumptions, Diagnostics, and Alternatives

The simple event-contribution calculation assumes a two-arm comparison, correctly defined risk sets, and distinct event times. The conditional null calculation uses the usual logrank setup: under \(H_0\), the event is equally likely to arise from any individual at risk, so its arm probability is determined by the risk-set proportions. Standard inferential use also depends on appropriate handling of censoring and the trial's randomization and follow-up structure. With tied failures, the variance calculation uses a tie adjustment rather than treating each tied event as an unrelated single event.

Before treating an event fraction as an information fraction, review whether the conditions supporting that approximation are plausible:

• Risk-set composition: Check the numbers at risk by arm at successive event times. Persistent imbalance lowers the ordinary logrank contribution per event relative to the balanced approximation.
• Analysis statistic: For a weighted logrank analysis, event contributions depend on the weights as well as the risk set. Counting every event equally may not reflect the information for that statistic.
• Event definition and data quality: Confirm that events are classified consistently and that censoring and follow-up are recorded correctly. Incorrect event dates or definitions can distort both the count and the risk sets.
• Calendar pattern: Compare event increments across calendar intervals rather than assuming that equal intervals contain equal numbers of events. A sudden change may reflect the changing trial population or follow-up, but should be investigated rather than automatically attributed to treatment.

There are two common operational alternatives. A calendar-time schedule sets analysis dates in advance; it offers predictable review windows but can encounter widely varying event totals at those dates. An event- or information-driven schedule triggers analysis when a target is met; it aligns monitoring more closely with accumulated evidence, but its calendar date is uncertain and must be managed operationally. Neither choice changes the underlying event data, and neither removes the need to specify the sequential monitoring procedure appropriately.

A further alternative is to use estimated analysis-specific information rather than the raw event count when the design and monitoring method support that approach. This can better reflect risk-set balance or weighting, but it requires a valid calculation for the particular statistic and design. An event count should not be relabelled “information” merely for convenience.

Common Mistake and Practical Limits

A common mistake is to assume that information rises linearly with calendar time: if a trial has run for half its planned duration, it must have accumulated half its planned information. That conclusion does not follow. A slow event period and a fast event period can occupy equal calendar time while contributing very different amounts of information, as the examples show.

The opposite mistake is to assume every observed event contributes exactly the same information. For an ordinary logrank statistic, the contribution depends on the risk set; for a weighted statistic, weights matter as well. Event targets are therefore practical design markers, not guarantees of an exact information amount. Actual information and its interpretation depend on the specified analysis, the observed risk sets, and the assumptions underlying inference. The examples use simple approximations to expose this structure, not to replace a trial-specific calculation.