Why Design Scale Matters
In a time-to-event trial, the number of participants enrolled is known early and is easy to communicate. Yet enrollment alone does not say how many endpoint events have occurred, how long participants have been followed, or how informative the observed risk sets are for comparing treatment groups. These distinctions matter when planning and monitoring a group sequential trial: a participant target is a recruitment quantity, whereas an event target is often a more direct approximation to the information available for the logrank comparison.
The two quantities are connected, but they are not interchangeable. A trial can enroll many participants and still have few observed events at a given analysis if follow-up has been short or the event is uncommon. Conversely, the same number of events can yield different logrank information if the treatment-group composition of the risk sets differs across event times. The earlier material on event-level information defines the logrank information through risk-set contributions; this tutorial focuses on how that information relates to enrollment and event counts as design scales.
How Events Contribute Information
Consider a two-group logrank analysis with no tied event times. Immediately before event time \(j\), let \(Y_{1j}\) and \(Y_{0j}\) be the numbers at risk in the treatment and control groups, and let \(Y_j=Y_{1j}+Y_{0j}\). Define \(q_j=Y_{1j}/Y_j\). Under the null hypothesis of equal event hazards, the event at that time is attributable to the treatment group with probability \(q_j\), conditional on one event occurring in that risk set. The contribution to the logrank score is the observed treatment-group event indicator minus \(q_j\); its conditional variance is \(q_j(1-q_j)\).
Summing those variances over the observed events gives the logrank information in this no-ties formulation. Because \(q_j(1-q_j)\leq 1/4\), an event contributes most when the risk set is evenly divided between the groups. The event count therefore provides a useful rough scale, but the information per event depends on the risk set at that event.
If risk sets remain approximately balanced, \(q_j\) is near \(1/2\), so each event contributes near \(1/4\) and \(I\) is approximately \(D/4\). This is an approximation, not an identity. For tied event times, the logrank variance uses the appropriate tied-event calculation rather than treating tied failures as a sequence of independent untied events. The examples below use untied events so that the contribution formula applies directly.
Worked Comparisons
Worked Example: Same Enrollment, Different Follow-Up
Suppose two hypothetical, equally allocated trials each enroll \(n=200\) participants. At a particular analysis, the first has observed \(D=48\) endpoint events after relatively short follow-up. The second has observed \(D=96\) events after longer follow-up. Assume that the risk sets are approximately balanced at each event and that event times are untied.
With balanced risk sets, each event contributes approximately \(0.5(1-0.5)=0.25\) information units. Thus, for the first trial,
For the second trial,
Both trials have enrolled 200 participants, but the second has twice as many observed events and, under the stated balance approximation, twice the logrank information. The difference is not caused by enrollment alone; it reflects the additional endpoint events observed by that analysis. These calculations are approximations because the actual risk-set proportions may depart from one-half.
Worked Example: Same Event Count, Different Risk-Set Balance
Consider two hypothetical analyses, each with \(D=100\) untied events. In the first, suppose \(q_j=0.50\) at every event. In the second, suppose \(q_j=0.80\) at every event. These are stipulated risk-set proportions for illustration; they describe different treatment-group compositions among participants at risk when events occur.
For the first analysis, each event contributes \(0.50(1-0.50)=0.25\), so
For the second analysis, each event contributes \(0.80(1-0.80)=0.16\), so
The event counts match, but the second analysis has less information under these risk-set conditions. The reason is that an event provides less information about the group contrast when relatively few participants at risk belong to one of the groups. Thus, \(D\) is a useful design scale, but it is not a complete substitute for calculating \(I\) from the risk sets.
Worked Example: Translating an Event Expectation Into Enrollment
A hypothetical planning calculation targets an expected 90 observed endpoint events by a specified analysis time. Based on assumed accrual, follow-up, and event experience, planners estimate that 45% of enrolled participants will have an observed event by that time. Using this average event fraction as a simple approximation, how many participants would be needed?
Let \(r=0.45\) be the assumed fraction of enrolled participants expected to contribute an observed event by the analysis. The approximate relationship is \(E(D)\approx nr\), where \(E(D)\) is the expected event count under the planning assumptions. Solving for \(n\) gives
The calculation gives an approximate enrollment of 200 participants. It does not guarantee 90 events: the realized event count varies, and the assumed 45% depends on the event process and follow-up pattern. Nor does it guarantee a particular information value, because the risk-set contributions may not all be \(0.25\). The number is a planning translation from an event expectation to enrollment, not an observed result.
Design Scale, Assumptions, and Interpretation
The approximation \(I\approx D/4\) requires risk sets to be close to balanced at event times. It also presumes the usual logrank setup: participants are assigned to the comparison groups, event and censoring times are recorded appropriately, and censoring is non-informative for the event process within the comparison being made. In practical terms, censoring should not selectively remove participants in a way that gives the observed risk sets a misleading representation of the event experience relevant to the test. Risk-set membership and endpoint status must be correctly determined at each analysis cutoff.
The event-to-enrollment relationship also depends on when participants enter and how long they are observed. A fixed enrollment can yield different event counts if follow-up differs, or if the event is more or less frequent than anticipated. Therefore a design that specifies only \(n\) does not uniquely determine the event count at an interim analysis. Similarly, specifying a target \(D\) does not uniquely determine \(n\) without assumptions about accrual, follow-up, and event occurrence.
When the logrank score has variance \(I\), its standardization uses the square root of that information as the score’s standard-error scale. This is why information, rather than enrollment alone, is central to the precision of the logrank comparison. Under approximately balanced risk sets, event count is a convenient proxy for that scale. If balance is poor or changes over time, direct information calculated from risk-set variances is more accurate than a fixed conversion such as one-quarter per event.
| Design quantity | What it counts or represents | What it does not determine by itself |
|---|---|---|
| Enrollment \(n\) | Participants enrolled by the specified time | How many endpoint events have occurred or the information in their risk sets |
| Event count \(D\) | Observed endpoint events included in the analysis | The precise risk-set variance contributed by each event |
| Logrank information \(I\) | Sum of the event-level risk-set variance contributions | The number of participants originally enrolled without additional design information |
Diagnostics, Alternatives, and Limitations
A basic diagnostic is to compare the observed information with the event-count approximation. For untied events, calculate \(I=\sum_j q_j(1-q_j)\) from the actual risk sets and compare it with \(D/4\). A substantial difference indicates that the risk sets were not consistently balanced. Also check that the event count includes only events eligible for the stated endpoint and analysis cutoff, and that participants censored before that cutoff are not mistakenly counted as events. If observed event accrual differs sharply from planning assumptions, the enrollment-to-event translation should be revisited rather than treated as a guaranteed conversion.
The main alternatives answer different planning questions. A participant-based target directly specifies recruitment but needs follow-up and event assumptions to predict information. An event-based target more directly describes how many endpoint observations are sought, and is often preferable when the logrank information is the relevant scale. Direct information-based monitoring is the most precise of these three descriptions when risk-set data are available, because it uses the actual variance contributions rather than relying on an average event contribution.
These scales have limits. Event count is only a proxy for information unless the risk-set contribution pattern is known or reasonably approximated. Enrollment is useful for operational planning but cannot alone specify the precision of a survival comparison. Finally, information is a measure of statistical precision, not a measure of clinical importance, and a given event target does not guarantee a desired result or a particular effect estimate.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- With balanced risk sets at every untied event, what is the approximate information from 64 observed events?
- Two analyses each have 40 events. One has \(q_j=0.50\) at every event; the other has \(q_j=0.75\) at every event. Calculate the information for each.
- Why can a study with a fixed enrollment have different event counts at two analysis times?
- What additional quantity is needed to calculate exact untied-event logrank information from an event count?
Solutions. (1) Each event contributes approximately \(0.25\), so \(I\approx64(0.25)=16\). (2) The first has \(I=40(0.25)=10\); the second has \(I=40(0.75)(0.25)=7.5\). (3) The amount of follow-up and resulting observed event experience can differ across analysis times, even with the same participants enrolled. (4) The risk-set proportion \(q_j\) at each event is needed, because \(I=\sum_j q_j(1-q_j)\).