Why Event Counts Can Track Information
In a time-to-event trial, an event count is often easier to monitor than a direct measure of statistical precision. When each event contributes about the same amount of information, the fraction of planned events observed is also about the fraction of planned information accumulated. But the number of events alone does not determine precision: the amount of information contributed by an event can vary as the risk sets change.
This tutorial makes that approximation precise for a standard two-arm log-rank comparison, or equivalently the treatment-coefficient score calculation in a Cox model. The key question is whether the events observed so far are contributing information at roughly the same average rate as the events still to come.
Model and Assumptions for the Approximation
Consider independent participants assigned to one of two treatment groups, with a binary treatment indicator. Suppose the treatment effect is assessed using the log-rank score or a Cox-model treatment coefficient. For the simple calculation below, assume event times are untied and use the usual risk-set information under the null hypothesis \(H_0\) of no treatment effect. Censoring must be noninformative for this analysis: conditional on the model and observed history, censoring must not selectively remove participants according to their unobserved event times. A proportional hazards model is a common additional assumption when planning power for a nonzero treatment effect, but the information-per-event calculation below concerns the null-model risk sets.
At an event time, let \(r\) be the proportion of participants at risk who are assigned to the treatment group. The corresponding control-group proportion is \(1-r\). Under the stated setup, that event contributes \(r(1-r)\) units of information for the treatment contrast. This contribution is largest when the risk set is evenly divided, and smaller when one group makes up a much larger share of the risk set.
Group events into blocks. For block \(j\), let \(d_j\) be its event count and \(v_j\) its average information per event. Then the block contributes \(d_jv_j\) information units. If the first \(D_1\) events have average contribution \(\bar{v}_1\), and the remaining \(D-D_1\) planned events have average contribution \(\bar{v}_2\), the information fraction is
Thus, the event fraction is exact if the average information per event is the same before and after the interim analysis. It is a reasonable approximation if those averages are close. The result does not require every individual event to contribute identically; it requires the average contribution in the observed portion to resemble the average in the remaining portion.
A Short Derivation of the Approximation Error
Write the event fraction as \(q=D_1/D\). Substituting \(D_1=qD\) into the information-fraction formula and dividing numerator and denominator by \(D\) gives
Subtracting \(q\), the event fraction, from this expression yields
The denominator is positive when events contribute positive information. Therefore, the information fraction exceeds the event fraction if the observed events have contributed more information per event, on average, than the future events are expected to contribute. It is smaller if the observed average is lower. The difference is zero when the two average contributions match. This gives both a justification for the approximation and a way to diagnose its direction of error.
Worked Examples
Worked Example: Balanced Risk Sets
Suppose a hypothetical two-arm trial plans to analyze 100 events. At each event, the risk set is evenly divided between treatment and control, so \(r=0.50\). By an interim analysis, 40 events have occurred. Calculate the event fraction and information fraction.
Each event contributes \(r(1-r)=0.50(0.50)=0.25\) information units. Therefore, the interim information is \(I_1=40(0.25)=10.0\) units, while planned total information is \(I=100(0.25)=25.0\) units.
Here the fractions agree exactly because the information contribution is constant. The result would also hold if risk-set proportions varied from event to event but the average contribution among the first 40 events equaled the average contribution across all 100 events.
Worked Example: Risk-Set Imbalance Reduces Future Information
Consider a hypothetical trial planned for 150 events, divided into three blocks of 50. Assume the treatment proportion in the risk set is \(0.50\) during the first block, \(0.70\) during the second, and \(0.90\) during the third. The interim analysis occurs after the first two blocks. Under the stated risk-set information formula, calculate both fractions.
The first block contributes \(50(0.50)(0.50)=12.50\) information units. The second contributes \(50(0.70)(0.30)=10.50\) units. The third contributes \(50(0.90)(0.10)=4.50\) units. The interim information is \(12.50+10.50=23.00\) units, and the planned total is \(12.50+10.50+4.50=27.50\) units.
The event fraction substantially understates the interim information fraction. The first 100 events came from risk sets with larger average treatment-control balance, so they contributed more information per event than the final block is expected to contribute. In particular, the third block adds about \(4.50\) information units, not \(5.00\).
Worked Example: A Smaller Change in Risk-Set Composition
In another hypothetical trial, suppose 60 events have occurred with risk-set treatment proportion \(0.50\), and 40 more events are planned with proportion \(0.25\). Assume the stated proportions are representative of their respective event periods. Compare the interim event and information fractions.
The first 60 events contribute \(60(0.50)(0.50)=15.00\) information units. Each future event contributes \(0.25(0.75)=0.1875\), so the remaining 40 events contribute \(40(0.1875)=7.50\) units. Total planned information is \(15.00+7.50=22.50\) units.
The event fraction is lower than the information fraction because the observed events came from more balanced risk sets than the future events. The discrepancy is less dramatic than in the previous example, but it is still large enough to matter if the interim information target is meant to be accurate.
Conditions, Diagnostics, and Practical Comparisons
The event-fraction approximation is most defensible when the trial design and observed data support a stable information contribution per event. Relevant conditions include:
- The same treatment contrast and analysis model are used at interim and final analyses.
- Risk-set treatment proportions, and therefore the event-level information contributions, do not change substantially over the event process.
- Censoring, treatment discontinuation, and recruitment patterns do not cause a major shift in which participants remain at risk.
- Any weighting or stratification in the analysis does not create large changes in the information contributed by events across periods.
Useful diagnostics include plotting or tabulating the risk-set treatment proportion over event time, calculating \(r(1-r)\) at event times or over event blocks, and comparing the mean contribution so far with the mean expected for later events. A pronounced change in risk-set balance is a warning that the event fraction may misstate the information fraction. So are changing weights, heavy or differential censoring, or treatment-specific depletion of the risk sets. These checks should be based on the analysis actually planned; event counts alone cannot reveal all changes in information.
When reliable standard errors or model-based information estimates are available, calculating information directly is preferable to assuming that event fraction is an adequate proxy. Earlier material on calculating information fractions from standard errors gives the corresponding direct approach. Enrollment fraction and calendar-time fraction are still more indirect: they track who has entered or how much time has passed, not how many informative events have occurred or how much information each event contributes. Event fraction is often more relevant than either of those measures in an event-driven analysis, but it remains an approximation to information fraction unless event contributions are sufficiently stable.
The approximation has limits even when risk sets look stable. The simple \(r(1-r)\) contribution assumes the stated two-arm, untied-event setup; ties, alternative weighting, stratification, or more complex models can change the relevant information calculation. Further, future risk sets are not observed at an interim analysis, so their contributions must be forecast using the design and available follow-up information. An event fraction can therefore be a useful operational summary without being a substitute for checking the assumptions behind it.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If 30 of 75 planned events have occurred and the average information per event is constant, what are the event fraction and information fraction?
- A trial has event fraction \(0.50\), with \(\bar{v}_1=0.20\) and \(\bar{v}_2=0.10\). Is the information fraction above or below \(0.50\)? Calculate it.
- Why can a high treatment proportion in a risk set reduce information per event in a two-arm comparison?
- Name two features of a trial that could make event fraction a poor proxy for information fraction.
Solutions. (1) The event fraction is \(30/75=0.400\); constant average information per event makes the information fraction \(0.400\) as well. (2) With \(q=0.50\), the information fraction is \([0.50(0.20)]/[0.50(0.20)+0.50(0.10)]=0.10/0.15=0.6667\), above \(0.50\), because observed events contributed more information per event. (3) The contribution is \(r(1-r)\), which is largest at \(r=0.50\) and declines as \(r\) approaches 1; an imbalanced risk set provides less information about the treatment contrast. (4) Examples include changing risk-set treatment balance, differential censoring or depletion, and changing event weights or analysis strata.