Why the Information Fraction Is Not Just an Event Fraction
In an event-driven survival trial, interim analyses are often scheduled around accumulating outcome data. The number of events is easy to count, but it does not by itself tell us how much logrank information those events contribute. Events occurring in risk sets with different treatment allocation can contribute different amounts of information. The information fraction therefore measures progress on the scale used by the logrank statistic, rather than simply progress in the event count.
This distinction matters when describing an interim look relative to the planned maximum analysis. If the look has accumulated 40% of the planned maximum information, its information fraction is 0.40. That value is not necessarily the same as the fraction of planned events that have occurred. The two fractions coincide only when the information-to-event relationship makes them coincide.
What Counts as Logrank Information?
For the two-sample logrank test, information is the variance of the logrank score under the null hypothesis, using the risk sets and analysis conventions for the trial. Write \(U_k\) for the cumulative score at look \(k\). Its null variance is \(I_k\), so the standardized logrank statistic is \(Z_k=U_k/\sqrt{I_k}\). The information fraction uses this variance quantity, not the score itself and not the squared observed \(Z\)-statistic.
With distinct event times and two treatment groups, the information contribution from an event at time \(j\), under the usual logrank variance calculation, is \(v_j=Y_{1j}Y_{0j}/(Y_{1j}+Y_{0j})^2\), where \(Y_{1j}\) and \(Y_{0j}\) are the numbers at risk in the two groups just before that event. In a stratified analysis, contributions are calculated within strata and combined according to the specified stratified logrank analysis. For tied event times or other variance conventions, use the corresponding logrank variance contribution; do not substitute the distinct-event formula without checking that it applies.
Information accumulates over analysis periods. If \(I_{k-1}\) is the information at the preceding look and \(\Delta I_k\) is the information added since then, then \(I_k=I_{k-1}+\Delta I_k\). Thus, a cumulative information fraction includes all information since the beginning of follow-up, not only information collected since the last analysis. The planned maximum information must refer to the same score and variance definition, population comparison, stratification, and analysis conventions as the interim numerator.
Worked Example: Calculating an Interim Information Fraction
Suppose a hypothetical trial has planned maximum logrank information \(I_{\max}=120\) and has accumulated \(I_k=42.6\) at an interim look. The trial has recorded 90 of its planned 240 events. Calculate both fractions.
The interim information fraction is $$ t_k=\frac{I_k}{I_{\max}}=\frac{42.6}{120}=0.355. $$ Thus, the look has accumulated 35.5% of the planned maximum information.
The event-count fraction is $$ \frac{D_k}{D_{\max}}=\frac{90}{240}=0.375. $$ So 37.5% of planned events have occurred. The fractions differ: the event count alone would overstate the information progress by 2.0 percentage points in this hypothetical calculation. Each fraction describes a different scale; neither result should be silently substituted for the other.
Why the Ratio Uses Information
The logrank score is standardized by its standard deviation under the null hypothesis. Since the variance of the cumulative score is the information, \(I_k\) determines the scale of the statistic at look \(k\). A ratio of information values therefore describes how much of the planned score-variance scale has accumulated. The derivation is a direct consequence of the definition of the standardized statistic.
At look \(k\), the variance of the cumulative logrank score under \(H_0\) is \(I_k\). At the planned maximum analysis, it is \(I_{\max}\).
The score standard deviations are \(\sqrt{I_k}\) and \(\sqrt{I_{\max}}\), respectively. Their squared ratio is \((\sqrt{I_k}/\sqrt{I_{\max}})^2=I_k/I_{\max}\).
By definition, \(t_k=I_k/I_{\max}\). It is a ratio of accumulated null variances, not a ratio of event counts or of observed score values.
This justification depends on using comparable information quantities at the numerator and denominator. In particular, both must correspond to the same test and compatible variance calculation. It does not establish that information is proportional to the number of events: that additional claim requires the information contribution per event to be sufficiently similar across the relevant parts of follow-up.
Worked Example: Building Cumulative Information From Increments
In a hypothetical trial, the information at the first look is \(I_1=18.4\). The next two analysis periods add \(\Delta I_2=20.1\) and \(\Delta I_3=19.5\). The planned maximum information is \(I_{\max}=145\). Find the cumulative information and fraction at look 3, and the information added during the third period as a fraction of the planned maximum.
Accumulated information is the sum of the first-look information and both subsequent increments: $$ I_3=I_1+\Delta I_2+\Delta I_3 =18.4+20.1+19.5=58.0. $$
The cumulative information fraction is $$ t_3=\frac{58.0}{145}=0.400. $$ The trial has reached 40.0% of planned maximum information at look 3.
The third period alone contributed $$ \frac{\Delta I_3}{I_{\max}}=\frac{19.5}{145}=0.1345. $$ This is the increase in the cumulative information fraction from look 2 to look 3, not the cumulative fraction at look 3. Indeed, \(I_2=18.4+20.1=38.5\), so \(I_2/I_{\max}=38.5/145=0.2655\), and \(0.2655+0.1345=0.4000\), up to rounding.
When Event Counts and Information Fractions Diverge
The per-event information depends on risk-set composition. In the distinct-event, two-group formula, a risk set split approximately evenly between groups contributes more information for that event than a risk set dominated by one group. As follow-up proceeds, the numbers at risk may change at different rates because of event timing, censoring, or the allocation pattern represented in the risk sets. Consequently, the average information contributed by an event before an interim look can differ from the average contribution over the full planned trial.
An event fraction can serve as a rough proxy when the design and observed data make information accrue nearly linearly with events. That is an approximation to assess, not an identity to assume. The relevant check is whether accumulated information per event is similar across the interim and planned maximum analyses. The event-count fraction on its own cannot establish this.
Worked Example: Same Event Fraction, Different Information Fraction
Consider a hypothetical trial planned for \(D_{\max}=200\) events, with maximum information \(I_{\max}=100\). At an interim look, 80 events have occurred and the logrank variance calculation gives \(I_k=35\). Calculate and compare the two fractions.
The event-count fraction is $$ \frac{D_k}{D_{\max}}=\frac{80}{200}=0.400. $$ The information fraction is $$ t_k=\frac{I_k}{I_{\max}}=\frac{35}{100}=0.350. $$ Therefore, 40.0% of the planned events have occurred, but only 35.0% of planned maximum information has accumulated.
In this hypothetical example, the average information per event by the interim look is \(35/80=0.4375\) information units per event. Across the full plan it is \(100/200=0.5000\) information units per event. The different averages explain why the fractions do not match. The discrepancy is not an arithmetic error; it indicates that event counts and logrank information are progressing at different rates.
Assumptions, Checks, and Limitations
A reported information fraction is meaningful only if the quantities in its ratio are well defined and comparable. Before calculating it, verify the following:
- Same analysis definition: The interim information and planned maximum information use the same logrank score, variance convention, and stratification approach.
- Correct accumulation: The interim numerator includes the information from all relevant event contributions up to that look. If information is supplied as increments, add them rather than using only the most recent increment.
- Correct maximum: The denominator is the planned maximum information for the analysis, not automatically the planned event count, participant count, or an information value from a different design calculation.
- Compatible data cutoff: The interim numerator is based on the data available at the specified cutoff and includes only contributions included in that analysis.
If all increments are nonnegative and \(I_{\max}\) is a positive fixed target, accumulated information should not decrease as looks progress, so the corresponding information fractions should also be nondecreasing. A fraction greater than 1 can occur if observed information exceeds the planned target; it should be reported as an overrun rather than truncated to 1. A negative increment, a falling cumulative information value, or an unexpectedly large fraction calls for checking data cutoffs, risk-set calculations, stratum combination, and the definition of \(I_{\max}\). Rounding can cause small discrepancies in displayed sums, but it should not explain a material inconsistency.
Information fractions summarize progress relative to a chosen maximum information target. They do not by themselves say how many events remain, how many participants must be enrolled, or what the treatment effect is. Nor do they repair a mismatch between the planned analysis and the interim variance calculation. Those questions require their own design or analysis quantities.
Comparing the Two Progress Measures
| Measure | Calculation | What it describes | Main caution |
|---|---|---|---|
| Information fraction | \(I_k/I_{\max}\) | Accumulated logrank variance relative to planned maximum information | Numerator and denominator must use compatible information definitions |
| Event-count fraction | \(D_k/D_{\max}\) | Events observed relative to planned maximum events | Does not account for differences in information contributed per event |
The information fraction is the appropriate quantity when progress is specifically being described on the logrank information scale. The event-count fraction is useful for describing event accrual. They answer related but distinct questions, and agreement between them should be checked rather than presumed.
Check Your Understanding
Answer each question by identifying the quantity being calculated and showing the arithmetic where numbers are supplied.
- A hypothetical interim analysis has \(I_k=54\) and \(I_{\max}=135\). What is the information fraction, and what does it mean?
- A trial has observed 72 of 180 planned events. Can you conclude that its information fraction is 0.40? Explain.
- At a look, cumulative information is 31.2, and the next period adds 16.8. If \(I_{\max}=120\), calculate the new cumulative fraction and the increase in fraction due to that period.
- Why should the interim and maximum information use the same variance convention and analysis definition?
- A calculated information fraction is 1.04. Should it be changed to 1.00? State an appropriate interpretation and one check to make.
Solutions. (1) \(54/135=0.400\), so 40.0% of planned maximum logrank information has accumulated. (2) No. The event fraction is \(72/180=0.400\), but the information fraction requires \(I_k/I_{\max}\); information per event need not be constant. (3) New cumulative information is \(31.2+16.8=48.0\), giving \(48.0/120=0.400\). The period adds \(16.8/120=0.140\) to the fraction. (4) Otherwise the numerator and denominator may represent different variance scales, so their ratio would not describe progress relative to the stated maximum. (5) Do not truncate it automatically: 1.04 indicates observed information is 4% above the planned maximum. Check that the interim numerator and planned denominator are based on compatible definitions and that the cutoff and accumulated contributions are correct.