Calculating Target Event Counts From Information

From Information to an Event Count

In an event-driven logrank design, the planned number of events is often used as a practical proxy for how much information will accumulate. But an event does not contribute a fixed amount of information in every trial: its contribution depends on the treatment composition of the risk set at that event time. Translating a target information level into an event count therefore requires an allocation assumption, and the calculation is only as good as that assumption.

This tutorial considers a two-group logrank score and works in the direction from a specified target information level to an approximate event count. It does not determine what information level is needed for a particular effect or power. That is a separate design question.

Definition: At an event time with one event and $Y_1$ participants at risk in the treatment group and $Y_0$ at risk in the control group, the event-level logrank information contribution under the null is $$ v=\frac{Y_1Y_0}{(Y_1+Y_0)^2}=q(1-q), $$ where $q=Y_1/(Y_1+Y_0)$ is the treatment fraction in that risk set. For $D$ single-event times, cumulative information is $I=\sum_{j=1}^{D}q_j(1-q_j)$.

The score and information here use the usual two-group logrank or Cox score-test convention at the null. The contribution is largest when the risk set is balanced: $q=0.5$ gives $v=0.25$. It becomes smaller as the risk-set allocation becomes more unequal, and is zero if a risk set contains participants from only one group.

For the event-count shortcut, assume that the trial has two groups, that the target is expressed in the same information units as the logrank score, and that the relevant risk-set treatment fraction is approximately stable across the events being counted. Also assume event times can be treated as single-event times, or that ties are sufficiently rare for this event-level approximation to be adequate. Under the null, the censoring and follow-up conditions needed for the logrank score and its information calculation must hold; in particular, censoring should not selectively distort comparisons of event experience within treatment groups. If allocation in the risk sets is not stable, use the contributions appropriate to the changing risk sets instead of assuming that the randomized allocation fraction applies throughout.

The Basic Conversion

If each event is expected to contribute approximately the same amount $v$, then $D$ events are expected to provide approximately $Dv$ information. Setting this equal to the target information $I^*$ and solving for $D$ gives the basic conversion.

I^*\approx Dv \qquad\Longrightarrow\qquad D\approx\frac{I^*}{v} =\frac{I^*}{q(1-q)}.

The result is an approximate event count, not an exact guarantee that the realized information will equal the target. Since events are counted in whole numbers, a design may round up when it needs to reach or exceed a target. The information obtained at the actual analysis should be calculated from the observed risk sets, not inferred only from the event count.

Worked Example: Balanced Risk Sets

Suppose a hypothetical two-group trial has a target logrank information level of $I^*=36$. Assume single-event times and that treatment and control participants remain equally represented in the risk sets, so $q=0.5$ throughout. The target is stated in the same logrank information units as the event-level formula.

The approximate contribution per event is $$ v=q(1-q)=0.5(1-0.5)=0.25. $$ Therefore, $$ D\approx\frac{36}{0.25}=144. $$ Under these assumptions, about 144 events are needed to accumulate information 36. This is an information-based approximation; it does not say that 144 participants are sufficient, because participants may not all experience an event during follow-up.

Worked Example: Unequal Randomization With Stable Risk Sets

Consider a hypothetical trial randomizing two participants to treatment for every one assigned to control. Suppose the corresponding treatment fraction in the risk sets is expected to remain close to $q=2/3$, event times are treated as single events, and the target information is $I^*=30$.

The expected information contribution per event is $$ v=\frac{2}{3}\left(1-\frac{2}{3}\right) =\frac{2}{3}\cdot\frac{1}{3} =\frac{2}{9}\approx0.2222. $$ The approximate event count is $$ D\approx\frac{30}{2/9}=135. $$ Thus about 135 events are needed under the stated stable-risk-set assumption. Compared with balanced allocation, each event contributes less information, so more events are needed for the same target. The key assumption is about the risk sets, not merely the original randomization ratio.

When Risk-Set Allocation Changes

The treatment fraction among participants still at risk can depart from the randomized fraction. For example, different event experience or different patterns of loss to follow-up can change which participants remain at risk. In that situation, a single $q$ may not describe all event times. The relevant total is instead the sum of the event-specific contributions.

I(D)=\sum_{j=1}^{D}q_j(1-q_j).

When planning rather than analyzing observed data, the $q_j$ values are not known exactly. A practical approximation is to specify expected risk-set fractions in stages, calculate the average contribution for each stage, and accumulate expected information until the target is reached. If the expected contribution in stage $k$ is $\bar v_k$ for $d_k$ events, that stage contributes approximately $d_k\bar v_k$.

Worked Example: A Change in the Expected Risk Sets

Suppose a hypothetical trial targets information $I^*=25$. Its planning assumptions are that the first 40 events occur with treatment fraction $q=0.5$, followed by events with treatment fraction $q=0.4$. Assume single-event times and that these fractions adequately describe the risk sets within each stage.

The first 40 events contribute approximately $$ 40(0.5)(0.5)=40(0.25)=10 $$ information units. Each later event contributes approximately $$ (0.4)(0.6)=0.24. $$ After a further 60 events, cumulative information is approximately $$ 10+60(0.24)=10+14.4=24.4. $$ The remaining information is $25-24.4=0.6$, requiring approximately $0.6/0.24=2.5$ more events. Because the event count is discrete and the target is to reach at least 25, round this stage’s requirement up to 3 events. The resulting approximate count is $40+60+3=103$ events. Its estimated information is $10+63(0.24)=25.12$, just above the target.

Using $q=0.5$ for all events would instead give $25/0.25=100$ events. That shortcut would understate the count under the stated changing-risk-set assumptions because later events contribute less information.

Justification and Practical Calculation

The conversion follows directly from the additive information definition when event contributions are treated as approximately constant. If $q_j=q$ for every event, then each of the $D$ terms in the cumulative sum equals $q(1-q)$, so

I(D)=\sum_{j=1}^{D}q(1-q)=Dq(1-q).

Solving $I(D)=I^*$ gives $D=I^*/\{q(1-q)\}$. This derivation also identifies the approximation: in a real design, the risk-set fractions may vary, and a target may be reached between integer event counts. If the fractions vary, calculate the cumulative sum using the expected stage-specific or event-specific contributions rather than applying the constant-$q$ formula mechanically.

State the target and the information convention.
Confirm that the target is logrank information on the same scale as the risk-set variance contributions.

Specify risk-set allocation.
Use a stable expected $q$ only when that is a defensible approximation; otherwise specify how $q$ changes over events or stages.

Calculate information per event.
For a stable risk-set fraction, calculate $q(1-q)$. For changing fractions, calculate each stage’s expected contribution.

Invert or accumulate.
Divide the target by the stable contribution, or add stage contributions until their total reaches the target. Round a required event count up when the design must meet or exceed the target.

Diagnostics, Alternatives, and Limitations

Before relying on a count, check whether the assumed allocation resembles the expected composition of the risk sets over follow-up. A planned randomization ratio alone does not establish that it will. Consider whether differential event experience, loss to follow-up, or administrative censoring can change the composition. In a stratified analysis, the total information is assembled from stratum-specific risk-set contributions; a single overall $q$ may conceal important differences between strata.

Tied events also require care. The simple formula $q(1-q)$ describes a single event at a risk set. With multiple events at the same time, the information contribution depends on the tie-handling method and the composition of the risk set and events. If ties are substantial, calculate information using the same method and assumptions as the planned logrank analysis rather than multiplying a single-event contribution by the number of tied events.

A useful alternative to a single closed-form conversion is a risk-set-based planning calculation: specify expected risk-set sizes and event timing, calculate expected information under those assumptions, and find the event count at which it reaches the target. This is more transparent when allocation changes, but it depends on more detailed assumptions. The simple conversion is easier to communicate and can be adequate when risk-set allocation is stable and events are not heavily tied. Neither approach removes uncertainty in the realized event process.

This calculation begins with an information target. It does not establish whether that target is appropriate for a particular treatment effect, significance level, or power requirement. Nor does it convert the event count into an enrollment target: how many participants must be enrolled depends on follow-up and event accrual. Keep these design quantities distinct.

Common Mistake

Common mistake: “The trial randomizes participants 2:1, so every event contributes $2/9$ information.” This assumes the risk-set treatment fraction stays at $2/3$. The correction is to use the fraction actually expected among participants at risk at event times. If that fraction changes, calculate the varying contributions $q_j(1-q_j)$, or justify why a stable average is adequate.

Check Your Understanding

Answer each question using the event-level information approximation. Solutions follow the questions.

With stable balanced risk sets, what event count approximately corresponds to target information 18?
With a stable risk-set treatment fraction of $q=0.25$, what is the information per event? Approximately how many events correspond to target information 15?
Why might a 1:1 randomization ratio fail to justify using $q=0.5$ for every event?
If the first 20 events contribute 5 information units and each subsequent event is expected to contribute 0.20, how many additional events are needed to reach target information 12?

Solutions. (1) Balanced risk sets contribute $0.5(0.5)=0.25$ per event, so $18/0.25=72$ events. (2) The contribution is $0.25(0.75)=0.1875$; $15/0.1875=80$ events. (3) Events, censoring, or follow-up can change the treatment fraction among those still at risk, so the risk-set fraction may differ from the randomized fraction. (4) The remaining information is $12-5=7$; $7/0.20=35$ additional events are needed, for 55 events total.