Event-Level Information In A Logrank Trial

Why an Event Does Not Always Add the Same Information

In a time-to-event trial, an event count alone does not tell us how much information the logrank score has accumulated. Each failure contributes according to the treatment-group composition of the risk set just before it occurs. An event with many participants at risk in both groups can be more informative about a treatment difference than an event occurring when one group is barely represented.

This tutorial defines information at the level of individual event times. The focus is the conditional variance of the logrank score increment under the null hypothesis, not a calculation of required sample size or a comparison of planned information across analysis looks. The earlier material on the two-sample logrank statistic and risk-set contributions supplies the score framework; here we identify and interpret the variance attached to each event.

Assumptions and the Event-Level Formula

For the basic two-group formula, consider a randomized comparison with a treatment and a control group. Participants are followed until failure or censoring, and the risk set at each event is defined immediately before that event. Under the null hypothesis of equal event hazards in the two groups, and conditional on one event occurring at a time when \(Y_j\) participants are at risk, each of those participants has the same instantaneous chance of being the one who fails. Thus the conditional probability that the event is in the treatment group is \(Y_{1j}/Y_j\).

This calculation assumes the usual conditions for the logrank comparison: event and censoring times are observed appropriately, and censoring is non-informative for the event process within the groups being compared. In practical terms, censoring must not selectively remove participants in a way that carries unaccounted information about their future event risk. Randomization alone does not repair severely informative loss to follow-up. The formula also presumes that the risk sets and event classifications have been constructed correctly.

Conditional on the risk set, \(D_{1j}\) is a Bernoulli variable under the null with success probability \(p_j=Y_{1j}/Y_j\). The score increment subtracts its conditional expectation, so its variance is the Bernoulli variance. Substitution gives the event-level formula:

The observed group in which the event occurs determines the score increment, but not this conditional variance: once the risk set is fixed, the formula for \(I_j\) is the same whichever group experiences the event. For distinct event times \(t_1,\ldots,t_m\), the information through the \(m\)th event is \(\sum_{j=1}^{m} I_j\). This accumulated variance is the logrank information used to scale the cumulative score; the corresponding standardized statistic has the familiar form \(U/\sqrt{I}\), where \(U\) is the cumulative score and \(I\) is its null variance.

Risk-Set Balance, Censoring, and Information

For one untied event, \(I_j=p_j(1-p_j)\), where \(p_j=Y_{1j}/Y_j\). This quantity is largest when the groups are equally represented in the risk set and becomes zero if either group has no one at risk. It is bounded by \(0.25\), since \((p_j-0.5)^2\geq 0\) implies \(p_j(1-p_j)\leq 0.25\). So, under this model, a single untied event cannot contribute more than \(0.25\) units of information.

Censoring is not itself a failure and therefore does not create a logrank score increment or a direct event-level information contribution. But censoring changes who remains at risk. If it changes the balance between groups before a later event, it can change that event’s information. This distinction is important: “no direct contribution” does not mean “no effect on subsequent information.”

Tied Event Times

The single-event formula treats one failure at a time. If \(d_j\) failures occur at the same recorded time, a tie-handling convention is needed. One standard conditional calculation treats the \(d_j\) failing participants as a sample drawn without replacement from the \(Y_j\) people at risk. If all such subsets are equally likely under the null, the number of treatment failures has a hypergeometric distribution. Its variance is the tied-time information:

The final factor is the finite-population correction for sampling without replacement. This is a particular tied-event convention; calculations using other approximations for tied times must use the variance consistent with that convention. A tie should not be handled by automatically applying the single-event formula \(d_j\) times to an unchanged risk set.

Checks, Alternatives, and Limitations

Before interpreting an event-level information calculation, check that each risk set refers to the instant just before the event, that event and censoring records have been distinguished, and that all participants who should be at risk are counted exactly once. For an untied event, useful arithmetic checks are \(I_j\geq 0\), \(I_j\leq 0.25\), and \(I_j=0\) if one group has no one at risk. If a purported single-event contribution exceeds \(0.25\), the risk-set denominator or arithmetic is wrong.

Logrank information is a variance measure, not an effect estimate. A large value does not show that treatment is beneficial, and it does not say how many events occurred without reference to the risk-set balance. Nor does the formula guarantee that the logrank comparison is appropriate: its standard interpretation depends on the event and censoring assumptions and on the chosen score procedure. The formula describes null information from the risk sets; it does not by itself establish that the treatment effects are proportional over time.

Event count is a simple alternative design or monitoring scale, but it assigns the same weight to events that may have very different risk-set compositions. Participant count is also not a substitute for score variance: a participant who has not yet had an event does not generate an event-level score increment, while participants entering or leaving the risk sets affect the composition at later failures. For analysis, the score and its variance must remain paired: the standardized cumulative score uses the logrank variance associated with the same risk sets and tie convention as the score.