The Logrank Variance From Risk-Set Allocation

Why the Risk Set Determines the Variance

At each event time, the logrank score compares the treatment group that experienced the event with the treatment group’s expected share of the event, given who was still at risk. Its variance is determined by that same allocation in the current risk set. It is not generally determined by the allocation at randomization: participants may have experienced an event or been censored before the event time being considered.

The earlier material on deriving the logrank score from the Cox model gives the score’s observed-minus-expected structure. Here the focus is its variance contribution, derived directly from the possible treatment allocation of a single event. The derivation assumes two groups, one event at each event time, and a null hypothesis of equal event hazards between groups. It also assumes censoring does not selectively remove participants who would otherwise be at higher or lower event risk, conditional on the model. Tied events require a different event-time calculation and are outside the single-event setup used here.

Deriving the Single-Event Contribution

Under the null hypothesis of equal hazards, the event is equally likely to come from any one of the \(Y_j\) people at risk, conditional on one event occurring at \(t_j\). Therefore the probability that the event is in the treatment group is the treatment fraction in that risk set:

Thus \(X_j\) is an indicator with a Bernoulli distribution, with mean \(p_j\) and variance \(p_j(1-p_j)\). The logrank contribution subtracts that expected treatment allocation from the observed allocation. Subtracting a constant changes the mean but not the variance, so

Substituting \(p_j=Y_{1j}/Y_j\) and \(1-p_j=Y_{0j}/Y_j\) gives the risk-set variance formula:

The event contributes \(1-Y_{1j}/Y_j\) to the score if it is in the treatment group, and \(-Y_{1j}/Y_j\) if it is in the control group. The variance does not depend on which group actually has the event: it depends on the risk-set allocation that existed just before it.

Accumulating Contributions Across Event Times

For \(D\) distinct event times, add the score contributions to obtain the cumulative logrank score. The usual null variance estimate is the sum of the corresponding risk-set contributions. The event-level formula above supplies each term; the risk set is updated as participants experience events or leave observation before later event times.

The variance is accumulated using the risk set at each event, rather than by multiplying the number of events by one fixed variance contribution. The risk-set composition can change during follow-up, so the contribution can change too. Under the usual logrank construction, \(\widehat V\) is the variance used to standardize the cumulative score under the null. The event-by-event derivation here explains its terms; it does not replace the separate justification of dependence across sequential score increments.

Risk-Set Balance and Information

For a fixed total \(Y_j\), the variance contribution is largest when the treatment and control counts are as balanced as possible. To see why, write the treatment fraction as \(p_j=Y_{1j}/Y_j\). Then \(V_j=p_j(1-p_j)\), and

Equality holds when \(p_j=1/2\). The contribution becomes small when one group is scarce in the risk set, and it is zero if either group has nobody at risk. This is why the relevant balance is the balance among participants still at risk at the event time, not merely the original randomization ratio.

Checks, Comparisons, and Limitations

A practical calculation should be checked against the risk-set table, not just the final sum. At each event time, verify that \(Y_j=Y_{1j}+Y_{0j}\), use the counts immediately before the event, and confirm that \(0\leq V_j\leq 0.25\). A contribution outside that range indicates an arithmetic or denominator error for this single-event setup.

The risk-set calculation is preferable to assigning every event the variance implied by the original allocation ratio: it uses the actual numbers still at risk at that event. It is also different from treating the observed group of the event as if that alone determined its variance. The observed group determines \(U_j\); the two group counts in the risk set determine \(V_j\). This variance is the one paired with the logrank score under the null, not a direct estimate of the variance of a hazard-ratio estimate.

The derivation has defined boundaries. It assumes a single event at each event time and uses the equal-hazards null to obtain the event’s treatment probability. It does not handle tied events, adjustment by strata, or a variance calculation under a specified alternative. It also relies on appropriate censoring conditions: if censoring selectively removes people according to their unobserved event risk, the observed risk sets may not support the intended comparison. These conditions should be checked against the trial design and follow-up process before interpreting a logrank result.

A common mistake is to use the original randomization fraction \(r\) at every event and set \(V_j=r(1-r)\). That is correct only if the current risk-set treatment fraction is in fact \(r\). For example, if randomization began at 1:1 but the risk set before an event contains 18 treatment and 42 control participants, the correct contribution is \(18(42)/60^2=0.210\), not \(0.25\). The correction is simple: recalculate the two group counts immediately before each event, then use those counts in both the expected score and variance formulas.