Weighted Log-Rank

Weighted Log-Rank

A weighted log-rank test compares time-to-event experience between groups by summing observed-minus-expected event differences at successive event times, with specified weights. The weights determine whether earlier or later risk-set differences contribute more to the test. This can improve sensitivity to an anticipated pattern such as delayed treatment benefit, but the test by itself neither estimates the size of that benefit nor identifies the date at which it begins.

What is compared at each event time?

At event time $t_j$, let $n_{1j}$ and $n_{0j}$ be the numbers at risk immediately before that time in groups 1 and 0. Let $d_{1j}$ and $d_{0j}$ be their event counts, and define $n_j=n_{1j}+n_{0j}$ and $d_j=d_{1j}+d_{0j}$. Under equal event experience at that risk set, the expected group-1 event count is $e_{1j}=n_{1j}d_j/n_j$.

The weighted score is

$$U_w=\sum_j w_j(d_{1j}-e_{1j})=\sum_jw_j\left(d_{1j}-\frac{n_{1j}d_j}{n_j}\right).$$

The sign depends on which arm is labeled group 1: if the event is undesirable, a negative score means fewer observed events in group 1 than expected at the times receiving weight. A positive score does not by itself measure a hazard ratio. When all $w_j=1$, this is the ordinary log-rank score. A common large-sample comparison uses $Z=U_w/\sqrt{V_w}$ and a specified one- or two-sided reference distribution; the variance must match the same weights and risk-set construction.

With a tied event count $d_j$ and a standard hypergeometric risk-set calculation, the variance contribution is

$$V_w=\sum_j w_j^2\frac{n_{1j}n_{0j}d_j(n_j-d_j)}{n_j^2(n_j-1)},$$

for risk sets with $n_j>1$ and relevant events. With one event at a time, the unweighted contribution reduces to $n_{1j}n_{0j}/n_j^2$. Software may use different tie handling or variance conventions; the implementation should be stated. These formulas describe an unstratified two-group illustration, with censoring and entry handled correctly in each risk set.

How the weights choose the time emphasis

One widely used family is Fleming–Harrington, with nonnegative parameters $\rho$ and $\gamma$ and pooled Kaplan–Meier survival $\widehat S(t_j^-)$ just before event time $t_j$:

$$w_j=\widehat S(t_j^-)^{\rho}\{1-\widehat S(t_j^-)\}^{\gamma},\qquad \rho,\gamma\geq0.$$

Using the pooled estimate and its pre-event value avoids defining the current event's weight from the post-event drop at that same time. The relative weights, rather than their absolute scale, drive the standardized score: multiplying every weight by the same positive constant scales both $U_w$ and $\sqrt{V_w}$ equally.

“Later” describes relative emphasis among observed event times, not a separate test restricted to a prespecified late window. A late-weighted score still uses the risk-set history and does not estimate survival only after a chosen date. Sparse late risk sets can make a late-emphasis analysis unstable despite their larger nominal weights.

Worked example with three event times

Consider a teaching dataset with five people initially at risk in each arm, one event at each of three distinct times, and no censoring or delayed entry before these events. Group 1 has an event at times 1 and 3, while group 0 has an event at time 2. Use Fleming–Harrington $(\rho,\gamma)=(0,1)$, so the weight is one minus pooled survival immediately before the event.

The pooled survival declines from 1 to 0.9 after time 1 and to 0.8 after time 2. The weighted score is $U_w=0-0.04444+0.10000\approx0.05556$. For these distinct single events, the variance is $V_w=0^2(0.25)+0.1^2(20/81)+0.2^2(0.25)\approx0.012469$, giving a numerical standardized score of about $0.50$. This tiny example explains the arithmetic; a large-sample normal $p$-value would not be reliable from only three events. The time-1 event has zero direct weight under this choice, but it affects the pooled survival used to weight later events.

When this test helps and when it misleads

The ordinary log-rank test is often effective when hazards are approximately proportional. A prespecified weighted test can gain power when the chosen weights align with a plausible nonproportional pattern, such as a delay before an immunotherapy effect appears. It can also lose power when reality differs from the planned pattern. Hazard curves that cross may create opposing contributions and cancellation, so simply selecting a late weight does not solve every crossing-hazard problem.

The validity of the comparison depends on correctly defined event times, risk sets and independent or otherwise appropriately handled censoring within the analysis design. A censored person contributes while at risk and then leaves the risk set; censoring is not an event. Delayed entry requires the person to enter the risk set at the proper time. Competing events need a clear endpoint and estimand: a cause-specific event analysis and a cumulative-incidence comparison answer different questions.

Choose the endpoint, direction, weight family and $\rho,\gamma$ before examining unblinded survival differences for a confirmatory claim. Choosing the most favorable weight after seeing the curves invalidates an unadjusted nominal test level. If several weighted tests are combined or multiple endpoints are analyzed, use a prespecified procedure that accounts for their dependence and multiplicity. A statistically significant weighted test should be interpreted alongside the complete survival curves, numbers at risk, absolute survival differences and a clinically meaningful effect summary.

Connecting a test result to health economic evaluation

Health economic models need absolute survival over time, and often extrapolation beyond trial follow-up, to estimate life-years, QALYs and costs. A weighted log-rank $p$-value supplies neither a survival curve nor a model for long-term hazards. A time-varying hazard ratio, flexible survival model, milestone survival difference or restricted mean survival time may help describe or estimate particular effects, with assumptions appropriate to the decision. The analysis should reconcile its estimand, follow-up window and population with what the economic model actually uses.

For example, a therapy with delayed benefit might generate more QALYs late in follow-up while causing early toxicity and costs. A late-weighted test may help assess evidence of a survival difference under its chosen weighting, but the costs, toxicity, health-state utilities and survival extrapolation must be evaluated separately. The test does not establish cost-effectiveness or prove that late benefits will persist indefinitely.

Reporting and verification checks

Report the endpoint and time origin, arms, risk-set and tie handling, censoring and entry rules, prespecified weight formula and parameters, pooled-survival convention, test direction, multiplicity approach and the event counts over follow-up. Provide survival curves and numbers at risk, explain how the test relates to the effect measure being reported, and distinguish an exploratory analysis from a confirmatory one.

• Check each risk set. At each time, only eligible people still under observation and event-free for the endpoint should be counted.
• Check timing of weights. Fleming–Harrington weights here use pooled survival immediately before, not after, the current event.
• Check tied events. Use a tie-aware variance and test implementation consistent with the displayed event counts.
• Check direction and scale. State which arm is group 1 and whether an event is adverse; rescaling all weights cannot change a properly standardized $Z$ statistic's magnitude.
• Check interpretation. A weighted log-rank test compares event experience under its weighting and does not quantify QALYs, a constant hazard ratio or the time of treatment-effect onset.

Sources and further reading

A methodological simulation comparison of survival tests under nonproportional hazards evaluates weighted log-rank tests alongside alternative estimands. The original research on improved log-rank-type tests using adaptive weights discusses weighting and time-varying alternatives. A group-sequential survival methods paper explains the Fleming–Harrington late-weighting interpretation and its limits. The risk-set example above is an original teaching calculation, not a clinical trial result.