Concept Architecture
Concept
Theoretically, the Weighted Log-Rank Test is a family of non-parametric statistical tests used to compare survival distributions while assigning different weights to event times. Unlike the standard log-rank test, which gives equal weight to all observed events, weighted log-rank tests emphasise specific portions of the survival curve, such as early or late events, depending on the chosen weighting function. This flexibility improves statistical power when treatment effects are expected to vary over time.
Mathematically, the weighted log-rank test modifies the standard log-rank statistic by multiplying the observed-minus-expected event difference at each event time by a predefined weight. The weighted differences are summed across all event times and standardised by their weighted variance to produce a test statistic that follows an approximate chi-square distribution under the null hypothesis of equal survival functions. Different weighting schemes produce well-known variants, including the Breslow?Gehan, Tarone-Ware and Fleming?Harrington tests.
In practice, weighted log-rank tests are selected according to the anticipated timing of treatment effects. Early-emphasis tests are useful when interventions produce rapid benefits, whereas late-emphasis tests are appropriate when survival curves separate after prolonged follow-up. These tests are widely applied in oncology trials and other studies that support health technology assessment and health economic modelling.
Purpose
Used to compare survival distributions while assigning greater importance to selected periods of follow-up, thereby improving statistical power when treatment effects are expected to occur predominantly early, late or within specific phases of survival.
Mathematical Formulae
Primary Formula
?� = (?w?(O? ? E?))� / ?w?�V?
where:
- w? = weight applied at event time i
- O? = observed events
- E? = expected events
- V? = variance of O? ? E?
Supporting Formulae
Standardised statistic:
Z = ?w?(O? ? E?) / �?w?�V?
Chi-square statistic:
?� = Z�
Examples of weighting functions:
Log-rank:
w? = 1
Tarone-Ware:
w? = �n?
Breslow?Gehan:
w? = n?
Fleming?Harrington:
w? = ?(t?)^? ? (1 ? ?(t?))^?
Related Mathematical Methods
- Log-Rank Test
- Tarone-Ware Test
- Breslow?Gehan Test
- Fleming?Harrington Test
- Kaplan?Meier Estimation
- Cox Proportional Hazards Model
Example
An oncology trial compares immunotherapy with chemotherapy. Because delayed treatment effects are anticipated, a Fleming?Harrington weighted log-rank test is selected to place greater emphasis on later events.
The weighted analysis produces:
?� = 6.84
p = 0.009
The weighted test detects a statistically significant survival difference that is more pronounced during later follow-up than would be identified using the standard log-rank test.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(Weights,ObservedMinusExpected) | Calculate the weighted numerator of the test statistic |
| SUMPRODUCT | =SUMPRODUCT((Weights^2),Variance) | Calculate the weighted variance |
| SQRT | =SQRT(B2) | Calculate weighting functions such as Tarone-Ware weights |
| CHISQ.DIST.RT | =CHISQ.DIST.RT(TestStatistic,1) | Calculate the p-value for the weighted log-rank test |
| IF | =IF(P2<0.05,"Significant","Not Significant") | Interpret statistical significance |
VBA (Optional)
A VBA procedure can automate weighted survival comparisons by calculating multiple weighted log-rank statistics and summarising results for alternative weighting schemes.
Sources
- Harrington DP, Fleming TR. A Class of Rank Test Procedures for Censored Survival Data. Biometrika. 1982.
- Tarone RE, Ware J. On Distribution-Free Tests for Equality of Survival Distributions. Biometrika. 1977.
- Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data.
- Collett D. Modelling Survival Data in Medical Research.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials – extrapolation with patient-level data — Nicholas R. Latimer, TSD 14 ed., 2013 (NICE Decision Support Unit (University of Sheffield))
The reference guidance on survival analysis for economic evaluation: fitting standard parametric models (exponential, Weibull, Gompertz, log-logistic, log-normal) to censored trial data and extrapolating to estimate lifetime survival benefit, with a process guide for model selection and justification.
Frequently Asked Questions (6)
What is a weighted log-rank test?
A class of tests comparing survival distributions using different weights at different follow-up times, including the log-rank, Peto, and Tarone-Ware tests.
Source: Fleming & Harrington 1991
What does varying the weights in a weighted log-rank test achieve?
The weights in a weighted log-rank test set how much each event time counts toward the comparison, and adjusting them concentrates the test's sensitivity where a difference between groups is expected. Weighting early events heavily makes the test alert to a difference that appears soon after treatment, while equal weighting spreads its attention across the whole curve. Choosing weights to match the anticipated pattern gives more power to detect that particular kind of difference. The family exists to allow this tuning. Klein and Moeschberger (2003) describe it.
Source: Klein & Moeschberger 2003
How do weighted log-rank tests work?
Weighted log-rank tests work by, at each event time, comparing the observed events in each group with those expected under equal survival, as the log-rank test does, but multiplying each comparison by a weight that may depend on the time or the number at risk, before summing across event times. Different weight functions emphasise events at different times. The weighted sum forms the test statistic, which under the null follows a known distribution, so the choice of weights tunes the test's sensitivity to the pattern of difference.
Source: Fleming & Harrington 1991
What tests belong to the weighted log-rank family?
The weighted log-rank family includes the standard log-rank test, with equal weights, most powerful under proportional hazards; the Wilcoxon, or Gehan, test, weighting by the number at risk to emphasise early events; the Tarone-Ware test, weighting by the square root of the number at risk, an intermediate compromise; the Peto test, using survival-based weights; and the Fleming-Harrington family, with weights that can emphasise early or late events. All compare survival with different weightings, suiting different patterns of difference.
Source: Mantel 1966
Why use weighted log-rank tests?
Weighted log-rank tests are used because the standard log-rank test, weighting all events equally, is most powerful under proportional hazards but can lose power when the survival difference is concentrated in a period, such as early or late. Weighting events to emphasise the expected period of difference can increase power in such cases. Weighted tests thus allow the comparison to be tuned to the anticipated pattern of the treatment effect over time, improving sensitivity where hazards are non-proportional or differences are time-localised.
Source: Fleming & Harrington 1991
How is the weighting chosen for a weighted log-rank test?
The weighting for a weighted log-rank test is chosen according to when a survival difference is expected to emerge: weighting early events suits early differences, weighting late events suits late ones, and equal weighting suits differences spread over time under proportional hazards. The choice, embodied in the particular test used, should be specified in advance based on the anticipated pattern, since selecting the weighting after seeing the data to maximise significance would invalidate the test. The weights determine the pattern of difference the test best detects.
Source: Fleming & Harrington 1991
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 24 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/weighted-log-rank
- Term code
- HE-EM-SM-092
Stable URI · Machine-readable · Resolvable · CC BY 4.0