Concept Architecture
Concept
Theoretically, the Fleming-Harrington Test is a weighted log-rank test used to compare survival distributions between two or more groups. It extends the standard log-rank test by applying weights to event times, allowing greater emphasis to be placed on early, middle or late differences in survival according to the research question. In health economics, the Fleming-Harrington test is used to evaluate treatment effects in survival analyses that inform economic evaluations and health technology assessments, particularly when proportional hazards may not hold.
Mathematically, the Fleming-Harrington test calculates a weighted comparison between the observed and expected numbers of events across groups. The weights are defined by powers of the Kaplan-Meier survival estimate, enabling the test to emphasise different regions of the survival curve. Under the null hypothesis of equal survival distributions, the test statistic follows an asymptotic chi-square distribution.
In practice, the Fleming-Harrington test is implemented using statistical software by specifying the weighting parameters (?, ?). Different parameter combinations are selected depending on whether early, late or intermediate survival differences are of greatest interest. The test is frequently reported alongside Kaplan-Meier curves and Cox proportional hazards models in clinical trials supporting health economic models.
Purpose
Used to compare survival distributions while placing greater emphasis on selected portions of the survival curve, particularly when treatment effects vary over time or proportional hazards assumptions are questionable.
Mathematical Formulae
Primary Formula
Z = U / �V
where:
U = ?w(t?)(O? ? E?)
and V is the variance of U.
Under H?:
Z� ~ ?�?
Supporting Formulae
w(t) = ?(t)? ? (1 ? ?(t))?
where:
- ?(t) = Kaplan-Meier estimate of survival
- ? � 0
- ? � 0
Common weighting schemes include:
- (? = 0, ? = 0): Log-rank test
- (? = 1, ? = 0): Early survival differences
- (? = 0, ? = 1): Late survival differences
- (? = 1, ? = 1): Intermediate survival differences
Related Mathematical Methods
- Log-rank test
- Kaplan-Meier estimator
- Cox proportional hazards model
- Weighted score tests
- Survival analysis
Example
An oncology trial compares overall survival between two treatments over five years. Visual inspection suggests that the survival curves diverge only after two years. A Fleming-Harrington test with weighting parameters (? = 0, ? = 1) is selected to place greater emphasis on late survival differences. The resulting ?� statistic indicates a statistically significant difference in long-term survival, supporting subsequent survival modelling for cost-effectiveness analysis.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| POWER | =POWER(B2,$F$1)*POWER(1-B2,$G$1) | Calculate Fleming-Harrington weights from estimated survival probabilities. |
| SUMPRODUCT | =SUMPRODUCT(WeightRange,ObservedRange-ExpectedRange) | Calculate the weighted numerator of the test statistic. |
| SQRT | =SQRT(B20) | Calculate the denominator of the standardised test statistic. |
| CHISQ.DIST.RT | =CHISQ.DIST.RT(B21,1) | Calculate the p-value from the ?� test statistic. |
VBA (Optional)
Automate weighted log-rank testing for multiple survival datasets using user-specified Fleming-Harrington weighting parameters.
Sources
- Fleming TR, Harrington DP. Counting Processes and Survival Analysis.
- Harrington DP, Fleming TR. A class of rank test procedures for censored survival data.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Collett D. Modelling Survival Data in Medical Research.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Tools & Resources
1
survHE — Survival Analysis for Health Economic Evaluation (R package) — Gianluca Baio, R package ed., 2023 (CRAN)
An R package for fitting and comparing parametric survival models for health economic evaluation, including Bayesian estimation, and for extrapolating time-to-event data to inform cost-effectiveness models.
Software (R package)View source →
Frequently Asked Questions (6)
What is the Fleming-Harrington test?
A hypothesis test comparing survival distributions between groups, generalising the log-rank test by allowing different weights for events at different follow-up times.
Source: Fleming & Harrington 1991
What does the Fleming-Harrington test compare?
The Fleming-Harrington test compares the survival experience of two or more groups to judge whether they differ, like the log-rank test, but it can emphasise differences at particular times by weighting early or late events more heavily. Setting the weights to treat all times equally recovers the ordinary log-rank test. This flexibility lets it detect a difference concentrated in one period, such as an early separation of curves, that an unweighted test might dilute. The weighting is chosen to match where a difference is expected. Klein and Moeschberger (2003) describe it.
Source: Klein & Moeschberger 2003
How does the Fleming-Harrington test generalise the log-rank test?
The Fleming-Harrington test generalises the log-rank test by introducing weights, governed by parameters, that determine how much events at different times contribute to the comparison. Setting the weights to be equal recovers the log-rank test, while other choices emphasise early or late differences. This family of weighted tests thus includes the log-rank test as a special case and extends it, allowing the comparison to focus on the period where a survival difference is expected, rather than treating all follow-up times alike.
Source: Fleming & Harrington 1991
When is the Fleming-Harrington test used?
The Fleming-Harrington test is used to compare survival between groups when the difference is expected to be concentrated in a particular period, such as early or late in follow-up, so that a weighted test is more powerful than the equally weighted log-rank test. For example, if a treatment's benefit emerges late, weighting late events increases sensitivity to it. The test is chosen when the pattern of the survival difference over time is anticipated, allowing the comparison to be tuned accordingly.
Source: Fleming & Harrington 1991
How are weights chosen in the Fleming-Harrington test?
Weights in the Fleming-Harrington test are chosen through parameters that control whether early or late events are emphasised, based on the expected pattern of the survival difference. Weighting early events suits differences that appear soon, and weighting late events suits differences that emerge later. To preserve the validity of the test, the weighting should ideally be specified in advance rather than chosen after seeing the data, since selecting the weights to maximise significance would inflate the chance of a spurious result.
Source: Fleming & Harrington 1991
How does the Fleming-Harrington test relate to non-proportional hazards?
The Fleming-Harrington test relates to non-proportional hazards because the log-rank test is most powerful when hazards are proportional, and loses power when the survival difference varies over time, as under non-proportional hazards. By weighting events at different times, the Fleming-Harrington test can be more powerful in such cases, focusing on the period where the difference is largest. It is therefore useful when hazards are not proportional and the treatment effect changes over follow-up, where the standard log-rank test may perform poorly.
Source: Kalbfleisch & Prentice 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 20 Oct 2025
Content version: 1.0.0
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