Concept Architecture
Concept
Theoretically, Stratified Log-Rank Test is a non-parametric hypothesis test used to compare survival distributions between treatment groups while controlling for one or more stratification variables. It extends the standard log-rank test by performing separate comparisons within predefined strata and combining the stratum-specific results into a single overall test statistic. The method exists to reduce confounding and improve statistical efficiency when prognostic factors used during randomisation or study design influence survival.
Mathematically, the stratified log-rank test calculates the observed and expected numbers of events within each stratum and sums these across all strata to form an overall test statistic. Under the null hypothesis of equal survival functions, the combined statistic follows an approximate chi-square distribution with one degree of freedom for two-group comparisons. Each stratum contributes independently to the overall comparison while preserving the proportional weighting of risk sets.
In practice, the stratified log-rank test is widely used in randomised clinical trials where participants are stratified by factors such as disease stage, age group or study centre. It is commonly specified as the primary hypothesis test in oncology trials that subsequently inform survival modelling and health economic evaluations for health technology assessment.
Purpose
Used to compare survival distributions between treatment groups while adjusting for predefined prognostic strata, thereby providing an unbiased assessment of treatment effects when important baseline characteristics influence survival.
Mathematical Formulae
Primary Formula
?� = (?(O? ? E?))� / ?V?
where:
- O? = observed events within stratum s
- E? = expected events within stratum s
- V? = variance within stratum s
Supporting Formulae
Overall observed events:
O = ?O?
Overall expected events:
E = ?E?
Variance:
V = ?V?
Test statistic:
Z = (O ? E) / �V
?� = Z�
Related Mathematical Methods
- Log-Rank Test
- Kaplan?Meier Estimation
- Cox Proportional Hazards Model
- Survival Analysis
- Mantel?Haenszel Methods
- Stratified Randomisation
Example
A multicentre oncology trial compares a new treatment with standard care. Patients are stratified by disease stage before randomisation.
| Stratum | Observed Events | Expected Events | Variance |
|---|---|---|---|
| Stage I?II | 22 | 27 | 6.5 |
| Stage III?IV | 38 | 44 | 9.5 |
Combined values:
O = 60
E = 71
V = 16
Test statistic:
?� = (60 ? 71)� � 16
?� = 121 � 16 = 7.56
The result provides evidence of a statistically significant difference in survival between treatment groups after accounting for disease stage.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUM | =SUM(B2:B3) | Sum observed events across strata |
| SUM | =SUM(C2:C3) | Sum expected events across strata |
| SUM | =SUM(D2:D3) | Sum variances across strata |
| SQRT | =SQRT(D5) | Calculate the standard error of the combined statistic |
| CHISQ.DIST.RT | =CHISQ.DIST.RT(((B5-C5)^2)/D5,1) | Calculate the p-value for the stratified log-rank test |
VBA (Optional)
A VBA procedure can automatically calculate stratified log-rank statistics across multiple subgroup analyses and generate summary tables for clinical trial reporting.
Sources
- Mantel N. Evaluation of Survival Data and Two New Rank Order Statistics Arising in Its Consideration. Cancer Chemotherapy Reports. 1966.
- Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data.
- Collett D. Modelling Survival Data in Medical Research.
- Kleinbaum DG, Klein M. Survival Analysis: A Self-Learning Text.
- 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 stratified log-rank test?
A variant of the log-rank test accounting for stratifying variables, comparing survival within each stratum before combining into an overall statistic.
Source: Mantel 1966
When is a stratified log-rank test used instead of the ordinary one?
An ordinary log-rank test comparing two treatments can be distorted if the groups differ in another factor that also affects survival, such as disease stage. A stratified log-rank test handles this by dividing patients into strata defined by that factor, comparing the treatments within each stratum where the confounder is held constant, and then combining the within-stratum comparisons into one overall test. This removes the confounder's influence from the treatment comparison. It is used when a known factor could otherwise bias the result. Collett (2015) describes it.
Source: Collett 2015
How does the stratified log-rank test work?
The stratified log-rank test works by dividing the data into strata defined by the stratifying variable, computing the log-rank comparison of observed versus expected events between groups within each stratum, and then summing the contributions across strata into a single statistic. This combines the evidence for a group difference from within each stratum, so that comparisons are made among individuals sharing the stratifying variable. The resulting statistic tests the group difference while controlling for the stratifying factor.
Source: Mantel 1966
Why use a stratified log-rank test?
A stratified log-rank test is used when a variable could confound or affect the comparison of survival between groups, such as study centre, disease stage, or another prognostic factor, and one wants to adjust for it without modelling it directly. By comparing within strata, the test controls for the stratifying variable, giving a fairer comparison of the groups. It is useful in trials with important prognostic factors or multiple centres, where an unadjusted comparison might be biased by imbalance in the stratifying factor.
Source: Kalbfleisch & Prentice 2002
How does stratification control for confounding?
Stratification controls for confounding by comparing groups only within strata that share the same value of the stratifying variable, so differences due to that variable cannot influence the within-stratum comparisons, and combining across strata gives an overall effect adjusted for it. Because the comparison is made among individuals alike on the stratifying factor, its confounding effect is removed. This makes the stratified log-rank test a way to compare survival controlling for a categorical confounder without including it in a regression model.
Source: Mantel 1966
What are the limitations of the stratified log-rank test?
The stratified log-rank test controls only for the categorical variables used for stratification, and stratifying on many variables or continuous ones creates small strata with little information, reducing power. It assumes the group effect is consistent across strata and does not quantify the effect size, for which a stratified Cox model is used. It also shares the log-rank test's reliance on proportional hazards for power. These limitations mean stratification suits a few important categorical confounders, with regression preferred for adjustment on many or continuous covariates.
Source: Kalbfleisch & Prentice 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 23 Oct 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/stratified-log-rank
- Term code
- HE-EM-SM-082
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