Concept Architecture
Concept
Theoretically, Peto Test is a non-parametric hypothesis test used in survival analysis to compare survival distributions between groups while giving greater weight to early events. It is a member of the weighted log-rank family of tests and is based on the Peto?Peto modification of the Gehan?Wilcoxon test. The concept exists because the standard log-rank test is most powerful under proportional hazards, whereas the Peto Test provides improved sensitivity when treatment effects occur predominantly during the early follow-up period.
Mathematically, the Peto Test compares observed and expected event counts at each event time using weighted differences. The recognised mathematical framework applies Peto?Peto survival weights to construct a test statistic that asymptotically follows a chi-square distribution with one degree of freedom for two-group comparisons. The weighting scheme increases the influence of earlier failures while reducing the contribution of later observations.
In practice, the Peto Test is calculated using individual patient survival data, event indicators and follow-up times. It is applied in clinical trials and health economic evaluations when comparing time-to-event outcomes such as overall survival or progression-free survival, particularly when early treatment effects are expected or proportional hazards may not hold.
Purpose
Used to compare survival distributions between groups by applying weighted log-rank methods that emphasise early survival differences in clinical and health economic studies.
Mathematical Formulae
Primary Formula
X� = (?w?(O? ? E?))� � ?w?�V?
where:
w? = Peto?Peto weight
O? = observed events
E? = expected events
V? = variance
Supporting Formulae
Peto?Peto weight:
w? = ?(t?)
Test statistic distribution:
X� ~ ?�?
Related Mathematical Methods
- Weighted Log-Rank Test
- Log-Rank Test
- Kaplan?Meier Estimation
- Survival Analysis
- Cox Proportional Hazards Model
- Wilcoxon-Type Survival Tests
Example
A randomised oncology trial compares overall survival between two treatments. Investigators expect most treatment benefit to occur during the first six months. A Peto Test is performed using Kaplan?Meier survival data and yields:
X� = 5.41
p = 0.020
The result indicates a statistically significant difference in survival, with the weighting scheme placing greater emphasis on early treatment effects.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(B2:B50,C2:C50) | Calculate weighted observed-minus-expected event totals |
| SUM | =SUM(D2:D50) | Calculate cumulative variance |
| CHISQ.DIST.RT | =CHISQ.DIST.RT(F2,1) | Calculate the p-value for the Peto test statistic |
| SQRT | =SQRT(G2) | Calculate the standard error from the variance |
VBA (Optional)
Automate weighted survival comparisons using the Peto Test across multiple treatment groups and sensitivity analyses.
Sources
- Peto R, Peto J. Asymptotically Efficient Rank Invariant Test Procedures. Journal of the Royal Statistical Society, Series A. 1972.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Collett D. Modelling Survival Data in Medical Research.
- Hosmer DW, Lemeshow S, May S. Applied Survival Analysis.
- NICE. Health Technology Evaluations: The 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 Peto test?
A variant of the log-rank test using a specific weighting scheme offering statistical efficiency advantages under certain patterns of censoring.
Source: Peto & Peto 1972
In which situations is the Peto test preferred?
The Peto test belongs to the weighted family of survival comparisons but places more weight on differences early in follow-up, when more patients remain at risk, and less on late events among the dwindling survivors. This makes it more sensitive when an effect appears mainly early and the later data are sparse or unreliable, whereas the log-rank test weights all times equally. Its early emphasis suits situations where late estimates carry little information. Peto and Peto (1972) set out the approach.
Source: Peto & Peto 1972
How does the Peto test differ from the log-rank test?
The Peto test differs from the log-rank test in its weighting of events over time: the log-rank test weights all events equally, while the Peto test applies weights, related to survival estimates, that give more importance to earlier events. This makes the Peto test more powerful when survival differences are concentrated early in follow-up, whereas the log-rank test is most powerful under proportional hazards. Both test whether survival distributions differ, but their differing weights suit different patterns of difference.
Source: Peto & Peto 1972
When is the Peto test used?
The Peto test is used to compare survival between groups when differences are expected to occur early in follow-up or under censoring patterns where its weighting is more efficient than the equal weighting of the log-rank test. It is one of the weighted rank tests chosen according to the anticipated timing of the survival difference. As with other weighted tests, the weighting should ideally be specified in advance, since choosing it after seeing the data to maximise significance would invalidate the test.
Source: Peto & Peto 1972
How does the Peto test weight events?
The Peto test weights events using weights based on an estimate of the survival function, so that earlier events, when more individuals are at risk and survival is higher, receive greater weight than later events. This emphasis on earlier events makes the test sensitive to differences arising early in follow-up. The weighting distinguishes the Peto test from the equally weighted log-rank test and from other weighted variants, and it determines the pattern of survival difference the test is most powerful at detecting.
Source: Mantel 1966
How does the Peto test relate to the log-rank family?
The Peto test is a member of the family of weighted log-rank tests, which generalise the log-rank test by assigning weights to events at different times. The log-rank test uses equal weights, the Peto test uses survival-based weights emphasising early events, and other members, such as the Fleming-Harrington tests, use different weighting schemes. All test whether survival distributions differ, differing in the pattern of difference they best detect. The Peto test is thus one option within this family, chosen for its weighting properties.
Source: Peto & Peto 1972
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/peto-test
- Term code
- HE-EM-SM-065
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