Peto Test

Peto Test

In this entry, Peto test means the Peto–Peto modification of the Gehan–Wilcoxon test for comparing time-to-event distributions between groups. It is a weighted log-rank type test that gives relatively more weight to earlier event times as pooled survival falls. The shortened label “Peto test” is ambiguous: papers also associate Peto's name with the ordinary log-rank test and with the Peto odds-ratio method for binary rare-event meta-analysis. Always name the actual statistic and implementation in a report.

Question and data structure

The test asks whether two groups have the same event-time distribution over the observed follow-up under a null hypothesis of no group difference. It uses each distinct observed event time, the number of individuals still at risk immediately beforehand, the observed events in each group, and a time-dependent weight. Right-censored participants contribute while at risk but not as events at censoring; event and censoring times, ties and delayed entry need consistent handling.

For group A at distinct event time $t_j$, let $n_{Aj}$ and $n_{Bj}$ be the numbers at risk, $n_j=n_{Aj}+n_{Bj}$, $d_{Aj}$ and $d_{Bj}$ the event counts, and $d_j=d_{Aj}+d_{Bj}$. Under the null, the expected group-A count conditional on the risk set and total events is $e_{Aj}=d_jn_{Aj}/n_j$. A generic weighted observed-minus-expected statistic is

$$U=\sum_j a_j(d_{Aj}-e_{Aj}),$$

where $a_j$ is a prespecified, nonnegative weight calculated from the pooled survival history. In a common weighted log-rank representation of the Peto–Peto family, $a_j$ is based on the pooled survival estimate near $t_j$; with an explicit pre-event convention, $a_j=\widehat S_{\mathrm{pool}}(t_j-)$. The exact Peto–Peto, Peto–Peto–Prentice and software implementations can differ in survival-score and variance conventions, especially with ties or censoring. State the implementation rather than assuming the shortened name determines every formula.

By contrast, the ordinary log-rank test assigns equal weight $a_j=1$ to each event-time observed-minus-expected contribution. The declining survival weights in the Peto–Peto variant emphasize differences appearing earlier in follow-up. “Earlier” refers to event time and available risk sets, not to a universal clinical priority; a late treatment benefit can be downweighted.

Variance and test decision

Under a conditional hypergeometric risk-set calculation, one contribution's variance is

$$v_j=\frac{n_{Aj}n_{Bj}d_j(n_j-d_j)}{n_j^2(n_j-1)},$$

for $n_j>1$, with zero variance when no information is available. The weighted variance in this representation is $V=\sum_j a_j^2v_j$. An approximate two-sided large-sample comparison uses $Z=U/\sqrt{V}$ or $Z^2=U^2/V$ against a one-degree-of-freedom chi-square reference. The approximation, variance handling, time weights and tie rules must match the chosen test; sparse events or highly unequal risk sets can make an asymptotic p-value unreliable.

The sign of $U$ depends on which group is labelled A and whether more events are beneficial or harmful. A p-value describes incompatibility with the null under the test assumptions; it does not estimate a hazard ratio, survival difference, life-years gained or causal effect. Report survival curves and an appropriate effect measure with uncertainty as well as the test result.

Worked risk-set example

Suppose ten people are at risk in each group initially, with no censoring before two distinct event times. At the first time, one event occurs in each group; immediately before the second, nine remain at risk in each. At the second time, both events occur in B. Using the pooled Kaplan–Meier survival just before each event time gives weights $1.00$ and $0.90$, because the first two events among 20 reduce pooled survival to $18/20=0.90$.

Thus $U=1(1-1)+0.90(0-1)=-0.90$. The corresponding conditional variances are $v_1=0.473684$ and $v_2=0.470588$; hence $V=0.473684+0.90^2(0.470588)\approx0.854861$. This produces $Z\approx-0.9734$ and a two-sided normal approximation $p\approx0.3304$. A negative statistic means fewer observed events in A than expected across the weighted comparison; it is not an estimated risk reduction. The numbers are a deliberately tiny, illustrative computation, for which a large-sample p-value is especially fragile; they are not a recommendation to use an asymptotic test with four events.

The example's explicit pre-event weighting makes its arithmetic reproducible. A software implementation using a modified pooled survival weight or a different tie/variance convention can give a different numerical statistic; that result should be labelled with its exact method. Do not report the example's p-value as if it were universal for every procedure called “Peto.”

When to choose it

A weighted test can be a reasonable prespecified choice when an early group difference is scientifically meaningful and expected. It may have better sensitivity to that pattern than an equal-weight log-rank test, but may be less sensitive to effects appearing late. If curves cross or hazards vary, a single weighted test can obscure direction over time; examine Kaplan–Meier curves, numbers at risk and appropriate time-specific or restricted-mean-survival summaries.

The method does not require assuming a constant hazard ratio merely to compute the weighted comparison. Interpretation as a single proportional-hazards effect, however, would need additional modelling and scrutiny. Avoid choosing whichever of several weighting schemes gives the smallest p-value after seeing the curves without accounting for that selection. If multiple tests are prespecified, explain the multiplicity strategy and the estimand each addresses.

As with other risk-set survival tests, censoring must be suitably independent of the event process conditional on relevant design and modelling assumptions for straightforward interpretation. Strong differences in censoring patterns, informative loss to follow-up, competing events or changes in treatment can complicate inference. The Peto–Peto modification was developed partly to address problems with the original Gehan weighting under unequal censoring, but it does not make informative censoring harmless.

Role in health economic analysis

A trial may use a Peto–Peto test to compare survival outcomes while an economic evaluation estimates life-years, quality-adjusted life-years (QALYs) and costs. A survival-curve p-value does not supply the whole survival distribution or the extrapolated mean survival needed for an economic model. Estimate clinically relevant effects over the appropriate time horizon and assess uncertainty, extrapolation and competing events separately.

An early survival advantage may matter considerably for patients yet not determine whether an intervention is cost-effective. Conversely, a nonsignificant weighted test does not prove equivalent lifetime outcomes. Use the test in the context of the clinical question, study design and decision model rather than as a substitute for incremental effects and resource consequences.

Naming and source guidance

The survival package documentation for survdiff distinguishes its ordinary log-rank option (rho = 0) from the Peto–Peto modification of the Gehan–Wilcoxon test (rho = 1), implemented through pooled-survival weights. The original Peto and Peto rank-invariant test procedures give the historical methodological background. The Cochrane Handbook chapter on meta-analysis describes the separate Peto method for pooling binary rare events; that odds-ratio method is not the survival test defined here. For reproducibility, report the software, version, weight and variance convention, null, censoring assumptions, tie handling and prespecified analysis plan.