Wilcoxon Test

Wilcoxon Test for Comparing Survival Curves

In survival analysis, the Wilcoxon test is the generalised Wilcoxon test of Gehan and Breslow, a rank test of whether trial arms share the same time-to-event experience once censoring is allowed for. It belongs to the weighted log-rank family, and its weight is the number of patients still under observation, so early deaths or progressions count for much more than later ones. Outside survival analysis the name usually means the rank-sum or signed-rank test, which appear here as the origin of the survival version and for their use with trial costs. Health economists meet the survival form when curves separate early and then converge. Its output is a chi-square statistic and a p-value, never a survival gain, so a cost-effectiveness model still needs estimated and extrapolated curves.

Rank tests before and after censoring

Mann and Whitney, in 1947, proposed a statistic $U$ based on the relative ranks of two samples for testing equality of their distributions, noting that Wilcoxon had proposed an equivalent test in 1945. The R documentation for wilcox.test records that definitions vary, since the rank sum can be reported with or without its minimum value subtracted, and that the adjusted form equals the number of pairs in which an observation from one sample is not greater than one from the other.

A censored survival time is only known to exceed the censoring time, so ordinary ranks fail. Gehan, in 1965, proposed a distribution-free two-sample test extending the Wilcoxon test to samples with arbitrary censoring on the right. Breslow, in 1970, extended this to $K$ samples and allowed the censoring distributions to differ between populations. The Stata manual describes sts test, wilcoxon as the generalised Wilcoxon test of Breslow (1970) and Gehan (1965).

The statistic as weighted observed minus expected deaths

At each distinct event time the test compares deaths observed in one arm with the number expected if both arms had the same hazard, given who is at risk. Tarone and Ware showed in 1977 that the log-rank and modified Wilcoxon statistics differ only in their weights, which are functions of the number at risk at each death. From the general form in the Stata manual, the two-arm Wilcoxon statistic is as follows.

$$U_{G} = \sum_{j=1}^{k} n_j \left( d_{1j} - e_{1j} \right), \qquad e_{1j} = \frac{n_{1j}\, d_j}{n_j}$$

where $k$ is the number of distinct event times, $n_j$ is the number at risk in both arms just before time $t_j$, $n_{1j}$ is the number at risk in arm 1, $d_j$ is the number of events at $t_j$, $d_{1j}$ is the number of those in arm 1, and $e_{1j}$ is the number expected in arm 1 under the null hypothesis of equal hazards.

$$V_{G} = \sum_{j=1}^{k} \frac{n_{1j}\, n_{0j}\, d_j \left( n_j - d_j \right)}{n_j - 1}, \qquad \chi^2_{G} = \frac{U_{G}^{2}}{V_{G}}$$

where $n_{0j}$ is the number at risk in arm 0, $V_{G}$ is the null variance of $U_{G}$ once the squared weight cancels the $n_j^2$ of the usual variance term, and $\chi^2_{G}$ is compared with a chi-square distribution with one degree of freedom (the Stata manual gives $r - 1$ for $r$ groups).

Each weighted term has a reading that links the test back to ranks.

$$n_j \left( d_{1j} - e_{1j} \right) = n_{0j}\, d_{1j} - n_{1j}\, d_{0j}$$

where $d_{0j}$ is the number of events in arm 0 at $t_j$. Each arm 1 death adds the number of arm 0 patients who outlived it and each arm 0 death subtracts the number of arm 1 patients who outlived it, so $U_{G}$ is the number of orderable pairs in which the arm 1 patient died first minus the number in which the arm 0 patient died first. Without censoring every pair is ordered, as in Mann and Whitney's $U$.

Early weight, censoring and the Peto alternative

The weight falls with every death and with every patient censored or lost. The Stata manual says the test suits hazards thought to vary other than proportionally when censoring patterns are similar across groups, and that its weighting makes it susceptible to differences in censoring between groups. The R survival documentation notes that Peto and Peto showed the Gehan-Wilcoxon test can be badly biased under different censoring patterns. Prentice and Marek, comparing cancer incidence in dogs given bone marrow transplants with controls, found a log-rank significance level of 0.01 against 0.76 for Gehan's test.

The Peto test weights by an estimate of pooled survival instead, and the Stata manual states that it is not affected by differences in censoring patterns, while the R documentation adds that for most data sets the two hardly differ. The Tarone-Ware test uses the square root of the number at risk and sits between the log-rank and Wilcoxon tests, and the Fleming-Harrington test can be tuned towards early or late differences.

Choosing the test before the curves are seen

The log-rank test weights every event equally and, in the Stata manual's words, is most appropriate when hazards are thought to be proportional. The Wilcoxon test gains statistical power over it when the difference lies early, where risk sets are largest, and loses power when curves separate late. Royston and Parmar, simulating nine tests in randomised trials (the Gehan-Wilcoxon test was not among them), found that tests aimed at a particular pattern performed well when the correct type of effect had been assumed, and that a low control-arm event rate reduced the power of weighted log-rank tests aimed at early effects.

A trial expecting early separation, for example from high perioperative mortality in one arm, can prespecify the Wilcoxon test as primary or as a sensitivity analysis. Picking the weighted test with the smallest p-value after seeing the curves inflates the false-positive rate unless multiplicity is controlled, for example by hierarchical testing.

Worked example with curves that meet at 12 months

All numbers in this example are illustrative. A trial randomises 10 patients to a new treatment (arm 1) and 10 to standard care (arm 0) for 12 months. Arm 0 deaths occur at months 1, 2, 3 and 4 and arm 1 deaths at months 6, 8, 9 and 10. The other six in each arm are censored alive at 12 months, so the Kaplan-Meier estimator gives 12-month survival of 0.60 in both arms.

1. Build the risk sets. Each time has a single death, so each variance term is $n_{1j} n_{0j}$.

2. Sum the score. Arm 1 had 4 deaths against 4.5050 expected, and the weighted score is $U_{G} = 4 \times (-10) + 4 \times 6 = -16$.

3. Check it as pairs. Each arm 0 death was outlived by all 10 arm 1 patients (40 pairs favouring arm 1), and each arm 1 death came while six arm 0 patients were alive (24 pairs favouring arm 0). The other 36 pairs are both censored and cannot be ordered, and 40 minus 24 gives 16.

4. Form the statistic. The variance is the sum of the last column, $V_{G} = 544$, so $\chi^2_{G} = (-16)^2 / 544 = 0.4706$, with a p-value of 0.49.

5. Compare the neighbours. The log-rank test gives $U = -0.5050$, $V = 1.9562$ and $\chi^2 = 0.1304$ (p = 0.72), and the Tarone-Ware test $\chi^2 = 0.2778$ (p = 0.60). The Wilcoxon statistic is over three times the log-rank one because all arm 0 deaths fall where its weights peak. The Peto-Peto-Prentice test also gives 0.4706, since with no censoring before the last death its weight is $n_j / 21$, a constant multiple of the Wilcoxon weight.

6. Change the censoring. If three arm 1 patients are instead lost at month 5, the score stays at minus 16, but the later weights shrink and the variance falls to 472, giving $\chi^2_{G} = 0.5424$ (p = 0.46). The Peto-Peto-Prentice statistic drops to 0.2205 and the log-rank statistic to 0.0020, so dropout alone has moved the tests apart.

7. Interpret. No test shows a difference with 20 patients, and none measures benefit. The model quantity is the restricted mean survival time to 12 months, $\text{RMST}_1 = 6 + 2(0.9) + 0.8 + 0.7 + 2(0.6) = 10.5$ months against $\text{RMST}_0 = 1 + 0.9 + 0.8 + 0.7 + 8(0.6) = 8.2$ months, a gain of 2.3 months although 12-month survival is identical.

What a cost-effectiveness model still needs

NICE Decision Support Unit Technical Support Document 14, on survival analysis for economic evaluation, explains that without extrapolation the survival benefit is restricted to the trial period, which is likely to underestimate the survival gain and the QALYs gained. It notes that mean survival equals the area under the survival curve, that parametric models are likely to be the preferred approach in most cases, and that standard survival methods suit only uninformative censoring.

A Wilcoxon p-value neither chooses a distribution nor supplies the curves a partitioned survival model needs. A significant Wilcoxon result beside a non-significant log-rank result is a signal to examine the proportional hazards assumption and how the extrapolation treats the period after early separation.

The rank-sum and signed-rank tests with costs

The R documentation describes the classic pair. The signed-rank test examines whether paired differences, such as utility before and after treatment, are symmetric about a stated value, and the rank-sum test is equivalent to the Mann-Whitney test. It warns that the two-sample location estimate is the median of the differences between an observation from each sample, not the difference in medians.

For costs these tests answer the wrong question. Mihaylova and colleagues note that costs are often positively skewed with heavy tails, that medical statistics has traditionally used rank methods for them, and that it is widely accepted in health economics that the population mean cost is the statistic of interest to policy makers. Thompson and Barber add that the Mann-Whitney U test is an overall comparison of distributions in shape and location, and cite a review of 45 trials published in 1995 in which only half the papers comparing costs used methods addressing arithmetic means. A permutation test or bootstrapping on the mean difference keeps the target the budget decision needs.

Software names and common misreadings

Stata runs the survival test with sts test, wilcoxon (stratified with strata()), and in its heart transplant example the Wilcoxon chi-square is 0.14 (p = 0.71) against 0.13 for the log-rank test. The statsmodels survdiff function takes weight_type="gb", documented as Gehan-Breslow weighting by the number at risk. R survdiff has no number-at-risk weight, and its rho = 1 option is documented as equivalent to the Peto and Peto modification of the Gehan-Wilcoxon test, so an R result labelled Wilcoxon may use the Peto weight.

1. Mixing up the tests that share the name. A rank-sum test on costs and a survival Wilcoxon test answer different questions, and a report should name the version.
2. Reading significance as a treatment effect. The test checks equality of survival distributions, and the size of any gain must come from survival estimates, a hazard ratio model or a decision model.
3. Ignoring the censoring pattern. Unequal dropout or accrual changes the weights, and the Peto-Peto-Prentice test is a check.
4. Choosing the weight after seeing the curves. Selecting the most favourable weighted test invalidates the nominal significance level.

Sources

• Breslow N. A generalized Kruskal-Wallis test for comparing K samples subject to unequal patterns of censorship. Biometrika. 1970;57(3):579-594. https://doi.org/10.1093/biomet/57.3.579
• Gehan EA. A generalized Wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika. 1965;52(1-2):203-223. https://doi.org/10.1093/biomet/52.1-2.203
• Latimer N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Decision Support Unit; June 2011, last updated March 2013. https://sheffield.ac.uk/sites/default/files/2022-02/TSD14-Survival-analysis.updated-March-2013.v2.pdf
• Mann HB, Whitney DR. On a test of whether one of two random variables is stochastically larger than the other. Annals of Mathematical Statistics. 1947;18(1):50-60. https://doi.org/10.1214/aoms/1177730491
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• Prentice RL, Marek P. A qualitative discrepancy between censored data rank tests. Biometrics. 1979;35(4):861-867. https://doi.org/10.2307/2530120
• R Core Team. wilcox.test: Wilcoxon rank sum and signed rank tests. R stats package help page, accessed 31 October 2026. https://stat.ethz.ch/R-manual/R-devel/library/stats/html/wilcox.test.html
• R survival package. survdiff: Test survival curve differences. Package survival version 3.8-6 help page, accessed 31 October 2026. https://stat.ethz.ch/R-manual/R-devel/library/survival/html/survdiff.html
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• StataCorp. sts test: Test equality of survivor functions. Stata Survival Analysis Reference Manual. StataCorp LLC; 2025. https://www.stata.com/manuals/stststest.pdf
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