Log-rank statistic for 2 groups

As it is stated in the literature, the Log-rank test for comparing survival (estimates of survival curves) in 2 groups (\(A\) and \(B\)) is based on the below statistic

\[LR = \frac{U^2}{V} \sim \chi(1),\]

where \[U = \sum_{i=1}^{T}w_{t_i}(o_{t_i}^A-e_{t_i}^A), \ \ \ \ \ \ \ \ V = Var(U) = \sum_{i=1}^{T}(w_{t_i}^2\frac{n_{t_i}^An_{t_i}^Bo_{t_i}(n_{t_i}-o_{t_i})}{n_{t_i}^2(n_{t_i}-1)})\] and

• \(t_i\) for \(i=1, \dots, T\) are possible event times,
• \(n_{t_i}\) is the overall risk set size on the time \(t_i\) (\(n_{t_i} = n_{t_i}^A+n_{t_i}^B\)),
• \(n_{t_i}^A\) is the risk set size on the time \(t_i\) in group \(A\),
• \(n_{t_i}^B\) is the risk set size on the time \(t_i\) in group \(B\),
• \(o_{t_i}\) overall observed events in the time \(t_i\) (\(o_{t_i} = o_{t_i}^A+o_{t_i}^B\)),
• \(o_{t_i}^A\) observed events in the time \(t_i\) in group \(A\),
• \(o_{t_i}^B\) observed events in the time \(t_i\) in group \(B\),
• \(e_{t_i}\) number of overall expected events in the time \(t_i\) (\(e_{t_i} = e_{t_i}^A+e_{t_i}^B\)),
• \(e_{t_i}^A\) number of expected events in the time \(t_i\) in group \(A\),
• \(e_{t_i}^B\) number of expected events in the time \(t_i\) in group \(B\),
• \(w_{t_i}\) is a weight for the statistic,

also remember about few notes

\[e_{t_i}^A = n_{t_i}^A \frac{o_{t_i}}{n_{t_i}}, \ \ \ \ \ \ \ \ \ \ e_{t_i}^B = n_{t_i}^B \frac{o_{t_i}}{n_{t_i}},\] \[e_{t_i}^A + e_{t_i}^B = o_{t_i}^A + o_{t_i}^B\]

that’s why we can substitute group \(A\) with \(B\) in \(U\) and receive same results.

Weighted Log-rank extensions

Regular Log-rank comparison uses \(w_{t_i} = 1\) but many modifications to that approach have been proposed. The most popular modifications, called weighted Log-rank tests, are

• n Gehan and Breslow proposed to use \(w_{t_i} = n_{t_i}\) (this is also called generalized Wilcoxon),
• srqtN Tharone and Ware proposed to use \(w_{t_i} = \sqrt{n_{t_i}}\),
• S1 Peto-Peto’s modified survival estimate \(w_{t_i} = S1({t_i}) = \prod_{i=1}^{T}(1-\frac{e_{t_i}}{n_{t_i}+1})\),
• S2 modified Peto-Peto (by Andersen) \(w_{t_i} = S2({t_i}) = \frac{S1({t_i})n_{t_i}}{n_{t_i}+1}\),
• FH Fleming-Harrington \(w_{t_i} = S(t_i)^p(1 - S(t_i))^q\).

Note: the Fleming-Harrington weight uses \(S(t^-)\) which is the left-continuous Kaplan-Meier estimate (the value just before time \(t\)).

Plots

library("survival")
data("lung")
fit <- survfit(Surv(time, status) ~ sex, data = lung)

After preparing a functionality for this GitHub’s issue Other tests than log-rank for testing survival curves and Log-rank test for trend we are now able to compute p-values for various Log-rank test in survminer package. Let as see below examples on executing all possible tests.

ggsurvplot(fit, data = lung, pval = TRUE, pval.method = TRUE)

ggsurvplot(fit, data = lung, pval = TRUE, pval.method = TRUE,
           log.rank.weights = "1")

ggsurvplot(fit, data = lung, pval = TRUE, pval.method = TRUE,
           log.rank.weights = "n", pval.method.coord = c(5, 0.1),
           pval.method.size = 3)

ggsurvplot(fit, data = lung, pval = TRUE, pval.method = TRUE,
           log.rank.weights = "sqrtN", pval.method.coord = c(3, 0.1),
           pval.method.size = 4)

ggsurvplot(fit, data = lung, pval = TRUE, pval.method = TRUE,
           log.rank.weights = "S1", pval.method.coord = c(5, 0.1),
           pval.method.size = 3)

ggsurvplot(fit, data = lung, pval = TRUE, pval.method = TRUE,
           log.rank.weights = "S2", pval.method.coord = c(5, 0.1),
           pval.method.size = 3)

ggsurvplot(fit, data = lung, pval = TRUE, pval.method = TRUE,
           log.rank.weights = "FH_p=1_q=1",
           pval.method.coord = c(5, 0.1),
           pval.method.size = 4)

The weighted_logrank() function

The examples above annotate a single weighted test on a plot. When you want the tests themselves – the statistic, its degrees of freedom and the p-value – as a tidy table, use weighted_logrank(). It takes vectors of rho and gamma, so you can compute a whole family of Fleming-Harrington G(rho, gamma) tests in one call. The classic three are FH(0, 0) (the standard log-rank), FH(1, 0) (early-emphasis, the Peto-Peto / Prentice weighting), and FH(0, 1) (late-emphasis):

weighted_logrank(Surv(time, status) ~ sex, data = lung,
                 rho = c(0, 1, 0), gamma = c(0, 0, 1))

    weight rho gamma statistic df      p.value
1 FH(0, 0)   0     0 10.326742  1 0.0013111645
2 FH(1, 0)   1     0 12.714151  1 0.0003628989
3 FH(0, 1)   0     1  3.459984  1 0.0628709170

Each row is one test with its weight, rho, gamma, chi-square statistic, df and p.value. Here the early-emphasis FH(1, 0) test gives the smallest p-value and the late-emphasis FH(0, 1) test the largest, which tells you the two survival curves separate more in early follow-up than late – the kind of shape a single log-rank p-value hides.

The same arbitrary (rho, gamma) are available inside ggsurvplot() through log.rank.weights = "FH" with rho = and gamma =, and surv_pvalue() now returns the test statistic and df alongside the p-value:

surv_pvalue(fit, data = lung, method = "FH", rho = 0, gamma = 1)

  variable   pval                              method  pval.txt statistic df
1      sex 0.0629 Fleming-Harrington (rho=0, gamma=1) p = 0.063  3.459984  1

References

• Gehan A. A Generalized Wilcoxon Test for Comparing Arbitrarily Singly-Censored Samples. Biometrika 1965 Jun. 52(1/2):203-23.

• Tarone RE, Ware J 1977 On Distribution-Free Tests for Equality of Survival Distributions. Biometrika;64(1):156-60.

• Peto R, Peto J 1972 Asymptotically Efficient Rank Invariant Test Procedures. J Royal Statistical Society 135(2):186-207.

• Fleming TR, Harrington DP, O’Sullivan M 1987 Supremum Versions of the Log-Rank and Generalized Wilcoxon Statistics. J American Statistical Association 82(397):312-20.

• Billingsly P 1999 Convergence of Probability Measures. New York: John Wiley & Sons.