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, …, T are possible event times,
• n_ti is the overall risk set size on the time t_i (n_ti = n_ti^A + n_ti^B),
• n_ti^A is the risk set size on the time t_i in group A,
• n_ti^B is the risk set size on the time t_i in group B,
• o_ti overall observed events in the time t_i (o_ti = o_ti^A + o_ti^B),
• o_ti^A observed events in the time t_i in group A,
• o_ti^B observed events in the time t_i in group B,
• e_ti number of overall expected events in the time t_i (e_ti = e_ti^A + e_ti^B),
• e_ti^A number of expected events in the time t_i in group A,
• e_ti^B number of expected events in the time t_i in group B,
• w_ti 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_ti^A + e_ti^B = o_ti^A + o_ti^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_ti = 1 but many modifications to that approach have been proposed. The most popular modifications, called weighted Log-rank tests, are available in ?survMisc::comp

• n Gehan and Breslow proposed to use w_ti = n_ti (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_ti = S(t_i)^p(1 − S(t_i))^q.

Watch out for FH as I submitted an info on survMisc repository where I think their mathematical notation is misleading for Fleming-Harrington.

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)

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.