Author: Marcin Kosinski

# Specifiyng weights in Log-rank comparisons

#### Marcin Kosinski

#### created 29-01-2017, revised 22-08-2018

    library("survminer")

> This vignette covers changes between versions 0.2.4 and 0.2.5 for specifiyng weights in the log-rank comparisons done in `ggsurvplot()`.

# 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 \(t\)).

## Why are they useful?

The regular Log-rank test is sensitive to detect differences in late survival times, where Gehan-Breslow and Tharone-Ware propositions might be used if one is interested in early differences in survival times. Peto-Peto modifications are also useful in early differences and are more robust (than Tharone-Whare or Gehan-Breslow) for situations where many observations are censored. The most flexible is Fleming-Harrington method for weights, where high `p` indicates detecting early differences and high `q` indicates detecting differences in late survival times. But there is always an issue on how to detect `p` and `q`.

> Remember that test selection should be performed at the research design level! Not after looking in the dataset.

# 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.

## Log-rank (survdiff)

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

## Log-rank (weighted)

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

## Gehan-Breslow (generalized Wilcoxon)

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

## Tharone-Ware

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

## Peto-Peto’s modified survival estimate

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

## modified Peto-Peto’s (by Andersen)

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

## Fleming-Harrington (p=1, q=1)

    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.