Comparing (Fancy) Survival Curves with Weighted Log-rank Tests We have just adopted weighted Log-rank tests to the survminer package, thanks to survMisc::comp. What are they and why they are useful? Read this blog post to find out. I used ggthemr to make the presentation a little bit more bizarre.

Log-rank statistic for 2 groups

Log-rank test, based on Log-rank statistic, is a popular tool that determines whether 2 (or more) estimates of survival curves differ significantly. 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

$U = \sum_{i=1}^{T}w_{t_i}(o_{t_i}^A-e_{t_i}^A), \ \ \ \ \ \ \ \ V = Var(U) = \sum_{i=1}^{T}(w_{t_i}^2\frac{n_{t_i}^An_{t_i}^Bd_i(n_{t_i}-o_{t_i})}{n_{t_i}^2(n_{t_i}-1)})$

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,

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

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 available in ?survMisc::comp

• 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 $$w_{t_i} = S(t_i)^p(1 - S(t_i))^q$$.

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

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

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

gghtemr

Let’s make it more interesting (or not) with ggthemr package that has many predefinied palettes.

After installation

one can set up a global ggplot2 palette/theme with

and check current colors with

Note: the first colour in a swatch is a special one. It is reserved for outlining boxplots, text etc. For color lines first color is not used.

Fleming-Harrington (p=1, q=1) + light theme

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

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

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

• 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. JSTOR

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