# Comparing (Fancy) Survival Curves with Weighted Log-rank Tests

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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 ( and ) is based on the below statistic

where

and

- for are possible event times,
- is the overall risk set size on the time (),
- is the risk set size on the time in group ,
- is the risk set size on the time in group ,
- overall observed events in the time (),
- observed events in the time in group ,
- observed events in the time in group ,
- number of overall expected events in the time (),
- number of expected events in the time in group ,
- number of expected events in the time in group ,
- is a weight for the statistic,

also remember about few notes

that’s why we can substitute group with in and receive same results.

# Weighted Log-rank extensions

Regular Log-rank comparison uses 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 (this is also called generalized Wilcoxon),`srqtN`

Tharone and Ware proposed to use ,`S1`

Peto-Peto’s modified survival estimate ,`S2`

modified Peto-Peto (by Andersen) ,`FH`

Fleming-Harrington .

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.

## Log-rank (survdiff) + sea theme

## Log-rank (comp) + dust theme

## Gehan-Breslow (generalized Wilcoxon) + flat dark theme

## Tharone-Ware + camoflauge

## Peto-Peto’s modified survival estimate + fresh theme

## modified Peto-Peto’s (by Andersen) + grass theme

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

# References

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)

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