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

by Marcin Kosiński · February 9, 2017

This article is originally published at

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,

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 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 $w_{t_i} = S1({t_i}) = \prod_{i=1}^{T}(\frac{1-e_{t_i}}{n_{t_i}+1})$ ,
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

library(survminer)
library(survival)
data("kidney", package="KMsurv")
fit <- survfit(Surv(time=time, event=delta) ~ type, data=kidney)

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

devtools::install_github('cttobin/ggthemr')

one can set up a global ggplot2 palette/theme with

library(ggthemr)

Error in library(ggthemr): nie ma pakietu o nazwie 'ggthemr'

ggthemr('dust')

Error in ggthemr("dust"): nie udało się znaleźć funkcji 'ggthemr'

and check current colors with

swatch()

Error in swatch(): nie udało się znaleźć funkcji 'swatch'

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

ggthemr("sea") # set ggthemr theme

Error in ggthemr("sea"): nie udało się znaleźć funkcji 'ggthemr'

ggsurvplot(
   fit, # fitted survfit object
   risk.table  = TRUE, # include risk table?
   conf.int    = TRUE, # add confidence intervals?
   pval        = TRUE, # add p-value to the plot?
   pval.method = TRUE, # write the name of the test  
                       # that was used compute the p-value?
   pval.method.coord = c(3, 0.1), # coordinates for the name
   pval.method.size = 4,          # size for the name of the test
   log.rank.weights = "survdiff", # type of weights in log-rank test
   
   ### few options are set by defualt in survminer
   ### we will need to turn them off to allow
   ### ggthemr to work in his full glory
   palette = swatch()[2:3],  # pass the active palette
   ggtheme      = NULL, # disable adding custom survminer theme
   font.x       = NULL, # disable adding custom survminer font for the x axis
   font.y       = NULL, # disable adding custom survminer font for the y axis
   font.main    = NULL, # disable adding custom survminer font for the title
   font.submain = NULL, # disable adding custom survminer font for the subtitle
   font.caption = NULL  # disable adding custom survminer font for the caption
)

Error in swatch(): nie udało się znaleźć funkcji 'swatch'

Log-rank (comp) + dust theme

ggthemr("dust") # set ggthemr theme

Error in ggthemr("dust"): nie udało się znaleźć funkcji 'ggthemr'

ggsurvplot(
   fit, # fitted survfit object
   risk.table  = TRUE, # include risk table?
   conf.int    = TRUE, # add confidence intervals?
   pval        = TRUE, # add p-value to the plot?
   pval.method = TRUE, # write the name of the test  
                       # that was used compute the p-value?
   pval.method.coord = c(3, 0.1), # coordinates for the name
   pval.method.size = 4,          # size for the name of the test
   log.rank.weights = "1", # type of weights in log-rank test
   
   ### few options are set by defualt in survminer
   ### we will need to turn them off to allow
   ### ggthemr to work in his full glory
   palette = swatch()[2:3],  # pass the active palette
   ggtheme      = NULL, # disable adding custom survminer theme
   font.x       = NULL, # disable adding custom survminer font for the x axis
   font.y       = NULL, # disable adding custom survminer font for the y axis
   font.main    = NULL, # disable adding custom survminer font for the title
   font.submain = NULL, # disable adding custom survminer font for the subtitle
   font.caption = NULL  # disable adding custom survminer font for the caption
)

Error in swatch(): nie udało się znaleźć funkcji 'swatch'

Gehan-Breslow (generalized Wilcoxon) + flat dark theme

ggthemr("flat dark") # set ggthemr theme

Error in ggthemr("flat dark"): nie udało się znaleźć funkcji 'ggthemr'

ggsurvplot(
   fit, # fitted survfit object
   risk.table  = TRUE, # include risk table?
   conf.int    = TRUE, # add confidence intervals?
   pval        = TRUE, # add p-value to the plot?
   pval.method = TRUE, # write the name of the test  
                       # that was used compute the p-value?
   pval.method.coord = c(5, 0.1), # coordinates for the name
   pval.method.size = 4,          # size for the name of the test
   log.rank.weights = "n", # type of weights in log-rank test
   
   ### few options are set by defualt in survminer
   ### we will need to turn them off to allow
   ### ggthemr to work in his full glory
   palette = swatch()[2:3],  # pass the active palette
   ggtheme      = NULL, # disable adding custom survminer theme
   font.x       = NULL, # disable adding custom survminer font for the x axis
   font.y       = NULL, # disable adding custom survminer font for the y axis
   font.main    = NULL, # disable adding custom survminer font for the title
   font.submain = NULL, # disable adding custom survminer font for the subtitle
   font.caption = NULL  # disable adding custom survminer font for the caption
)

Error in swatch(): nie udało się znaleźć funkcji 'swatch'

Tharone-Ware + camoflauge

ggthemr("camoflauge") # set ggthemr theme

Error in ggthemr("camoflauge"): nie udało się znaleźć funkcji 'ggthemr'

ggsurvplot(
   fit, # fitted survfit object
   risk.table  = TRUE, # include risk table?
   conf.int    = TRUE, # add confidence intervals?
   pval        = TRUE, # add p-value to the plot?
   pval.method = TRUE, # write the name of the test  
                       # that was used compute the p-value?
   pval.method.coord = c(3, 0.1), # coordinates for the name
   pval.method.size = 4,          # size for the name of the test
   log.rank.weights = "sqrtN", # type of weights in log-rank test
   
   ### few options are set by defualt in survminer
   ### we will need to turn them off to allow
   ### ggthemr to work in his full glory
   palette = swatch()[2:3],  # pass the active palette
   ggtheme      = NULL, # disable adding custom survminer theme
   font.x       = NULL, # disable adding custom survminer font for the x axis
   font.y       = NULL, # disable adding custom survminer font for the y axis
   font.main    = NULL, # disable adding custom survminer font for the title
   font.submain = NULL, # disable adding custom survminer font for the subtitle
   font.caption = NULL  # disable adding custom survminer font for the caption
)

Error in swatch(): nie udało się znaleźć funkcji 'swatch'

Peto-Peto’s modified survival estimate + fresh theme

ggthemr("fresh") # set ggthemr theme

Error in ggthemr("fresh"): nie udało się znaleźć funkcji 'ggthemr'

ggsurvplot(
   fit, # fitted survfit object
   risk.table  = TRUE, # include risk table?
   conf.int    = TRUE, # add confidence intervals?
   pval        = TRUE, # add p-value to the plot?
   pval.method = TRUE, # write the name of the test  
                       # that was used compute the p-value?
   pval.method.coord = c(5, 0.1), # coordinates for the name
   pval.method.size = 4,          # size for the name of the test
   log.rank.weights = "S1", # type of weights in log-rank test
   
   ### few options are set by defualt in survminer
   ### we will need to turn them off to allow
   ### ggthemr to work in his full glory
   palette = swatch()[2:3],  # pass the active palette
   ggtheme      = NULL, # disable adding custom survminer theme
   font.x       = NULL, # disable adding custom survminer font for the x axis
   font.y       = NULL, # disable adding custom survminer font for the y axis
   font.main    = NULL, # disable adding custom survminer font for the title
   font.submain = NULL, # disable adding custom survminer font for the subtitle
   font.caption = NULL  # disable adding custom survminer font for the caption
)

Error in swatch(): nie udało się znaleźć funkcji 'swatch'

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

ggthemr("grass") # set ggthemr theme

Error in ggthemr("grass"): nie udało się znaleźć funkcji 'ggthemr'

ggsurvplot(
   fit, # fitted survfit object
   risk.table  = TRUE, # include risk table?
   conf.int    = TRUE, # add confidence intervals?
   pval        = TRUE, # add p-value to the plot?
   pval.method = TRUE, # write the name of the test  
                       # that was used compute the p-value?
   pval.method.coord = c(5, 0.1), # coordinates for the name
   pval.method.size = 4,          # size for the name of the test
   log.rank.weights = "S2", # type of weights in log-rank test
   
   ### few options are set by defualt in survminer
   ### we will need to turn them off to allow
   ### ggthemr to work in his full glory
   palette = swatch()[2:3],  # pass the active palette
   ggtheme      = NULL, # disable adding custom survminer theme
   font.x       = NULL, # disable adding custom survminer font for the x axis
   font.y       = NULL, # disable adding custom survminer font for the y axis
   font.main    = NULL, # disable adding custom survminer font for the title
   font.submain = NULL, # disable adding custom survminer font for the subtitle
   font.caption = NULL  # disable adding custom survminer font for the caption
)

Error in swatch(): nie udało się znaleźć funkcji 'swatch'

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

ggthemr("light") # set ggthemr theme

Error in ggthemr("light"): nie udało się znaleźć funkcji 'ggthemr'

ggsurvplot(
   fit, # fitted survfit object
   risk.table  = TRUE, # include risk table?
   conf.int    = TRUE, # add confidence intervals?
   pval        = TRUE, # add p-value to the plot?
   pval.method = TRUE, # write the name of the test  
                       # that was used compute the p-value?
   pval.method.coord = c(5, 0.1), # coordinates for the name
   pval.method.size = 4,          # size for the name of the test
   log.rank.weights = "FH_p=1_q=1", # type of weights in log-rank test
   
   ### few options are set by defualt in survminer
   ### we will need to turn them off to allow
   ### ggthemr to work in his full glory
   palette = swatch()[2:3],  # pass the active palette
   ggtheme      = NULL, # disable adding custom survminer theme
   font.x       = NULL, # disable adding custom survminer font for the x axis
   font.y       = NULL, # disable adding custom survminer font for the y axis
   font.main    = NULL, # disable adding custom survminer font for the title
   font.submain = NULL, # disable adding custom survminer font for the subtitle
   font.caption = NULL  # disable adding custom survminer font for the caption
)

Error in swatch(): nie udało się znaleźć funkcji 'swatch'

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

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Categories

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

Log-rank statistic for 2 groups

Weighted Log-rank extensions

Why are they useful?

Plots

gghtemr

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

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