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Merge pull request #127 from MarcinKosinski/log-comp
use survMisc::comp to enable using weighted Log-rank tests - fix #17
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vignettes/Specifiyng_weights_in_log-ran_comparisons.Rmd
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| --- | ||
| title: Specifiyng weights in Log-rank comparisons | ||
| author: Marcin Kosinski | ||
| date: "29-01-2017" | ||
| output: | ||
| html_document: | ||
| mathjax: default | ||
| fig_caption: true | ||
| toc: true | ||
| section_numbering: true | ||
| css: ggsci.css | ||
| vignette: > | ||
| %\VignetteEngine{knitr::rmarkdown} | ||
| %\VignetteIndexEntry{Specifiyng weights in Log-rank comparisons} | ||
| --- | ||
| ```{r include = FALSE} | ||
| library(knitr) | ||
| opts_chunk$set( | ||
| comment = "", | ||
| fig.width = 12, | ||
| message = FALSE, | ||
| warning = FALSE, | ||
| tidy.opts = list( | ||
| keep.blank.line = TRUE, | ||
| width.cutoff = 150 | ||
| ), | ||
| options(width = 150), | ||
| eval = TRUE | ||
| ) | ||
| ``` | ||
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| ```{r} | ||
| library("survminer") | ||
| ``` | ||
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| > This vignette covers changes between versions 0.2.4 and 0.2.5 for specifiyng weights in the log-rank comparisons done in `ggsurvplot()`. | ||
| # Log-rank statistic for 2 groups | ||
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| As it is stated in the literature, the Log-rank test for comparing survival (estimates of survival curves) in 2 groups ($A$ and $B$) is based on the below statistic | ||
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| $$LR = \frac{U^2}{V} \sim \chi(1),$$ | ||
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| 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 | ||
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| - $t_i$ 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, | ||
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| also remember about few notes | ||
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| $$e_{t_i}^A = n_{t_i}^A \frac{o_{t_i}}{n_{t_i}}, \ \ \ \ \ \ \ \ \ \ e_{t_i}^B = n_{t_i}^B \frac{o_{t_i}}{n_{t_i}},$$ | ||
| $$e_{t_i}^A + e_{t_i}^B = o_{t_i}^A + o_{t_i}^B$$ | ||
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| that's why we can substitute group $A$ with $B$ in $U$ and receive same results. | ||
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| # Weighted Log-rank extensions | ||
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| 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` | ||
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| - `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 $w_{t_i} = S(t_i)^p(1 - S(t_i))^q$. | ||
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| > Watch out for `FH` as I [submitted an info on survMisc repository](https://github.com/dardisco/survMisc/issues/15) where I think their mathematical notation is misleading for Fleming-Harrington. | ||
| ## Why are they useful? | ||
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| 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`. | ||
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| > Remember that test selection should be performed at the research design level! Not after looking in the dataset. | ||
| # Plots | ||
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| ```{r} | ||
| library("survival") | ||
| data("kidney", package="KMsurv") | ||
| fit <- survfit(Surv(time=time, event=delta) ~ type, data=kidney) | ||
| ``` | ||
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| After preparing a functionality for this GitHub's issue [Other tests than log-rank for testing survival curves and Log-rank test for trend](https://github.com/kassambara/survminer/issues/17) 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. | ||
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| ## Log-rank (survdiff) | ||
| ```{r} | ||
| ggsurvplot(fit, pval = TRUE, pval.method = TRUE) | ||
| ``` | ||
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| ## Log-rank (comp) | ||
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| ```{r} | ||
| ggsurvplot(fit, pval = TRUE, pval.method = TRUE, | ||
| log.rank.weights = "1") | ||
| ``` | ||
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| ## Gehan-Breslow (generalized Wilcoxon) | ||
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| ```{r} | ||
| ggsurvplot(fit, pval = TRUE, pval.method = TRUE, | ||
| log.rank.weights = "n", pval.method.coord = c(200, 0.1), | ||
| pval.method.size = 3) | ||
| ``` | ||
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| ## Tharone-Ware | ||
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| ```{r} | ||
| ggsurvplot(fit, pval = TRUE, pval.method = TRUE, | ||
| log.rank.weights = "sqrtN", pval.method.coord = c(100, 0.1), | ||
| pval.method.size = 4) | ||
| ``` | ||
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| ## Peto-Peto's modified survival estimate | ||
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| ```{r} | ||
| ggsurvplot(fit, pval = TRUE, pval.method = TRUE, | ||
| log.rank.weights = "S1", pval.method.coord = c(200, 0.1), | ||
| pval.method.size = 3) | ||
| ``` | ||
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| ## modified Peto-Peto's (by Andersen) | ||
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| ```{r} | ||
| ggsurvplot(fit, pval = TRUE, pval.method = TRUE, | ||
| log.rank.weights = "S2", pval.method.coord = c(200, 0.1), | ||
| pval.method.size = 3) | ||
| ``` | ||
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| ## Fleming-Harrington (p=1, q=1) | ||
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| ```{r} | ||
| ggsurvplot(fit, pval = TRUE, pval.method = TRUE, | ||
| log.rank.weights = "FH_p=1_q=1", | ||
| pval.method.coord = c(200, 0.1), pval.method.size = 4) | ||
| ``` | ||
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| # References | ||
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| - Gehan A. A Generalized Wilcoxon Test for Comparing Arbitrarily Singly-Censored Samples. Biometrika 1965 Jun. 52(1/2):203–23. [JSTOR](www.jstor.org/stable/2333825) | ||
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| - Tarone RE, Ware J 1977 On Distribution-Free Tests for Equality of Survival Distributions. Biometrika;64(1):156–60. [JSTOR](http://www.jstor.org/stable/2335790) | ||
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| - Peto R, Peto J 1972 Asymptotically Efficient Rank Invariant Test Procedures. J Royal Statistical Society 135(2):186–207. [JSTOR](http://www.jstor.org/stable/2344317) | ||
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| - 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](http://www.jstor.org/stable/2289169) | ||
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| - Billingsly P 1999 Convergence of Probability Measures. New York: John Wiley & Sons. [Wiley (paywall)](dx.doi.org/10.1002/9780470316962) |