---
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
)
```

```{r}
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 $$U = \sum_{i=1}^{T}w_{t_i}(o_{t_i}^A-e_{t_i}^A), \ \ \ \ \ \ \ \ V = Var(U) = \sum_{i=1}^{T}(w_{t_i}^2\frac{n_{t_i}^An_{t_i}^Bd_i(n_{t_i}-o_{t_i})}{n_{t_i}^2(n_{t_i}-1)})$$
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](https://github.com/dardisco/survMisc/issues/15) 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 

```{r}
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 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.

## Log-rank (survdiff)
```{r}
ggsurvplot(fit, data = kidney, pval = TRUE, pval.method = TRUE)
```

## Log-rank (comp)

```{r}
ggsurvplot(fit, data = kidney, pval = TRUE, pval.method = TRUE,
           log.rank.weights = "1")
```

## Gehan-Breslow (generalized Wilcoxon)

```{r}
ggsurvplot(fit, data = kidney, pval = TRUE, pval.method = TRUE,
           log.rank.weights = "n", pval.method.coord = c(5, 0.1),
           pval.method.size = 3)
```

## Tharone-Ware

```{r}
ggsurvplot(fit, data = kidney, 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

```{r}
ggsurvplot(fit, data = kidney, 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)

```{r}
ggsurvplot(fit, data = kidney, 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)

```{r}
ggsurvplot(fit, data = kidney, 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. [JSTOR](www.jstor.org/stable/2333825)

- 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)

- 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)

- 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)

- Billingsly P 1999 Convergence of Probability Measures. New York: John Wiley & Sons. [Wiley (paywall)](dx.doi.org/10.1002/9780470316962)
