Survival Analysis Using R

The most commonly used survival analysis methods in clinical trials include:

Kaplan-Meier (KM) estimators: non-parametric statistics utilized for estimating the survival function
Log-rank test: a non-parametric test for comparing the survival functions across two or more groups
Cox proportional hazards (PH) model: a semi-parametric model often used to assess the relationship between the survival time and explanatory variables

Additionally, other methods for analyzing time-to-event data are available, such as:

Parametric survival model
Accelerated failure time model
Competing risk model
Restricted mean survival time
Time-dependent Cox model

While these models may be explored in a separate document, this particular document focuses solely on the three most prevalent methods: KM estimators, log-rank test and Cox PH model.

Analysis of Time-to-event Data

Below is a standard mock-up for survival analysis in clinical trials.

Example Data

Data source: https://stats.idre.ucla.edu/sas/seminars/sas-survival/

The data include 500 subjects from the Worcester Heart Attack Study. This study examined several factors, such as age, gender and BMI, that may influence survival time after heart attack. Follow up time for all participants begins at the time of hospital admission after heart attack and ends with death or loss to follow up (censoring). The variables used here are:

lenfol: length of followup, terminated either by death or censoring - time variable
fstat: loss to followup = 0, death = 1 - censoring variable
afb: atrial fibrillation, no = 0, 1 = yes - explanatory variable
gender: males = 0, females = 1 - stratification factor

library(tidyverse)
library(haven)
library(survival)
library(survminer)
library(broom)
library(knitr)
knitr::opts_chunk$set(echo = TRUE)

dat <- read_sas(file.path("../data/whas500.sas7bdat")) %>%
  mutate(LENFOLY = round(LENFOL/365.25, 2), ## change follow-up days to years for better visualization
         AFB = factor(AFB, levels = c(1, 0))) ## change AFB order to use "Yes" as the reference group to be consistent with SAS

The Non-stratified Model

First we try a non-stratified analysis following the mock-up above to describe the association between survival time and afb (atrial fibrillation).

The KM estimators are from survival::survfit function, the log-rank test uses survminer::surv_pvalue, and Cox PH model is conducted using survival::coxph function. Numerous R packages and functions are available for performing survival analysis. The author has selected survival and survminer for use in this context, but alternative options can also be employed for survival analysis.

KM estimators

fit.km <- survfit(Surv(LENFOLY, FSTAT) ~ AFB, data = dat)

## quantile estimates
quantile(fit.km, probs = c(0.25, 0.5, 0.75))

$quantile
        25   50   75
AFB=1 0.26 2.37 6.43
AFB=0 0.94 5.91 6.44

$lower
        25   50   75
AFB=1 0.05 1.27 4.24
AFB=0 0.55 4.32 6.44

$upper
        25   50 75
AFB=1 1.11 4.24 NA
AFB=0 1.47   NA NA

## landmark estimates at 1, 3, 5-year
summary(fit.km, times = c(1, 3, 5))

Call: survfit(formula = Surv(LENFOLY, FSTAT) ~ AFB, data = dat)

                AFB=1 
 time n.risk n.event survival std.err lower 95% CI upper 95% CI
    1     50      28    0.641  0.0543        0.543        0.757
    3     27      12    0.455  0.0599        0.351        0.589
    5     11       6    0.315  0.0643        0.211        0.470

                AFB=0 
 time n.risk n.event survival std.err lower 95% CI upper 95% CI
    1    312     110    0.739  0.0214        0.699        0.782
    3    199      33    0.642  0.0245        0.595        0.691
    5     77      20    0.530  0.0311        0.472        0.595

Log-rank test

survminer::surv_pvalue(fit.km, data = dat)

  variable         pval   method    pval.txt
1      AFB 0.0009646027 Log-rank p = 0.00096

Cox PH model

fit.cox <- coxph(Surv(LENFOLY, FSTAT) ~ AFB, data = dat)
fit.cox %>% 
  tidy(exponentiate = TRUE, conf.int = TRUE, conf.level = 0.95) %>%
  select(term, estimate, conf.low, conf.high)

# A tibble: 1 × 4
  term  estimate conf.low conf.high
  <chr>    <dbl>    <dbl>     <dbl>
1 AFB0     0.583    0.421     0.806

The Stratified Model

In a stratified model, the Kaplan-Meier estimators remain the same as those in the non-stratified model. To implement stratified log-rank tests and Cox proportional hazards models, simply include the strata() function within the model formula.

Stratified Log-rank test

fit.km.str <- survfit(Surv(LENFOLY, FSTAT) ~ AFB + strata(GENDER), data = dat)

survminer::surv_pvalue(fit.km.str, data = dat)

            variable        pval   method   pval.txt
1 AFB+strata(GENDER) 0.001506607 Log-rank p = 0.0015

Stratified Cox PH model

fit.cox.str <- coxph(Surv(LENFOLY, FSTAT) ~ AFB + strata(GENDER), data = dat)
fit.cox.str %>% 
  tidy(exponentiate = TRUE, conf.int = TRUE, conf.level = 0.95) %>%
  select(term, estimate, conf.low, conf.high)

# A tibble: 1 × 4
  term  estimate conf.low conf.high
  <chr>    <dbl>    <dbl>     <dbl>
1 AFB0     0.594    0.430     0.823