Survival Analysis

Survival analysis focuses on the time until an event of interest occurs, such as death, relapse, or device failure. It accounts for censored data, where the event has not occurred for some subjects during the study period.

---

1. Key Concepts

• Survival Time: Time from a defined starting point to the occurrence of the event

• Censoring: When the exact survival time is unknown (e.g., lost to follow-up)

• Hazard Function: The instantaneous rate of the event occurring at time t

---

2. Kaplan-Meier Estimator

Non-parametric method to estimate survival probabilities over time.

# Kaplan-Meier curve in R
library(survival)
km_fit <- survfit(Surv(time, status) ~ group, data = data)
plot(km_fit, xlab = "Time", ylab = "Survival Probability")

• time: follow-up time

• status: 1 = event occurred, 0 = censored

---

3. Log-rank Test

Used to compare survival curves between groups.

survdiff(Surv(time, status) ~ group, data = data)

---

4. Cox Proportional Hazards Model

Semi-parametric model that estimates hazard ratios while adjusting for covariates.

Model

[ h(t|X) = h_0(t) \cdot \exp(\beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_k X_k) ]

cox_model <- coxph(Surv(time, status) ~ age + sex + treatment, data = data)
summary(cox_model)

Assumptions

• Proportional hazards: the ratio of hazards is constant over time

• Linearity in log-hazard

---

5. Visualization

• Kaplan-Meier plots for survival function

• Forest plots for hazard ratios

---

6. Summary

Survival analysis is essential for time-to-event data. Use Kaplan-Meier for visualization, log-rank test for group comparison, and Cox models for multivariable adjustment.