Survival Analysis: Kaplan–Meier Estimation and Cox Models in Epidemiology

This chapter introduces statistical methods for analyzing time-to-event data in epidemiological and clinical research. Unlike conventional approaches that focus only on whether an event occurs, survival analysis considers both the time until the event and the event status at the end of observation. Students learn fundamental concepts, including event definition, time origin, right censoring, and left truncation, using examples such as time to death, disease recurrence, first infection, and treatment failure. The chapter also introduces the survival function, S(t), which represents the probability of remaining event-free beyond a specified time, and the hazard function, h(t), which describes the instantaneous event rate among individuals who remain at risk. Understanding these concepts helps Statistics students recognize the importance of follow-up time and censoring when analyzing health outcomes and distinguish survival analysis from logistic and linear regression.

The second major focus is Kaplan–Meier estimation and comparing survival between groups. Students learn how the Kaplan–Meier estimator calculates survival probabilities while incorporating censored observations, producing step-shaped survival curves that decrease when events occur. The chapter explains how to interpret survival probabilities, median survival time, standard errors, and 95% confidence intervals to assess the uncertainty of survival estimates. Students also explore the log-rank test, a nonparametric method for comparing survival distributions between two or more groups, such as patients receiving different treatments or populations with different health characteristics. Through practical examples and R programming using the survival and survminer packages, students gain experience estimating survival curves, visualizing differences between groups, and interpreting statistical test results. The chapter emphasizes that Kaplan–Meier analysis and the log-rank test describe and compare survival patterns but do not provide adjusted estimates of exposure effects when multiple covariates must be considered simultaneously.

The final component introduces the Cox proportional hazards model and evaluates its assumptions. Students learn how the Cox model relates the hazard of an event to explanatory variables, including age, exposure status, treatment, and other potential confounders, while leaving the baseline hazard unspecified. Regression coefficients are interpreted as hazard ratios (HRs), which quantify relative differences in event rates between individuals or groups with different covariate values while holding other variables constant. The chapter also discusses model estimation, confidence intervals, p-values, and the importance of assessing the proportional hazards assumption, which requires hazard ratios to remain constant over time. Diagnostic approaches include log-minus-log survival plots, Schoenfeld residuals, and the cox.zph() function in R. In addition, the material introduces time-varying covariates and the implications of changes in exposure effects over follow-up. Overall, this chapter equips Statistics students with the conceptual knowledge and computational skills to analyze censored time-to-event data, compare survival experiences, estimate adjusted hazard ratios, evaluate model assumptions, and communicate reliable epidemiological findings for clinical and public health decision-making.