Estimation and Comparison of Survival Curves

A central question in empirical research, exemplified by clinical trials, is whether a significant effect, difference, or relationship exists when comparing groups or variables. This is determined by the statistical method of hypothesis testing. This chapter focuses on analyzing time-to-event data. It introduces the core functions governing a time-to-event variable: the density, cumulative distribution, survival, hazard, and cumulative hazard functions. The text describes and illustrates the phenomenon of right censoring. The Kaplan-Meier estimator for the survival function and the Nelson-Aalen estimator for the cumulative hazard function are heuristically derived, along with their variance estimators. The log-rank test, stratified log-rank test, and weighted log-rank tests are also heuristically derived for nonparametric testing of the equality of two survival functions. Important concepts such as the oneto- one relationship between the survival and cumulative hazard functions, stochastic ordering preservation, the independent censoring model, and the notations of counting process and Lebesgue-Stieltjes integrals are also presented and illustrated.