Survival Analysis: Kaplan-Meier and Cox Regression

Censored Data

In survival analysis, the event of interest (death, claim, lapse) may not be observed for all subjects. Right censoring occurs when a subject's observation ends before the event (e.g., a policyholder is still active when the study ends). The censoring time C and event time T produce the observed time Y = min(T, C) and a censoring indicator delta = I(T <= C). Ignoring censoring leads to biased estimates; survival analysis methods properly account for it.

Kaplan-Meier Estimator

The Kaplan-Meier (product-limit) estimator of the survival function is S_hat(t) = product over all event times t_i <= t of (1 - d_i/n_i), where d_i is the number of events at time t_i and n_i is the number at risk just before t_i. The KM estimator is a step function that decreases at each observed event time. Censored observations reduce the risk set but do not cause a step. The Greenwood formula provides the variance: Var(S_hat(t)) = S_hat(t)^2 * sum of d_i / (n_i * (n_i - d_i)).

The log-rank test compares survival curves between two groups using a chi-squared statistic based on observed versus expected events at each event time.

Cox Proportional Hazards Model

The Cox model specifies the hazard function as h(t|X) = h_0(t) * exp(beta'X), where h_0(t) is the unspecified baseline hazard. The model is semi-parametric: the beta coefficients are estimated without specifying h_0(t) using partial likelihood. exp(beta_j) is the hazard ratio for a one-unit increase in Xj, holding other predictors constant. The proportional hazards assumption means hazard ratios are constant over time. For Exam SRM, understand the interpretation of hazard ratios, the role of the baseline hazard, and how the Cox model handles censoring.