Survival Analysis and Multilevel Modeling

Survival analysis (or time-to-event analysis) is used when the outcome of interest is the time until a specific event occurs — death, equipment failure, disease recurrence, customer churn, time to employment. The challenge: many participants have not experienced the event by the end of the study period (right censoring) — they were still 'event-free' at last follow-up. Censored observations cannot simply be excluded (this introduces bias) or treated as non-events (this ignores the partial information that the event had not occurred by a certain time). The Kaplan-Meier (KM) estimator is the standard nonparametric method for estimating the survival function S(t) — the probability of surviving past time t. KM accounts for censoring by computing survival probability at each event time using only participants still at risk. The KM curve is a step function that drops at each event time. Log-rank test: compares KM survival curves between two or more groups (e.g., treatment vs. control). H₀: survival curves are identical. The log-rank test gives equal weight to early and late differences; the Wilcoxon test weights early differences more heavily. Median survival time (time at which S(t) = 0.50) is the primary summary statistic — more informative than mean survival time, which is distorted by censored observations and cannot be estimated when fewer than 50% of participants have experienced the event.