Time-to-Event Data, Censoring, and Hazard Functions

A pump is still running when a maintenance study ends. Its recorded duration is twelve months, but its lifetime is not twelve months: the lifetime is greater than twelve months. Treating that record as a failure moves an event to a time when no event occurred. Dropping it loses twelve months of evidence that the pump kept working. Survival analysis starts by preserving that distinction.

Probability Distributions and Descriptive Statistics provide the tools for describing durations. If every event time were observed, we could fit a distribution or summarize the complete durations directly. With incomplete follow-up, some pumps are still running at the administrative cutoff, and we know only a lower bound on their event times. This is right censoring. Ignoring the distinction between an event and a censoring time can distort estimates of how long things last.

This chapter defines the observed duration and event indicator, then connects the survival function, density, and hazard for a positive, absolutely continuous event time. The identity $S(t)=\exp(-H(t))$ connects survival probabilities to cumulative hazards. Under independent censoring and the other assumptions stated below, these functions give a likelihood that uses both exact event times and survival up to a censoring time.

By the end, you should be able to interpret right-censored records, construct their parametric likelihood, and explain why ordinary regression on observed durations can target the wrong outcome. The next chapters develop non-parametric and regression estimators:

• Kaplan–Meier and Nelson–Aalen provide the non-parametric view.
• The Cox proportional hazards model provides a semi-parametric regression.
• Accelerated failure-time models provide fully parametric alternatives.