Survival analysis studies how long it takes for an event to happen, such as equipment failure, disease recurrence, or the time until a customer cancels a subscription. Unlike ordinary statistics, it is designed for data where some subjects have not had the event by the end of the study. These incomplete observations are called censored data, and they are still useful.
The main goal is to estimate the chance that a subject survives past a time t.
Key Facts
- Survival function: S(t) = P(T > t), where T is the event time.
- Hazard function: h(t) is the instantaneous event rate at time t among those still at risk.
- Kaplan-Meier estimate: S(t) = product over event times ti <= t of (1 - di / ni).
- At an event time ti, di is the number of events and ni is the number at risk just before ti.
- Right-censored observations reduce the number at risk after their censoring time but do not cause a drop in S(t).
- Median survival time is the time when S(t) first reaches or falls below 0.5.
Vocabulary
- Survival analysis
- A set of statistical methods for analyzing the time until a defined event occurs.
- Event
- The outcome being timed, such as failure, death, relapse, recovery, or cancellation.
- Censoring
- A situation where the exact event time is unknown, but partial information about survival time is available.
- Kaplan-Meier curve
- A step-shaped graph that estimates the survival probability over time using observed event and censoring information.
- Hazard
- The rate at which events occur at a given time among subjects who have not yet had the event.
Common Mistakes to Avoid
- Treating censored observations as event times is wrong because censoring means the event was not observed at that time.
- Ignoring censored observations is wrong because it throws away valid information about how long subjects were known to survive.
- Interpreting the Kaplan-Meier curve as a smooth trend is wrong because it changes only at observed event times and stays flat between them.
- Confusing survival probability with hazard is wrong because S(t) gives the probability of surviving past time t, while hazard describes the event rate among those still at risk.
Practice Questions
- 1 A study begins with 20 patients. At month 3, 2 patients have the event and none are censored before then. What is the Kaplan-Meier survival estimate immediately after month 3?
- 2 At month 6, 18 patients are still at risk and 3 have the event. If the survival estimate just before month 6 is 0.90, what is the Kaplan-Meier survival estimate immediately after month 6?
- 3 In a survival study, one participant leaves the study at month 8 without having the event. Explain how this censored observation affects the number at risk and why the survival curve does not drop at month 8.