In survival analysis, what is the survival function S(t)?

• A. S(t) = P(T > t), the probability that survival time exceeds time t.
• B. S(t) = P(T <= t)
• C. S(t) = hazard at time t.
• D. S(t) = cumulative incidence.

Answer: (A)

The survival function describes the probability of surviving beyond a given time t. If T is the time to the event, S(t) is defined as P(T > t), which is the complement of the cumulative distribution F(t) = P(T ≤ t). So S(t) = 1 − F(t). This function starts at S(0) = 1 (everyone is alive at time zero) and typically decreases over time, approaching zero as time grows large when the event eventually occurs for individuals.

The hazard function h(t) is a different concept: it is the instantaneous risk of the event at time t given survival to that moment, not the probability of surviving. They relate through h(t) = f(t) / S(t), where f(t) = dF/dt is the density and f(t) = −dS/dt. Also, cumulative incidence refers to the probability of experiencing the event by time t in the presence of competing risks, which is not the same as the survival function.