What is the survival function S(t)?

• A. The probability of surviving beyond time t.
• B. The probability of failing by time t.
• C. The density of event times.
• D. The hazard at time t.

Answer: (A)

The survival function S(t) is the probability of not yet experiencing the event by time t. In other words, it answers “what is the chance of surviving beyond time t?” It’s the complement of the cumulative distribution function F(t) = P(T ≤ t), since F(t) = 1 − S(t). This function starts at S(0) = 1 (everyone has survived up to the starting time) and typically decreases as t increases, approaching zero if the event occurs for everyone given enough time.

So the correct concept is the probability of surviving beyond time t. The other ideas describe related but different quantities: the probability of failing by time t is 1 − S(t) (that’s F(t)); the density of event times is f(t) = dF/dt; and the hazard at time t is the instantaneous event rate given survival to t, h(t) = f(t)/S(t).