Which statement correctly describes survival analysis functions?

• A. The survivor function S(t) is the probability of failing by time t; the failure function F(t) is the probability of surviving past time t; the probability density f(t) is the derivative of S(t).
• B. The survivor function S(t) is the probability that survival time exceeds t; the failure function F(t) is the probability of failing by time t; the probability density f(t) is the derivative of F(t).
• C. The survivor function S(t) equals F(t).
• D. The pdf f(t) is the derivative of the survival function S(t).

Answer: (B)

In survival analysis, the survival function S(t) represents the probability that the event of interest (failure) has not occurred by time t, i.e., S(t) = P(T > t). The failure function F(t) is the complement, giving the probability that failure has occurred by time t, F(t) = P(T ≤ t) = 1 − S(t). The probability density function f(t) is the rate of failure at time t and is defined as the derivative of the failure function, f(t) = dF(t)/dt. Since F(t) = 1 − S(t), you also have f(t) = − dS(t)/dt.

Therefore, the statement that S(t) is the probability survival time exceeds t, F(t) is the probability of failing by time t, and f(t) is the derivative of F(t) aligns with the standard definitions. The other options mix up these relationships or misstate the derivatives (for example, f(t) is not the derivative of S(t) but the negative derivative of S(t)).