Which statement about the cumulative hazard function is correct?

• A. The derivative of the survival function.
• B. It is the integral of the hazard over time.
• C. It is the survival probability at time t.
• D. It is the hazard at time t.

Answer: (B)

The main concept here is how the cumulative hazard function accumulates the instantaneous failure risk over time and how it connects to survival and the hazard itself. The cumulative hazard H(t) is defined as the integral of the hazard rate h(u) from 0 to t: H(t) = ∫0^t h(u) du. This makes sense intuitively: at each moment, there is a small amount of risk h(u) per unit time, and integrating those rates up to time t sums all that risk into a total amount of “hazard exposure” by time t. Because, when you look at the relationship with survival, the survival function satisfies S(t) = exp(-H(t)). So, if the hazard is constant, say h, then H(t) = h t and S(t) = exp(-h t). The statement that the cumulative hazard is the integral of the hazard over time is therefore the correct one.

Understanding what the other options describe helps avoid confusion: the derivative of the survival function is not the cumulative hazard; it is dS/dt = -h(t) S(t), showing the instantaneous hazard moderates how quickly survival declines. The survival probability at time t is S(t), not H(t). The hazard at time t is h(t), not H(t). The cumulative hazard is a non-probabilistic quantity with units of hazard times time, whereas survival is a probability between 0 and 1.