Which statement about the relationship between hazard and survival is true?

• A. The hazard equals derivative of the survival function.
• B. The survival equals the integral of the hazard over time.
• C. The derivative of the survival function equals minus hazard times survival: dS/dt = -h(t) S(t).
• D. The hazard is equal to the survival probability.

Answer: (C)

Hazard is the instantaneous risk of the event at time t for those still at risk, while the survival function S(t) is the probability of remaining event-free up to time t. The key link is that the rate at which the survival probability decreases is proportional to both the number still at risk (S(t)) and the instantaneous hazard h(t). This gives the ordinary differential equation dS/dt = -h(t) S(t). Put simply, the slope of the survival curve at any time t is determined by how high the hazard is and how many people are still at risk.

This relationship also explains why the survival function can be expressed in terms of the cumulative hazard H(t) = ∫0^t h(u) du: S(t) = exp(-H(t)). If the hazard is constant, S(t) declines exponentially as S(t) = e^{-h t}, assuming S(0) = 1.

Why the other statements don’t fit: the hazard is not simply the derivative of survival; in fact, the derivative of survival is -h(t) times S(t). The survival function is not the integral of hazard; that integral yields the cumulative hazard, not survival. And hazard is not equal to the survival probability, since they measure different things with different units.