In survival analysis, what is the hazard function and what does the Cox proportional hazards model assume?

• A. The instantaneous risk of the event; the model assumes proportional hazards with hazard ratios constant over time.
• B. The cumulative probability of the event by time t; the model assumes hazard ratios vary with time.
• C. The average time until the event occurs; the model assumes constant hazard across individuals.
• D. The likelihood of censoring; the model assumes independence between censoring and survival times.

Answer: (A)

Hazard function describes the instantaneous risk of the event at time t, given that the subject has survived up to t. The Cox proportional hazards model expresses the hazard for an individual with covariates x as h(t|x) = h0(t) · exp(β′x), where h0(t) is an unspecified baseline hazard. The key assumption is proportional hazards: the hazard ratio between any two individuals is constant over time, i.e., h(t|x1)/h(t|x2) = exp(β′(x1 − x2)) and does not depend on t. This means covariates shift risk multiplicatively but do not change the way risk evolves over time.

So the statement that the hazard is the instantaneous risk and that the model assumes hazards are proportional with constant hazard ratios over time best captures the concept. The alternative ideas mix up what is being measured (cumulative probability vs. instantaneous hazard), or imply time-varying effects, which the proportional hazards framework does not permit. The model focuses on how covariates affect relative risk over time, not on censoring likelihood or a fixed average time to event.