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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Estimation procedures for lifetime distributions

In the Cox proportional hazards model h(t; z) = h0(t) exp(βz), what is the key property that makes the model 'proportional hazards'?

The defining property is that the ratio of hazards for two individuals with fixed covariates is constant over time, because the baseline hazard cancels and leaves exp of beta times the covariate difference. The baseline hazard itself can have any shape.

  1. AThe ratio of hazards for two individuals with fixed covariates is constant over timeCorrect
  2. BThe baseline hazard h0(t) must be constant over time
  3. CThe difference between the hazards of two individuals is constant over time
  4. DThe survival functions of any two individuals are parallel lines
  5. The covariates must change in proportion to time

Explanation

For two individuals with covariates z1 and z2, h(t;z1)/h(t;z2) = exp(β(z1−z2)), because the baseline hazard cancels. This ratio does not depend on t. The baseline hazard may take any shape, so a constant baseline is not required, and it is the ratio, not the difference, that is constant.

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