Statistics for Maintenance Modelling: Regression Modelling

# Proportional Hazards Models¤

The second type of regression model commonly used in maintenance and reliability modelling is the proportional hazards models. In the proportional hazards models the hazard function satisfies:

h(t;x) = h0(tx) where:

- h(t;x) is the hazard under the conditions x
- h0(t;x) is some baseline hazard function
- x is some positive function of x.

The name 'proportional hazards models' comes from the simple mathematical fact that since h(t;x1)/h(t;x2) = x1 / x2 is independent of t, the hazards at different values of x are in constant proportion over time.

For this model, the survival function satisfies:

S(t;x) = S0(t;x) where S0 is the baseline survival function corresponding to h0.

This is simple to prove since:

ln S(t;x) = -h(t;x) dt = - x h0 dt = x ln S0(t)