Statistics for Maintenance Modelling: Proportional Hazards and Accelerated Life Models

# Example using a Weibull Distributed Baseline Survival Function¤

Suppose for some and S0(t) = , then since S(t; y) = {S0(t)}g(y) (from screen 9), S(t; y) = , which is also a proportional hazard model with a Weibull distributed baseline survival function using (see The Reliability Function for the Weibull Distribution screen in the Continuous Lifetime Distributions 1 session). This shows that if the baseline survival function is Weibull, then we can produce the same result either by an accelerated life model or by a proportional hazards model.

In the accelerated life model, the scaled life time t' = t • g(y) has survival function S0. Equivalently, the log lifetime may be expressed as ln t = ln t' - ln g(y) where t' has survival function S0(t'). Thus the distribution of ln t is just that of ln t' subject to a location shift of -ln g(y), and is some times referred to as location-scale modelling.

As an initial guide to whether an accelerated life model is appropriate, a plot of observed log times against y can provide information on the form of g(y), and can yield rough estimates of its parameters. Since ln t = ln t' - ln g(y), a plot of ln t against y is effectively a plot of -ln g(y) against y, ignoring the "error" term ln t'.