Statistics for Maintenance Modelling: Proportional Hazards and Accelerated Life Models

# Analysis of the Semi-Parametric Proportional Hazards Model¤

In reliability and maintenance analysis, the function for h0(t) is usually specified or has to be specified. However, in other applications the functional form h0(t) does not need to be completely specified, such as in medical statistics. If h0(t) is not specified, or the application is only focused on the meaning of , a semi-parametric form of h(t; y) can be analysed and estimated.

Suppose that a random sample of n items yields r distinct lifetimes and n - r censored times (see the Censoring screen in the Model Selection session). Let the ordered life times be t(1), ..., t(j), ..., t(r) where t(1) is the shortest life time and t(r) is the longest among these r items. Let Rj be the set of items with t t(j), where t may be either an observed lifetime or a censored time. Here Rj is the risk set at time t(j), that is those items whose lifetimes are longer than t(j) and could yet fail. Estimation of may then be based on the 'likelihood function':

L()

where yi(j) is the ith observed covariate associated with the item observed to fail at time t(j), and yil is the ith observed covariate associated with item l among the risk set at time t(j).