Statistics for Maintenance Modelling: Regression Modelling

# Models Based on the Weibull Distribution¤

The three models introduced so far, namely the accelerated life model, the proportional hazards model and the proportional odds model, can be adapted and generalised. First described by Lawless (1986), these generalisations will not be discussed here, although it is useful to be aware that they exist, and where to get more information about them from.

In the broadest sense, regression based on the normal distribution serves as the most useful model in all of statistics. However, as a consequence of the nature of reliability data, regression models based on the Weibull distribution are of particular interest here.

The general premise for models based on the Weibull distribution is to let the distribution parameters vary according to some relationship with a covariate, x. For instance, the Weibull survivor function is given by:

S(t;x) =
where is the Weibull shape parameter (independent of x), and x is a parameter whose exact value is dependent on the covariate x, for example, ln x = xT, where is a vector of regression coefficients.

If we let S0(t) = then we can write S(t;x) = S0(tax) where ax = 1/x and we have an accelerated life model. Alternatively, if we let S(t;x) = S0(t) where px = x-, we have a proportional hazards model. Clearly the models based on the Weibull distribution provide a wide ranging and powerful tool for modelling reliability data.

The models can be further generalised by letting the shape parameter also depend on a covariate. This may be appropriate when initial exploratory analysis of the data suggests that the Weibull shape parameter is not the same everywhere.