The practical motivation for the work described in this thesis arose from the development of a new Jaguar car engine. Development tests on prototype engines led to multiple failure time data which are modelled as a non-homogeneous Poisson process in its log-linear form. Initial analysis of the data using failure time plots showed considerable differences between prototype engines and suggested the use of models incorporating random effects for the engine effects. These models were fitted using the method of maximum likelihood. Two random effects have been considered: a proportional effect and a time dependent effect. In each case a simulation study showed the method of maximum likelihood to produce good estimates of the parameters and standard errors. There is also shown to be a bias in the estimate of the random effect, especially in smaller samples. The likelihood ratio test has been shown to be valid in assessing the statistical significance of the random effect, and a simulation exercise has demonstrated this in practical terms. Applying this test to the models fitted to the Jaguar data gives the proportional random effect to be significant while the time dependent random effect is not found to be significantly different from zero. This test has also been demonstrated to be of use in distinguishing between the two models and again the proportional random effect model is found to be more suitable for the Jaguar data. Residual analysis is performed to aid model validation Covariates are included, in various forms, in the proportional random effect model and the inclusion of these in the time dependent model is briefly discussed. The use of these models is demonstrated for the Jaguar data by including the type of test an engine performed as a covariate. The covariate models have also been used to compare engine phases. A framework for extending the models for interval censored data is developed. Finally this thesis discusses possible extensions of the work summarised in the previous paragraphs. This includes work on alternative models, Bayesian methods and experimental design.

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