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The aim of this research is to analyze parametric survival models in the presence of left, right, interval and doubly interval censored data with time-dependent covariates. In this research we utilize and extend two important parametric survival models, the Gompertz and the exponential, to accommodate these censoring schemes and time-dependent covariates. The analysis starts with the extension of the Gompertz model to incorporate time-dependent covariates in the presence of right-censored data. Then, the performance of the model is compared with the fixed covariate model. Following that, comparison is made when a fixed covariate model was fitted wrongly to a data set with time-dependent covariate. In addition, two methods of constructing confidence intervals, the Wald and jackknife are explored for the parameters of this model. Conclusions are drawn based on the coverage probability study. In the next step, the Gompertz model is further extended to incorporate time dependent covariates with left, right and interval censored data as well as uncensored data. The model is then investigated thoroughly at dependent and independent covariate levels through a comprehensive simulation study. Following that, the model is compared with a fixed covariate model. Then, two methods of constructing confidence intervals the Wald and likelihood ratio are investigated for the parameters of the model and conclusions are drawn based on the coverage probability study. Finally, a parametric survival model that accommodates doubly interval-censored data with time-dependent covariates is developed and studied. In order to formulate this censoring scheme let V and W be the times of two related consecutive events where both of them are interval-censored and V ≤ W. Then, the survival time of interest could be defined as, T = W −V . Here it is assumed that the time to the first event, V , and the survival time, T, follow the exponential distribution (special case for Gompertz distribution). In

This research looks at the simulation of interval censored data when the survivor function of the survival time is known and attendance probability of the subjects for follow-ups can take any number between 0 to 1. Interval censored data often arise in the medical and biological follow-up studies where the event of interest occurs somewhere between two known times. Regardless of the methods used to analyze these types of data, simulation of interval censored data is an important and challenging step toward model building and prediction of survival time. The simulation itself is rather tedious and very computer intensive due to the interval monitoring of subjects at prescheduled times and subject's incomplete attendance to follow-ups. In this paper the simulated data by the proposed method were assessed using the bias, standard error and root mean square error (RMSE) of the parameter estimates where the survival time T is assumed to follow the Gompertz distribution function.

In this paper we extend a repairable system model that incorporates both time trend and renewal-type behavior to include a time dependent covariate. We calculated the bias, standard error andrmse of the parameter estimates of this model at different sample sizes using simulated data.Following that, we studied several alternative computer intensive methods of constructingconfidence interval estimates for the parameters of the general model. Alternative methodsrelieve us from making assumptions and having to depend solely on the traditional methods derived from asymptotic statistical theory. In addition, the high capability of modern daycomputers makes these methods easily applicable and practical. Several parametric bootstrap methods and jackknife confidence interval procedures were compared to the Wald interval via coverage probability study. The results clearly show that the B-t and jackknife techniques work much better than other methods when sample sizes are moderate and low. The Wald intervals was found to be highly asymmetrical and only starts to work when sample sizes are rather large.