Bivariate Semi-Parametric Model: Bayesian Inference

The motivation of this paper came from two bivariate diabetic retinopathy data sets. The main aim is to test whether the laser treatment delays the onset of blindness compared to the traditional treatment. The first data set is of 197 patients and it provides the onset of blindness of the two eyes. The observed data are heavily censored on both the variables. The second data set is of 91 patients, and it indicates the minimum time of the onset of blindness between the two eyes. In both the cases there is a significant proportion where the onset of blindness has occured on both the eyes simultaneously, i.e. there is a significant proportion where both the variables are equal. Hence, the two variables cannot be treated as independent. We have used a Marshall-Olkin bivariate exponential or Weibull type a bivariate semi-parametric model, where the base line distribution is more flexible than any parametric distribution. The base line distribution is assumed to have a piecewise constant hazard function and that makes the Bayes line hazard function to be very flexible. Marshall-Olkin bivariate exponential distribution becomes a special case of the model. The maximum likelihood estimators may not always exist, and we have considered the Bayesian inference of the unknown parameters. We have used importance sampling technique to compute Bayes estimators and the associated credible intervals. We have addressed some testing of hypothesis problem also based on Bayes factors. Finally, both the data sets have been analyzed, and it is observed for one data set the laser treatment has a significantly different effect than the traditional treatment, where as for the other data set no significantly different effect has been observed.