We present a method for model selection based on a proper reference prior. The choice of prior is somewhat arbitrary so Bayesian sensitivity analysis plays an important role in the analysis. We illustrate the methods in the context of a case study. We consider survival times (e.g., time to recurrence of depression) from a clinical trial. Because of the nature of the application we consider a mixture model that allows for a "surviving fraction." A Bayesian treatment of this model has been considered previously by Chen, Hill, Greenhouse and Fayos (1985), Greenhouse and Paul (1995) and Stangl (1991). In this paper, we are concerned with the question: does treatment effect both the probability of being a survivor and the survival times of "non-survivors"? The question is cast as a model selection problem. Reference priors give rise to improper posteriors and, moreover, do not lead to well defined Bayes factors. We adapt the idea of Kass and Wasserman (1995) who proposed "unit information priors." These priors are somewhat ad-hoc. To address this concern, we perform a sensitivity analysis with respect to the priors. We also consider case influence. Our conclusion is that treatment is important for determining long term survival but, among short term survivors, treatment may be less predictive of survival time.