Bayesian estimation of survival functions under stochastic precedence

When estimating the distributions of two random variables, X and Y; investigators often have prior information that Y tends to be bigger than X. To formalize this prior belief, one could potentially assume stochastic ordering between X and Y; which implies Pr(X less than or equal to z) greater than or equal to Pr(Y less than or equal to z) for all z in the domain of X and Y: Stochastic ordering is quite restrictive, though, and this article focuses instead on Bayesian estimation of the distribution functions of X and Y under the weaker stochastic precedence constraint, Pr(X less than or equal to Y) greater than or equal to 0.5. We consider the case where both X and Y are categorical variables with common support and develop a Gibbs sampling algorithm for posterior computation. The method is then generalized to the case where X and Y are survival times. The proposed approach is illustrated using data on survival after tumor removal for patients with malignant melanoma.