| Ranking methods are important in performance comparison of a group of units and in identifying outlying units. Examples of the former are to rank health services providers or educational institutions; examples of this later are to identify regions with elevated disease incidence and to identify differentially expressed genes. When (posterior) distributions of the parameters of interest are stochastically ordered, all reasonable ranking methods should lead to same result. However, when these distributions are not stochastically ordered, the performance of ranks based on traditional statistics (e.g., Maximum likelihood estimates, Bayes Posterior Means, hypothesis test statistics) are usually not optimal; since these statistics were not designed to produce effective ranks.; In this thesis, we consider loss function based ranking methods. With loss functions as guides, we use both parametric and semi-parametric hierarchical models to produce ranks and evaluate them by both mathematical analysis and computer simulation. We find that estimates that minimize Squared Error Loss for ranks (e.g., the posterior mean ranks) are effective, but in many applications interest focuses on identifying the relatively good (e.g. 7 in the upper 10%) or relatively poor performers. Therefore, we construct loss functions and optimizing rank estimates that address these goals and evaluate these and other candidate estimates. We apply our new ranking methods to two applications: ranking dialysis providers based on standardized mortality ratios using data from the United States Renal Data System and selection of the most differentially expressed genes using data on two groups of lung cancer patients. We compare results to traditional analyses of these data. |
