Multistage Ranking Models
将排序过程分解为多个阶段，在每个阶段采用概率模型，并假设各阶段选择的准确性独立，从而构建层次化模型来分析人群对项目的偏好和共识。
Abstract Suppose that a sample of people independently examines a fixed set of k items and then ranks these items according to personal judgment. The process of ranking the items is decomposed into k −1 stages. In the forward model, the most preferred item is selected at the first stage, the best of the remaining items is selected at the second stage, and so on until the least preferred item is selected by default. Various probability models are adopted at each stage, and properties of the resulting models for randomly sampled rankings are investigated. Luce (1959) first proposed such a modeling scheme, where each item i was thought to have an intrinsic value θi , and the probability of choosing a particular item i at any stage, conditional on the set S of items not previously chosen, was given by I{i ∈ S}θ i /Σ j∈S θ j , where I{} is an indicator function. Plackett (1975) began with the same model but added interaction terms that would theoretically extend the usefulness of this approach when the basic model did not adequately describe the data. The major difficulty with these formulations was the analytic intractability of the models, especially when the number k of items under consideration was more than a few. Our approach avoids analytic difficulties by assuming that the “accuracy” of the choice made at any stage is independent of the accuracies at the other stages. Accuracy is assessed with respect to a central ranking that indicates the general opinion of the population regarding the ordering of the items. Once a central ranking has been determined, a multistage model is parameterized by the probabilities of the different degrees of accuracy at each stage. Our multistage models form a hierarchy with three levels. At the most general level, no assumption is made about the probabilities of the possible degrees of accuracy at each stage. At the second level, these probabilities are assumed to increase with the degree of accuracy. The third level of the hierarchy contains models whose probabilities are linked across stages. Examples of models at this level include the ϕ model (Mallows 1957) and its generalization, the ϕ-component model (Fligner and Verducci 1986a). This hierarchical structure is used to make inferences about item preferences via notions of consensus. For models at the second level of the hierarchy, the central ranking represents a simple consensus of the population. Necessary and sufficient conditions are developed under which a subclass of the third hierarchy exhibits a stronger notion of consensus. A sample of rankings of the social prestige of 10 occupations is analyzed to illustrate the kinds of inferences that may be made using these models.