A new statistical model combining strength and binary choice, with applications to paired comparison problems
Paired comparison experiments are frequently used to rank a set of items, such as a class of products in consumer preference studies or sports teams in athletic competitions. Most of the literature focuses on binary response models, which confine their attention to summarizing the preferences and ignore any supplementary information contained in the degree of preference. Models of this type include the well-known Bradley-Terry and Thurstone-Mosteller models.
In some experiments, however, there is more information at the analyst's disposal than merely which items are deemed preferable to which others. Usually this information is in the form of accrued scores or worth. Consequently, another approach, not well commented upon in the literature seeks to model these observed worths (usually considered latent to the binary response formulation) by estimating two merits for each item: one relating to its ability to accumulate worth in a competition, and one describing its ability to prevent its adversaries from accumulating worth against it. These dual merits may again be used to determine a rank order or predict the outcome of future comparisons.
Each of these model families has its deficiencies. While the binary class of ranking models ignores available worth information, the accumulated worth models decouple the observed worths and, in effect, ignore the determination of preference. These two paradigms are explored, and their virtues and shortcomings detailed. In conclusion, a new statistical model is proposed which overcomes the deficiencies of its predecessor. It can be viewed as a synthesis of the binary response and accumulated worth approaches, incorporating information of both types. Its improvement over existing methods is demonstrated through a simulation study and applied to real data in Chapter 5.