A principled approach to cyclicality and intransitivity in cardinal paired comparison data is developed within the framework of graphical linear models. Fundamental to our developments is a detailed understanding and study of the parameter space which accommodates cyclicality and intransitivity. In particular, the relationships between the reduced, completely transitive model, the full, not necessarily transitive model, and all manner of intermediate models are explored for both complete and incomplete paired comparison graphs. It is shown that identifying cyclicality and intransitivity reduces to a model selection problem and a new method for model selection employing geometrical insights, unique to the problem at hand, is proposed. The large sample properties of the estimators as well as guarantees on the selected model are provided. It is thus shown that in large samples all cyclicalities and intransitivities can be identified. The method is exemplified using simulations and the analysis of an illustrative example.

翻译：本文在图形线性模型的框架内，针对基数型配对比较数据中的循环性与非传递性提出了一种原理性分析方法。我们研究的基础是对容纳循环性与非传递性的参数空间进行深入理解与系统探究。特别地，本文针对完整与非完整的配对比较图，探讨了简约的完全传递模型、非必然传递的完整模型以及各类中间模型之间的关系。研究表明，识别循环性与非传递性可归结为模型选择问题，并提出了一种基于几何视角的新型模型选择方法，该方法充分利用了本问题特有的几何特性。研究给出了估计量的大样本性质及所选模型的统计保证，从而证明在大样本条件下所有循环性与非传递性均可被识别。最后通过模拟实验与示例分析展示了该方法的应用。