Thurstonian Covariance and Correlation Structures for Multiple Judgment Paired Comparison Data

Thurstone's (1927) is not a proper model for multiple judgment paired comparison data as it assigns zero probabilities to all intransitive patterns. To obtain a proper model, Takane (1987) extended Thurstone's model by adding a vector of pair specific random errors. We investigate an unrestricted Thurstone-Takane model when (a) the variances of the paired specific errors are assumed to be equal, and (b) when they are assumed to be unequal.We also consider a new model, an unrestricted Thurstonian mean and correlation structure model. As the Thurstone-Takane model is a mean and covariance structure model, and its covariance structure is not scale invariant, Thurstonian correlation and covariance structure models are not equivalent, except in some restricted cases. In particular, this correlation structure model is equivalent to the mean and covariance structure model considered in Maydeu-Olivares (2001). Yet, their substantive interpretation differ