Attitudes, order and quantity: deterministic and direct probabilistic tests of unidimensional unfolding.

: This article is the final in the series on unidimensional unfolding. The investigations of Kyngdon (2006b) and Michell (1994) were extended to include direct probabilistic tests of the quantitative and ordinal components of unfolding theory with the multinomial Dirichlet model (Karabatsos 2005); and tests of the higher order axiomatic conjoint measurement (ACM, Krantz, Luce, Suppes and Tversky (KLST) 1971) condition of triple cancellation. Strong Dirichlet model support for both the ordinal and quantitative components of unfolding was only found in datasets that satisfied at least double cancellation. In contrast, the Item Response Theory (IRT) simple hyperbolic cosine model for pairwise preferences (SHCMpp, Andrich 1995) fitted all datasets. The paper concluded the SHCMpp is suited to the instrumental rather than scientific task (Michell 2000) of psychological measurement; with the caveat of the problematic chi square fit statistic. The paper also presents original work by the second author on coherent tests of triple cancellation.