This dissertation consists of three papers on a central topic: the application of scale type theory to the Rasch and 2PL models. Each paper uses a different framework or set of frameworks for defining a typology of scales.
In the first paper, I begin the analysis of scale types through the Stevens (1946) typology. I introduce the notion of difference scales using a formalization of this typology. In applying this formalization to the Rasch and 2PL models, I discuss alternate paradigms for conceptualizing the restrictive assumptions of the Rasch model.
In the second paper, I apply the Suppes (1958) typology to the two IRT models. The conclusions here echo and reinforce those of the first chapter. In the second half of this paper, I examine other perspectives on connecting scale type theory and the Rasch model, focusing on the case of additive conjoint measurement theory.
The first two papers consider the scale types of the odds and logit forms of the models separately. In the third paper, I look at typologies in which the model form is not a factor. The first of these is the axiomatic system of Hölder (1901), which identifies two types of quantities: magnitudes, and points on a line. The second is the homogeneity-uniqueness typology of Narens and Luce (1986), which classifies scales by the types of choices that are possible during the process of numeral assignment.
Together, the three papers form an argument for considering psychometric properties behaving as predicted by the Rasch model as having the scale type of ratio scale attributes, while those that behave as predicted by the 2PL model have the scale type of interval scale attributes.