An Expanded Derivation of the Threshold Structure of the Polytomous Rasch Model That Dispels Any "Threshold Disorder Controversy"

Responses to items with formats in more than two ordered categories are ubiquitous in education and the social sciences. Because the putative ordering of the categories reflects an understanding of what it means to have more of the variable, it seems mandatory that the ordering of the categories is an empirical property of the assessments and not merely a property of the model used to analyze them. To provide an unequivocal interpretation of category ordering in rating formats, this article expands the original derivation of the polytomous Rasch model for ordered categories. To do so, it integrates a complex of mathematical relationships among response spaces from which a space of experimentally independent Bernoulli variables, characterized by Rasch's simple logistic model, can be inferred. From this inference, the article establishes the necessary and sufficient evidence to test the hypothesis that the required ordering of the categories is an empirical property of the assessments. This expanded derivation, which exposes how Adams, Wu, and Wilson (2012) misconstrue the model and its implications, is intended to dispel the so-called disordered threshold controversy they claim exists.