Modelling ordinal assessments: fit is not sufficient. Issue 12 (18th June 2019) - Andrich, David; Pedler, Pender
Abstract: Assessments in ordered categories are ubiquitous in the social sciences. These assessments are assigned ordinal counts and analyzed with probabilistic models. If the counts fit the model, it is assumed that no unaccounted for factors govern the distribution and that it is a random error distribution. However, because tests of fit utilize parameter estimates from the data, the data may fit the model even when the modeled distributions cannot be random error distributions. This paper applies the additional criterion of strict unimodality, common to all random error distributions, to decide if a modeled distribution is not a random error distribution. However, not only are common random error distributions strictly unimodal, they are also strictly log-concave, a stronger form of unimodality which ensures smooth transitions between probabilities of adjacent counts. The paper shows that the operation for determining the strict unimodality also ensures that the distribution is locally strictly log-concave around the measure of the entity of assessment.