A law of ordinal random error: The Rasch measurement model and random error distributions of ordinal assessments

Abstract

Assessments in ordered categories are ubiquitous in educational, social and health sciences. These assessments are analogous to measurements in the natural sciences in that an idealised linear continuum is partitioned by successive thresholds into contiguous, ordered categories. In advanced analyses, the ordinal assessments are characterised with a probabilistic model as a function of a vector of threshold parameters defining the categories and a scalar parameter for the entity of measurement which is taken to be a measurement on an interval scale with an arbitrary origin and unit. One such model is the Rasch measurement model. If the ordinal assessments fit the model the probability distribution is taken to be a random error distribution of inferred replicated assessments. Therefore, it is analogous to the Gaussian random error distribution of replicated measurements known as the law of error. However, the Gaussian distribution is strictly log-concave which makes it unimodal with a smooth transition between probabilities of adjacent measurements. Such a distribution, referred to as randomly unimodal, ensures there is no evidence that unknown factors have produced systematic errors, and in turn justifies the mean as an estimate of the measure of the entity. The paper establishes that random unimodality arises from the natural ordering of the thresholds in the Rasch measurement model. Then by analogy to the Gaussian law of error, a distribution of ordinal assessments that has its thresholds in the natural order and fits the Rasch model may be said to satisfy the law of ordinal error. Again by analogy to Gaussian distribution with respect to replicated measurements, the law of ordinal error ensures that no unaccounted-for factors have produced systematic errors and that the estimates of the scalar parameter of the Rasch model can be taken as an estimate of the measure of the entity assessed.