On a law of ordinal error

When no systematic factor disturbs replicated measurements of the same entity with the same instrument, the observed or inferred distribution is expected to satisfy the Gaussian law of measurement error. A characteristic of this distribution, which ensures it is unimodal with a smooth transition between adjacent probabilities, is that it is strictly log-concave. Many assessments in the social sciences begin by analogy to measurements in that an idealised linear continuum is partitioned by successive thresholds into contiguous, ordered categories. However, the distances between successive thresholds, generally finite, are not equal and assessments remain ordinal. The paper establishes that if the thresholds in the probabilistic Rasch measurement model used to transform ordinal assessments into measurements are in their natural order, then the distribution is also strictly log-concave. Therefore it is proposed that the Rasch model with ordered thresholds be referred to as the law of ordinal error. Accordingly, by analogy to the expectation that the distribution of replicated measurements satisfy the law of measurement error, it is proposed that the observed or inferred distribution of replicated ordinal assessments be expected to satisfy the proposed law of ordinal error.