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

机译：序数随机误差定律：序数评估的Rasch测量模型和随机误差分布

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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. (C) 2018 Elsevier Ltd. All rights reserved.
机译：有序类别的评估在教育，社会和健康科学中普遍存在。这些评估类似于自然科学中的测量，因为所以通过连续的阈值分配成连续的，有序类别的理想线性连续体。在高级分析中，序数评估具有概率模型，作为阈值参数的向量的函数，该阈值参数的函数定义了测量实体的实体的标量参数，该标量被视为具有任意起点的间隔刻度的测量值。单元。一个这样的模型是Rasch测量模型。如果序列评估适合模型，则概率分布被认为是推断复制评估的随机误差分布。因此，它类似于称为误差法的复制测量的高斯随机误差分布。然而，高斯分布严格对数凹陷，其使其在相邻测量的概率之间具有平滑的过渡。这种分布，称为随机单峰，确保没有证据表明未知因素产生了系统错误，并且反过来证明平均值是实体衡量标准的估计。本文建立了随机单变性，从Rasch测量模型中的阈值的自然排序产生。然后通过比喻对高斯的误差定律，可以说是在自然秩序中具有其阈值的序数评估的分布，并可拟合Rasch模型来满足序数误差的定律。再次通过比较重复测量的高斯分布来再次，序号误差的定律确保不计算的因素产生系统误差，并且RASCH模型的标量参数的估计可以被视为对测量的估计实体评估。 （c）2018年elestvier有限公司保留所有权利。

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