On the lack of comonotonicity between likert scores and Rasch-based measures

The Rating Scale Model (RSM) and the Partial Credit Model (PCM) are fairly well-known examples of Rasch models for polytomously scored items. In addition to a number of threshold parameters, both the models contain two scalar parameters characterizing item and person location on a common interval-level scale. The rank order of items and persons defined by the Likert summative scores (i.e. the raw total scores) is compared with that obtained from the Rasch-based measures (i.e. the maximum likelihood estimates of person and item parameters). It is proved that: 1) the property of comonotonicity between Likert summative scores and Rasch-based measures holds for both the person and item parameters of the RSM; 2) the property of comonotonicity between Likert summative scores and Rasch-based measures holds for the PCM only with reference to the person parameters; 3) violations of comonotonicity are possible, for particular datasets, for the item parameters of the PCM.