# On the Structure of Ordered Latent Trait Models

## TL;DR

This paper explores the structure of ordered latent trait models, clarifies the role of binary models within them, and discusses threshold ordering, enhancing understanding of model construction and interpretation.

## Contribution

It provides a detailed analysis of how binary models underpin various ordinal models and clarifies the interpretation of parameters, especially in the partial credit model.

## Key findings

- Binary models are integral to ordinal models.
- Threshold ordering is not essential for the partial credit model.
- All models exhibit a Guttman space structure.

## Abstract

Ordered item response models that are in common use can be divided into three groups, cumulative, sequential and adjacent categories model. The derivation and motivation of the models is typically based on the assumed presence of latent traits or underlying process models. In the construction frequently binary models play an important role. The objective of this paper is to give motivations for the models and to clarify the role of the binary models for the various types of ordinal models. It is investigated which binary models are included in an ordinal model but also how the models can be constructed from a sequence of binary models. In all the models one finds a Guttman space structure, which has previously been investigated in particular for the partial credit model. The consideration of the binary models adds to the interpretation of model parameters, which is helpful, in particular, in the case of the partial credit model, for which interpretation is less straightforward than for the other models. A specific topic that is addressed is the ordering of thresholds in the partial credit model because for some researchers reversed ordering is an anomaly, others disagree. It is argued that the ordering of thresholds is not a constitutive element of the partial credit model.