# Optimal Designs for the Generalized Partial Credit Model

## TL;DR

This paper investigates optimal test designs for the generalized partial credit model (GPCM) in educational assessments, deriving conditions for local and Bayesian optimality, and showing that optimal designs often reduce to simple dichotomous models.

## Contribution

It introduces new optimal design strategies for the GPCM, including conditions for Bayesian optimality and reductions to the 2PL model, enhancing test efficiency in educational measurement.

## Key findings

- Local optimality achieved by using only first and last categories.
- Bayesian optimality conditions derived for symmetric weight distributions.
- Results applicable to the 2PL model as a special case.

## Abstract

Analyzing ordinal data becomes increasingly important in psychology, especially in the context of item response theory. The generalized partial credit model (GPCM) is probably the most widely used ordinal model and finds application in many large scale educational assessment studies such as PISA. In the present paper, optimal test designs are investigated for estimating persons' abilities with the GPCM for calibrated tests when item parameters are known from previous studies. We will derive that local optimality may be achieved by assigning non-zero probability only to the first and last category independently of a person's ability. That is, when using such a design, the GPCM reduces to the dichotomous 2PL model. Since locally optimal designs require the true ability to be known, we consider alternative Bayesian design criteria using weight distributions over the ability parameter space. For symmetric weight distributions, we derive necessary conditions for the optimal one-point design of two response categories to be Bayes optimal. Furthermore, we discuss examples of common symmetric weight distributions and investigate, in which cases the necessary conditions are also sufficient. Since the 2PL model is a special case of the GPCM, all of these results hold for the 2PL model as well.