# Bayesian Thurstonian IRT Modeling: Logical Dependencies as an Accurate Reflection of Thurstone’s Law of Comparative Judgment

## TL;DR

This paper introduces a Bayesian Thurstonian IRT model that accurately accounts for logical and stochastic dependencies in forced-choice data.

## Contribution

The novel approach reduces computational effort by respecting logical and stochastic dependencies in Thurstonian IRT.

## Key findings

- The Bayesian implementation respects both logical and stochastic dependencies in the Thurstonian IRT model.
- A simulation shows a significant reduction in computational effort with the new approach.
- The model ensures correct parameter estimation by eliminating impossible answer patterns.

## Abstract

Thurstonian item response theory (Thurstonian IRT) is a well-established approach to latent trait estimation with forced choice data of arbitrary block lengths. In the forced choice format, test takers rank statements within each block. This rank is coded with binary variables. Since each rank is awarded exactly once per block, stochastic dependencies arise, for example, when options A and B have ranks 1 and 3, C must have rank 2 in a block of length 3. Although the original implementation of the Thurstonian IRT model can recover parameters well, it is not completely true to the mathematical model and Thurstone’s law of comparative judgment, as impossible binary answer patterns have a positive probability. We refer to this problem as stochastic dependencies and it is due to unconstrained item intercepts. In addition, there are redundant binary comparisons resulting in what we call logical dependencies, for example, if within a block
A<B
and
B<C
, then
A<C
must follow and a binary variable for
A<C
is not needed. Since current Markov Chain Monte Carlo approaches to Bayesian computation are flexible and at the same time promise correct small sample inference, we investigate an alternative Bayesian implementation of the Thurstonian IRT model considering both stochastic and logical dependencies. We show analytically that the same parameters maximize the posterior likelihood, regardless of the presence or absence of redundant binary comparisons. A comparative simulation reveals a large reduction in computational effort for the alternative implementation, which is due to respecting both dependencies. Therefore, this investigation suggests that when fitting the Thurstonian IRT model, all dependencies should be considered.