Singularities of Multidimensional Item Response Models and the Rank of the Fisher Information Matrix
Abstract
Statistical models in item response theory are known to be singular. This singularity manifests itself as a loss of rank in the Fisher information matrix, and the usual theories underlying AIC and BIC, which assume regularity, cannot be applied without further justification. Chen and colleagues analyzed this degeneracy and nonregular likelihood ratio tests, but their analysis does not identify all the directions that produce degeneracy or their dimensions. In this study, we analyze the entire kernel of the Fisher information matrix as a linear space for multidimensional item response models, separating the degeneracy due to rotations from that due to an over-specified latent dimension without double counting. We determine the dimension of each component and the exact rank of the information matrix, and prove a formula for computing, from the item parameters, the dimension of the degeneracy that remains after these components are removed. These results go beyond establishing the existence of singularity to identifying its causes and extent, providing a foundation for inference and model comparison in IRT.
Related
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BibTeX
@misc{suzuki2026singularities,
title = {{Singularities of Multidimensional Item Response Models and the Rank of the Fisher Information Matrix}},
author = {Joe Suzuki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26144},
eprint = {2610.26144},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26144},
}
RIS
TY - UNPB
TI - Singularities of Multidimensional Item Response Models and the Rank of the Fisher Information Matrix
AU - Joe Suzuki
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26144
UR - https://arxiv.org/abs/2610.26144
AB - Statistical models in item response theory are known to be singular. This singularity manifests itself as a loss of rank in the Fisher information matrix, and the usual theories underlying AIC and BIC, which assume regularity, cannot be applied without further justification. Chen and colleagues analyzed this degeneracy and nonregular likelihood ratio tests, but their analysis does not identify all the directions that produce degeneracy or their dimensions. In this study, we analyze the entire kernel of the Fisher information matrix as a linear space for multidimensional item response models, separating the degeneracy due to rotations from that due to an over-specified latent dimension without double counting. We determine the dimension of each component and the exact rank of the information matrix, and prove a formula for computing, from the item parameters, the dimension of the degeneracy that remains after these components are removed. These results go beyond establishing the existence of singularity to identifying its causes and extent, providing a foundation for inference and model comparison in IRT.
ER -