Institute of Statistics, RWTH Aachen University, Aachen, Germany.
Psychometrika. 2022 Sep;87(3):1146-1172. doi: 10.1007/s11336-021-09838-2. Epub 2022 Feb 11.
The asymptotic posterior normality (APN) of the latent variable vector in an item response theory (IRT) model is a crucial argument in IRT modeling approaches. In case of a single latent trait and under general assumptions, Chang and Stout (Psychometrika, 58(1):37-52, 1993) proved the APN for a broad class of latent trait models for binary items. Under the same setup, they also showed the consistency of the latent trait's maximum likelihood estimator (MLE). Since then, several modeling approaches have been developed that consider multivariate latent traits and assume their APN, a conjecture which has not been proved so far. We fill this theoretical gap by extending the results of Chang and Stout for multivariate latent traits. Further, we discuss the existence and consistency of MLEs, maximum a-posteriori and expected a-posteriori estimators for the latent traits under the same broad class of latent trait models.
在项目反应理论（IRT）模型中，潜在变量向量的渐近后正态性（APN）是 IRT 建模方法的一个关键论点。在单个潜在特质的情况下，并在一般假设下，Chang 和 Stout（Psychometrika，58（1）：37-52，1993）证明了广泛的二项式潜在特质模型的 APN。在相同的设置下，他们还证明了潜在特质最大似然估计量（MLE）的一致性。从那时起，已经开发了几种考虑多元潜在特质并假设其 APN 的建模方法，这一假设迄今为止尚未得到证明。我们通过将 Chang 和 Stout 的结果扩展到多元潜在特质来填补这一理论空白。此外，我们还在相同的广泛潜在特质模型下讨论了潜在特质的 MLE、最大后验和期望后验估计量的存在性和一致性。
