An efficient adaptive dimension selection algorithm for multidimensional probit graded response models

Determining the appropriate number of latent dimensions in multidimensional item response theory (MIRT) models is a persistent challenge. Traditional approaches, such as fitting multiple models and comparing them via information criteria (e.g., AIC, BIC) or cross-validation, are computationally expensive and often ignore uncertainty in dimensionality. This paper addresses this by developing an adaptive Bayesian framework that treats the latent dimension as an estimable parameter.

The authors employ a multidimensional probit graded response model (MGRM) and introduce a cumulative ordered spike-and-slab (COSS) prior on the item loading matrix. This prior imposes an ordered structure where higher-indexed latent dimensions are increasingly likely to be shrunk toward zero. By using Albert-Chib latent response augmentation, the authors transform the non-linear ordinal probit likelihood into a series of conditionally Gaussian updates. This allows for the implementation of an efficient Gibbs sampler that dynamically learns the effective latent dimensionality during the estimation process.

Simulation studies demonstrate that the proposed algorithm accurately recovers the true latent structure and parameter values across various scenarios. Compared to conventional fixed-dimensional estimation procedures, this approach avoids the need for repeated model fitting, significantly reducing computational overhead. The authors also illustrate the method's practical utility by applying it to real psychological assessment data, where it successfully identifies an interpretable latent structure.

This research provides a robust, automated solution for psychometricians and educational researchers who need to determine the complexity of latent constructs in ordinal data. By integrating dimension selection directly into the Bayesian estimation process, the method improves both computational efficiency and the interpretability of latent factor models, helping to prevent overfitting and the inclusion of redundant dimensions.