MultiLCIRT: An R package for multidimensional latent class item response models

We illustrate a class of Item Response Theory (IRT) models for binary and ordinal polythomous items and we describe an R package for dealing with these models, which is named MultiLCIRT. The models at issue extend traditional IRT models allowing for (i) multidimensionality and (ii) discreteness of latent traits. This class of models also allows for different parameterizations for the conditional distribution of the response variables given the latent traits, depending on both the type of link function and the constraints imposed on the discriminating and the difficulty item parameters. We illustrate how the proposed class of models may be estimated by the maximum likelihood approach via an Expectation-Maximization algorithm, which is implemented in the MultiLCIRT package, and we discuss in detail issues related to model selection. In order to illustrate this package, we analyze two datasets: one concerning binary items and referred to the measurement of ability in mathematics and the other one coming from the administration of ordinal polythomous items for the assessment of anxiety and depression. In the first application, we illustrate how aggregating items in homogeneous groups through a model-based hierarchical clustering procedure which is implemented in the proposed package. In the second application, we describe the steps to select a specific model having the best fit in our class of IRT models.

💡 Research Summary

This paper introduces a comprehensive class of Item Response Theory (IRT) models designed for binary and ordinal polytomous items, implemented in the R package named MultiLCIRT. The proposed models significantly extend traditional IRT frameworks by incorporating two key features: (i) multidimensionality, allowing different items to measure distinct latent traits (between-item multidimensionality), and (ii) discreteness of latent traits, assuming the latent variable vector follows a discrete distribution with a finite number of support points (latent classes). This discrete assumption facilitates a semi-parametric estimation and avoids the computational complexity of evaluating multidimensional integrals required in continuous latent trait models.

The core of the model class is defined by a generalized equation for the conditional probability of an item response, which can be tailored through three specification choices: the type of link function (global/cumulative logits for Graded Response Models or local/adjacent-category logits for Partial Credit Models), constraints on the discrimination parameters (free for a 2PL-type or constrained to be equal across items for a Rasch-type), and formulation of the difficulty parameters (free or constrained under a rating scale assumption where thresholds are equally spaced). The combination of these choices generates a wide array of specific models, including multidimensional latent class versions of the Graded Response Model, Rating Scale Model, 2PL model, and Rasch model, unifying them under a single framework.

Parameter estimation is performed via the maximum likelihood method, implemented through an Expectation-Maximization (EM) algorithm. The paper details the model selection procedure, which is crucial for determining the optimal number of latent classes (k), the number of dimensions (s), the allocation of items to dimensions, and the specific parameterization (link function and constraints). This selection relies on statistical indices such as AIC and BIC. The package also includes a model-based hierarchical clustering procedure to help explore and aggregate items into homogeneous groups corresponding to potential dimensions.

The functionality of the MultiLCIRT package is described, highlighting main functions for model fitting (multiLC) and item aggregation (aggr). The paper demonstrates the practical utility of the package through two detailed applications. The first application uses binary item data from a mathematics ability assessment (NAEP project). It illustrates how the model-based clustering procedure can automatically group items measuring distinct latent traits, revealing the underlying multidimensional structure of the test. The second application analyzes ordinal polytomous data from the Hospital Anxiety and Depression Scale (HADS). It walks through the step-by-step model selection process to identify the best-fitting model (e.g., a two-dimensional model with a specific number of classes and a Graded Response parameterization) and subsequently interprets the characteristics of each resulting latent class in terms of estimated anxiety and depression levels, providing clinically meaningful profiles of patients.

In conclusion, the MultiLCIRT package provides a powerful and flexible tool for estimating a broad family of multidimensional latent class IRT models. It addresses limitations of traditional models, offers computational efficiency via the EM algorithm, and includes sophisticated procedures for model selection and exploratory dimensionality analysis, making it valuable for researchers in psychometrics, educational testing, and health assessment.