Bi-factor and second-order copula models for item response data

TL;DR

This paper introduces copula-based bi-factor and second-order models for item response data, allowing for more flexible dependence structures and improved data fit over traditional Gaussian models.

Contribution

It develops a general copula-based framework for bi-factor and second-order models, including estimation, model selection, and fit assessment techniques.

Findings

Copula models improve fit over Gaussian models.

Models capture tail dependencies and heterogeneous dependence.

Demonstrated with Toronto Alexithymia Scale data.

Abstract

Bi-factor and second-order models based on copulas are proposed for item response data, where the items can be split into non-overlapping groups such that there is a homogeneous dependence within each group. Our general models include the Gaussian bi-factor and second-order models as special cases and can lead to more probability in the joint upper or lower tail compared with the Gaussian bi-factor and second-order models. Details on maximum likelihood estimation of parameters for the bi-factor and second-order copula models are given, as well as model selection and goodness-of-fit techniques. Our general methodology is demonstrated with an extensive simulation study and illustrated for the Toronto Alexithymia Scale. Our studies suggest that there can be a substantial improvement over the Gaussian bi-factor and second-order models both conceptually, as the items can have interpretations of latent maxima/minima or mixtures of means in comparison with latent means, and in fit to data.