Factor tree copula models for item response data

TL;DR

This paper introduces factor tree copula models that combine factor and vine copula approaches to better capture dependencies in item response data, improving interpretability and fit while addressing limitations of existing models.

Contribution

The paper proposes a novel combined factor tree copula model that integrates factor and vine copula structures, along with algorithms for model selection and estimation.

Findings

Model effectively captures residual dependencies in item response data.

Simulation studies demonstrate improved fit over traditional models.

Application to PTSD data illustrates practical utility.

Abstract

Factor copula models for item response data are more interpretable and fit better than (truncated) vine copula models when dependence can be explained through latent variables, but are not robust to violations of conditional independence. To circumvent these issues, truncated vines and factor copula models for item response data are joined to define a combined model, the so-called factor tree copula model, with individual benefits from each of the two approaches. Rather than adding factors and causing computational problems and difficulties in interpretation and identification, a truncated vine structure is assumed on the residuals conditional on one or two latent variables. This structure can be better explained as a conditional dependence given a few interpretable latent variables. On the one hand, the parsimonious feature of factor models remains intact and any residual dependencies are being taken into account on the other. We discuss estimation along with model selection. In particular, we propose model selection algorithms to choose a plausible factor tree copula model to capture the (residual) dependencies among the item responses. Our general methodology is demonstrated with an extensive simulation study and illustrated by analyzing Post Traumatic Stress Disorder.