A Hybrid Latent-Class Item Response Model for Detecting Measurement Non-Invariance in Ordinal Scales

TL;DR

This work addresses measurement non-invariance in ordinal scales by introducing a hybrid latent-class item response framework that treats group membership as latent and responses via a proportional-odds model. DIF is captured through class-specific intercept and slope shifts, accommodating both uniform and non-uniform DIF, while an $L_1$ penalty induces sparsity to identify non-invariant items without predefined anchors; estimation is performed with a tailored EM algorithm and a two-stage BIC-based model selection. Empirical analysis on the Fisher Temperament Inventory demonstrates a small latent two-class structure with distinct DIF patterns, and background variables show limited association with class membership, highlighting the method's ability to separate latent heterogeneity from item bias. Simulation studies confirm accurate parameter recovery, strong respondent/classification performance, and effective DIF detection across varying design sizes, underscoring the framework's potential for robust measurement invariance assessment in complex, unlabeled populations.

Abstract

Measurement non-invariance arises when the psychometric properties of a scale differ across subgroups, undermining the validity of group comparisons. At the item level, such non-invariance manifests as differential item functioning (DIF), which occurs when the conditional distribution of an item response differs across groups after controlling for the latent trait. This paper introduces a statistical framework for detecting DIF in ordinal scales without requiring known group labels or anchor items. We propose a hybrid latent-class item response model to ordinal data using a proportional-odds formulation, assigning individuals probabilistically to latent classes. DIF is captured through class-specific shifts in item intercepts and slopes, allowing for both uniform and non-uniform DIF. The identification of DIF effects is achieved via an $L_1$-penalised marginal likelihood function under a sparsity assumption, and model estimation is implemented using a tailored EM algorithm. Simulation studies demonstrate strong recovery of item parameters and both uniform and non-uniform types of DIF. An empirical application to a personality test reveals latent subgroups with distinct response patterns and identifies items that may bias group comparisons. The proposed framework provides a flexible approach to assessing measurement invariance in ordinal scales when comparison groups are unobserved or poorly defined.

Figures (13)

• Figure 1: Path diagram of the hybrid latent‐class IRT model. Solid arrows represent the baseline proportional‐odds parameters ($\tau_{jm},a_j$), and dashed arrows represent DIF effects ($\delta_{1j,\xi_i},\delta_{2j,\xi_i}$).
• Figure 2: RMSE of item parameter estimates for $N=1,000$ and $J=15$ under the two-class case.
• Figure 3: RMSE of item parameter estimates for $N=1,000$ and $J=25$ under the two-class case.
• Figure 4: RMSE of item parameter estimates for $N=1,000$ and $J=50$ under the two-class case.
• Figure 5: RMSE of item parameter estimates for $N=5,000$ and $J=15$ under the two-class case.