Mixture IRT Model With a Higher-Order Structure for Latent Traits

Abstract

Mixture item response theory (IRT) models have been suggested as an efficient method of detecting the different response patterns derived from latent classes when developing a test. In testing situations, multiple latent traits measured by a battery of tests can exhibit a higher-order structure, and mixtures of latent classes may occur on different orders and influence the item responses of examinees from different classes. This study aims to develop a new class of higher-order mixture IRT models by integrating mixture IRT models and higher-order IRT models to address these practical concerns. The proposed higher-order mixture IRT models can accommodate both linear and nonlinear models for latent traits and incorporate diverse item response functions. The Rasch model was selected as the item response function, metric invariance was assumed in the first simulation study, and multiparameter IRT models without an assumption of metric invariance were used in the second simulation study. The results show that the parameters can be recovered fairly well using WinBUGS with Bayesian estimation. A larger sample size resulted in a better estimate of the model parameters, and a longer test length yielded better individual ability recovery and latent class membership recovery. The linear approach outperformed the nonlinear approach in the estimation of first-order latent traits, whereas the opposite was true for the estimation of the second-order latent trait. Additionally, imposing identical factor loadings between the second- and first-order latent traits by fitting the mixture bifactor model resulted in biased estimates of the first-order latent traits and item parameters. Finally, two empirical analyses are provided as an example to illustrate the applications and implications of the new models.

Mixture Random-Effect IRT Models for Controlling Extreme Response Style on Rating Scales

“…A fixed-length test was used in this study, so we did not expect that the precision of the latent trait estimation could reach an excellent level for all respondents, such as in computerized adaptive testing (CAT). Nevertheless, our results were comparable to those of previous studies relative to mixture IRT modeling when similar sample sizes and test lengths were used (e.g., Huang, 2016). …”

“…The threshold parameters were set to −0.6, 0, and 0.6 for the 4-category items and −0.8, −0.4, 0, 0.4, and 0.8 for the 6-category items. The specifications of the model parameters were consistent with those that are commonly found in practice and similar to previous research (e.g., Li et al, 2006; Morren et al, 2012; Jin and Wang, 2014a; Huang, 2014, 2015, 2016). Each condition was replicated 30 times.…”

“…The most common simulation settings in previous studies that were relevant to the mixture Rasch model were the use of more than 25 items and more than 500 examinees (e.g., Meyer, 2010; Dai, 2013; Choi and Wilson, 2015). When a multi-parameter mixture IRT model was used, the number of examinees and items had to be increased substantially because of its complex nature (e.g., Cho et al, 2014; Huang, 2016). Our developed models belong to the family of multi-parameter IRT models and must identify three different latent classes, so the use of extremely small numbers of individuals and test items is not practical and therefore not recommended.…”

Investigation of adolescents’ developmental stages in deductive reasoning: An application of a specialized confirmatory mixture IRT approach

“…Multidimensional mixture IRT models have been applied in several contexts, for instance, for large-scale crosscountry data analysis (e.g., De Jong & Steenkamp 2010), longitudinal data analysis (e.g., Cho, Cohen, Kim, & Bottge, 2010;Cho, Cohen, & Bottge, 2013;von Davier, Xu, & Carstensen, 2011), and differential item functioning investigations (e.g., De Boeck, Cho, & Wilson, 2011). In addition, Cho, Cohen, and Kim (2014) and Huang (2016) presented extended mixture IRT models for bifactor and higher-order test structures, respectively. Finch and Finch (2013) applied a multilevel and multidimensional extension of a mixture IRT model and Choi and Wilson (2015) presented a mixture random weights linear logistic test model, which can be seen as a mixture extension of the Rasch model with item features and multidimensionality.…”

A Constrained Confirmatory Mixture IRT Model: Extensions and Estimation of the Saltus model using Mplus

“…Various mixture IRT models have been utilized to examine subjects' heterogeneity in depression/anxiety (Wanders et al, 2016), extreme response styles (H.-Y. Huang, 2016), at-risk unhealthy behaviors (Finch & Pierson, 2011), and item solving strategies (Mislevy & Verhelst, 1990). Other applications of mixture IRT models also exist, e. g., for studying test speededness (Bolt, Cohen, & Wollack, 2002), models' goodness of fit (Rost, 1990), and differential item functioning (Cho & Cohen, 2010;Cohen & Bolt, 2005).…”