Abstract
The estimation of high-dimensional latent regression item response theory (IRT) models is difficult because of the need to approximate integrals in the likelihood function. Proposed solutions in the literature include using stochastic approximations, adaptive quadrature, and Laplace approximations. We propose using a second-order Laplace approximation of the likelihood to estimate IRT latent regression models with categorical observed variables and fixed covariates where all parameters are estimated simultaneously. The method applies when the IRT model has a simple structure, meaning that each observed variable loads on only one latent variable. Through simulations using a latent regression model with binary and ordinal observed variables, we show that the proposed method is a substantial improvement over the first-order Laplace approximation with respect to the bias. In addition, the approach is equally or more precise to alternative methods for estimation of multidimensional IRT models when the number of items per dimension is moderately high. Simultaneously, the method is highly computationally efficient in the high-dimensional settings investigated. The results imply that estimation of simple-structure IRT models with very high dimensions is feasible in practice and that the direct estimation of high-dimensional latent regression IRT models is tractable even with large sample sizes and large numbers of items.
Highlights
Item response theory (IRT) is a class of models for categorical observed variables where an underlying latent variable is assumed to generate the observedAndersson and Xin variables
A second-order Laplace approximation was introduced for the estimation of multidimensional simple structure item response models with a latent regression component
Through numerical illustrations using realistic settings in education and psychology, it was shown that the estimation method gave a substantial improvement over the first-order Laplace approximation for the estimation of latent regression models and multidimensional item response theory (IRT) models with both binary and ordinal data
Summary
Item response theory (IRT) is a class of models for categorical observed variables where an underlying latent variable is assumed to generate the observedAndersson and Xin variables. As the dimension increases, the computational expense grows exponentially which makes the fixed quadrature methods excessively computationally and memory intensive when the dimension is higher than three. The adaptive quadrature methods reduce the required number of quadrature points per dimension, but the computational expense still increases exponentially with higher dimensions, making the approach prohibitively computationally demanding in very high dimensions. The computational demand of the first-order Laplace approximation grows only linearly with increasing dimensionality and as a result, in high-dimensional models, the first-order Laplace approximation is by far the most computationally efficient method out of the four referenced. To improve the computational accuracy, higher order Laplace approximations can be pursued, which has been done with generalized linear models (Raudenbush et al, 2000), generalized linear latent variable models with ordinal data (Bianconcini & Cagnone, 2012), and confirmatory factor analysis with ordinal data and a probit link (Jin et al, 2017). A higher order Laplace approximation requires a substantial amount of higher order derivatives and greatly increases the computational expense, especially for high-dimensional models (Bianconcini, 2014)
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