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.

由于需要在似然函数中近似积分，因此难以估计高维潜在回归项响应理论（IRT）模型。文献中提出的解决方案包括使用随机逼近，自适应正交和拉普拉斯逼近。我们建议使用可能性的二阶Laplace近似值来估计具有类别观测变量和固定协变量的IRT潜在回归模型，其中同时估计所有参数。该方法适用于IRT模型具有简单结构的情况，这意味着每个观察到的变量仅加载一个潜在变量。通过使用具有二元和有序观测变量的潜在回归模型进行模拟，我们表明，相对于一阶拉普拉斯逼近而言，所提出的方法是一项实质性的改进。此外，当每个维度的项目数量适度较高时，该方法与用于估计多维IRT模型的替代方法同等或更精确。同时，该方法在所研究的高维环境中具有很高的计算效率。结果表明，在实践中估算具有非常高维度的简单结构IRT模型是可行的，并且即使在样本量较大且项目数量众多的情况下，对高维潜在回归IRT模型的直接估算也是很容易做到的。当每个维度的项目数量适度较高时，该方法与用于估算多维IRT模型的替代方法同等或更精确。同时，该方法在所研究的高维环境中具有很高的计算效率。结果表明，在实践中估算具有非常高维度的简单结构IRT模型是可行的，并且即使在样本量较大且项目数量众多的情况下，对高维潜在回归IRT模型的直接估算也是很容易做到的。当每个维度的项目数量适度较高时，该方法与用于估算多维IRT模型的替代方法同等或更精确。同时，该方法在所研究的高维环境中具有很高的计算效率。结果表明，在实践中估算具有非常高维度的简单结构IRT模型是可行的，并且即使在样本量较大且项目数量众多的情况下，对高维潜在回归IRT模型的直接估算也是很容易做到的。