Shaobo Jin

Different data types often occur in psychological and educational measurement such as computer‐based assessments that record performance and process data (e.g., response times and the number of actions). Modelling such data requires specific models for each data type and accommodating complex dependencies between multiple variables. Generalized linear latent variable models are suitable for modelling mixed data simultaneously, but estimation can be computationally demanding. A fast solution is to use Laplace approximations, but existing implementations of joint modelling of mixed data types are limited to ordinal and continuous data. To address this limitation, we derive an efficient estimation method that uses first‐ or second‐order Laplace approximations to simultaneously model ordinal data, continuous data, and count data. We illustrate the approach with an example and conduct simulations to evaluate the performance of the method in terms of estimation efficiency, convergence, and parameter recovery. The results suggest that the second‐order Laplace approximation achieves a higher convergence rate and produces accurate yet fast parameter estimates compared to the first‐order Laplace approximation, while the time cost increases with higher model complexity. Additionally, models that consider the dependence of variables from the same stimulus fit the empirical data substantially better than models that disregarded the dependence.

In the context of structural equation modeling, the model-implied instrumental variable (MIIV) approach has been shown to be more robust against model misspecification than the systemwide approaches (e.g., maximum likelihood and least squares). Besides the goodness-of-fit tests that test the fit of the entire hypothesized covariance structure, the overidentification tests for MIIV can be used to test model specification on an equation-by-equation basis. However, it is known in the econometrics literature that the overidentification tests are inconsistent against general misspecification, if it is used to test a zero correlation between the instrumental variables and the error terms. In this paper, we show that such inconsistency can also occur for the MIIV approach. Numerical examples where the powers of the tests converge to the size are presented. Theoretical results are proved to support the numerical findings. Implications on when the overidentification tests are consistent are also presented.

This study develops a new limited information estimator for random intercept Multilevel Structural Equation Models (MSEM). It is based on the Model Implied Instrumental Variable Two-Stage Least Squares (MIIV-2SLS) estimator, which has been shown to be an excellent alternative or supplement to maximum likelihood (ML) in SEMs (Bollen, 1996 Bollen, K. A. (1996). An alternative two stage least squares (2SLS) estimator for latent variable equations. Psychometrika, 61, 109–121. [Crossref], [Web of Science ®] , [Google Scholar] ). We also develop a multilevel overidentification test statistic that applies to equations at the within or between levels. Our Monte Carlo simulation analysis suggests that MIIV-2SLS is more robust than ML to misspecification at within or between levels, performs well given fewer than 100 clusters, and shows that our multilevel overidentification test for equations performs well at both levels of the model.

The model-implied instrumental variable (MIIV) estimator is an equation-by-equation estimator of structural equation models that is more robust to structural misspecifications than full information estimators. Previous studies have concentrated on endogenous variables that are all continuous (MIIV-2SLS) or all ordinal. We develop a unified MIIV approach that applies to a mixture of binary, ordinal, censored, or continuous endogenous observed variables. We include estimates of factor loadings, regression coefficients, variances, and covariances along with their asymptotic standard errors. In addition, we create new goodness of fit tests of the model and overidentification tests of single equations. Our simulation study shows that the proposed MIIV approach is more robust to structural misspecifications than diagonally weighted least squares (DWLS) and that both the goodness of fit model tests and the overidentification equations tests can detect structural misspecifications. We also find that the bias in asymptotic standard errors for the MIIV estimators of factor loadings and regression coefficients are often lower than the DWLS ones, though the differences are small in large samples. Our analysis shows that scaling indicators with low reliability can adversely affect the MIIV estimators. Also, using a small subset of MIIVs reduces small sample bias of coefficient estimates, but can lower the power of overidentification tests of equations.

The literature on non-linear structural equation modeling is plentiful. Despite this fact, few studies consider interactions between exogenous and endogenous latent variables. Further, it is well known that treating ordinal data as continuous produces bias, a problem which is enhanced when non-linear relationships between latent variables are incorporated. A marginal maximum likelihood-based approach is proposed in order to fit a non-linear structural equation model including interactions between exogenous and endogenous latent variables in the presence of ordinal data. In this approach, the exact gradient of the approximated observed log-likelihood is calculated in order to attain the approximated maximum likelihood estimator. A simulation study shows that the proposed method provides estimates with low bias and accurate coverage probabilities.