Use of the Lagrange Multiplier Test for Assessing Measurement Invariance Under Model Misspecification

Abstract

This article studies the Type I error, false positive rates, and power of four versions of the Lagrange multiplier test to detect measurement noninvariance in item response theory (IRT) models for binary data under model misspecification. The tests considered are the Lagrange multiplier test computed with the Hessian and cross-product approach, the generalized Lagrange multiplier test and the generalized jackknife score test. The two model misspecifications are those of local dependence among items and nonnormal distribution of the latent variable. The power of the tests is computed in two ways, empirically through Monte Carlo simulation methods and asymptotically, using the asymptotic distribution of each test under the alternative hypothesis. The performance of these tests is evaluated by means of a simulation study. The results highlight that, under mild model misspecification, all tests have good performance while, under strong model misspecification, the tests performance deteriorates, especially for false positive rates under local dependence and power for small sample size under misspecification of the latent variable distribution. In general, the Lagrange multiplier test computed with the Hessian approach and the generalized Lagrange multiplier test have better performance in terms of false positive rates while the Lagrange multiplier test computed with the cross-product approach has the highest power for small sample sizes. The asymptotic power turns out to be a good alternative to the classic empirical power because it is less time consuming. The Lagrange tests studied here have been also applied to a real data set.

Power Analysis for the Wald, LR, Score, and Gradient Tests in a Marginal Maximum Likelihood Framework: Applications in IRT

“…6. Similar approaches were presented by Gudicha et al (2017) and Guastadisegni et al (2021). It builds upon the assumption of asymptotically χ 2 -distributed statistics, the proportionality of the noncentrality parameter with the sample size, λ(β, n) = nλ(β, 1), as well as the asymptotic convergence of the ML estimators.…”

“…The LR test (Silvey, 1959) is known for testing different overall models against each other (Bock & Lieberman, 1970), but can also be used to test for DIF (Irwin et al, 2010;Thissen et al, 1986) . The score test (Rao, 1948) is very flexible since the calculation of the score statistic is less complex than for the Wald and LR statistics (Guastadisegni et al, 2021). There is ample research on utilizing score statistics to test for general model fit (Haberman, 2006), model violations based on item characteristic curves, violations of local independence (Glas, 1999;Glas & Falcón, 2003;Liu & Maydeu-Olivares, 2013), person fit (Glas & Dagohoy, 2007), or DIF (Glas, 1998).…”

“…As a third contribution, we therefore present a sampling-based method. Although it has been already presented in a similar form (Guastadisegni et al, 2021;Gudicha et al, 2017), we apply a different computation with the aim of providing a more accurate result. Lastly, we provide an R package that implements the power analysis for user-defined parameter sets and hypotheses at https://github.com/flxzimmer/ irtpwr.…”

Research on Risk Measurement in Chinese Stock Market - Based on GARCH-VaR Modeling

“…The following Table 3 shows the results of the ARCHLM Lagrange multiplier test [10], showing the 112th order ARCH test, it can be found that the P-value of the LM test is smaller than the significance level of 0.05, so it can be deduced that the variance of the series is non-uniform, there is an ARCH effect, and the ARCH model can be established:…”

Detecting Latent Variable Non-normality Through the Generalized Hausman Test

“…However, assuming normality of the latent variable(s) when the true distribution has a different shape can lead to biased parameter estimates, especially with binary outcomes (Ma and Genton, 2010). Furthermore, assuming an incorrect distribution of the latent variable can lead to erroneous conclusions when conducting hypothesis testing (Guastadisegni et al, 2022). In the literature of the generalized linear latent variable models (GLLVM) (Bartholomew et al, 2011, Skrondal andRabe-Hesketh, 2004) and IRT models, several methods that assume a different form for the distribution of the latent variable have been proposed.…”