UCLA

Nonlinear multilevel latent variable modeling has been suggested as an alternative to traditional hierarchical linear modeling to more properly handle measurement error and sampling error issues in contextual effects modeling. However, a nonlinear multilevel latent variable model requires significant computational effort because the estimation process involves high dimensional numerical integration, particularly when the number of latent variables is large. The main purpose of this study is to improve estimation efficiency in obtaining full-information maximum likelihood (FIML) estimates of contextual effects by adopting the Metropolis-Hastings Robbins-Monro algorithm (MH-RM; Cai, 2008, 2010a, 2010b). This study considers contextual effects not only as compositional effects but also as cross-level interactions, in which latent variables are measured by categorical manifest variables. R programs (R Core Team, 2012) implementing the MH-RM algorithm were produced to fit nonlinear multilevel latent variable models. Computational efficiency and parameter recovery were assessed by comparing results with an EM algorithm that uses adaptive Gauss-Hermite quadrature for numerical integration. Results indicate that the MH-RM algorithm can produce FIML estimates and their standard errors efficiently, and the efficiency of MH-RM was more prominent for a cross-level interaction model, which requires 5-dimensional integration. Simulations, with various sampling and measurement structure conditions, were conducted to obtain information about the performance of nonlinear multilevel latent variable modeling compared to traditional hierarchical linear modeling. Results suggest that nonlinear multilevel latent variable modeling can more properly estimate and detect a contextual effect and a cross-level

interaction than the traditional approach. As empirical illustrations, two subsets of data extracted from Programme for International Student Assessment (PISA; OECD, 2000) were used. A negative contextual effect was found from the U.S. data in terms of the relationship between reading literacy and self-concept about reading, supporting results from previous studies. A negative, but not statistically significant, cross-level interaction was found between reading literacy and co-operative learning preference from the analysis of data collected in Korea.