Purpose. Data from international educational assessments conducted in many countries are mostly analyzed using item response theory. The assumption that all items behave the same in all countries is often not tenable. The variability of item parameters across countries can be taken into account by assuming that the item parameters are random effects (De Jong et al., 2010). However, the complex latent structure of such a model, with latent variables both at the item and the person level, renders maximum likelihood estimation computationally challenging. We describe a variational estimation technique that consists of approximating the likelihood function by a computationally tractable lower bound. Methods. A mean field approximation to the posterior distribution of the latent variables was used. The update equations were derived for the specific case of discrete random effects and implemented in a Maximization Maximization algorithm (Neal and Hinton,1998). Parameter recovery was investigated in a simulation study. The method was also applied to the Progress in International Reading Study of 2006. Results. The model parameters were recovered well under all conditions of the simulation study. In the application, the estimated variances of the random item effects showed a high positive correlation with traditional measures for the lack of item invariance across groups. Conclusions. The mean field approximation and variational methods in general offer a computationally tractable alternative to exact maximum likelihood estimation.
