Using latent class models with random effects for investigating local dependence

In psychometric latent variable modeling approaches such as item response theory one of the most central assumptions is local independence (LI), i.e. stochastic independence of test items given a latent ability variable (e.g., Hambleton et al., Fundamentals of item response theory, 1991). This strong assumption, however, is often violated in practice resulting, for instance, in biased parameter estimation. To visualize the local item dependencies, we derive a measure quantifying the degree of such dependence for pairs of items. This measure can be viewed as a dissimilarity function in the sense of psychophysical scaling (Dzhafarov and Colonius, Journal of Mathematical Psychology 51:290–304, 2007), which allows us to represent the local dependencies graphically in the Euclidean 2D space. To avoid problems caused by violation of the local independence assumption, in this paper, we apply a more general concept of “local independence” to psychometric items. Latent class models with random effects (LCMRE; Qu et al., Biometrics 52:797–810, 1996) are used to formulate a generalized local independence (GLI) assumption held more frequently in reality. It includes LI as a special case. We illustrate our approach by investigating the local dependence structures in item types and instances of large scale assessment data from the Programme for International Student Assessment (PISA; OECD, PISA 2009 Technical Report, 2012).