Goodness-of-fit of multilevel latent class models for categorical data

In the context of multilevel latent class models, the goodness-of-fit depends on multiple aspects, among which are two local independence assumptions. However, because of the lack of local fit statistics, the model and any issues relating to model fit can only be inspected jointly through global fit statistics. This hinders the search for model improvements, as it cannot be determined where misfit originates and which of the many model adjustments may improve its fit. Also, when relying solely on global fit statistics, assumption violations may become obscured, leading to wrong substantive results. In this paper, two local fit statistics are proposed to improve the understanding of the model, allow individual testing of the local independence assumptions, and inspect the fit of the higher level of the model. Through an application in which the local fit statistics group-variable residual and paired-case residual are used as guidance, it is shown that they pinpoint misfit, enhance the search for model improvements, provide substantive insight, and lead to a model with different substantive conclusions, which would likely not have been found when relying on global information criteria. Both residuals can be obtained in the user-friendly Latent GOLD 5.0 software package.