  The paper proposes a novel model assessment paradigm aiming to address
shortcoming of posterior predictive $p-$values, which provide the default
metric of fit for Bayesian structural equation modelling (BSEM). The model
framework of the paper focuses on the approximate zero approach, according to
which parameters that would before set to zero (e.g. factor loadings) are now
formulated to be approximate zero via informative priors (Muthen and
Asparouhov, 2012). The introduced model assessment procedure monitors the
out-of-sample predictive performance of the fitted model, and together with a
list of guidelines we provide, one can investigate whether the hypothesised
model is supported by the data. We incorporate scoring rules and
cross-validation to supplement existing model assessment metrics for Bayesian
SEM. The proposed tools can be applied to models for both categorical and
continuous data. The modelling of categorical and non-normally distributed
continuous data is facilitated with the introduction of an item-individual
random effect that can also be used for outlier detection. We study the
performance of the proposed methodology via simulations. The factor model for
continuous and binary data is fitted to data on the `Big-5' personality scale
and the Fagerstrom test for nicotine dependence respectively.
