Generalized AIC and chi-squared statistics for path models consistent with directed acyclic graphs

We explain how to obtain a generalized maximum-likelihood chi-square statistic, $$ , and a full-model Akaike Information Criterion (AIC) statistic for piecewise structural equation modeling (SEM); that is, structural equations without latent variables whose causal topology can be represented as a directed acyclic graph (DAG). The full piecewise SEM is decomposed into submodels as a Markov network, each of which can have different distributional assumptions or functional links and that can be modeled by any method that produces maximum-likelihood parameter estimates. The generalized $$ is a function of the difference in the maximum likelihoods of the model and its saturated equivalent and the full-model AIC is calculated by summing the AIC statistics of each of the submodels.