There are some structural assumptions to path analysis that are not difficult ascertain. They are (a) no loops (b) no going forward and backward (c) a maximum of one curved arrow per path.
I am aware that path analysis assumes multivariate normality if the dependent variable is continuous. How do we check for this? What is the path model includes categorical variable as one of the dependent variables or independent variables? What are the additional assumptions?
It's important to distinguish causal or conceptual assumptions from statistical assumptions here. Reverse causality, for instance, has no test, since statistically $A \rightarrow B$ leads to a probability model for $P(A, B)$ that is identical to $B \rightarrow A$ without control for additional causal moderators.
Unlike most regression techniques, SEM requires normality of the residuals. There is one reason for this: only for multivariate normal RVs does no correlation imply independence.
Path models run sequences of regression models, and each regression leads to a corresponding error term (when specified in the model). You can graphically inspect the distribution of error terms and latent variables using their predicted values and a QQplot, which plots the quantiles of the observed distribution against the quantiles of an expected distribution (in this case normal). A good distributional fit will produce a straight calibration line. S shaped curves, or bathtub shaped curves indicate poor fit.
SEM in MPlus allows estimation of some models using robust standard errors, where the error terms are allowed to have non-normal, and non-heteroscedastic fit. This is the MLR... which I find a little disingenuous because the theory of robust standard errors does not jive with maximum likelihood. Differences in these fits and their inference would indicate problems with the path model and normality. See the Mplus discussion here. Lastly, even with MLR, any latent variables would still need to be checked for distributional assumptions, because they are obtained from a maximum likelihood estimate as plug-in estimates for other regression models. It's not as strictly robust as the Muthens' description would lead one to think. Use the QQPlot to assess.
I never in analyses or reporting use distributional tests, statistical testing is a poor framework for verifying assumptions. Graphical inspection is much cleaner, clearer and can actually give you some insight into exactly how small departures from assumptions may affect subsequent analyses.
We have four assumptions: 1 : linearity of model 2: The exogenous variables not correlated with disturbances. 3 : The disturbance are not correlated. 4 : Recursive model See Bollen 1989 for more details and linear causal modeling book.
