Comparing path analysis models: mediation vs. (residual) covariance

[Jeanne Sinclair]

Greetings,

I am using lavaan to run path analysis to predict PC from WA and LC. These are multi-group models. Described here is the first step (identifying best-fitting paths); it is  followed by constraining parameters to be equal across the three groups (group.equal = c("intercepts", "regressions")), which is not described here.

My question has to do with the equivalence of models. A reviewer of the manuscript indicated that these three models should be equivalent, but I am not sure. Please advise.

Model 1 models WA and LC predictors separately but allows (residual) covariance.

Mod_1  <- '
PC ~ c(c1, c2, c3) * WA  + c(d1, d2, d3) * LC  
WA ~~ LC
'

Model 2 models an indirect effect via WA.

Mod_2  <- '
PC ~ c(c1, c2, c3) * WA  + c(d1, d2, d3) * LC  
WA ~ c(a1, a2, a3) * LC 
'

Model 3 models an indirect effect via LC.

Mod_3  <- '
PC ~ c(c1, c2, c3) * WA  + c(d1, d2, d3) * LC  
LC ~ c(a1, a2, a3) * WA 
'

All three models were run with this syntax (models would not converge without intercepts being constrained):

fit<- sem(mod, data = new_df,  group = "grade", missing="ML",

               group.equal = c("intercepts")) 

Here are the model fit statistics: 

The models appear to be different. I have read that allowing exogenous variables to covary can be equivalent to modeling a path between them (see for example this article), but these results appear to indicate otherwise. Plus, in the mediated model, there is only 1 exogenous variable, so I am not sure how that applies to these models.

With regard to models 2 and 3, the only difference is directionality of the indirect effect. The only estimated parameter that differs between them is whether WA is predicted by LC (Model 2), or vice versa (Model 3). Across groups, the estimates of the slopes for WA~LC are greater for model 2 (.25-.32) than LC~WA in model 3 (.20-.23). Given that the other slope estimates are equal across models, I believe the model with greater variance explained will have better fit statistics. Yet, the reviewer indicates that these models are equivalent.

I would appreciate any clarification  about the equivalence of models.

Many thanks,

Jeanne Sinclair