I want to compare 2 models. Model 1 has a latent dependent variable (4 indicators) and two binary independent variables (CFI = 1.00). Model 2 is the same as above, but also has several covariates (CFI = 0.97). I read on previous threads that you must use the full set of variables in each model if you wish to compare them and that you can fix the paths that you don't want to zero in the model "without covariates." However, when I run it this way, it no longer has good model fit (CFI = 0.937) even though it does achieve good fit without covariates and with covariates when they are not set to zero. Why is this? Is there any other way to compare the models or to improve the fit of Model 1 with covariates set to 0?
I used the following where d1eee and d1zz are IVs and the rest are covariates:
You say that your Model 2 fits when the covariates are included in the model and their slopes not fixed at zero, but doesn't fit when their slopes are fixed at zero. That sounds like the covariates should be in the model. The fact that your model fits well when they are not included obscures the fact that in that model the residual of the factor is correlated with the factor - because the residual contains those left-out covariates.
But as a general statement, you can do BIC comparisons between models that have the same observed DVs, but different covariates because the likelihood is considered for DVs conditional on IVs.