I am attempting to identify profiles based on 10 variables (personal value ratings). In order to control for style biases (as routinely done by other researchers in that area), I ran latent profile analyses as well as factor mixture models that include a continuous style factor (its loadings are fixed to equality across classes). The fit indices are best for the FMMs (vs. the LPAs), and point to a 2-profile + 1 style factor solution. So far so good.
When I try to relax the assumption of equal variances by freeing the residual variances across classes, things get messy, however. Only the 2-profile/1-factor solution converges (the models with more classes don't), and even then, the 2-profile/1-factor solution with unequal variances results in a model that is very different from the one with equal variances (i.e., the factor loadings are different, the profiles don't look the same at all, some items now have very small error variances in one class vs. the other, etc.).
I'm confused as to which final model to choose, the one with equal variances or the one with unequal variances? All of the indices of fit suggest the model with unequal variances is superior, but I'm puzzled by the inconsistent and vastly different results between the two models.
Relaxing residual variances in an already quite flexible model can cause these problems. Small variances can cause spikes in the likelihood without providing a useful solution. You may want to let only the factor variance differ across classes. Or, if you want to vary the residual variances, you may want the classes to differ with respect to only a scale difference in residual variances - so only 1 parameter different per class(may not be very useful, however).