Model to base comparisons on for chi-square goodness-of-fit?

I fitting twins data and comparing an ACE model (say) to the fully saturated model, using the chi-square from this as a measure of goodness-of-fit.

I have ten parameters in the fully saturated model (2 x 3 covariance parameters and 2 x 2 mean parameters) and four in the ACE model (a,c & e plus overall mean), so am comparing to a chi(6).

I am worried that in comparing this models as a goodness-of-fit test I am to some extent testing whether an overall mean should be fitted (as distinct from MZ mean 1, MZ mean 2, DZ mean 1, DZ mean 2) rather than just whether the ACE model is a good fit.

Would it be better to test whether the means can be equated and, if so, equate them and compare the difference in deviance between a fully-saturated-apart-from-means model with seven parameters and the four-parameter ACE model to a chi(3)?

Thankyou

Karin