Hello Heidi,

Assuming your principal concern (research question) is whether the factor structure holds for this batch of respondents over the length of the course in question, then:

1. Your illustration would correspond to the restricted model, in which respective variable-factor (and second-order) loadings were constrained to be equal. In the unrestricted model, you would remove such constraints. As well, in the unrestricted model, you'd likely not start by assuming correlated error terms.

2. I don't see that inclusion of covariates would help...again, the driving force here is the specific research question you're trying to address.

3. Yes, N makes a difference (that's how one arrives at a value of 11,814). However, given N and your data, your results would appear to suggest that: (a) the constrained model (equal structures) is significantly different from the unrestricted model; (b) both models appear to fit well enough (to your implied criteria) that one could consider the difference to be statistically significant but not of such magnitude as to be of practical significance.

The fact of the matter is, sample to sample variance may often be the only real "culprit" for why one research team claims support for structure "A" for a measure while another claims support for structure "B."

4. See #3.

Males vs. females: Was this a research question? I couldn't tell.

Good luck with your work.