Are any pairs of these four structural models nested in another? I think the second-order single factor model is nested in the second-order two factor model (second order single factor model is when covariance between the two second order factors in second-order two factor model is 1) but I can't determine whether other pairs are nested or not.
Best Answer
a) is the saturated model. All models are nested within the saturated model. That's the basis of the chi-square test. You're testing against the saturated model, which has 0 df and 0 chi-square.
c) and d) are equivalent models. d) isn't identified without a constraint on the loadings, so there is only one parameter to estimate.
All that we are left with is the relationship between b), and c)/d). These are nested, but not (quite) in the usual way. Constraing the correlation to 1 in model b) gives you nested models, but you can't test these nested models using the chi-square test, because the parameter is on the boundary of the permitted space. In a sense it's a one-tailed test - when you relax the correlation in model c) we know that the parameter will change, but it must get smaller, not larger. The nested model chi-square test assumes that the parameter can change in any direction. It's surprisingly tricky. Here's a paper: http://psycnet.apa.org/?&fa=main.doiLanding&doi=10.1037/1082-989X.13.2.150