What is the difference between a nested and non-nested model in CFA?

Question

I thought it had to do with varying numbers of factors but based on the literature I have read it seems to do more with fixing/freeing of parameters. Any info would be greatly appreciated!

Answer 1

Model A is nested in model B if some of the coefficients in B, or their combinations, can be restricted to obtain model A.

In case of CFA models, the number of factors is a moderately complicated example of nesting. Consider a two-factor model (taken from UCLA ATS website)

A simpler one-factor model will be nested in it with a linear constraint $${\rm Corr}[{\rm visual}, {\rm verbal}]=1.$$ Alternatively, we can say that it is nested with a nonlinear constraint that $${\rm Cov}({\rm visual},{\rm verbal}) = \mathbf{V}[{\rm visual}]\cdot\mathbf{V}[{\rm verbal}].$$ Depending on how exactly the model is parameterized, there may be small differences in the results (every 1 matters!). Conversely, you can move from 1-factor to 2-factor model by freeing the correlation coefficient, although you still had to present the model as a 2-factor model to begin with.

Note that testing one vs. two factors may give rise to complications: the natural range of the correlation parameter is $[-1,1]$, so it is tempting to restrict it to this range (as EQS does) to avoid Heywood cases. Vika Savalei and I argued that this is generally a poor testing strategy, because it produces weird likelihood ratio distributions that are not the traditional beloved $\chi^2$.

Moreover, in some factor presence testing situations (e.g., MTMM), the likelihood ratio has a wrong null distribution even if you do not restrict the range of the parameter: if you have zero variance of say the methods factor, then its loadings are not identified, so you get an effective mix of $\chi^2$s with different degrees of freedom asymptotically. This is a poorly understood topic, overall, not only in SEM, but in other areas of application of statistics.