Re: Second-order CFA - Covariances between first-order factors?

[Edward Rigdon]

Elena--

     The second order factor constrains covariances among the first order factors to conform to a certain proportionality, which is a function of the loadings of the first order factors on the second order factor. Allowing residual covariance between two first order factors relaxes the proportionality constraint.

     The modification index does not tell you that you should relax this constraint. It only says that a different model, which relaxes that constraint, would fit your data better in a chi-square sense. The large modification index is a rejection of your model, but there may be multiple other models that would also fit your data better. Some of those may be equivalent to this model with the relaxed constraint. Some may fit even better than that.

     Some people approach model fit assessment not in terms of, is my model strictly consistent with the data?, but in terms of, is there any better alternative? The modification index suggests that there is a better alternative, evaluated strictly in terms of fit to your data. But also keep in mind that this result is purely data driven. It is not necessarily true that an independent random sample taken under identical conditions would yield the same kind of result.

On Sun, May 14, 2023 at 5:43 PM Elena O. <elenaort...@gmail.com> wrote:

Hi everyone!  

I am currently working on a second-order CFA model. The model works fine, but when I ask for the modification indices, one of the largest MI corresponds to the covariance between two first-order latent factors. 

Why is this? 

I thought that the introduction of a second-order latent factor accounts for the covariances between the first-order factors, so that these are no longer necessary. 

Even if I employ a second-order CFA, should I still  include the covariance between the first-order latent factors?

This is my model:

pos_affect =~            enjlf + enrglot + absddng + fltpcfl
 neg_affect =~           fltanx + fltsd + fltdpr + cldgng +  flteeff + flttrd + fltbrd
 life_satisfaction =~  stflife + stflfsf + stfsdlv

wellbeing =~               pos_affect + abs_neg_affect + life_satisfaction

And this is the output I obtain when I call for the modification indices:

                                                            MI            RIV           FMI               EPC           SEPC.LV

pos_affect ~~ neg_affect 50.425 0.013 0.013 0.089 0.655

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[Elena O.]

Hi Edward,

Thank you very much for your answer! 

If I wanted my model to fit the data better

* Would it be correct to still model the second order factor and relax the proportionality constraint by including the covariance between the two first-order factors?

* Or should I then get rid of the second-order factor?  

My first order factors (positive emotions, absence of negative emotions, and life satisfaction) are all correlated, and this makes sense theoretically. This is why I model the second-order factor. 

Yet, it does make sense that two of the first-order factors (positive emotions, and absence of negative emotions) are more strongly correlated than, for example, positive emotions and lkfe satisfaction. 

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[Edward Rigdon]

Elena--

     I have attached some slides from one of my classes--maybe they will help to explain the basic concept of proportionality constraints. These slides present this at the level of observed variables loading on a factor, but the first-order / second order situation is entirely analogous.

     It is a common misunderstanding that the factor model is all about strong covariances, when it is really about a pattern of covariances. In exploratory factor analysis, where there may be several factors linked to a given observed variable, the pattern is obscured--not so obvious to see.

     Your second order factor model implies proportional covariances among not just the three first order factors but between them and any "downstream" variable that is directly or indirectly dependent on the second order factor (a "descendent," in the language of causal inference) and does not have any other relation with the first order factors. So it could be that the real violation of proportionality has to do with relations between the first order factors and some downstream variable.

     It also could be that the real problem is the assumption of linear relations between first order and second order. Maybe the correct relation is exponential or quadratic.

     Or maybe you want to capture the failure of proportionality by creating another second order factor. Even with only 3 first order factors, you might be able to estimate such a model if you fix the loadings of two first order factors on this one to equality and you make this second order uncorrelated with the other second order.

     What is the "best" thing to do? That depends on your goals and your prior knowledge. If your goal is to publish, then look at practice in the venues where you want to publish. If you have prior information about some of these variables, then you might favor a model where those variables behave like they are supposed to. Without specific goals and without prior knowledge it is hard to say what is best. Sorry.

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